Quantum Physics - Matter as Discrete Spherical Standing Waves, Light as Discrete Resonant Coupling

Physical foundation

WSM Postulates

WSM Wave Action: the mathematics already in hand

Abstract / Summary

Real spherical matter and discrete resonant light coupling

Quantum physics began when experiments forced physicists to join continuous waves to discrete changes. Its equations became extraordinarily accurate. WSM asks what physically waves, why matter has stable discrete modes, how a finite change travels between atoms, why a receiver records one result, and how spin, antimatter, relativity and entanglement arise within one continuous Space.

What is calculated and what remains open. This page gives exact conditional receiver models that convert specified quadratic channel weights into Born frequencies. Deriving those weights, the receiver’s invariant readiness distribution, exclusive recording and the full Bell and GHZ correlations from the same real-wave dynamics remains the physical programme.

Further calculations connect the free de Broglie phase to centre guidance, demonstrate deterministic approach towards Born statistics in a multimode envelope, and identify stable receiver dynamics that preserve channel weights. Matched capture, finite sea memory and source coherence make the connection from travelling curves to records more quantitative.

A stationary electron is a spherical, directionally equal e-sphere: it has the same \(E_d\), \(c'\), wavelength and frequency in every direction and is therefore continually rebuilt at the same physical position in Space. Motion requires a three-dimensional wave egg with an elongated lower-\(E_d\) front and a flattened higher-\(E_d\) rear. H-M1 assigns the axial reconstruction rates \(c'_{\rm rear}=c_0+v\) and \(c'_{\rm lead}=c_0-v\); over one rest-reference interval \(T_0\) they determine travel distances, not automatically simultaneous crest spacings. The complete e-sphere retains one intrinsic recurrence identity, while resonant stability requires no phase slip among the waves actually arriving at the moving centre. In the axial H-M2 representation, that condition fixes the opposed fixed-Space frequency ratio; H-M3’s geometric-mean rest-frequency closure fixes the remaining scale and gives \(\omega_\pm=\gamma\omega_e(1\pm\beta)\). Exact factorisation then gives the Lorentz–de Broglie phase relation and the common centre-encounter rate \(\omega_e/\gamma\). Electron and positron remain locked to the same incoming sea relation and exactly \(\pi\) apart. In the Bohr ground state, \(\alpha=\Delta X_0/\lambda_e\) is the centre displacement during the rest-reference interval, while a complete moving-centre cycle lasts \(T_v=\gamma T_0\).

A bound transition changes the position of the half-sphere curve imprinted on each successive background plane wave passing through the bound e-sphere. The finite sequence of changed curves carries corresponding changes of Space’s displacement, longitudinal velocity, strain and phase. A resonant receiver is progressively reshaped until it settles into another stable standing-wave mode. The propagation is continuous; the source and receiver changes are discrete because only stable recurrent organisations persist.

The electron and positron supply P3’s two opposite radial phases. WSM associates two proposed spherical phase-rotation hands with the two spin channels; deriving their \(4\pi\) transport and physical Dirac representation is the task of §19 and §20. In the WSM proton model, neutral hydrogen has the balanced phase inventory \((++-)_{\mu}+(-)_e=++--\): opposite-phase roles are bound inside the proton. The two radial phases write oppositely oriented forward and rear curves, producing equal and opposite charge-like reconstruction pushes in neutral matter. After either curved portion leaves its e-sphere, however, it spreads over greater area, has lower \(E_d\) and lower \(c'\) than the flatter carrying plane wave, and widens, flattens and lags. The opposite charge-like pushes cancel while this common delay remains, so the resultant plane wave is slightly delayed and changes other e-spheres so their centres reconstruct toward the source—the proposed gravitational relation. Entangled pairs are source-created joint curve and phase relations; analyser settings define the available receiver modes, while the joint probabilities must reproduce Born’s rule, Bell violation and no-signalling.

“In the year nineteen hundred… the law of radiation of bodies as a function of temperature could not be derived solely from the laws of Maxwellian electrodynamics. To arrive at results consistent with the relevant experiments, radiation of a given frequency had to be treated as though it consisted of energy atoms of the individual energy (h f)…”
— Albert Einstein, “Considerations Concerning the Fundaments of Theoretical Physics” (1940)

“This discovery became the basis of all twentieth-century research in physics… and set science a fresh task: that of finding a new conceptual basis for all physics.”
— Albert Einstein, “Max Planck in Memoriam” (1948)

In 1951, after half a century of work, Einstein wrote to Michele Besso:

“All these fifty years of conscious brooding have brought me no nearer to the answer to the question, ‘What are light quanta?’ Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.”

Essential WSM glossary

Extended WSM reference: the complete glossary, definitions and research notes.

Claim-status key

Part I · The quantum is born

Long before the quantum crisis, Huygens and Maxwell insisted that visible propagation and apparently empty fields should be sought in real physical action. Their models were not WSM, but their questions belong at the beginning of the journey.

“Visible rays, coming from an infinitude of diverse places, cross one another without hindering one another in any way.”
— Christiaan Huygens, Treatise on Light (1690)

“…where we find these lines of force, some physical state or action must exist in sufficient energy to produce the actual phenomena.”
— James Clerk Maxwell, “On Physical Lines of Force” (1861)

1. Planck: the furnace that would not obey classical physics

The first quantum problem was not an electron circling a nucleus. It was heat.

Imagine an enclosed cavity with a tiny opening. Its walls absorb and emit radiation until matter and light reach thermal equilibrium. The spectrum escaping the opening depends only on the temperature. At low frequencies classical reasoning worked reasonably well; at high frequencies it predicted an impossible flood of energy—the later-named ultraviolet catastrophe.

Planck modelled the matter in the cavity walls as ideal resonators of frequency \(\nu\). To reproduce the measured spectrum, he counted their energies in elements

\[ \varepsilon=h\nu, \qquad E_n=nh\nu, \qquad n=0,1,2,\ldots \]

and obtained

\[ u(\nu,T)=\frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/kT}-1}. \]

Planck had introduced a new universal constant, \(h\), and a new kind of relation between energy and frequency. The quantum first entered physics through material resonators. Planck did not begin by proving that light was a shower of tiny particles moving through empty space. His calculation said that exchanges between matter and radiation occurred in frequency-linked elements.

“In this case the quantum of action must play a fundamental role in physics, and here was something entirely new, never before heard of.”
— Max Planck, Nobel lecture, “The Genesis and Present State of Development of the Quantum Theory”

For WSM this historical order matters. A standing-wave system naturally admits discrete mode labels because only particular phase-closing patterns remain stable. Planck’s resonators therefore point toward matter as organised wave structure. But a discrete set of allowed frequencies does not, by itself, prove discrete amplitude, discrete action or the numerical value of \(h\). The universal action unit remains a D-tier derivation.

The first lesson is nevertheless clear:

The quantum did not enter as a little object. It entered as a constraint on what a resonator could exchange.

2. Einstein: the quantum leaves the furnace

Planck had quantised the energy bookkeeping of the resonators. Einstein made the light quantum travel into the laboratory.

In 1905 he considered several phenomena that classical wave theory could not explain together, especially the photoelectric effect. When light strikes a suitable material, electrons can be emitted. Increasing the light intensity increases the number of events, but the maximum energy of an emitted electron is governed primarily by the light frequency:

\[ K_{\max}=h\nu-\Phi, \]

where \(\Phi\) is the material’s work function.

Below a threshold frequency, greater intensity does not rescue the event. Above it, energy appears in complete frequency-dependent transfers. Einstein proposed that, in emission and absorption, radiation behaved as if it were composed of localised energy quanta \(h\nu\).

“…the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta.”
— Albert Einstein, “On a Heuristic Point of View Concerning the Production and Transformation of Light” (1905)

This was bolder than Planck. It was also only the beginning. Einstein later analysed spontaneous emission, stimulated emission and the momentum carried in radiative exchange. The laser would eventually grow from this framework. Yet the interference of extremely weak light remained wave-like. A source can be attenuated until detector records arrive one at a time, while the accumulated pattern still depends on phase differences between alternatives.

That is the enduring puzzle. The detector record is local and complete; the wave process determining its distribution is extended and coherent.

WSM’s proposed answer begins by separating three statements that are often blended together:

1. Matter makes local, discrete records. A
2. Each complete spectral transfer obeys \(E=h\nu\). A
3. A microscopic pellet carrying that energy followed one definite path between source and detector. Not established.

In the WSM picture, a bound source is already a repeating standing-wave organisation of Space. During a transition, its changing pattern writes an ordered, finite disturbance into the travelling waves that pass through it. A resonant receiver responds to that complete wave relation and eventually recloses into a new stable state. The photon is the finite source-written curve train carried by longitudinal waves between stable source and receiver organisations; it is not a second substance or a tiny bullet independent of those waves.

This is a B/C-tier identification, not yet the final receiver-reclosure dynamics. It takes Einstein’s question literally: frequency belongs to the physical wave process, while discreteness belongs to the stable source and receiver modes.

3. Bohr: an atom that could stand still

Rutherford’s nuclear atom created an immediate crisis. In classical electrodynamics, an orbiting electron accelerates, radiates energy and spirals into the nucleus. Ordinary matter should collapse.

In 1913 Niels Bohr proposed that the atom possessed special stationary states in which this did not occur. Radiation was associated not with continuous motion around a classical orbit, but with a transition between allowed energies:

\[ E_m-E_n=h\nu_{mn}. \]

“…the lines correspond to a radiation emitted during the passing of the system between two different stationary states.”
— Niels Bohr, “On the Constitution of Atoms and Molecules” (1913)

For hydrogen the model reproduced the observed spectral pattern with astonishing success. The classical orbit was still present as scaffolding, but the crucial ideas were new: stable states, discrete labels and transition frequencies.

Bohr’s “quantum jump” was a rule, not a mechanical account of how an entire atom changed. That incompleteness became one of the central tensions of quantum theory. Yet his stable-state insight survived every later formulation.

Translated into WSM language:

• a stationary state is a stable repeating standing-wave organisation of an atom;
• a spectral transition is a finite reorganisation between two such patterns;
• \(\nu_{mn}\) is the frequency of their difference relation;
• the apparatus matters because it supplies real boundary conditions and possible receiver channels.

Bohr later insisted that atomic phenomena must be described with reference to the complete experimental arrangement. That insight does not require human consciousness to create reality. Turning an analyser or inserting a which-path detector physically changes the couplings, phase relations and available records. The observer chooses the question by constructing the apparatus. The observer does not choose the answer.

4. The laboratory rebellion: atoms, recoil and divided beams

The quantum was not born from philosophy alone. Different experiments, built for different purposes, began pointing toward the same unfamiliar structure.

In 1914 James Franck and Gustav Hertz accelerated electrons through mercury vapour. The electrons surrendered energy to the atoms in sharply repeated steps rather than through an arbitrary continuum. When the mercury atoms relaxed, they emitted radiation at the corresponding spectral frequency. Bohr’s stationary states had acquired a direct collision experiment.

In 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms through an inhomogeneous magnetic field. A classical distribution of tiny magnetic orientations should have spread continuously. Instead the beam divided into discrete components. The full electron-spin interpretation came later, but Nature had already displayed an irreducible two-valued response to an apparatus-defined axis.

In 1923 Arthur Compton scattered X-rays from electrons and measured an angle-dependent wavelength shift,

\[ \lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta). \]

The result joined wave frequency to directed momentum and relativistic recoil. A viable wave account of light must therefore explain not only interference and diffraction, but complete energy–momentum transfer and the Klein–Nishina angular distribution.

Then Davisson and Germer observed electron diffraction. Matter made localised detector records while its distribution was governed by extended phase. Later single-quantum double-slit and interferometer experiments sharpened the same fact: records arrive one at a time, yet the accumulated pattern depends on coherent alternatives,

\[ P=|\psi_1+\psi_2|^2 =|\psi_1|^2+|\psi_2|^2 +2\operatorname{Re}(\psi_1^*\psi_2). \]

If path information becomes physically available, the cross-term disappears. No human mind is required; the apparatus has changed the real relation.

5. De Broglie: the integers were telling us something

Louis de Broglie saw the asymmetry in the young theory. Light, long known as a wave, had acquired particle-like quanta. Matter, long treated as particles, might therefore possess wave-like phase.

His argument went deeper than analogy. A purely particle theory contained no natural frequency with which to write \(E=h\nu\). Atomic integers, meanwhile, resembled the integers found in interference and normal modes. In his Nobel lecture he emphasised precisely that clue: the earlier physical phenomena involving integers were “interference and normal modes of vibration.”

“On the one hand the quantum theory of light cannot be considered satisfactory since it defines the energy of a light particle by the equation \(E=hf\), containing the frequency \(f\). Now a purely particle theory contains nothing that enables us to define a frequency… On the other hand, determination of the stable motion of electrons in the atom introduces integers, and up to this point the only phenomena involving integers in physics were those of interference and of normal modes of vibration. This fact suggested to me the idea that electrons too could not be considered simply as particles, but that frequency must be assigned to them also.”
— Louis de Broglie, Nobel lecture (1929)

De Broglie proposed

\[ E=\hbar\omega, \qquad \mathbf p=\hbar\mathbf k, \qquad \lambda=\frac{h}{p}. \]

An electron could no longer be understood as a featureless point. Its motion carried a phase, and stable atomic states could be associated with standing-wave closure. Einstein immediately recognised the importance of the proposal:

De Broglie “made intelligible to some extent the discrete character of Bohr’s ‘permitted’ paths by the stationary character of the corresponding waves.”
— Albert Einstein, “Considerations Concerning the Fundaments of Theoretical Physics” (1940)

Electron diffraction, observed by Davisson and Germer in 1927, made the wavelength experimentally undeniable.

De Broglie initially retained both a particle and an accompanying wave. WSM makes the more radical identification: the matter wave is not an accessory attached to matter; the organised wave is the matter. What appears particle-like is the concentrated, repeatedly rebuilt centre of an extended standing-wave structure.

The distinction between the full Compton wavelength and its reduced form will matter later:

\[ \lambda_C=\frac{h}{mc}, \qquad \bar\lambda_C=\frac{\hbar}{mc}=\frac{\lambda_C}{2\pi}. \]

They are different phase ledgers, not interchangeable numerical conventions.

6. Heisenberg: abandon the orbit and keep the transitions

In 1925 Werner Heisenberg refused to build atomic theory from electron trajectories that no experiment could reveal. Spectroscopy gave frequencies and transition strengths, so he organised those quantities directly.

“The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.”
— Werner Heisenberg, “Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations” (1925)

Max Born recognised that Heisenberg’s arrays obeyed matrix multiplication. With Pascual Jordan, the new matrix mechanics was developed rapidly. A physical quantity became an operator with transition elements

\[ A_{mn}=\langle m|\hat A|n\rangle. \]

Schrödinger’s continuous waves and Heisenberg’s discontinuous-looking matrices initially seemed like rival worlds. They were soon shown to be mathematically equivalent representations of the same quantum structure.

Heisenberg’s uncertainty relation followed in 1927:

\[ \Delta x\,\Delta p\geq\frac{\hbar}{2}. \]

This is often reduced to the claim that measurement disturbs a small particle. Measurement can certainly disturb a system, but the inequality is deeper. Position and momentum are Fourier-related descriptions, and their operators do not commute:

\[ [\hat x,\hat p]=i\hbar. \]

A finite wave cannot be made arbitrarily narrow in position and arbitrarily narrow in wave number at once.

In a WSM reading, Heisenberg’s matrices describe relations among stable wave patterns, not secret classical orbits hidden underneath them. The element \(A_{mn}\) is naturally read as a coupling between complete bound organisations \(n\) and \(m\). The formalism was right to place transitions at the centre of atomic physics even while remaining silent about the substance undergoing them.

7. Schrödinger: the atom begins to sing

Heisenberg had kept the observable transitions and abandoned the imagined orbit. In 1926 Erwin Schrödinger approached the same atom from de Broglie’s other direction: let it be a wave eigenproblem.

\[ i\hbar\frac{\partial\psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2+V\right)\psi, \qquad \hat H\psi_n=E_n\psi_n. \]

Boundary conditions allow only particular eigenfunctions and energies. The mysterious quantum numbers of the older atom reappeared as labels of standing-wave patterns. Hydrogen’s spectrum followed without a tiny planet circling the nucleus. Schrödinger then showed that his apparently continuous wave mechanics and Heisenberg’s apparently discontinuous matrix mechanics were equivalent representations.

This was one of the great moments in the history of thought. An abstract list of stationary states had become a geometry of modes, and two paths that looked opposed met in the same mathematics.

“Let me say at the outset that in this discourse I am opposing not a few special statements of quantum mechanics held today. I am opposing, as it were, the whole of it. I am opposing its basic views that have been shaped twenty-five years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody.”
— Erwin Schrödinger, lectures on the interpretation of quantum mechanics

Schrödinger hoped the wave was physically real and returned repeatedly to the possibility that apparent jumps arose from continuous resonance between modes. WSM keeps that instinct: discrete stable endpoints need not imply a discontinuous substance between them.

The many-body wavefunction generally lives on configuration space rather than ordinary three-dimensional space. WSM therefore reads \(\psi\) as an extraordinarily powerful collective amplitude or demodulated envelope of deeper real-space standing-wave relations—not as a second substance.

The reciprocal Doppler pair gives a conditional route. H-M1 supplies the physical \(c_0\pm v\) reconstruction rates. Separately, H-M2 centre phase matching fixes the opposed fixed-Space frequency ratio, and H-M3 geometric-mean rest-frequency closure fixes its scale: \(K_\pm=\gamma(1\pm\beta)=e^{\pm s}\). These reciprocal frequency factors are not additional grades or replacement speed laws. If the moving e-sphere supplies the composite dispersion

\[ \omega_{\rm ph}^2-c^2k_{\rm dB}^2=\omega_e^2, \]

then write the real composite phase as

\[ \Phi_{\rm eff}=\operatorname{Re}\!\left[\psi(\mathbf x,t)e^{-i\omega_et}\right]. \]

The slow-envelope limit gives

\[ iJ_*\partial_t\psi=-\frac{J_*^2}{2m}\nabla^2\psi, \qquad m=\frac{J_*\omega_e}{c^2}, \]

which is Schrödinger’s free equation when the universal wave-action scale is \(J_*=\hbar\).

8. Pauli, Bose and Fermi: spin, exclusion and quantum identity

While de Broglie, Heisenberg and Schrödinger were rebuilding the single electron, another revolution was changing the meaning of many quanta.

In 1924 Satyendra Nath Bose derived Planck’s radiation law by counting light quanta as fundamentally indistinguishable. Einstein recognised the importance immediately and extended the method to material particles. The result predicted Bose–Einstein condensation: many integral-spin quanta may occupy one common mode.

Electrons obey the opposite rule. In 1925 Wolfgang Pauli formulated the exclusion principle: no two electrons in an atom can occupy the same complete set of quantum numbers. In 1926 Enrico Fermi and Paul Dirac developed the corresponding statistics. Exchange of two identical electrons changes the sign of the joint amplitude; two electrons cannot enter the same one-particle state.

“Already in my initial paper, I especially emphasized the fact that I could not find a logical substantiation for the exclusion principle.”
— Wolfgang Pauli, Nobel lecture, “Exclusion Principle and Quantum Mechanics” (1946)

This is why atoms possess shells, why the periodic table has its structure, why matter resists collapse and why chemistry exists. Quantum identity is not classical similarity. There is no additional hidden label saying “electron 1” and “electron 2” after the complete state has been formed.

Pauli’s other great clue was two-valuedness. Spectra and Stern–Gerlach splitting required a degree of freedom that was later recognised as electron spin. Pauli’s matrices gave its compact algebra:

\[ \sigma_x= \begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \sigma_y= \begin{pmatrix}0&-i\\i&0\end{pmatrix}, \quad \sigma_z= \begin{pmatrix}1&0\\0&-1\end{pmatrix}. \]

The spin operators are

\[ S_i=\frac{\hbar}{2}\sigma_i, \qquad \mathbf S^2=\frac34\hbar^2, \]

so any chosen axis has the two projections \(\pm\hbar/2\). A rotation through physical angle \(\theta\) is represented by

\[ U(\theta,\hat{\mathbf n}) = \exp\!\left(-\frac{i\theta}{2}\hat{\mathbf n}\!\cdot\!\boldsymbol\sigma\right), \]

which contains the half-angle, changes sign after \(2\pi\), and returns after \(4\pi\). Sequential analyser probabilities therefore contain \(\cos^2(\theta/2)\) and \(\sin^2(\theta/2)\).

Spin is not well pictured as a solid bead rotating about one little axle. A spinor transforms by a half-angle and returns to the identical complex state after \(4\pi\), while an ordinary vector closes after \(2\pi\). WSM proposes a radial standing-wave vibration carrying spherical phase organisation; §19 separates that physical construction from the still-required derivation of its half-angle transport.

9. Born: the wave becomes a probability law

Schrödinger’s equation evolved amplitudes, but an experiment ended in a particular record. What connected the two?

In 1926 Max Born, studying scattering, proposed that the squared magnitude of the wave amplitude gives the probability density of an outcome:

\[ P(x)\,dx=|\psi(x)|^2dx. \]

For a projective measurement with alternatives \(|j\rangle\),

\[ P(j)=|\langle j|\psi\rangle|^2. \]

More generally, modern quantum theory writes

\[ P(j)=\operatorname{Tr}(\rho E_j), \]

where \(\rho\) is the state and \(E_j\) represents a physical measurement channel.

“Here, I would like to try to give a third interpretation and probe its utility in collision processes… I would then like to pursue the following idea heuristically… only a probability that a certain path will be followed will be determined by the function \(\psi\). One can perhaps summarize this, somewhat paradoxically, as: the motion of the particle follows the laws of probability, but the probability itself propagates in accord with causal laws.”
— Max Born, “Quantum Mechanics of Collision Processes” (1926)

Born’s rule works. It is among the most thoroughly tested rules in science. Yet it joins two different things: a continuously evolving amplitude and a discrete realised detector record. The formalism assigns their frequencies; it does not, by that fact alone, specify the microscopic process by which one receiver channel becomes the stable recorded mode.

Schrödinger objected because he wanted a real wave process. Einstein objected because he doubted that irreducible chance was the final layer of Nature. Bohr and Heisenberg emphasised that the theory concerned what could be said and measured under a complete experimental arrangement. The disagreement was not over the laboratory predictions. It was over what sort of world made them true.

What complex conjugation does to a real vibration

Write a real monochromatic compression using two quadratures:

The conjugate forms a quadratic intensity independent of the common clock origin. It does not remove a physical transverse rotation from Space. Relative phases remain in the interference term \(2\operatorname{Re}(z_1z_2^*)\) when two real wave contributions are added. The energy coefficient and action norm must still be calculated from the named compression and displacement variables.

Four physical questions remain distinct. Which incoming curves couple to each receiver mode, with what weight? Which distribution of unresolved receiver states turns those weights into observed frequencies? What prevents additional records after the first crossing? What produces the setting-dependent correlations between separated receivers? A square of an amplitude answers none of these questions by itself.

Quadratic receiver coupling and its metric

WSM offers a physical location for Born’s square. Let a travelling source pattern be \(\Xi\), and let receiver channel \(j\) possess a phase-matched standing-wave mode \(D_j\). Its coherent coupling is

\[ g_j=\langle D_j,\Xi\rangle_G. \]

Here \(G\) is the proposed positive action metric on physical receiver-mode coordinates. The Action page must derive the coordinates and this metric. A null direction is justified only when it changes no reconstructed physical field. Invariance under a common phase convention does not by itself make the phase direction a zero-norm vector; relative phase between core and sea remains physical. Stored energy, absorbed power and scattering flux must follow consistently from the same dynamics, with their coordinate and normalization maps shown. Their operators are not automatically identical.

For a passive linear receiver, with the force/response normalization fixed, cycle-averaged absorbed power is quadratic:

\[ P_{\mathrm{abs},j} = \frac{\omega}{2} \langle\Xi,\operatorname{Im}\chi_j\,\Xi\rangle \propto |g_j|^2 \]

near an isolated phase-matched receiver mode. This is an A-tier response result under its stated assumptions. It explains why a squared overlap, rather than a linear signed amplitude, belongs naturally to energy uptake.

Absorbed work is the time integral of this power. For a fixed-frequency transfer on the linear-action branch, the transferred action is that work divided by \(\omega\). Quadratic energy uptake is a response result; identifying normalized uptake with exclusive record frequencies is a further physical step.

Refinement theorem. If a physical receiver-basin measure is continuous, nonnegative and additive under lossless refinement of the same measurement, it is linear in action. These assumptions give the Born functional form; obtaining that basin measure from the waves remains open.

Action is the proposed invariant currency. On a harmonic or linear-action branch, a monochromatic mode obeys

\[ J=\frac E\omega=\frac1{2\pi}\oint p\,dq. \]

Under a boost, \(E'=K_{\hat n}E\) and \(\omega'=K_{\hat n}\omega\), so \(J'=J\). Probabilities must be frame independent; WSM should therefore formulate both its sea normalization and its receiver-transition action budget in action. The conventional zero-point reference is \(J_*=\hbar/2\), not “unit energy for every observer.”

In the receiver calculations below, \(J_0\) denotes the variance in projected, action-normalized mode coordinates and can be kept symbolic. The comparison with a Lorentz-invariant zero-point spectrum sets the reference modal action \(J_*=\hbar/2\) when matched to the usual quantum normalization. The physical sea spectrum and core response determine whether the receiver’s projected \(J_0\) has that value.

For a general periodic nonlinear family, canonical action obeys \(\omega=\partial E/\partial J\); the stronger equality \(E=\omega J\) requires the branch condition just stated. The displayed boost relation applies to the wave component carrying the stated Doppler factor.

Neutral selection theorem. Bounded action shares that remain martingales and eventually reach one exclusive absorbing vertex have final frequencies equal to their initial shares. This establishes a conditional conversion rule, including a sharp test of drift neutrality.

Readiness race and fold. One canonical quadrature pair per channel with the declared uniform action-shell measure gives the normalized first-winner law \(P_j=w_j/\sum_k w_k\). The fold has threshold \(F_c=A/4\). Neither result derives the physical shell or the mechanism that prevents other channels recording later.

A Conditional probability and threshold theorems. C Proposed receiver models. D Derive the wave coupling, invariant readiness measure and physical exclusivity, including any coordination between separated receivers.

The surrounding matter supplies WSM’s incoming waves and a physical route to calculating the sea’s spectrum, directional balance and reference frame. The universal intrinsic recurrence fixes a common frequency standard; the received frequencies also depend on source motion and propagation. A specified matter-network state may select the time slicing used by joint guidance. Its wave response and clock comparisons can then be tested together.

Incoming waves can also write a much slower pattern: nearby frequencies repeatedly move into and out of step, so their combined energy beats. The cycle-energy candidate makes wave speed follow those beats; the receiver’s response determines how they enter its own modes.

Stable receivers and wave-guided centres

Two complementary calculations now sharpen this picture. A stable receiver can preserve the incoming channel ratios while its configurations relax. Deterministic motion of centres can also bring a nonequilibrium ensemble towards the Born distribution through spatial mixing, without adding random kicks to its guidance law.

The observer has no supernatural role. A polariser, slit, screen, magnetic gradient or detector changes the physical decomposition \(\{D_j\}\). That determines which alternatives are available. It does not allow the experimenter to command which one occurs.

10. Solvay and Copenhagen: what can physics say?

At the 1927 Solvay Conference the founders confronted what they had made. The equations agreed with experiment, but the old picture of independently possessed properties had dissolved. Einstein tested the new theory with ingenious thought experiments. Bohr answered by examining the complete apparatus and showing that different experimental arrangements make different complementary quantities definite.

“Copenhagen” was never one perfectly uniform creed. Born supplied the probability rule. Heisenberg emphasised uncertainty, potentiality and state reduction. Bohr emphasised complementarity and the indivisibility of the phenomenon from the conditions under which it is recorded. Later textbooks often condensed this plurality into a recipe: prepare a state, evolve it, choose an observable, apply Born’s rule and update the state after a result.

The sophistication is important. Bohr was not saying that a human gaze magically manufactures an atom. He was saying that position, momentum, path and polarisation are not context-free labels read by an apparatus that leaves their meaning unchanged. The apparatus is part of the physical phenomenon.

Einstein accepted the predictions and rejected the elevation of the probability algorithm into a complete ontology:

“I cannot but confess that I attach only a transitory importance to this interpretation. I still believe in the possibility of a model of reality—that is to say, of a theory which represents things themselves and not merely the probability of their occurrence. On the other hand, it seems to me certain that we must give up the idea of complete localization of the particle in a theoretical model.”
— Albert Einstein, Herbert Spencer Lecture (1933), published as “On the Method of Theoretical Physics” (1934)

WSM accepts the contextual lesson while rejecting the need for an unknowable substrate. A slit, polariser or magnet changes the real mode decomposition and receiver boundary conditions. The physical question changes. One Vibrating Space continues to exist and move throughout the experiment.

11. Von Neumann, mixtures, decoherence and one actual record

John von Neumann gave quantum theory its rigorous state-and-measurement architecture. Pure states are rays in Hilbert space; observables are operators; statistical mixtures are density matrices; composite systems use tensor products; ideal outcomes are represented by projections. Modern measurements generalise the projectors to positive operators \(E_j\), giving

\[ P(j)=\operatorname{Tr}(\rho E_j). \]

This language answers an essential question that ordinary words blur. A coherent alternative is not the same thing as ignorance about a fixed alternative. For the two photon modes \(|HV\rangle\) and \(|VH\rangle\), compare

\[ \rho_{\rm ent}=|\Psi^-\rangle\langle\Psi^-|, \qquad |\Psi^-\rangle=\frac{|HV\rangle-|VH\rangle}{\sqrt2}, \]

with the classical mixture

\[ \rho_{\rm mix}=\frac12|HV\rangle\langle HV| +\frac12|VH\rangle\langle VH|. \]

The entangled state contains off-diagonal phase relations; the mixture does not. Interference and rotated-basis coincidence experiments distinguish them. Yet either wing of the singlet alone has the reduced state

\[ \rho_A=\rho_B=\frac I2. \]

The whole relation is pure while each local description is maximally mixed. This is the mathematical heart of entanglement.

Measurement then exposes the famous dual structure. Between records, the quantum state evolves linearly and unitarily. At a record, the textbook state is conditioned on one outcome. Decoherence explains much of the intervening physics: the apparatus becomes entangled with an enormous environment, relative phases between macroscopically distinct records become inaccessible, and a stable pointer basis emerges. It explains why alternatives cease to interfere in practice. It does not, by itself, say why this experiment contains one actual record rather than the unobserved sum of all records.

“In the case of the waves of wave mechanics we have no idea what is waving… and do not ask the question… There is no hint in the mathematics that the actual phenomenon is a minute flash at some particular point… It is only in applying the rule, relating the probable location of the flash to the intensity of the wave, that indeterminism enters the theory.”
— John S. Bell, “Six Possible Worlds of Quantum Mechanics”

WSM supplies a visual candidate for the missing physical step. The state vector records real wave relations; decoherence is differential phase scrambling into uncontrolled e-spheres; each detector channel is a possible stable standing-wave mode. Incoming curves deform the receiver, and a stable nonlinear reclosure would be the persistent record. Under the readiness and selection assumptions of §9, the reduced models reproduce normalized Born frequencies. A first threshold crossing and exclusion of every later competing record are distinct requirements. D Produce both from the same real-wave dynamics.

Part I continued · The unfinished argument about reality

12. Einstein, Podolsky and Rosen: is the wavefunction complete?

By 1935 quantum mechanics was already powerful. Einstein, Boris Podolsky and Nathan Rosen asked whether it was complete.

Their original thought experiment concerned two systems prepared with correlated position and momentum. After separation, a measurement on one side allows a corresponding property of the other to be predicted. If the distant measurement cannot physically disturb the remote system, EPR argued, then the remote property must already be an element of reality. Quantum mechanics did not assign simultaneous sharp values to both quantities, so they concluded that its description was incomplete.

“One is thus led to conclude that the description of reality as given by a wave function is not complete.”
— Albert Einstein, Boris Podolsky and Nathan Rosen (1935)

Schrödinger responded that year by naming the central feature entanglement. The state of the whole may be definite while neither part has its own independent pure state. This was not a small correction to classical probability. It was a new kind of composition.

“By the interaction the two representatives (or ψ-functions) have become entangled.”
— Erwin Schrödinger, “Discussion of Probability Relations between Separated Systems” (1935)

The later Bohm spin version makes the issue transparent. A pair is prepared in the singlet relation

\[ |\Psi^-\rangle = \frac{|+\rangle_A|-\rangle_B-|-\rangle_A|+\rangle_B}{\sqrt2}. \]

If both analysers are aligned, the singlet outcomes are always opposite, yet each local outcome is individually random. Perfect anticorrelation for aligned analysers alone does not exclude every shared-direction model. For a shared unit direction \(\hat\lambda\), the local rules \(r=\operatorname{sgn}(\mathbf a\cdot\hat\lambda)\) and \(s=-\operatorname{sgn}(\mathbf b\cdot\hat\lambda)\) reproduce that feature. With uniformly distributed \(\hat\lambda\), however, different analyser axes give \(E(\theta)=-1+2\theta/\pi\), whereas the singlet gives \(E(\theta)=-\cos\theta\). At \(60^\circ\) these are \(-1/3\) and \(-1/2\). Bell’s inequality tests the incompatibility across several setting pairs.

EPR did not discover a faster-than-light message. They exposed a dilemma about separability, locality and completeness.

13. Bohm: reality returns, but locality does not

David Bohm transformed the debate twice. He recast EPR in terms of two-valued spin measurements, giving the form later used by Bell. Then, in 1952, he constructed an explicitly causal and deterministic interpretation of nonrelativistic quantum mechanics.

In Bohm’s theory, particles have positions and are guided by a real wavefunction. The theory reproduces ordinary quantum predictions, but the guiding law for many particles is nonlocal: the configuration of the whole enters the motion of each part.

This was historically decisive. It showed that quantum statistics did not logically force the abandonment of objective reality or determinism. It also showed the price of retaining them in that form: nonlocality was already present.

“But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be introduced… with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the ‘observer,’ could be eliminated.”
— John S. Bell, “On the Impossible Pilot Wave” (1982)

“Why is the pilot wave picture ignored in text books? Should it not be taught… to show that vagueness, subjectivity, and indeterminism are not forced on us by experimental facts, but by deliberate theoretical choice?”
— John S. Bell (1982)

Bohm later spoke of wholeness and the danger of treating the world as independently existing fragments. WSM shares that relational instinct but proposes a different ontology. It seeks to remove the remaining point particles and the fundamental configuration-space wave, replacing both with finite centres and extended relations of one real Space.

The joint information remains essential. A relation such as \(\Gamma_{AB}\) records how the two ends belong to one preparation and respond to their analysers. WSM seeks its physical support in the connected waves of Space. Writing this information on configuration space need not make that mathematical space another substance; the joint-guidance construction below states explicitly how such information could enter centre motion.

“The notion that all these fragments are separately existent is evidently an illusion.”
— David Bohm, Wholeness and the Implicate Order (1980)

The WSM connection can now be made more precise. The free de Broglie phase and the centre’s intrinsic recurrence select the same uniform velocity. For an interfering wave, the amplitude profile contributes to the changing guidance; the open core’s incoming and outgoing action also enters its centre balance.

That ambition raises a harder question than simply declaring everything connected. Can a real-space wave theory reproduce the exact nonlocal correlations without reducing them to local hidden instructions?

John Bell supplied the test.

14. Bell: the line Nature crossed

Bell’s 1964 theorem did not say that every realistic theory was impossible. Bohm’s model was already a counterexample. Bell showed that a broad class of locally causal completions cannot reproduce all quantum predictions.

“…there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote.”
— John S. Bell, “On the Einstein Podolsky Rosen Paradox” (1964)

If a common cause \(\lambda\) is distributed independently of later settings \(\mathbf a,\mathbf b\), and each outcome depends only on its local setting and relevant past data, the joint probability factorises:

\[ P(r,s|\mathbf a,\mathbf b) = \int d\lambda\,\rho(\lambda) P_A(r|\mathbf a,\lambda) P_B(s|\mathbf b,\lambda). \]

This structure implies a Clauser–Horne–Shimony–Holt bound

\[ |S|\leq2. \]

For a spin singlet, quantum theory predicts

\[ P(r,s|\mathbf a,\mathbf b) = \frac14\left(1-rs\,\mathbf a\cdot\mathbf b\right), \]

\[ E(\mathbf a,\mathbf b) = \sum_{r,s=\pm1}rs\,P(r,s|\mathbf a,\mathbf b) = -\mathbf a\cdot\mathbf b, \]

and suitable settings give

\[ |S|=2\sqrt2. \]

For photon polarisation, physical analyser angles are represented by doubled angles on the Bloch sphere, yielding correlations of the form

\[ E(\alpha,\beta)=\pm\cos 2(\alpha-\beta), \]

with the sign fixed by the prepared Bell state.

Bell’s bound is not a generic “straight line” that quantum theory bends. It follows from a precise causal factorisation. Shared ancient waves, local conservation laws, common classical orientations and two independent detector races all remain Bell-bounded if they enter only through \(\lambda\) and local response functions.

Experiments violate Bell inequalities while preserving random local marginals. What they exclude is a setting-independent table of local answers carried from the source, under the ordinary independence assumptions. They do not by themselves prove consciousness, controllable faster-than-light signalling, irreducible randomness or the literal splitting of worlds.

Bell made the reality question experimental.

15. What the laboratories actually create

The phrase “two entangled particles” can hide the most important physical fact: the preparation creates one coherent relation with two separated ends.

Entangled photons

A common modern source uses spontaneous parametric down-conversion. A pump wave enters a nonlinear crystal. With low probability, the crystal transfers one pump-frequency excitation into a signal–idler pair of allowed modes. Energy and phase matching constrain the whole creation process:

\[ \omega_p=\omega_s+\omega_i, \qquad \mathbf k_p\approx\mathbf k_s+\mathbf k_i, \]

with the finite crystal supplying a calculable phase-matching width rather than an infinitely sharp equality.

The corresponding classical three-wave action balance is the Manley–Rowe relation

\[ -\dot J_p=\dot J_s=\dot J_i. \]

Thus one complete conversion has \(\Delta J_p=-\hbar\) and \(\Delta J_s=\Delta J_i=+\hbar\), while \(\omega_p=\omega_s+\omega_i\) preserves energy. The pair is one coherently created bilocal relation, but a successful pair measurement contains two endpoint absorptions. Exclusivity means that only one joint channel pair \((r,s)\) completes—not that one quantum of action is divided between two detectors.

Entanglement is created when two or more alternatives are made coherent and no physical record distinguishes them. A polarisation source may prepare, ideally,

\[ |\Psi^-\rangle = \frac{|H\rangle_A|V\rangle_B-|V\rangle_A|H\rangle_B}{\sqrt2}. \]

This is not a statistical mixture in which one photon was secretly horizontal and the other secretly vertical. The relative phase between the two creation alternatives is experimentally active. Change it, preserve it or destroy it, and the later coincidence table changes.

The source can be written without picturing two pre-labelled pellets. Let \(G^T\) be the transverse response from a creation point \(z\) to the two output regions and let \(\chi^{(2)}\) be the crystal’s nonlinear response. The pair kernel has the form

\[ \Xi_{ij}(x_A,x_B) = \int d^4z\, G^T_{im}(x_A,z) G^T_{jn}(x_B,z) \chi^{(2)}_{pmn}(z)\,\mathcal E_p(z). \]

Projection onto the two allowed polarisations at each end gives the \(2\times2\) source matrix

\[ C_{\lambda\mu} = \epsilon_{\lambda}^{i*}\, \Xi_{ij}\, \epsilon_{\mu}^{j*}. \]

A pure two-polarisation pair is separable exactly when this matrix has rank one; it is entangled when its rank exceeds one—equivalently, for a nonzero \(2\times2\) pair, when \(\det C\neq0\). The antisymmetric Bell source is represented by

\[ C=\frac1{\sqrt2} \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}. \]

For analyser vectors \(u_+(a)=(\cos a,\sin a)\) and \(u_-(a)=(-\sin a,\cos a)\), the joint amplitude is \(M_{rs}=u_r(a)^TCu_s(b)\). Born weighting gives

\[ P_{++}=P_{--}=\frac12\sin^2(a-b), \qquad P_{+-}=P_{-+}=\frac12\cos^2(a-b), \]

and therefore \(E(a,b)=-\cos2(a-b)\). This calculation identifies exactly what is connected: the coherent source tensor and its two endpoint projections. Experiments that spectrally resolve SPDC identify imbalance between the two decay paths as a direct cause of reduced polarisation-entanglement quality.

At each wing, wave plates and a polarising beam splitter define a measurement basis. The two output ports feed detectors. The apparatus records a setting, an output label and a time tag. Only after the two local records are compared are paired trials identified and the four joint frequencies estimated:

\[ P(++|a,b),\quad P(+-|a,b),\quad P(-+|a,b),\quad P(--|a,b). \]

Quantum-optical controls probe the detector-response structure itself. An ideal single-photon source sent to two outputs is antibunched, with \(g^{(2)}(0)=0\): one prepared excitation does not become two simultaneous complete detections. Conversely, two indistinguishable photons arriving at a balanced beam splitter show Hong–Ou–Mandel interference and leave together. These are not decorative paradoxes; they test exclusivity, indistinguishability and coherent two-mode addition.

No isolated click displays “nonlocality.” The nonlocal fact is the setting-dependent structure of this joint table together with the impossibility of decomposing it into Bell’s local form.

Entangled electrons

Electron spins can be entangled through interactions that conserve total angular momentum, through exchange coupling in neighbouring quantum systems, or by entanglement swapping. In a landmark loophole-free experiment, remote electron spins associated with diamond defects were first entangled with emitted photons. Interference and a heralding detection then projected the distant electron spins into one entangled relation. Each spin was read locally using a separately chosen basis.

The physical implementation differs from a photon polariser, but the logical measurement is the same: prepare a pair relation, choose two local decompositions, produce two records, and test their joint statistics.

For massive particles, a Stern–Gerlach magnetic gradient is the historical model of a spin analyser. Modern electron-spin experiments often rotate the spin with microwave pulses and convert its state into a local fluorescence or charge record. “Spin up” never means absolutely up. It means the output channel selected relative to the physical analyser axis used in that trial.

The experimental arc

Freedman and Clauser observed a Bell-inequality violation with photons in 1972. Aspect, Dalibard and Roger changed analyser conditions during the photons’ flight in 1982. In 2015 independent experiments closed the principal detection and locality loopholes in the same test. In 2017 entangled photons distributed from the Micius satellite violated a Bell inequality across roughly 1,200 kilometres.

The experiments do not show that one detector sends a readable instruction to the other. The ideal maximally entangled law has fair random local marginals:

\[ P(r|\mathbf a,\mathbf b)=\frac12, \qquad P(s|\mathbf a,\mathbf b)=\frac12. \]

The remote setting changes no controllable local frequency. The correlation appears only when records are later compared through ordinary communication.

This is what any WSM account must explain: not a mystical influence visible at one detector, but one relation that produces two local records and refuses local factorisation.

16. Were the two outcomes genuinely open—or merely unknown?

The usual alternatives are often stated too crudely.

If “fixed” means that each member carried a local answer for every possible analyser direction, Bell experiments exclude that model under measurement independence. If “open” means that nothing physical existed until a conscious person looked, the experiments do not establish that either.

Kochen–Specker contextuality sharpens the point even without a distant partner: quantum observables cannot all possess one noncontextual table of predetermined values that preserves their required relations across every compatible measurement arrangement. This does not abolish reality. It says the actual property belongs to the real system–apparatus relation, not to an isolated object carrying answers to every unperformed question.

A more precise possibility is contextual realism:

• the source prepares a real invariant relation, not two complete answer tables;
• the analysers physically define the channels available in the chosen bases;
• those channels are genuine possible receiver-reclosure modes;
• one joint receiver branch becomes the actual pair of records;
• consciousness neither creates the channels nor chooses the branch.

Two causal versions remain possible. The complete global wave state and boundary data may select the branch deterministically, while no outcome was locally preassigned for unrealised settings. Or nonlinear receiver reclosure may contain irreducible stochasticity. Bell tests distinguish neither. They constrain the causal structure, not every possible ontology beneath it.

The measurement setting determines which physical question is implemented. It does not determine which answer the experimenter receives.

Part II · Spherical matter in one Vibrating Space

17. One Vibrating Space and the e-sphere

Real longitudinal compression waves

“To deny the ether is ultimately to assume that empty space has no physical qualities whatever.”
— Albert Einstein, “Ether and the Theory of Relativity” (1920)

“Physical objects are not in space, but these objects are spatially extended. In this way the concept ‘empty space’ loses its meaning.”
— Albert Einstein, note dated 9 June 1952 to the fifteenth edition of Relativity: The Special and the General Theory

P1 states that Space is the one nearly rigid, slightly elastic wave medium and that its primitive motions are longitudinal plane waves. Because no second substance exists to bound, create or interrupt it, Space is necessarily infinite, eternal and continuous. Its regions remain connected to their neighbouring regions and vibrate backwards and forwards through small distances. A compression travels because each region compresses the next region and then returns towards its balanced position.

We represent this real motion with longitudinal plane waves travelling in every direction. A plane wave has a flat plane of equal phase. The compression advances at right angles to that plane, and Space vibrates backwards and forwards in exactly the same direction that the wave travels:

The complete motion is the overlap of these waves over every direction:

Every term is a vibration of the same Space. The enduring connection of neighbouring regions makes Space solid in the ontological sense. Ordinary material solids are later structures made from bound e-spheres; their atomistic shear modes do not define the primitive vibration of Space.

How all-direction plane waves form an e-sphere

An electron is an open, stable, spherical standing-wave organisation formed where longitudinal plane waves from every direction continually interact. The waves converge towards the wave centre, cross the centre and continue. A wave approaching from one side becomes a wave receding on the other side because it has passed through the centre; it remains the same longitudinal compression wave.

For any chosen e-sphere, its finite observable Huygens sphere is the all-direction relation through which other e-spheres’ out-waves collectively form its in-waves. Every e-sphere is the centre of its own changing sphere. These spheres overlap, and matter and organised structure continue beyond every one of them; the observable boundary is neither an edge of matter nor an edge of Space. If matter ended, boundary e-spheres would lose equal all-direction support and an isolated finite domain would collapse. External matter therefore supplies the continuing Mach–Huygens support of every local sphere. This is the WSM physical origin proposed for the non-collapsing large-scale effect called dark energy; its magnitude is a D calculation. As the waves enter and cross the e-sphere core, directional \(E_d\) changes their \(c'\), travel time, curve and accumulated phase so the spherical vibration continually recloses. Wavelength follows only after the relevant fixed-position frequency has been specified.

Why a passing plane wave leaves with a half-sphere curve

Take an e-sphere of radius \(R_e\) and a plane wave advancing through it at the balanced exterior speed \(c_0\). For a ray at impact parameter \(b\), the half-chord is

Start the clock when the flat wavefront is tangent to the front of the e-sphere. That ray reaches the near surface after

When the undisturbed exterior plane has advanced one radius and reached the centre plane, the total time is \(T=R_e/c_0\). The ray therefore has only \(x_b/c_0\) inside the e-sphere, yet it must cross the complete chord \(2x_b\). Its chord-effective speed is consequently

At that same instant every ray reaches the corresponding point \(+x_b\) on the far hemisphere. The initially flat wavefront has therefore acquired an exact half-sphere curve. For the normalized e-sphere \(R_*=\sqrt3/2\), the central ray crosses the diameter \(\sqrt3\) while the exterior carrier advances only the radius \(\sqrt3/2\).

The result fixes a travel-time condition, not a demand that every point inside have one uniform local speed. If \(c'\) varies along a chord, the exact condition is

Thus the harmonic path-average of \(c'\) is \(2c_0\), and the One Law turns the half-sphere boundary into a precise directional \(E_d\) constraint. After the curve leaves the e-sphere, its greater area spreads the ordered wave energy: \(E_d\) and \(c'\) fall below those of the flatter carrying plane, so the curve becomes retarded, wider and flatter as it travels. The fast chord formation and the slower far-field curve are successive parts of one physical process.

The equal all-direction phase sum gives

The spherical compression and extension can be written

with its linked quarter-cycle longitudinal motion

At maximum compression or extension, the local velocity is zero. A quarter-cycle later, the regions of Space pass through their balanced positions with maximum radial velocity. Another quarter-cycle reverses the compression. These are successive phases of one real spherical vibration.

A These are uniform-frequency linear-wave identities. They do not yet establish the autonomous variable-speed core at P3’s radius. Its nonlinear profile and stability must follow from the all-direction recurrence, without a prescribed core source or reflecting shell.

Why the spherical e-sphere remains stationary

The stationary construction uses balanced all-direction timing, with no preferred reconstruction displacement of the centre. Its simplest local ansatz takes \(E_d\) to be direction-independent at each radius. Spherical symmetry alone is weaker: a rotationally covariant distribution can depend on \(r\) and \(\hat{\mathbf n}\cdot\hat{\mathbf r}\). Under the stronger direction-independent ansatz, the One Law

then gives the same local propagation speed \(c'\) in every direction at that radius. With a common fixed-position crest frequency \(f_e\) in the same coordinates,

equal \(c'\) and equal frequency give equal wavelength in every direction:

Every directional wave returns to the same phase relation at the same centre. Every directional change is balanced by the corresponding change from the opposite direction. Successive waves therefore rebuild the e-sphere at the same position in Space. Its spherical equality is the physical cause of its stationary centre.

A The \(j_0/j_1\) identities are exact under the linear-wave premises. B A stable stationary e-sphere must restore its balanced all-direction phase relation. C Local directional isotropy is the displayed profile ansatz. D Derive the self-consistent profile and its stability.

18. Matter, antimatter and universal clocks

Opposite radial phase

An electron and positron are the same kind of spherical standing-wave organisation held in opposite phase relative to the background plane waves:

When the electron pattern is compressing, the positron pattern is stretching. When regions of Space in the electron recurrence move radially inward, the corresponding regions in the positron recurrence move radially outward. Half a cycle later both motions have reversed while their opposition remains.

The phase difference is physical because both e-spheres are continually measured against and maintained by the same background sea of plane waves. It is a relation between complete physical wave organisations, not an arbitrary sign attached to an isolated symbol.

A real core–sea phase diagnostic

For matching monochromatic templates, the normalized cycle correlation records the radial lock:

P3’s opposed branches give opposite signs of this diagnostic. The measured electromagnetic current and its phase-odd coupling must still be derived from the outgoing curves. An angular \(2\pi\) operation must leave the physical charge unchanged: it cannot simply exchange the electron and positron radial locks.

A cycle-averaged bilinear is not the instantaneous compression. For a single harmonic \(\psi=u e^{-i\omega t}\), \(\bar\psi\psi=\bar u u\) has lost the carrier oscillation. Relative phases that affect later interference must remain in the real core–sea pattern. Neither that bilinear nor a count of phase labels supplies a conserved charge current by itself.

Universal cosmic clock: intrinsic identity and local sea lock

The common background wave sea maintains one intrinsic rest-frequency identity \(\omega_e\) for every e-sphere. Electron and positron are the two opposite radial-phase branches of that same recurrence. Motion cannot create another sea or release either branch from its phase opposition:

The physical lock is to the incoming wave relation actually encountered at the e-sphere centre. Resonant stability requires the waves arriving from every direction to maintain the phase relation that reconstructs one complete recurrence. A drift among their centre-encountered phases would destroy the common spherical organisation.

This universal intrinsic identity does not require every component of a fixed-Space Fourier decomposition to have the same coordinate frequency. In the axial construction below, the two fixed-Space Fourier frequencies are unequal, while both are encountered at the moving centre at the same rate \(\omega_e/\gamma\). Electron and positron share that motion-dependent rate and remain exactly \(\pi\) apart.

Experiments constrain this construction without directly observing WSM radial phase. Electron and positron \(g\)-factors agree to about two parts in \(10^{12}\); positive and negative muons circulating at \(\gamma=29.33\) display the same relativistic lifetime behaviour; and proton and antiproton charge-to-mass magnitudes agree to 16 parts per trillion. These results strongly constrain any matter–antimatter difference under motion while leaving the WSM wave mechanism to be calculated and tested. Electron–positron comparison · moving \(\mu^+\) and \(\mu^-\) · proton–antiproton comparison.

Every free electron therefore carries the same resonantly selected rest-frequency identity. Atomic clocks ultimately compare stable recurrences made from these same e-spheres. Motion changes the rate of complete recurrence measured in background-Space time; it does not change which radial-phase branch exists.

Where the opposite phase is bound inside neutral matter

In the WSM proton model, a proton is an inseparable muonic-scale three-lobed recurrence

with two positive radial-phase lobes and one negative radial-phase lobe. This proposed bound pattern is intended to reproduce net charge \(+e\); the physical value and conserved current depend on the coupling of all three lobes. A neutral hydrogen atom adds the electron recurrence \((-)_e\):

Within this proton construction, neutral hydrogen has two positive and two negative radial-phase roles. They are bound roles in one proposed recurrence, not a population of free electrons and positrons. Interpreting opposite-phase roles as bound antimatter remains subject to the current, mass, magnetic-moment and stability calculations on the QCD and hadron page.

The two radial phases write oppositely oriented forward and rear curves onto the plane waves crossing the atom. The two orientations produce equal and opposite charge-like reconstruction pushes in neutral matter. Once the curved portions leave the e-spheres, however, both orientations spread over greater area, both have lower \(E_d\) and lower \(c'\) than their flatter carrying plane waves, and both widen, flatten and lag. The opposite charge-like pushes cancel, but the common delay does not. The resultant plane wave is therefore slightly delayed and changes other e-spheres so their centres reconstruct towards the neutral source—the proposed WSM origin of gravity.

P Opposite background-relative radial phase is P3’s electron–positron identification. D Derive its conserved charge current and measured coupling from the real waves.

19. A spherical phase-wave construction for spin

The complete sphere changes its phase organisation

The radial compression is not the complete proposed electron pattern. Longitudinal plane waves arriving from all directions overlap throughout the spherical region. Their moving equal-phase intersections can form a collective phase pattern. WSM seeks the electron’s angular behaviour in this spherical organisation and its lock to the incoming sea.

Space itself does not flow or rotate. Each constituent wave displaces neighbouring regions back and forth along its own propagation direction. Spherical phase rotation names the changing three-dimensional relation among those waves, not a rigid surface or circular bodily motion about an intrinsic axis.

A phase intersection can move faster than the constituent waves. It does not thereby carry a newly imposed change. The latter must be calculated from the physical propagation and response law.

Two proposed angular hands

The candidate angular organisation has two opposite hands \(h=\pm1\), represented schematically by

Here \(I_{\hat r}=I\hat{\mathbf r}\) denotes an oriented plane element, with \(I^2=-1\) in the stated three-dimensional geometric algebra. These symbols represent relations among real longitudinal waves. The physical proposal requires an angular-hand change to leave P3’s radial branch unchanged. Its identification with the two measured spin channels must follow from the receiver response.

The first-order hand-field identity

Within the uniform-\(k\) spherical control, define

The spherical Bessel identities give these first-order eigensectors. On the corresponding Helmholtz solution space, \(P_h\) are projectors and squaring \(I\nabla\) returns \(-\nabla^2\). This exact algebra does not by itself derive how the physical wave pattern transforms when an analyser is rotated.

The half-angle transport that the waves must realise

On a candidate two-component angular mode space, use one rotation operator acting on both channels:

The axis \(\hat{\mathbf s}\) describes a chosen physical orientation operation; it is not an intrinsic axle of the e-sphere. Battey-Pratt and Racey developed spherical-rotation geometry related to the Dirac equation. The displayed half-angle and sign identities are exact once this representation is chosen. The WSM construction must derive that transport from the incoming directional phases, the core and the analyser.

The sign after a nontrivial \(2\pi\) rotation loop must be protected under deformations within the same rotation class, leave the charge current unchanged and produce the observed relative-phase interference. Ordinary scalar or vector all-direction superposition alone does not establish this double-valued transport. A designed sign-changing mode along one path is a useful control, not yet the electron’s response under arbitrary physical rotations.

If the wave-derived doublet has the action calibration \(J_*=\hbar\), its generators would give

A The Bessel identity and stated \(SU(2)\) geometry are exact mathematical controls. C The physical candidate is spherical phase organisation of longitudinal waves. D Derive its independent angular modes, protected half-angle transport, charge-preserving coupling and stability from the same wave response.

20. The real-wave foundation of the Dirac equation

The candidate real-wave structure. P3 fixes the finite e-sphere core and opposite background-relative radial phases. Longitudinal waves arrive from every direction, cross the core and continue. Their uniform-wave control gives \(j_0/j_1\); §19 proposes the additional spherical angular organisation. Two radial branches and two proposed angular modes provide four candidate phase–hand configurations. Their independent amplitudes, first-order coupling, conserved current and stability must be derived before these coordinates constitute a physical Dirac theory.

Four candidate phase–hand configurations

The proposed coordinate construction combines two physical distinctions whose independent dynamical realisation remains to be checked:

1. two opposite radial phases relative to the background, \(q=+1\) and \(q=-1\);
2. two opposite directions of spherical phase rotation, \(h=+1\) and \(h=-1\).

A candidate Clifford factorisation

Introduce two two-component coordinate factors, with \(\tau_i\) and \(\sigma_i\) acting on them. They are intended to represent the radial and angular distinctions, but that physical identification requires a reconstruction map. Their proposed coordinate space is

The explicit bridge to Dirac’s matrices is

These matrices supply the following isotropic first-order candidate and its exact squared dispersion:

The Clifford algebra is exact. It does not by itself identify its component grading with the conserved charge of a real wave pattern. In particular, taking the physical charge-sign operator to be \(Q=\tau_3\otimes I_2\) would give

which is nonzero at general momentum. A rest-basis association cannot therefore make these two component blocks conserved electron and positron labels under the displayed free evolution. WSM must derive the real-wave reconstruction, conserved current and charge-conjugation map; P3’s opposite radial phases remain the physical premise to represent.

Why the notation is complex

The intended configurations are real wave organisations with compression and motion phases. Complex numbers conveniently record two real quadratures:

The mathematical \(i\) records a real quarter-cycle phase relation. A proposed rest-basis ledger may be written as

A moving or interacting e-sphere is one complete changing pattern. These labels are a proposed rest-basis bookkeeping scheme; their map to the displayed Dirac components must be established, rather than assuming that arbitrary component mixing changes physical charge. The entries are coordinates, not pieces from which an electron is assembled.

A real first-order wave factorisation

A one-dimensional slice through the all-direction recurrence makes the first-order algebra visible. Let \(a_{\rightarrow}\) and \(a_{\leftarrow}\) record the real compression and quarter-cycle motion of the longitudinal wave relations travelling in the two opposed directions along that slice. Let \(\mathsf J\) be the real ninety-degree operation between those two phases, with \(\mathsf J^2=-I\). Then

Applying the opposite directional operator and eliminating the other amplitude gives

With \(E=\hbar\omega\), \(p=\hbar k\) and \(mc_0^2=\hbar\omega_e\), this is

The opposed arrows name propagation directions within one real spherical vibration; they are not “reciprocal reconstruction grades” and do not multiply the four WSM configurations. The full three-dimensional factorisation replaces the one-axis sign by the Pauli–Clifford spherical rotation algebra.

What the Dirac operator describes

The intended WSM interpretation joins three real processes, provided their common generator is derived:

• the continuing rest-frequency compression and extension of the e-sphere;
• the changing directional phases of the longitudinal plane waves when the e-sphere moves;
• the two spherical rotations of the phase-intersection pattern.

The Pauli matrices supply the chosen angular algebra. The spatial Dirac matrices are a candidate coupling between the two coordinate factors. The rest calibration is

The Clifford anticommutation relations ensure that squaring the first-order equation returns

A The displayed Clifford representation and its factorisation are exact. Measured Dirac behaviour is a target for the wave construction. P P3 fixes opposite radial branches. C Two proposed angular modes give four candidate configurations. D Derive the independent modes, physical coordinate map, first-order coupling, conserved current, charge conjugation and \(g=2\) from the longitudinal waves.

21. Motion: the spherical e-sphere becomes a wave egg

Spherical equality fixes the resting centre

An exactly spherical e-sphere cannot translate while its directional wave conditions remain identical. Equal \(E_d\), equal \(c'\), equal wavelength and equal frequency continually rebuild the same spherical centre. Translation requires a three-dimensional directional asymmetry.

The moving wave egg

A moving e-sphere is rebuilt as an egg-shaped standing-wave organisation:

H-M1: axial reconstruction. During the rest-reference interval \(T_0=1/f_e\), the centre moves \(\Delta\mathbf X_0=\mathbf vT_0\). Axially, the rear and leading waves must reconstruct through the speeds

Their midpoint advances at \(v\), while their closing rate remains \(2c_0\):

For a wave direction \(\hat{\mathbf n}\), H-M1’s first-order all-direction extension is the centre-displacement projection

Exactly orthogonal directions are unchanged at first order; oblique directions interpolate continuously between the axial limits. The corresponding distances travelled during \(T_0\) are

where \(\beta=v/c_0\) and \(\lambda_e=c_0T_0\). These are reconstruction distances, not automatically simultaneous crest spacings. Under the additional common fixed-position frequency hypothesis H-M1F they also become wavelength ratios; that optional identification is not used below to obtain the reciprocal Doppler pair.

Speed limit and the unknown second-order shape

Physical wave-energy density and longitudinal wave speed must remain positive in every direction. In the leading axial direction H-M1 gives

As \(|v|\to c_0\), the leading reconstruction speed and energy density approach zero; beyond \(c_0\) their signs reverse. A finite e-sphere rebuilt by this all-direction reconstruction therefore cannot outrun the waves that sustain it.

The first-order projection has no spherical monopole change because \(\langle\mu\rangle_{S^2}=0\), where \(\mu=\cos\theta\):

Its first-order angular content is \(P_0+\beta P_1\). It does not determine the physical second-order egg. Write that still-unknown response as

If H-M1F additionally assigns a common fixed-position frequency, then

The trial choice \(a_2=0\) contains \(\mu^2=\tfrac13P_0+\tfrac23P_2\). This shows how an \(O(\beta^2)\) quadrupole can arise; it does not establish the physical second-order wave-egg solution. D Calculate \(a_2(\mu)\), the higher harmonics and their complete three-dimensional reclosure.

Universal intrinsic frequency and local sea lock

The complete e-sphere retains one intrinsic rest-frequency identity \(f_e\). Resonant stability requires the incoming waves from all directions to arrive with the phase relation that reconstructs one common moving centre. If \(\Theta_{\rm in}\) denotes the complete incoming sea relation encountered at \(\mathbf X(t)\), then electron and positron remain on its two opposite radial-phase branches:

This universal intrinsic identity does not require every component in a fixed-Space Fourier decomposition to possess the same coordinate frequency. Changed \(E_d\) entails changed \(c'\), hence changed travel time and accumulated phase. Wavelength follows only after the relevant fixed-position frequency has been specified.

De Broglie phase and the Lorentz factor

H-M2: phase-matched opposed-wave representation. Represent the egg’s axial recurrence by two coherent Fourier components propagating through balanced background Space at \(c_0\):

Resonant stability requires no phase slip at the centre \(x=vt\). The phase rates encountered there must therefore agree:

This is the same axial ratio as H-M1, but not the same kind of quantity: H-M1 states reconstruction speeds, whereas H-M2 fixes the ratio of the two fixed-Space Fourier frequencies.

Phase matching fixes the ratio but not the absolute scale. H-M3: geometric-mean rest-frequency closure supplies the remaining normalization:

H-M2 and H-M3 then give

The raw identity \((c_0+v)(c_0-v)=c_0^2(1-\beta^2)\) contains the same square root algebraically. It does not, by itself, select H-M3’s frequency normalization. Once H-M2 and H-M3 are stated, the wave sum factorises exactly:

The first factor reconstructs the wave centre at velocity \(v\). The second is the de Broglie phase modulation:

The phase wave may move faster than \(c_0\) because it is the changing position of an interference relation. No region of Space moves at that phase speed; every new physical change remains connected by the underlying longitudinal waves.

H-Q1: empirical energy–momentum calibration. To express the calculated phase quantities in conventional measured units, identify

It then follows that

At the moving centre, both incoming components are encountered at the same rate:

A complete centre-followed cycle therefore lasts \(T_v=\gamma T_0\). Electron and positron remain locked to this same incoming sea relation and exactly \(\pi\) apart; their common slowing in background-Space time does not change their opposition.

The fine-structure constant: rest-period displacement and moving-cycle closure

The full electron Compton wavelength and its rest-reference period are

In the ideal Bohr ground state, \(v=\alpha c_0\). During one rest-reference interval, its centre is therefore rebuilt a distance

Under the Bohr identification, the fine-structure constant is the fraction of one full electron rest wavelength by which the orbiting e-sphere centre advances during one rest-reference period. Using the 2022 CODATA values,

The same rest-interval displacement appears in H-M1’s axial reconstruction distances:

These are wavelengths only under the additional common fixed-position frequency hypothesis H-M1F.

Writing \(\bar\lambda_e=\lambda_e/(2\pi)=\hbar/(m_ec_0)\), the Bohr radius and circumference are

The number of rest-reference intervals in one Bohr orbit and the change of orbital direction during each such interval are therefore

The complete moving-centre phase cycle lasts slightly longer in background-Space time:

The interval-by-interval geometry is therefore precise: \(\alpha\) fixes the rest-period displacement, while the phase-matched moving clock supplies the additional \(\gamma_\alpha\) factor for one complete moving-centre cycle. These identities are exact under the stated ideal Bohr ground-state relations \(v=\alpha c_0\) and \(a_0=\bar\lambda_e/\alpha\). D Recover why the hydrogen bound state selects those relations rather than inserting the measured coupling by hand.

Reverse-engineering the curve between two e-spheres

For two electron-phase e-spheres separated by \(R\), the measured Coulomb force fixes the magnitude of the curve that one must write onto the plane waves reaching the other:

During one electron recurrence, either receiving e-sphere acquires the velocity change

Let \(\zeta(R)\) be the longitudinal displacement of the shallow curve relative to its carrying plane wave. In the small-slope limit the curve’s directional change is \(|d\zeta/dR|\). Equating that slope to the measured per-cycle velocity change and setting \(\zeta(\infty)=0\) gives

This is a physically motivated shallow-curve response ansatz: an incoming longitudinal curve slope changes the receiver’s directional reconstruction by the measured one-cycle velocity increment. Coulomb’s measured force fixes the required target but does not by itself prove that this is the unique response law. D Derive the relation and its coefficient.

The curve orientation supplies the sign: equal radial phases reconstruct apart and opposite radial phases reconstruct together. At \(R=1\ {\rm m}\), the magnitude ledger for either electron is

The real-wave chain is now explicit:

How the source half-sphere becomes a distant shallow curve

The e-sphere first writes a half-sphere curve by the \(2c_0\) chord condition. During free propagation the curve spreads across a growing transverse half-width \(w(R)\), so its height, slope and transverse curvature fall at different rates. For a shallow spherical cap,

When that half-width grows with propagation distance, \(w(R)\simeq R\), the Coulomb-matched curve gives

At one metre, \(\kappa_\perp=1.36743949\times10^{-26}\ {\rm m^{-1}}\), corresponding to a curvature radius \(7.31294\times10^{25}\ {\rm m}\). The curve has become extraordinarily flat and wide, yet within the declared response ansatz its remaining slope is exactly the directional push required by the Coulomb acceleration.

In calm Space the background longitudinal carrier retains \(c_0,\lambda_0,f_0\), plane order and plane spacing with distance. What changes is the finite source writing upon it: successive curves widen and flatten, their effect on a bound receiver decreases, and the common overlap of the source and receiver Huygens spheres diminishes. The receiver therefore recloses through the smaller energy gap and lower reciprocal history rate \(K=1/(1+z)\). This complete source-written curve → Huygens spreading → declining common overlap → receiver reclosure chain is the WSM redshift mechanism, not mere dimming and not stretching Space. D Calculate one universal \(K(D)\) for spectral lines, complete event envelopes, distance scaling and energy transfer.

The Schrödinger limit

The envelope step uses linear superposition of slow modes on the free branch, \(\Omega-\omega_e=c_0^2K^2/(2\omega_e)+O(K^4)\), with \(c_0K\ll\omega_e\).

At speeds small compared with \(c\), remove the rapid rest-frequency recurrence and retain the slowly changing de Broglie envelope. With \(mc^2=\hbar\omega_e\), the envelope obeys the free Schrödinger equation:

The wavefunction records the phase and amplitude of the extended moving e-sphere and its possible relations to other standing-wave structures.

A Under H-M2 and H-M3, centre phase matching and geometric-mean rest-frequency closure give the Lorentz–de Broglie factors exactly. B/C H-M1 supplies the physical \(c_0\pm v\) reconstruction and first-order all-direction projection. D Calculate the full side-sector solution, its \(O(\beta^2)\) shape and the complete clock and energy–momentum response.

Part III · Light, receivers and entanglement

22. Light: changing half-sphere curves on real plane waves

A stable bound state repeats one curve pattern

Atoms are bound standing-wave organisations of e-spheres. As a background plane wave crosses a bound e-sphere, the directional change in \(E_d\) changes \(c'\), crossing time and accumulated phase. Wavelength follows only after the relevant fixed-position frequency has been specified. The plane wavefront leaves with a real half-sphere curve imprinted upon it.

The source geometry is exact: while the exterior carrier advances one e-sphere radius at \(c_0\), each internal ray crosses its complete chord with harmonic path-average speed \(2c_0\) and reaches the far hemisphere. Once outside, the curved portion has greater area, lower \(E_d\) and lower \(c'\) than the flatter carrier, so it lags, widens and flattens as it propagates.

In a stable bound state, successive plane waves receive the same half-sphere curve in the same repeating position. The complete bound structure therefore reconstructs the same mode cycle after cycle.

A transition moves the curve on successive wavefronts

When the bound structure changes from one stable mode to another, it changes the position of the half-sphere curve imprinted on each successive plane wave passing through it.

The first wavefront receives the curve in the old position. The next receives it slightly displaced. Further wavefronts carry the ordered sequence until the new stable atomic pattern is established. This finite sequence of changing half-sphere curves is the emitted light train.

Because each curve is a real deformation of Space, its changing position on successive plane waves carries a corresponding change in:

• the displacement of Space on the wavefront;
• the velocity of that displacement;
• the compression and extension of neighbouring Space;
• the accumulated phase and, once the fixed-position frequency is specified, wavelength;
• the energy and wave action of the whole train.

One photon

A photon is one finite train of changing curves written during one complete bound-state transition. When its frequency, phase and curves match an allowed receiver mode, the train can rebuild that receiver into the corresponding stable standing-wave organisation.

Its frequency is the repetition rate of the curve pattern. Its energy is

when the complete train carries the universal wave-action unit. A free e-sphere continually repeats its stable curve pattern and does not emit a discrete train merely by existing. Radiation is written when a bound relationship changes.

Two photon helicities from longitudinal waves

The individual plane waves carrying the train remain longitudinal compression waves. Across the complete directional collection, the curve and phase pattern can change around the propagation direction in either of two handed orders. These are the two photon helicities:

The first angular harmonic on the Huygens ring supplies exactly those two hands. Its transverse geometry describes the arrangement of curves across many longitudinal plane waves; the vibration of every individual plane-wave component remains longitudinal.

The ring therefore supplies the candidate \(m=\pm1\) light hands. By itself it does not establish phase-reference redundancy, masslessness or the absence of a radiative longitudinal pole; those are the sharper D Hessian tests developed in the directional construction below.

Maxwell’s fields as organised curve relations

Take the two real transverse moments of the travelling curve pattern as \(\mathbf X\) and \(\mathbf Y\). They are the two independent first-moment components of the changing half-sphere curve across the Huygens ring, not added substances or primitive transverse vibrations of Space. Locality, isotropy, opposite parity and positive wave energy identify the following paired evolution as the minimal Maxwell target:

These are Maxwell’s free-wave equations after identifying the two moments with the electric and magnetic field description. The field vectors record the changing geometry of one real train carried by the waves of Space.

A The two ring harmonics and Maxwell algebra are exact inside the stated directional construction. B/C The changing half-sphere train is their real-wave interpretation. D The Space Hessian must derive the phase-reference null direction, Gauss constraint, two positive transverse poles and no additional radiative longitudinal pole.

Exact and conditional construction

23. Directional phase, two light hands and Maxwell geometry

Each symbol in this section refers to the curve, timing or phase relations written on real longitudinal plane waves. The individual waves continue to move Space backwards and forwards in their own directions of travel. The scalar-plus-vector connection, electric and magnetic fields and two helicities describe the collective relation among many such directions. Tier A marks the exact angular and field mathematics; tiers C and D mark the task of deriving its physical coefficients.

Two photon helicities from collective longitudinal Space waves

The two observed photon helicities are not transverse vibrations of individual Space waves. The light train is collective: every constituent Space wave remains longitudinal along its own direction, while the all-direction phase and curve relation contains the two transverse geometrical hands resolved by an analyser.

Let \(a(x,\hat{\mathbf n})\) denote a small phase-reference or curve perturbation associated with the Space waves arriving from direction \(\hat{\mathbf n}\). Its four-dimensional coherent sector can be identified both by scalar and vector angular moments and, more strongly, by local phase additivity.

It need not be selected merely by inspection. Let \(\alpha_x(v)\) be the phase increment assigned at \(x\) to an infinitesimal displacement \(v\). If local phase transport is continuous, additive and homogeneous,

\[ \alpha_x(v+w)=\alpha_x(v)+\alpha_x(w), \qquad \alpha_x(\lambda v)=\lambda\alpha_x(v), \]

then it is a spacetime one-form: \(\alpha_x(v)=A_\mu(x)v^\mu\). Restrict it to the future-null directions \(\ell_{\hat n}^{\mu}=(1,\hat{\mathbf n})\):

\[ \boxed{ a_{\hat n}=A_\mu\ell_{\hat n}^{\mu} =A_0+\hat{\mathbf n}\cdot\mathbf A. } \]

Thus local additive phase implies a spacetime one-form and hence exactly \(l=0\oplus1\) directional content. Higher harmonics describe finite-path curvature, multipolar sea distortion or internal e-sphere deformation rather than extra electromagnetic components. This is exact mathematics conditional on the additive local-phase premise. D Make that premise physical.

The connection also fits WSM’s existing Doppler geometry. Under a Lorentz boost a null direction transforms projectively as

\[ \Lambda\ell_{\hat n}=K_{\hat n}\ell_{\hat n'}, \qquad K_{\hat n}=W-P\,\hat{\mathbf n}\cdot\hat{\mathbf v} =\gamma(1-\boldsymbol\beta\cdot\hat{\mathbf n}), \]

with \(a'_{\hat n'}=K_{\hat n}^{-1}a_{\hat n}\) and aberration measure \(d\Omega'=K_{\hat n}^{-2}d\Omega\). The coefficients transform as a Lorentz four-potential one-form—or, after raising the index, as a four-vector. The same reciprocal identity \(W^2-P^2=1\) that generated the Lorentz–de Broglie factor now supplies a Lorentz-covariant Huygens connection.

\[ A_0(x)=\frac1{4\pi}\int_{S^2}a(x,\hat{\mathbf n})\,d\Omega, \qquad \mathbf A(x)=\frac3{4\pi}\int_{S^2}\hat{\mathbf n}\,a(x,\hat{\mathbf n})\,d\Omega. \]

The corresponding \(l=0\oplus l=1\) projection is \((P_{01}a)(\hat{\mathbf n})=A_0+\hat{\mathbf n}\cdot\mathbf A\). Orthogonality of spherical harmonics gives the exact positive complement

\[ \boxed{ Q_{01}[a] =\frac{\mu_*^2}{2} \left[ \int a^2d\Omega -\frac1{4\pi}\left(\int a\,d\Omega\right)^2 -\frac3{4\pi}\left|\int\hat{\mathbf n}a\,d\Omega\right|^2 \right]\ge0. } \]

Equality holds exactly for \(a=A_0+\hat{\mathbf n}\cdot\mathbf A\). Thus a scalar directional field possesses a natural four-dimensional coherent sector and a positive residual measuring everything that cannot be represented by one scalar plus one ordinary vector. A local angular operator with the same kernel is

\[ K_\Omega=\Lambda^2L^2(L^2-2), \qquad L^2=-\Delta_{S^2}, \]

whose eigenvalues vanish for \(l=0,1\) and are positive for \(l\ge2\). The projector and operator are exact mathematics; identifying either as the Hessian of Vibrating Space is a C-tier action ansatz.

The ordinary spherical-harmonic projector \(P_{01}\) is orthogonal in the chosen e-sphere or sea rest frame. It does not commute with boosts on an arbitrary directional field: aberration can move higher harmonics into lower spherical moments. What is Lorentz closed is the additive connection sector itself, because \(a'_{\hat n'}=K_{\hat n}^{-1}a_{\hat n}\) preserves the one-form relation. Thus the connection is covariant, while an arbitrary split \(a=P_{01}a+(I-P_{01})a\) is not automatically observer independent.

A covariant angular-content diagnostic is the trace-free spherical Hessian

\[ \mathcal S_{AB}[a] =\left(\nabla_A\nabla_Ba\right)^{\rm TF}, \]

for which

\[ \boxed{ \int_{S^2}|\mathcal S_{AB}[a]|^2d\Omega =\frac12\left\langle a,L^2(L^2-2)a \right\rangle . } \]

Its kernel is exactly \(l=0\oplus1\). This is exact angular mathematics for the appropriate conformal weight. D Derive the Lorentz-covariant spacetime kinetic law and its coupling to higher content.

A crucial distinction follows. An \(l=1\) deformation of a particular spherical solution can be a translation zero mode of that solution. Translation symmetry alone does not turn it into a photon, does not create local gauge redundancy and does not eliminate a longitudinal radiative pole. That shortcut is Q-tier. The photon requires the spacetime connection and constraint structure below.

The underlying curvature calculation remains useful. For a normal displacement \(\Gamma(\hat{\mathbf n})\) of a geometric sphere of radius \(R\), the first variation of mean curvature is

\[ \delta H=-\frac1{R^2}(\Delta_{S^2}+2)\Gamma. \]

A preferred-curvature energy therefore has Hessian \((\Delta_{S^2}+2)^2\) and eigenvalues proportional to \([l(l+1)-2]^2\). It selects the full \(l=1\) translation triplet exactly. That is a real spherical-geometry result and a useful stiffness control; it is not the electromagnetic derivation. The physical e-sphere, an expanding Huygens wavefront and the dimensionless sphere of directions also need not share the same radius.

The complete direction sphere remains the physical construction. Finite angular quadratures provide numerical controls, with their symmetry limits stated explicitly.

Relational phase, cross-direction curvature and Maxwell

The stronger origin of gauge structure is not centre translation but relational phase. Absolute labels assigned to neighbouring Huygens reconstructions are conventional; phase mismatch around a closed chain is physical. Define

\[ D_{\hat{\mathbf n}} =\partial_0+\hat{\mathbf n}\cdot\nabla, \qquad \partial_0=\frac1c\partial_t, \]

and impose the local reference change

\[ \boxed{ a_{\hat{\mathbf n}} \longrightarrow a_{\hat{\mathbf n}}+D_{\hat{\mathbf n}}\chi. } \]

Because \(D_{\hat{\mathbf n}}\chi\) contains only \(l=0\) and \(l=1\) angular content, its moments transform as

\[ A_0\rightarrow A_0+\partial_0\chi, \qquad \mathbf A\rightarrow\mathbf A+\nabla\chi. \]

This transformation changes the zero reference used to describe relational phase. It does not relabel physical regions of Space and introduces no motion or flow. If matter couples through \(S_{\rm int}=\int d^4x\,d\Omega\,j_{\hat n}a_{\hat n}\), invariance of the mathematical representation under this phase-reference change requires

\[ \int d\Omega\,D_{\hat n}^{\dagger}j_{\hat n}=0, \]

which reduces in the scalar-plus-vector sector to the ordinary continuity equation \(\partial_0\rho+\nabla\cdot\mathbf J=0\). Thus the same phase-reference symmetry that removes the free longitudinal photon also forces conserved source coupling and, at the effective-field level, the Ward identity.

The boost weight of the directional source is then fixed rather than guessed. Since

\[ a'_{\hat n'}=K_{\hat n}^{-1}a_{\hat n}, \qquad d\Omega'=K_{\hat n}^{-2}d\Omega, \]

invariance of \(\int j_{\hat n}a_{\hat n}d\Omega\) requires

\[ \boxed{j'_{\hat n'}=K_{\hat n}^{3}j_{\hat n}.} \]

Consequently \(J^\mu=\int j_{\hat n}\ell_{\hat n}^{\mu}d\Omega\) transforms as a four-current. WSM now has a precise source target: the e-sphere’s directional in/out action imbalance must possess this boost weight and satisfy \(\partial_\mu J^\mu=0\).

The same reasoning gives a complete tensor hierarchy. A rank-\(m\) null-direction moment

\[ T^{\mu_1\cdots\mu_m} =\int f_{(m)}(\hat n) \ell^{\mu_1}\cdots\ell^{\mu_m}d\Omega \]

is Lorentz covariant when \(f'_{(m)}=K^{m+2}f_{(m)}\). Thus an energy-direction density has weight four,

\[ t'_{\hat n'}=K_{\hat n}^4t_{\hat n}, \qquad \boxed{ T^{\mu\nu}=\int t_{\hat n}\ell^\mu\ell^\nu d\Omega. } \]

Since \(\omega'=K\omega\), the economical relation \(t_{\hat n}=\omega_{\hat n}j_{\hat n}\) has exactly the required weight. Directional action transport generates current through its first null-direction moment and energy–momentum through its second. A sum of future-null wave-direction moments is traceless, \(T^\mu{}_{\mu}=0\); massive e-sphere stress-energy must also contain standing-wave interference, internal compression and extension or other non-null coherence. Null-direction moments alone cannot make rest mass.

The invariant failure of two directional phase reconstructions to agree is

\[ \boxed{ C_{\hat{\mathbf n}\hat{\mathbf n}'} =D_{\hat{\mathbf n}}a_{\hat{\mathbf n}'} -D_{\hat{\mathbf n}'}a_{\hat{\mathbf n}}. } \]

The directional derivatives commute, so this curvature is exactly unchanged by the local reference transformation. In the \(P_0\oplus P_1\) sector, define

\[ \mathbf E=\nabla A_0-\partial_0\mathbf A, \qquad \mathbf B=\nabla\times\mathbf A. \]

Direct expansion gives

\[ \boxed{ C_{\hat{\mathbf n}\hat{\mathbf n}'} =(\hat{\mathbf n}-\hat{\mathbf n}')\cdot\mathbf E +(\hat{\mathbf n}\times\hat{\mathbf n}')\cdot\mathbf B. } \]

The electric-like field is differential temporal phase slope. The magnetic-like field is oriented cross-direction curvature. No fundamental transverse or axial Space wave has been inserted: two longitudinal directions define the oriented area from which the curl emerges.

The construction is invertible:

\[ \mathbf E=\frac{3}{32\pi^2} \int d\Omega\,d\Omega' (\hat{\mathbf n}-\hat{\mathbf n}')C_{\hat{\mathbf n}\hat{\mathbf n}'}, \]

\[ \mathbf B=\frac{9}{32\pi^2} \int d\Omega\,d\Omega' (\hat{\mathbf n}\times\hat{\mathbf n}')C_{\hat{\mathbf n}\hat{\mathbf n}'}. \]

Therefore \(C_{\hat n\hat n'}=0\) for every pair if and only if \(F_{\mu\nu}=0\); locally the connection then contains only a removable phase reference. If \(\nabla_{\hat n}=D_{\hat n}-i(q/J_*)a_{\hat n}\), its commutator is proportional to \(C_{\hat n\hat n'}\). The Jacobi identity gives \(\partial_{[\lambda}F_{\mu\nu]}=0\), equivalently

\[ \nabla\cdot\mathbf B=0, \qquad \partial_0\mathbf B+\nabla\times\mathbf E=0, \qquad \partial_0=\frac1c\partial_t. \]

Half of Maxwell is therefore geometrical: these homogeneous equations express the consistency of reconstructing one relational phase through different Huygens directions. Varying the Maxwell action supplies the dynamical half, \(\partial_\mu F^{\mu\nu}=J^\nu\).

This coupling is nonlocal only on the internal sphere of directions: both directional components meet at the same physical spacetime event. It is therefore an ordinary local field construction, not Bell nonlocality. The pair-wide probability and boundary problem remains the separate task of §25 and §26.

Now take the isotropic quadratic closure action

\[ S_C=\frac\kappa2 \int d^4x\,d\Omega\,d\Omega'\, W(\mu)C_{\hat{\mathbf n}\hat{\mathbf n}'}^2, \qquad \mu=\hat{\mathbf n}\cdot\hat{\mathbf n}'. \]

Parity removes the mixed \(\mathbf E\cdot\mathbf B\) term. Angular integration returns

\[ S_C=\frac\kappa2\int d^4x \left(C_E\mathbf E^2+C_B\mathbf B^2\right), \]

\[ C_E=\frac{16\pi^2}{3} \int_{-1}^{1}W(\mu)(1-\mu)\,d\mu, \qquad C_B=\frac{8\pi^2}{3} \int_{-1}^{1}W(\mu)(1-\mu^2)\,d\mu. \]

For the minimal affine kernel \(W(\mu)=w_0+w_1\mu\), Maxwell’s relative coefficient \(C_B=-C_E\) is equivalent to one number:

\[ \boxed{\frac{w_1}{w_0}=4.} \]

The minimal choice \(W(\mu)=-(1+4\mu)\) gives

\[ S_C\propto\int d^4x\,(\mathbf E^2-\mathbf B^2) =-\frac12\int d^4x\,F_{\mu\nu}F^{\mu\nu}. \]

The sign-indefinite directional weight reconstructs the ordinary Lorentzian field Lagrangian inside the projected Maxwell sector, whose Hamiltonian density is proportional to \(\mathbf E^2+\mathbf B^2\). It cannot, however, be extended unchanged to every directional harmonic. If

\[ \widehat W_l=\frac12\int_{-1}^{1}W(\mu)P_l(\mu)\,d\mu, \qquad q_l=2(\widehat W_0-\widehat W_l), \]

then the Maxwell condition forces

\[ \boxed{q_2=-3q_1.} \]

A positive photon kinetic term would therefore make the unrestricted quadrupole sector a negative-kinetic-energy ghost; adding a positive mass would not cure it. The affine \(WC^2\) expression is an exact low-sector reconstruction, not a fundamental all-harmonic energy.

The clean ghost-free effective split is

\[ \bar a=P_{01}a, \qquad b=(I-P_{01})a, \]

\[ \boxed{ S^{(2)}=-\frac Z4\int F_{\mu\nu}[\bar a]F^{\mu\nu}[\bar a]\,d^4x +S_{\rm high}[b], } \]

where \(S_{\rm high}\) has separately positive time kinetics and the positive angular operator \(K_\Omega=\Lambda^2L^2(L^2-2)\), with \(\lambda_l=\Lambda^2(l-1)l(l+1)(l+2)\) for \(l\ge2\). Since \(D_{\hat n}\chi\) lies entirely in \(l=0\oplus1\), the high sector is separately invariant under the phase-reference change. This block decomposition is a ghost-free rest-frame effective ansatz. An exact Lorentz completion must use the covariant angular diagnostic or introduce appropriate spacetime fields. D Derive or replace it.

The same condition survives the eight cube-corner sum exactly:

\[ C_E^{\rm cube}=\frac29(3w_0-w_1), \qquad C_B^{\rm cube}=\frac29w_0. \]

The cube does not derive the number four, but it preserves the continuum Maxwell test without approximation. More generally the kernel predicts

\[ \frac{c_\gamma^2}{c^2} =-\frac{C_B}{C_E} =-\frac{w_0}{3w_0-w_1}. \]

Thus \(w_1/w_0=4\) is simultaneously the Maxwell and invariant-light-speed condition. It is a sharp D-tier target for the actual Space Hessian, not a coefficient to hide inside the finished theory.

Once the Maxwell form is obtained, \(A_0\) has no independent radiative kinetic term and imposes Gauss’s law; the longitudinal part of \(\mathbf A\) is reference redundancy. The quotient leaves two transverse light-cone poles. In the minimal unbroken connection sector a photon mass term is forbidden by local rephasing—not by ordinary translation symmetry. The quaternion–Clifford packaging then follows exactly: with normalised polar and axial fields \(\mathbf X,\mathbf Y\),

\[ \mathcal F_h=\mathbf X+hI\mathbf Y, \qquad \frac{hI}{c}\partial_t\mathcal F_h =\nabla_q\mathcal F_h, \qquad h=\pm1, \]

whose scalar part contains the divergence constraints and whose vector part contains the two Maxwell curls.

The ring is the polarisation slice—not the axial carrier

For propagation along \(\hat{\mathbf k}\), choose transverse gauge and parameterise the orthogonal Huygens circle by

\[ \hat{\mathbf n}(\varphi) =\mathbf e_1\cos\varphi+\mathbf e_2\sin\varphi. \]

The full-sphere vector moment restricts to

\[ a(\varphi) =\hat{\mathbf n}(\varphi)\cdot\mathbf A_\perp =A_1\cos\varphi+A_2\sin\varphi. \]

Circular combinations \(\mathbf A_h=(\mathbf e_1+ih\mathbf e_2)/\sqrt2\) give \(a_h\propto e^{ih\varphi}\): exactly the two spin-one helicities. But if each orthogonal constituent were asked to carry an axial modulation by its individual plane-wave propagation equation, \(\hat{\mathbf n}\cdot\mathbf k=0\) would force \(\omega=0\). The ring is therefore the photon’s transverse phase boundary; axial energy and momentum belong to the complete null all-direction solution.

At fixed \(\mathbf k\), the two complex transverse amplitudes form a Jones vector \(z\in\mathbb C^2\). Removing intensity and common phase gives the Poincaré sphere, \(\mathbb{CP}^1\simeq S^2\). Linear-polarisation angle \(\alpha\) appears as Stokes angle \(2\alpha\), producing Malus’s law and the doubled angle in photon Bell correlations. This Poincaré sphere, the physical e-sphere and the direction sphere \(S^2_{\hat n}\) are different geometries and must not be conflated.

As \(\hat{\mathbf k}\) ranges over momentum directions, the two circular-helicity line bundles have first Chern numbers \(\mp2\). A complete WSM photon must reproduce this global gluing. The topology tests the two-helicity structure; it does not by itself create masslessness or quantisation.

From a Maxwell mode to one photon

For a finite normal mode with action variable \(\mathcal J\), time translation, spatial translation and rotation about the propagation axis give the exact Noether relations

\[ E=\omega\mathcal J, \qquad \mathbf p=\mathbf k\mathcal J, \qquad J_{\hat{\mathbf k}}=h\mathcal J. \]

One derived wave-action step \(\Delta\mathcal J=\hbar\) would therefore produce, together rather than separately,

\[ E=\hbar\omega, \qquad \mathbf p=\hbar\mathbf k, \qquad J_{\hat{\mathbf k}}=h\hbar. \]

The Maxwell mode and its two helicities are the linear field problem. The universal action step and one stable receiver reclosure are the nonlinear receiver-reclosure problem. Neither should be used to pretend the other has already been solved.

The reduced receiver supplies an exact conditional threshold and first-winner rule; a physical latch or action-exhaustion mechanism preventing further records remains to be derived. The combined D-tier task is to derive the directional connection, Maxwell kernel ratio, action budget \(J_*=\hbar\), train profile, readiness shell and nonlinear coefficients from the living e-sphere rather than choosing them at the field or receiver level.

The decisive blind photon calculation is the quadratic expansion

\[ S^{(2)} =\frac12\int d^4k\,d\Omega\,d\Omega'\, \delta a_{-k,\hat n}\, \mathcal K(k;\hat n,\hat n')\, \delta a_{k,\hat n'}. \]

The complete off-shell Hessian must possess the reference-null vector

\[ \boxed{ R_{\hat n}(k)=i(k_0+\hat n\cdot\mathbf k), \qquad \int d\Omega'\, \mathcal K(k;\hat n,\hat n')R_{\hat n'}(k)=0 } \]

for every \(k\), not merely on a light-wave solution. After the higher angular modes are integrated out, the phase-reference quotient and Gauss constraint must leave exactly two positive-residue zeros at \(\omega^2=c^2k^2\). This single calculation decides whether the candidate connection is truly Maxwell or only a three-component elastic vector wave.

24. Absorption, Born probabilities and one receiver reclosure

Receiver deformation

A light train reaches many possible receivers because its constituent plane waves extend through Space. At each receiver, the changing half-sphere curves begin altering the receiver’s own standing-wave pattern.

A receiver whose frequency, phase and geometry match the arriving train can be driven from one stable bound mode towards another. Absorption is complete when the full sequence of incoming curves has rebuilt that receiver as the new stable standing-wave organisation.

The train’s temporal shape matters as well as its frequency and geometry. A matched rising wave can cancel outgoing leakage during loading and leave more action in the receiver. The same response calculation determines how well the receiver’s existing vibration provides a reference for the arriving sea.

Continuous receiver deformations, one stable reclosure

Many receivers or detector channels can begin small continuous deformations. Their phase matches and amplitudes determine how strongly the arriving curve train changes each possible receiver standing wave. The apparatus physically creates those possible standing-wave outcomes.

WSM identifies a discrete record with one stable receiver reclosure. Obtaining exactly one such record requires physical competition, inhibition or exhaustion of the other allowed channels. The conditional constructions in §9 specify this task; they do not make a unique first crossing equivalent to complete exclusivity.

A guided centre offers a direct connection between the extended wave and a particular receiver. The wave enters the available branches while one localized wave organisation follows one history. If capture absorbs that organisation, or exhausts its capacity to trigger this event, only one receiver completes the associated record. Guidance sets the branch population; receiver dynamics supplies capture, timing and the lasting change. For a centre-free light train, the corresponding allocating feature belongs in the joint source–train–receiver calculation.

For a harmonic or linear-action mode of energy \(E\) and angular frequency \(\omega\), write

The complex amplitude records two real wave quadratures. Given the quadratic action coupling, normalization and physical selection assumptions of §9, \(J_j\propto|g_j|^2\) and the conditional outcome law is

A bounded zero-drift action-share martingale with eventual exclusive absorption makes its initial shares the final probabilities. The stated invariant-shell race also gives those first-winner frequencies. Under their respective assumptions, repeated preparations reproduce Born’s rule. The same wave dynamics must still supply the distribution, neutral selection and inhibition or exhaustion that leave one stable record.

Why one click does not reveal a travelling pellet

A discrete detector record is one receiver rebuilt as a new stable standing-wave organisation. The light between source and receiver remains the extended finite train of changing curves. The local receiver reclosure and the extended causal wave train are two stages of one process.

A Born frequencies and discrete records are experimental constraints; the displayed selection theorems are conditional mathematics. C WSM locates recording in stable nonlinear receiver reclosure. D Derive its weights, readiness measure and exclusivity from the physical response.

25. Entanglement: one source-created wave relation

The source writes the pair

An entangled pair begins as one source-created pattern of curves, phases and spherical hands written into the shared waves of Space. The two eventual measurement regions are parts of that one extended relation.

The universal background supplies the physical connection among regions. The source supplies the phase relation belonging to this particular pair. Background connection alone does not determine the experimental answers.

Analysers create the possible reclosure channels

Each analyser changes the local standing-wave conditions and selects a basis in which the incoming spherical phase relation can rebuild a stable detector state. For a spin singlet, the axis-free total relation is represented by

Given the physical angular doublet, the stated singlet source and the quadratic outcome measure, the half-angle calculation gives

The measured result consists of two local records with a joint distribution that violates Bell’s local factorisation under its independence assumptions. WSM proposes to explain them through the shared source-created wave relation. A conditional calculation over joint alternatives does not yet derive the physical selection of those records.

No-signalling

For the normalized singlet joint weights, changing the distant analyser changes the joint decomposition while summing over its complete alternatives leaves the local marginal unchanged. The proposed whole-boundary reclosure must derive those joint weights, preserve complete local channels and reproduce this basis-independent marginal; only then does it yield Bell correlations without a controllable message channel.

A Bell inequalities, singlet mathematics, their experimental violation and no-signalling are established. B/C WSM locates the pair in one source-written extended wave relation. D The common action must generate its nonfactorisable joint reclosure and no-signalling marginals.

Exact and conditional construction

26. Shared Huygens boundaries, joint receiver channels and the singlet

The source creates a pair-specific phase and curve relation with two receiver ends. The following calculations distinguish exact shared-wave geometry and conditional quantum target laws from the still-required physical mode construction, measure and recording dynamics. A shared background connection is necessary to the WSM picture; it is not by itself a derivation of Bell correlations.

One vibrating system: settled pattern and new setting

Source, analysers and receivers are organisations of one vibrating Space. Their incoming and outgoing waves can carry mutually consistent phases. In a settled apparatus, a joint standing pattern can therefore encode the arrangement that has had time to affect it. This is the physical content of a coherently connected whole.

A new analyser change is a further wave event. In a retarded model it reaches a distant receiver only after the earliest permitted propagation delay. Calling an arriving wave the receiver’s future in-wave does not place an as-yet-unarrived change in its present compression. The coordinate \(t\) labels the order of real wave changes; it does not introduce time as another substance.

Write the all-direction network in named wave variables:

\(F_i\) names real incoming patterns; \(G_i\) outgoing curves; \(H_{ij}\) their propagation and angular mixing; \(\mathcal R_i\) the core response and any stable record; \(d_i\) the analyser hardware; and \(\zeta_i\) unresolved incident support. These are readings of the same vibrating Space. They must be derived from one physical state and action, not assigned as independent substances.

If complete common wave data \(\lambda\) are independent of the later settings, each record depends only on its local setting and arrived waves, and the measurement windows lie outside each other’s domains of influence, then

The data \(\lambda\) may contain the complete relevant Huygens sea, its source relations and correlated receiver readiness. Its size does not change this implication. The issue is conditional causal factorisation, not whether the physical medium can be cut into independent pieces. Any successful Bell account must show which premise fails and calculate its probabilities.

Propagation must use P2’s actual characteristics. Retarded means that a change arrives after its propagation delay; it does not by itself set a maximum speed \(c_0\) when \(c'\) varies. Compare the model’s earliest possible arrival with the actual setting and record times. Newly changed settings and a long-settled apparatus are distinct tests.

Shared geometry. The all-direction wave sea provides common phase surfaces. The following exact uniform-wave controls explain that support and its limits; they do not create knowledge of an independently changed remote setting.

Technical geometry: shared phase surfaces and curved background waves

The orthogonal Huygens family as a shared boundary

The same orthogonal geometry now appears in its second role. In the photon construction the ring is perpendicular to the propagation direction \(\hat{\mathbf k}\) and carries the two polarisation coordinates. In a separated source–receiver or two-centre problem the relevant great circle is perpendicular to the separation \(\mathbf L\) and supplies common wavefronts. These roles coincide when propagation and separation are parallel, but the concepts remain distinct.

Place two possible receiver reclosures at \(A\) and \(B\), with spacelike separation \(\Delta x^\mu=(\Delta t,\mathbf L)\). An ideal directional background component has null wave-vector \(q^\mu=\omega(1,\hat{\mathbf n})\) and phase

\[ \theta_{\hat{\mathbf n}}(\mathbf x,t) = k\hat{\mathbf n}\cdot\mathbf x-\omega t+\alpha_{\hat{\mathbf n}}. \]

For the uniform-background null-wave control, zero unwrapped phase difference between the two events occurs when

\[ q_\mu\Delta x^\mu=0 \quad\Longleftrightarrow\quad \hat{\mathbf n}\cdot\mathbf L=\Delta t \]

in units with \(c=1\). For spacelike separation this defines a complete circle \(\mathcal C_{AB}\cong S^1\) on the sphere of null directions. In the frame where the events are simultaneous it becomes Geoffrey’s orthogonal great circle:

\[ \boxed{ \hat{\mathbf n}\perp\mathbf L \quad\Longrightarrow\quad \Delta\theta_{\hat{\mathbf n}}=0. } \]

Under a boost this equator becomes another circle, while the invariant condition \(q\cdot\Delta x=0\) remains true. For this zero-difference family, a spacelike pair has a circle of null directions; at null separation it contracts to one ray; at timelike separation no nonzero null direction satisfies the condition. Equality of phase modulo \(2\pi\) also permits \(q\cdot\Delta x=2\pi m\); those other phase sheets are not included in this classification. Nothing has to cross from \(A\) to \(B\) along that circle at measurement time. The shared wavefront relation is already extended across both events.

The circle has a Lorentz-invariant induced measure. With \(\ell'=\Lambda\ell/K\), \(\omega'=K\omega\), \(d\Omega'=K^{-2}d\Omega\) and \(\delta(\ell'\cdot\Delta x')=K\delta(\ell\cdot\Delta x)\),

\[ \boxed{ d\mu_{AB} =\omega\,d\Omega\, \delta(\ell\cdot\Delta x) } \]

is invariant. A covariant two-centre boundary functional may therefore be written

\[ \mathcal B_{AB}[\Gamma] =\int\omega\,d\Omega\, \delta(\ell\cdot\Delta x)\,\Gamma(\omega,\hat n). \]

A finite train replaces the exact delta function by a source-dependent coherence window. The universal sea supplies the invariant surface; the spectral envelope \(\Gamma\) supplies pair identity and finite bandwidth.

There is also a spatially uniform phase sum in the chosen background rest frame. For equal-frequency, exactly counterpropagating waves \(\hat{\mathbf n},-\hat{\mathbf n}\), the sum

\[ \Theta_{\hat{\mathbf n}} = \theta_{\hat{\mathbf n}}+\theta_{-\hat{\mathbf n}} = \alpha_{\hat{\mathbf n}}+\alpha_{-\hat{\mathbf n}}-2\omega t \]

has no spatial term. Under these premises its summed phase is spatially common at equal background time. This identity does not establish a dynamical core–sea lock or independence from a future setting choice.

These identities are exact under the ideal plane-wave premises. They reveal how an all-direction standing-wave substrate can possess shared phase surfaces without a new messenger being launched at measurement time. But the exact circle has zero area in the full direction sphere, and its finite-width coherence band scales roughly as \(1/(kL)\). An ordinary statistical average over unrelated background waves cannot therefore produce a strong Bell correlation from this circle alone. Curved wavefronts also replace absolute phase labels by reference-independent differential phase and closed holonomy: large common distortions cancel, while differential curvature remains observable.

Its exact moments expose why it cannot be the answer table. Writing \(\rho=\Delta t/L\),

\[ \langle\hat n\rangle=\rho\hat L, \qquad \langle n_in_j\rangle =\rho^2\hat L_i\hat L_j +\frac{1-\rho^2}{2}(\delta_{ij}-\hat L_i\hat L_j). \]

Any direct hidden-orientation rule built from these directions retains the separation axis \(\hat L\) and cannot equal the isotropic singlet law. The Huygens circle supplies boundary support; the axis-free source tensor must supply rotationally invariant entanglement.

But the orthogonal family is not itself an answer table. It knows \(\mathbf L\); it does not know the independently chosen analyser axes \(\mathbf a\) and \(\mathbf b\). Used only as common prior information, it remains a Bell-local hidden variable. Its more powerful role is geometrical: it supplies the continuously available surface on which one two-centre boundary relation may reclose.

The universal sea supplies the substrate. The source-written train supplies pair identity.

A curved cosmic sea: common history and differential history

The background waves of an infinite active Space will not be perfectly flat. Every star, atom, nuclear transition and organised structure changes the waves passing through it. The local sea carries a vast superposition of old curvature histories.

That need not wash out quantum coherence. Suppose transport through the sea rotates the two ends of a singlet relation by \(R_A\) and \(R_B\). The singlet is invariant under a common rotation, so only the relative transport

\[ R_\Delta=R_A R_B^{-1} \]

can change the measured correlation:

\[ \boxed{ E(\mathbf a,\mathbf b) = -\mathbf a\cdot R_\Delta\mathbf b. } \]

Enormous common curvature may therefore be invisible to the pair. Differential curvature causes loss of visibility, phase drift or decoherence.

For a small statistically isotropic random relative rotation with variance \(\sigma_\Delta^2\), the leading visibility is

\[ v\simeq1-\frac{\sigma_\Delta^2}{3}, \qquad E\simeq-v\,\mathbf a\cdot\mathbf b, \qquad S_{\max}=2\sqrt2\,v. \]

This is a useful WSM prediction template. D Supply the spectrum and correlation length of background curvature, then calculate entanglement degradation rather than merely assert universal connection.

The anisotropic result is also exact. Let \(\overline R=\langle R_AR_B^{-1}\rangle\). Then

\[ E(\mathbf a,\mathbf b) =-\mathbf a^T\overline R\,\mathbf b. \]

If \(s_1,s_2\) are the two largest singular values of \(\overline R\), the optimized Bell value is

\[ \boxed{S_{\max}=2\sqrt{s_1^2+s_2^2}.} \]

For isotropic curvature \(\overline R=vI\), with \(v=[1+2\langle\cos\phi\rangle]/3\), this reduces to \(S_{\max}=2\sqrt2|v|\). Bell violation survives precisely when \(|v|>1/\sqrt2\). Unequal singular values would predict an orientation-dependent loss of entanglement and are therefore tightly constrained by existing rotational tests.

The distinction is fundamental:

\[ \boxed{ \text{universal ancient sea} \neq \text{pair-specific entangled train}. } \]

The sea makes connection possible everywhere. The source transition selects which changed modes belong to this pair.

Joint receiver constructions and their physical requirements

A pair created at a source is most naturally represented in WSM not as two independent objects carrying complementary instructions, but as one changed wave relation \(\Gamma_{AB}\) with two receiver ends.

The complete joint receiver relation has the schematic form

\[ (r,s)=\mathcal C[z;\Gamma_{AB},\mathbf a,\mathbf b], \]

This notation allows a dependence on both settings; it does not yet state how the real waves implement it. For a setting-independent complete \(\lambda\), local deterministic response functions of the form

\[ \big( \mathcal C_A[\lambda,\mathbf a], \mathcal C_B[\lambda,\mathbf b] \big). \]

would satisfy the Bell bound; the stochastic statement is the conditional factorisation above. A nonfactorisable wave shape or source tensor alone does not establish a failure of that causal condition. Conversely, a setting-dependent common-past distribution is a different route and must be displayed as such, with the setting hardware included. The global notation is a requirement to explain, not an explanation of its physical implementation.

For a pure two-channel pair, represent the source-written train by a \(2\times2\) tensor \(C\) with \(\mathcal N=\operatorname{tr}(CC^\dagger)\). Local analyser choices act as basis changes,

\[ M(\mathbf a,\mathbf b) =U_A^\dagger(\mathbf a)\,C\,U_B^*(\mathbf b). \]

The train carries the analyser-independent entanglement invariant

\[ \boxed{ \mathscr C= \frac{2|\det C|}{\operatorname{tr}(CC^\dagger)}. } \]

Here \(\mathscr C=0\) means a rank-one separable train, while equal singular values give \(\mathscr C=1\). For a pure two-qubit relation,

\[ \boxed{S_{\max}=2\sqrt{1+\mathscr C^2}.} \]

If \(q=\sigma_2/\sigma_1\le1\) is the singular-value ratio, then \(\mathscr C=2q/(1+q^2)\). Production-path imbalance therefore predicts the maximum Bell strength. Equivalently, with \(\rho_A=CC^\dagger/\mathcal N\),

\[ \mathscr C^2 =2\left(1-\operatorname{tr}\rho_A^2\right). \]

The more completely the whole train is entangled, the less pure either endpoint is by itself.

Given the normalized quadratic joint law \(P_{rs}=|M_{rs}|^2/\mathcal N\) and complete local analyser channels, no-signalling follows by summation. Since \(U_B^*U_B^T=I\),

\[ \boxed{ \sum_sP_{rs} =\frac{(U_A^\dagger CC^\dagger U_A)_{rr}}{\mathcal N}, } \]

which contains no remote analyser \(U_B\). The remote setting changes the joint decomposition but cancels from the complete local sum. For maximal entanglement \(CC^\dagger=(\mathcal N/2)I\), giving both local outcomes probability \(1/2\).

The Born selector and Bell relation can now be written as one conditional object. For the four joint channels \(j=(r,s)\), define

\[ w_{rs}(\mathbf a,\mathbf b) = \left| \left\langle D_r^A(\mathbf a)\otimes D_s^B(\mathbf b), \Gamma_{AB} \right\rangle \right|^2. \]

One Haar–Liouville readiness shell over these joint alternatives gives

\[ (r,s)_* = \arg\min_{r,s}\frac{Y_{rs}}{w_{rs}}, \qquad P(r,s|\mathbf a,\mathbf b) = \frac{w_{rs}}{\sum_{r',s'}w_{r's'}}. \]

The sampler chooses one joint label intended to describe two endpoint records. Changing either analyser changes its joint weights. It reproduces the supplied law, but it is not yet a spacetime process producing the records. The physical locations of its joint readiness coordinates, their distribution, and the allowed causal dependence on both settings remain to be derived.

The reciprocal-action diffusion of §9 gives the same conditional law when it acts on the four joint shares \(x_{rs}=w_{rs}/\sum w\). These two constructions use joint alternatives. They do not prove that every successful model must implement a four-way race: sequential conditional sampling can also reproduce the table, but its shared dependence and individual physical history still need a mechanism. Independent local samplers with setting-independent common data remain Bell-bounded.

Tetrode’s absorber idea and Wheeler–Feynman theory are precedents for exploring source–absorber boundary descriptions. They are not consequences of P1–P3. A periodic or standing-wave boundary problem does not itself establish dependence on future analyser settings. Any two-time history law requires its own well-posed equations and measure, together with an explicit compatibility statement or revision of Action §14’s rule that no future endpoint changes an earlier physical wave.

Two kernels remain distinct. Retarded response vanishes outside the actual domain of dependence of the specified wave law. Shared covariance can persist between separated regions without transmitting a new setting change. The reference scale \(J_*=\hbar/2\) fixes a conventional per-mode normalization; it does not determine the full frequency spectrum, angular content or cross-correlations of the sea. A match to a vacuum symmetric correlator must be calculated. Shared Gaussian noise remains Bell-local when its distribution is setting-independent and the responses satisfy the local condition.

WSM’s Huygens sphere suggests such a whole-boundary reading:

• the source writes the pair-specific joint train;
• the two analysers impose the available final decompositions;
• the orthogonal and opposed-wave families supply candidate geometrical support; existence, stability and a physical measure on complete solutions remain to be proved;
• a specified physical selection law must produce the joint records and their observed probabilities;
• two local receivers then make two local records.

For the normalized singlet weights, no usable signal follows because the local marginal remains complete:

\[ \sum_s P(r,s|\mathbf a,\mathbf b)=\frac12, \qquad \sum_r P(r,s|\mathbf a,\mathbf b)=\frac12. \]

The same conditional identity is not restricted to a singlet. With the product-mode metric \(G\) specified and \(\langle\Gamma_{AB}|\Gamma_{AB}\rangle_G=1\), remote analyser completeness gives

\[ \sum_s |D_s^B(\mathbf b)\rangle \langle D_s^B(\mathbf b)|_G =I_B, \]

and therefore

\[ \boxed{ \sum_sP(r,s|\mathbf a,\mathbf b) = \langle\Gamma_{AB}| P_r^A(\mathbf a)\otimes I_B |\Gamma_{AB}\rangle_G, } \]

independent of the remote basis \(\mathbf b\). This proves no-signalling for the granted quadratic measure and complete channels. A physical recording model must derive that measure and preserve the cancellation. It constrains ensemble marginals; it does not imply that every hidden complete history is independent of the remote setting.

A vital counting point follows. One entangled photon pair is not “one \(\hbar\) shared by two detectors.” Successful pair detection produces two local energy transfers. What is single is the selected joint branch, not the number of endpoint absorptions.

D The physical Bell calculation must identify the failed premise of the local model and derive the replacement. Possibilities to investigate include setting-dependent common-past wave data, an explicitly nonlocal constraint, or a whole-history boundary law. No one of these is deduced merely by naming one Space or an extended standing pattern. Each requires equations, an energy and probability account, and agreement with local marginals and multipartite measurements.

The conditional singlet law

Suppose first that the e-sphere’s \(4\pi\) hand structure forms a genuine \(SU(2)\) doublet. An axis-free total-hand-zero source tensor must obey

\[ UCU^T=C \qquad\text{for every }U\in SU(2). \]

The only nonzero solution, up to phase and scale, is the invariant antisymmetric tensor

\[ \boxed{ C\propto\varepsilon= \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}. } \]

Thus once the physical doublet and axis-free total-zero condition exist, the singlet tensor is forced rather than fitted.

Now let the analyser channels obey the ordinary spinor half-angle overlap. If receiver \(A\) recloses in its \(+\) channel along \(\mathbf a\), the joint null relation places the corresponding \(B\) mode opposite to \(\mathbf a\). An analyser at \(B\) along \(\mathbf b\), separated by angle \(\theta\), then gives

\[ P(B=+|A=+) = \sin^2\frac{\theta}{2}, \qquad P(B=-|A=+) = \cos^2\frac{\theta}{2}. \]

Hence

\[ E(\mathbf a,\mathbf b) = \sin^2\frac\theta2-\cos^2\frac\theta2 = -\cos\theta = -\mathbf a\cdot\mathbf b. \]

With the standard CHSH angle choices, \(|S|=2\sqrt2\).

This short derivation shows exactly where the work resides. Once an axis-free joint null and the SU(2) half-angle measure are present, the singlet law and Tsirelson value follow. The D-tier task is to derive those structures without inserting the quantum overlap as an unexplained rule.

The analyser does not reveal a hidden ordinary arrow. It chooses a physical spinor decomposition of one invariant relation.

27. Exclusion, creation and annihilation

Pauli exclusion as a joint standing-wave restriction

Every electron in an atom is an extended e-sphere participating in one shared collection of plane waves. The complete atomic state is one joint standing-wave recurrence.

Two identical electron patterns cannot form the same complete bound recurrence with identical mode, radial phase and analyser-resolved spherical rotation. The allowed shell arrangements are the joint modes in which the full wave organisation recloses. Antisymmetric exchange is the mathematical statement of this restriction.

Electron–positron annihilation

An electron and positron have opposite radial phase. When their e-spheres overlap, the compression of one coincides with the extension of the other, and their corresponding radial motions are opposed:

This equation states the local cancellation condition and endpoint. It is not by itself the nonlinear annihilation dynamics.

The two spherical standing-wave patterns destructively interfere. As cancellation completes, the repeated half-sphere curves that defined the electron and positron disappear from the background plane waves. Those waves no longer reconstruct either e-sphere centre.

During the cancellation, the changing curve pattern is written onto successive passing plane waves as the observed outgoing gamma-ray trains. After those trains have left the interaction region, neither e-sphere remains there. The underlying sea of longitudinal plane waves remains.

Destructive interference removes the recurring e-sphere patterns; it does not remove their energy. During the nonlinear changing overlap, the conserved energy, momentum and angular phase relation are transferred into the outgoing curve trains. The four-momentum ledger is

Pair creation

Pair creation is the reciprocal process. A sufficiently concentrated and correctly organised incoming wave train, interacting with an appropriate standing-wave structure, establishes two new spherical recurrences locked into opposite radial phase. Each new e-sphere can take either spherical rotation while the complete creation process preserves energy, momentum and angular phase relation.

B/C Exclusion is joint e-sphere closure; opposite radial phase supplies the physical cancellation mechanism; creation forms the phase-opposed pair. D Calculate the annihilation and creation dynamics, including the allowed two- and three-photon channels and conservation of action, energy, momentum and angular phase.

Part IV · Relativistic fields and tests

Relativistic wave equations, quantum fields and the QED tribunal

28. Relativity refuses to stop at Schrödinger

Schrödinger’s equation transformed atomic physics, but it treats time and space differently and uses the nonrelativistic energy \(E=p^2/2m\). A relativistic wave equation should encode

\[ E^2=c^2\mathbf p^2+m^2c^4. \]

Applying the quantum substitutions gives the Klein–Gordon equation,

\[ \left(\frac1{c^2}\partial_t^2-\nabla^2+\frac{m^2c^2}{\hbar^2}\right)\phi=0. \]

It is a correct relativistic equation for spin-zero fields. It was not the electron equation: it did not contain the electron’s two spin channels, and its natural conserved density could not serve as a positive single-particle probability density in the way Schrödinger’s did.

29. Dirac: the electron equation opens the future

Paul Dirac would not abandon relativity, spin or the first-order time evolution of quantum mechanics. In 1927 he had already quantised the radiation field and written emission and absorption using operators that change the number of light quanta. In 1928 he sought an electron Hamiltonian linear in momentum,

\[ H=c\,\boldsymbol\alpha\!\cdot\!\mathbf p+\beta mc^2. \]

Demanding that its square reproduce the relativistic energy relation forces

\[ \{\alpha_i,\alpha_j\}=2\delta_{ij}, \qquad \{\alpha_i,\beta\}=0, \qquad \beta^2=1. \]

The matrices were not decorative complications. The Clifford algebra was forced by the linear factorisation. In covariant form,

\[ \boxed{(i\hbar c\,\gamma^\mu\partial_\mu-mc^2)\psi=0}. \]

Suddenly several mysteries became one structure. The electron required four complex components. Rotations acted spinorially. A positive conserved density \(\rho=\psi^\dagger\psi\) accompanied the current \(j^\mu=c\bar\psi\gamma^\mu\psi\). With electromagnetic minimal coupling, the nonrelativistic limit produced Pauli’s spin interaction and the tree-level magnetic factor \(g=2\).

For charge \(q_e\), that limit is

\[ i\hbar\partial_t\varphi= \left[ \frac{(\mathbf p-q_e\mathbf A)^2}{2m} +q_e\Phi -\frac{q_e\hbar}{2m}\boldsymbol\sigma\!\cdot\!\mathbf B \right]\varphi. \]

The magnetic coefficient is not obtained by numerology from \(4\pi/2\pi\). It follows from the Clifford/Pauli algebra joined to the electromagnetic connection. This is the exact conventional target for the WSM spherical phase dynamics.

The equation also contained negative-energy solutions. Dirac first struggled to interpret them through a filled sea of states; the deeper message was particle–antiparticle symmetry. The positron was discovered by Carl Anderson in 1932. Mathematics had revealed a new form of matter before experiment found it.

“…a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron.”
— P. A. M. Dirac, “Quantised Singularities in the Electromagnetic Field” (1931)

The four solution sectors have a simple physical classification: two electron spin states and two positron spin states. But they are not four little substances. A moving Dirac spinor is a coupled amplitude in which the electron/positron and two spin sectors are coupled according to momentum, rotation and interaction.

Dirac completed the conceptual arc from de Broglie to Pauli. Matter possesses a rest-frequency phase; motion changes its phase by Lorentz geometry; three-dimensional orientation requires a noncommuting spinor algebra; charge-conjugate sectors represent antimatter; WSM maps this physical distinction to the opposite background-relative radial phase. The equation did not merely add relativity to quantum mechanics. It showed what the hidden wave anatomy had to be capable of doing.

30. Why quantum field theory had to be born

Relativity does not permit a permanent fixed number of particles. Energy creates matter–antimatter pairs; excited atoms emit radiation; radiation is absorbed; identical quanta cannot be tracked as classical individuals. Dirac’s negative-energy solutions and radiation theory were signs that “one-particle quantum mechanics” had reached its boundary.

The natural language became a field decomposed into normal modes. For bosonic modes the oscillator ledger is

\[ H=\sum_{\mathbf k,s}\hbar\omega_{\mathbf k} \left(a^\dagger_{\mathbf k,s}a_{\mathbf k,s}+\frac12\right). \]

The operator \(a^\dagger\) raises a mode occupation; \(a\) lowers it. Fock space collects states with zero, one, two or any allowed number of excitations. For bosons,

\[ [a_i,a_j^\dagger]=\delta_{ij}; \]

for fermions,

\[ \{b_i,b_j^\dagger\}=\delta_{ij}, \qquad (b_i^\dagger)^2=0. \]

The last identity is Pauli exclusion in occupation language. The spin–statistics theorem joins integral spin to bosonic symmetry and half-integral spin to fermionic antisymmetry under the requirements of relativistic local field theory.

Born, Heisenberg and Jordan had already quantised wave fields; Bose had changed the counting of light; Dirac made emission and absorption operator dynamics; Pauli and Jordan developed field anticommutation; Fock organised variable particle number. QFT was not born because physicists disliked pictures. It was born because modes, identity, relativity and changing occupation demanded one common calculus.

WSM preserves the calculus while changing its ontology. A field mode is a mode of one Vibrating Space or an effective mode of its recurrent e-spheres. A creation operator raises a field-mode occupation; in the WSM reading this must correspond to formation or an allowed increase of a stable wave organisation. An annihilation operator lowers the occupation; electron–positron annihilation physically removes the two opposed e-sphere recurrences by destructive interference. Fock space is the exact occupation ledger of possible multi-transition relations, not proof that a separate abstract field substance inhabits every point.

The gain is conceptual economy, but also a precise obligation. One Space must reduce to bosonic light modes, fermionic matter modes, tensor products, vacuum correlations, causal propagators and the measured scattering amplitudes. The universal sea supplies a physical candidate for the background state; the pair-specific source train supplies the changed excitation; nonlinear receiver reclosure supplies the persistent detector record. Their effective operator algebra must be derived, not chosen separately for each sector.

QED: the precision threshold

Quantum electrodynamics joins the Dirac field to the electromagnetic field through local phase symmetry. Its calculations can be organised as electron propagation, radiation propagation and interaction vertices; Feynman’s path integral reorganises the same amplitudes as a sum over histories. Early point-field calculations produced divergent self-energies. The measured Lamb shift and electron magnetic anomaly made the problem unavoidable.

Tomonaga, Schwinger and Feynman built renormalised formulations; Dyson showed their equivalence. QED then predicted scattering, spectral shifts and magnetic moments with astonishing precision. This is one of humanity’s greatest mathematical achievements.

Dirac and Feynman nevertheless remained dissatisfied with the foundational infinities:

“I must say that I am very dissatisfied with the situation, because this so-called good theory does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics.”
— Paul Dirac, lectures delivered in 1975; published in Directions in Physics (1978)

“But no matter how clever the word, it is what I call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent… I suspect that renormalisation is not mathematically legitimate.”
— Richard Feynman, QED: The Strange Theory of Light and Matter

Their dissatisfaction must not be turned into a false claim that modern renormalisation is arbitrary nonsense. Renormalised QED encodes real scale dependence, phase-reference identities and finite relations repeatedly confirmed by experiment. A deeper finite wave theory must explain those successes, not discard them.

WSM gives its familiar structures physical ancestors:

• propagator → Green response of the one wave medium;
• vertex → nonlinear change of source–Space–receiver connection;
• virtual line → an intermediate modal term, not necessarily a hidden pellet;
• loop → recursive self-response of the finite in-wave, centre and out-wave organisation;
• vacuum correlation → the ever-present wave sea;
• point electron → a finite e-sphere with form factor and internal phase history.

Fine-structure scale from the e-sphere response

Under the ideal Bohr ground-state identification \(v=\alpha c_0\), the centre advances \(\Delta X_0=\alpha\lambda_e\) during one rest-reference interval \(T_0=\lambda_e/c_0\). Under §21’s moving-clock construction, a complete centre-followed recurrence lasts \(\gamma_\alpha T_0\), giving displacement \(\gamma_\alpha\alpha\lambda_e\). The distant curve law \(|d\zeta/dR|=2\pi\alpha\bar\lambda_e^2/R^2\) is the Coulomb-matched shallow-curve response ansatz specified there. These relations express the required measured coupling in real-wave terms; they do not independently predict its value.

In the normalized cube–sphere construction, the e-sphere boundary radius, enclosed volume and circumference are

\[ r_*=\frac{\sqrt{3}}{2}, \qquad V_*=\frac{4\pi}{3}r_*^3=\frac{\pi\sqrt{3}}{2}, \qquad C_e=2\pi r_*=\pi\sqrt{3}=2V_*. \]

In the declared static spherical-response ansatz, a curved incoming plane wave excites the e-sphere’s first angular, \(\ell=1\), response. The dimensionless returned-plane-wave response is

\[ E_{\rm rp}^{(0)} =\frac{r_*^3}{2} =\frac{3\sqrt{3}}{16} =0.324759526419\ldots . \]

The WSM coupling normalization is then

\[ \alpha_0 =\frac{E_{\rm rp}^{(0)}}{6V_*^{\,2}} =\frac{2E_{\rm rp}^{(0)}}{3C_e^{\,2}} =\frac{1}{8\pi^2\sqrt{3}}, \]

and therefore

\[ \boxed{ \alpha_0^{-1} =8\pi^2\sqrt{3} =136.757250186\ldots }. \]

The conventional position should be stated by the same standard. QED takes the renormalized value of \(\alpha\) at a reference scale from experiment and then predicts—with extraordinary accuracy—how measured processes depend on it and how the coupling runs with scale. It does not derive from its own dynamics why the low-energy coupling has this numerical value. The wider Standard Model likewise uses around twenty-five or more independently measured parameters under common counts that include neutrino masses and mixing, with the exact number depending on convention. This is a triumph of quantitative organisation, but it is distinct from a physical derivation of the inputs. The WSM MDL audit applies that same accounting to both frameworks.

WSM must declare its inputs and ansätze plainly. Here \(E_{\rm rp}^{(0)}=r_*^3/2\) and normalization by \(6V_*^2\) are the proposed first-angular response model. Its estimate reaches the measured inverse fine-structure constant to about \(0.2034\%\) without a continuously adjustable parameter. The selected response and normalization nevertheless count as modelling inputs. D Derive them and any corrections independently of the measured value, then compare with experiment. The discrepancy is a test, not a correction that may be chosen to enforce agreement.

The measured low-energy inverse coupling listed in the 2022 CODATA adjustment is \(\alpha^{-1}=137.035999177(21)\). The static result is lower by \(0.278748991\ldots\), or approximately \(0.2034\%\). Equivalently, the measured coupling requires

\[ E_{\rm rp}^{\rm required} =6V_*^{\,2}\alpha =0.324098923434\ldots, \qquad \delta E_{\rm rp} =-0.000660602986\ldots . \]

In dimensionless normalized variables, the D schematic first-angular response equation to be justified and completed is

\[ \frac1{r^2}\frac{d}{dr} \left(r^2E_d\,u'\right) -\frac{2E_d}{r^2}u +\frac{\omega^2}{E_d}u =S_1(r), \]

with the e-sphere background, regular central behaviour and outgoing-wave radiation condition fixed independently. D Define the dimensional normalization and coefficients, then derive the low-energy correction and the momentum-dependent running \(\alpha(q^2)\).

A The cube–sphere arithmetic and the conventional experimental value. C The normalized cube–sphere reference, declared static dipole-response ansatz and its \(136.757250186\ldots\) result. D Derive the response, nonlinear correction, running and QED precision.

The static e-sphere geometry already gives the physically motivated constrained skeleton

\[ \alpha_0^{-1}=8\pi^2\sqrt{3}=136.757250\ldots, \]

within \(0.2034\%\) of the measured inverse fine-structure constant without a continuously adjustable fit. The QED programme has also isolated exact target forms for the one-loop Pauli response and finite returned-wave self-interaction. These are meaningful mathematical structures awaiting D completion.

The most severe photon gate is now sharper. The full-sphere directional construction supplies a candidate \(P_0\oplus P_1\) connection; antisymmetric cross-direction phase curvature supplies electric and magnetic fields; and the minimal affine angular kernel gives Maxwell exactly when \(w_1/w_0=4\). Its orthogonal restriction gives the two helicities. This is a concrete reverse-engineered target for the second variation of WSM Action.

Higher angular modes give a further quantitative test: the same wave action must recover the measured nonlinear electromagnetic response. The following coefficient comparison is a conditional control.

Quantum-field language in real-wave terms

Quantum field theory is the exact mode and transformation ledger. The physical interpretation required by WSM is:

The measured QED results and the equations that calculate them are quantitative constraints; particle and field ontology is a theoretical interpretation of those records. WSM gives the electron, spinor and interaction a real-wave physical meaning. D Derive the currents, propagators, statistics, Ward relations, two photon helicities and precision coefficients from stable e-spheres and source-written longitudinal-wave relations.

The finite-core response appendix takes a concrete step into this calculation: it joins the Darwin spatial moment, the Zemach pair moment and the angular mode overlap in one response problem.

Appendix · How a finite wave core responds

An e-sphere continually rebuilds itself from waves crossing its centre. An arriving change in their curves alters the directional energy, wave speed and phase of that reconstruction; the changed core writes a response onto outgoing waves. Darwin and Zemach moments ask how that response samples its neighbourhood. The calculations below connect a spatial response profile to measurable coefficients, then show how coupled phase and hand modes bring spatial and angular response together.

Part V · Knowledge and calculation ledger

Knowledge ledger

Established and exact structures

• A All-direction plane-wave integration produces the spherical \(j_0\) pattern and linked radial \(j_1\) motion.
• A The e-sphere travel-time geometry gives an exact half-sphere wavefront when every chord has harmonic path-average speed \(2c_0\).
• A The hand fields \(F_h\) are first-order eigensectors whose square returns the Helmholtz equation.
• A \(SU(2)\), Pauli matrices, the half-angle and \(4\pi\) closure give the exact geometry of spin-\(\tfrac12\).
• A Under H-M1, the pair \(c_0\pm v\) moves the opposed-wave midpoint at \(v\) while preserving closing rate \(2c_0\). Under H-M2 and H-M3, centre phase matching plus geometric-mean rest-frequency closure gives the reciprocal pair \(e^{\pm s}=\gamma(1\pm\beta)\), the Lorentz algebra, de Broglie phase and slow Schrödinger envelope.
• A Under the Bohr ground-state relations, \(\alpha=\Delta X_0/\lambda_e\), \(N_{\rm ref}=1/\alpha^2\) and \(\Delta\theta_{\rm ref}=2\pi\alpha^2\) are exact rest-reference identities; the complete moving-centre cycle adds the stated \(\gamma_\alpha\) factor.
• A The first Huygens-ring harmonics give two handed photon patterns; the paired real moments reproduce free Maxwell evolution under the stated premises.
• A Born frequencies, antibunching, Bell violations, no-signalling, Dirac dynamics and QED precision are experimental constraints.
• A Under the stated free-motion premises, intrinsic clock matching selects the de Broglie group velocity. Transported scalar angular coefficients give a matching diagnostic current, and the exact open-core centre balance retains the surface-flux moment.
• A Whole-generator rate scaling preserves Born first-candidate ratios for independent, commonly prepared quasi-stationary receivers. Matched temporal capture and the background-reference covariance give explicit response tests.
• C The specified deterministic 16-mode centre calculation shows substantial coarse-grained approach to Born, with an equivariant control. The separable recurrence identifies the role of mixing.
• A Within the displayed two-mode probability model, retained source coherence gives \(S_{\max}=2\sqrt{1+|\eta|^2}\); the receiver delay identity separately tests the allocation of actual records.
• A The fixed-action readiness shell fixes both Born winner ratios and a finite-support timing law. The background and bandwidth controls quantify which receiver properties preserve those ratios.
• A Matched averaging of density and current preserves the resolved density; response-scale bounds connect coarse mixing to detector probabilities. Absorbing allocation joins branch populations to one completed record under the stated capture conditions.
• A The Darwin coefficient fixes a second response moment. In the displayed static Coulomb model, first- and second-order response combine into a third Zemach moment; the free-mode overlap joins Darwin and spin–orbit terms in one benchmark.
• A The uniform fixed-action shell makes winner identity independent of first-crossing time, preserving its ratios under a common gate. The reversible receiver’s survival spectrum gives the displayed preparation-error identity and bound.
• A Under the stated readiness measure, common total-action fluctuations preserve first-winner ratios while changing timing. Specified sea filters give exact readiness variance and memory, and reference bounds.

WSM physical deductions and constructions

• B For each e-sphere, the out-waves of the other matter in its finite Huygens sphere collectively form its all-direction in-waves; the chosen e-sphere’s own directional \(E_d\) then changes their \(c'\), travel time, curve and accumulated phase as the spherical vibration recloses. Wavelength follows only after the relevant fixed-position frequency has been specified.
• P P3 fixes finite e-sphere cores and opposite radial phases. A The stated uniform-wave Huygens sum gives the \(j_0/j_1\) control. C Further spherical phase organisation is the proposed angular construction.
• B Electron and positron are sea-locked opposite radial-phase branches with one universal intrinsic rest-frequency identity. Under motion they remain exactly \(\pi\) apart and share the same centre-encountered recurrence; fixed-Space Fourier component frequencies are distinct quantities.
• C Within the proposed proton recurrence \(p=(++-)_{\mu}\), adding \((-)_e\) gives the balanced phase inventory \(++--\). This exact bookkeeping is conditional on the construction; its physical charge follows from the collective current, not the count alone.
• C Spherical phase organisation of all-direction longitudinal waves is the proposed physical spin construction. D Derive its protected half-angle transport, charge-preserving current and coupling; the four candidate phase–hand labels alone do not derive Dirac dynamics.
• B/C Spherical equality fixes the stationary centre; for motion, H-M1 gives axial reconstruction rates \(c_0\pm v\), while the projection \(\hat{\mathbf n}\!\cdot\!\mathbf v\) gives unchanged orthogonal directions at first order and smooth oblique interpolation. The second-order all-direction response remains uncalculated.
• B/C Physical \(c'\) changes with directional \(E_d\), while the linked changes of wave signals, e-sphere clocks and bound rulers preserve the same locally measured \(c\).
• C The declared shallow-curve response ansatz \(|d\zeta/dR|=\Delta v/c_0\), matched to the measured Coulomb acceleration, gives \(|d\zeta/dR|=2\pi\alpha\bar\lambda_e^2/R^2\) and \(|\zeta|=2\pi\alpha\bar\lambda_e^2/R\), linking incoming curve slope to the next \(c_0\pm v\) e-sphere reconstruction.
• B/C A photon is a finite source-written sequence of changing half-sphere curves carried by longitudinal plane waves.
• C WSM identifies a detector record with stable receiver reclosure. Its weights, readiness measure and exclusivity have distinct physical derivations.
• C WSM locates entanglement in a source-created extended wave relation. A joint probability sampler is not yet the physical production of its distant records.
• B/C Opposite radial phase supplies the electron–positron destructive-interference mechanism; the complete annihilation dynamics must transfer conserved action, energy, momentum and angular phase into the allowed outgoing gamma curve trains.
• B/C In neutral hydrogen, \((++-)_{\mu}+(-)_e=++--\) supplies equal forward and rear curve orientations. Their opposite charge-like reconstruction pushes cancel. After leaving the e-spheres, both orientations spread, acquire lower far-field \(E_d\) and lower \(c'\), and lag the flatter carrying plane wave; this common delay remains and is the proposed gravitational relation.
• B/C In calm Space the carrier keeps \(c_0,\lambda_0,f_0\), order and spacing. WSM redshift is the connected decay of its source-written curves, decreasing common Huygens overlap and lower-gap/lower-rate receiver reclosure.

Required quantitative outputs

• D Stable stationary and moving electron and positron solutions, including nonlinear stability of the complete all-direction phase envelope and its projection and reclosure geometry.
• D The unknown \(O(\beta^2)\) response \(a_2(\mu)\), including whether it produces the trial \(P_2\) term, and the higher-harmonic reclosure of the moving egg, together with either a derived cancellation or a predicted magnitude for orientation- and sidereal-dependent clock and spin signals relative to the background wave sea.
• D The universal values of \(\hbar\), electron mass, electric charge and the e-sphere scale.
• D The exact \(3+1\) Dirac projection, conserved current, charge conjugation and \(g=2\).
• D Maxwell’s equations and precisely two photon helicities as the unique free radiative curve patterns.
• D Exclusion and the distinct receiver and correlation calculations listed below, all from the same wave dynamics.
• D Derive the static fine-structure response and any independent corrections, then test the value, running, Lamb shift, vacuum polarisation and anomalous magnetic moments at QED precision.
• D The universal numerical redshift factor—including spectra, line widths, complete light-curve time dilation and energy transfer—the cosmic microwave background, Mach–Huygens support, large-scale flow and structure, and the quantitative population distribution within the unbounded matter network.

Separate physical derivations

Failure ledger: excluded inferences and ontology errors

The following constructions are removed from the WSM foundation and retained here only so future work does not silently recreate them:

• Q Active volume-preserving relabelling of Space, which imposed perfect-fluid symmetry.
• Q A determinant-only constitutive law \(W(\det F)\) promoted as the complete action.
• Q Chaplygin, pentamode, material-flow, vorticity and acoustic-fluid interpretations of Space.
• Q Introducing independent transverse displacements into Space and then trying to stiffen, reinterpret or gauge them away.
• Q Importing the transverse modes of atomistic material solids into the primitive vibration of Space.
• Q Describing the \(j_1\) radial motion as substance flowing through the e-sphere.
• Q Describing spin as circular rotation, a vortex, circulation or a small body turning about an axis.
• Q Inventing “reciprocal reconstruction grades” to obtain four Dirac components by abstract counting.
• Q Identifying electron and positron with negative frequency, negative energy or an unproved Skyrmion winding number.
• Q Treating scalar jets, Skyrmions or Finkelstein–Rubinstein controls as the physical derivation of the electron.
• Q Treating the Huygens sphere as a local reflecting shell or a pre-written boundary detached from the other e-spheres whose out-waves form the in-waves.
• Q Reversing the moving-egg geometry. The canonical assignment is elongated lower-\(E_d\) front and flattened higher-\(E_d\) rear.
• Q Treating the moving egg’s phase envelope as a separate material surface or importing primitive transverse waves. Its side-sector longitudinal strain and \(O(\beta^2)\) shape remain D outputs.
• Q Rejecting H-M1’s axial \(c_0\pm v\) reconstruction rates or replacing them with \(e^{\pm s}\). The reciprocal factors describe fixed-Space Fourier frequencies and follow only from H-M2 centre phase matching plus H-M3 geometric-mean rest-frequency closure.
• Q Confusing the balanced phase inventory \((++-)_{\mu}+(-)_e=++--\) with equal populations of free electrons and positrons.
• Q Confusing the chord-effective \(2c_0\) that writes the source half-sphere with the lower \(c'\) of the widened far-field curve after it leaves the e-sphere.
• Q Describing a photon as a pellet or as a primitive transverse vibration of Space.
• Q Saying annihilation converts two particle substances into outgoing photon substances, or treating \(\chi_e+\chi_p=0\) as the complete dynamics. The e-spheres disappear by destructive interference; the nonlinear changing cancellation must transfer the conserved quantities into outgoing curve trains.
• Q Calling a quantum interaction merely a “completed event” instead of naming its source transition, changing curve train, receiver deformation and new stable mode.

Ontology contract

Every physical term must name an actual compression, extension, vibration, wavelength, speed, frequency, phase, spherical curve, standing-wave recurrence or interference pattern of the one Space.

One connected quantum problem

Planck found that material resonances exchange complete frequency-linked changes. Einstein asked what a light quantum physically is. Bohr found stable modes. De Broglie found matter wavelength. Schrödinger calculated recurring wave patterns. Born connected squared amplitude to detector-record frequencies. Pauli found two-valued spherical behaviour and exclusion. Dirac joined spin, antimatter, phase and relativity. Bell proved that the joint quantum relation cannot be divided into two independent local answer mechanisms.

WSM reads these discoveries through one physical foundation: longitudinal plane waves in one Space forming continually reconstructed matter and changing source–receiver relations. P3 fixes opposite radial phases; moving-wave geometry links reconstruction to de Broglie phase; bound changes write finite curve trains. The proposed angular organisation, receiver selection and distant joint records must be obtained from that same foundation.

A light quantum is one complete change carried by the waves of Space between stable organisations of Space.

The destination is a quantum physics whose equations retain their measured success while every symbol returns to a visible change of real Space.

Present status. WSM has a declared longitudinal-wave ontology and exact conditional constructions linking wave geometry, response and quantum targets. These now include Born ratios under a shared gate, a calculated receiver-preparation bound, deterministic centre transport and absorbing capture. Joint guidance is a candidate extension, and the Darwin–Zemach appendix joins spatial and angular response targets for the same core. The receiver models convert specified action shares into Born frequencies under stated assumptions; they do not yet derive those assumptions from the medium. The spin construction specifies the required half-angle transport, while its stable physical realisation remains open. Shared incoming waves provide real connection across an experiment; the current calculations do not yet derive the Bell and GHZ laws from that connection. The next decisive result is one action producing the stable core, its response, its record measure and the relevant joint correlations.

Selected historical and experimental sources

• Christiaan Huygens, Treatise on Light (1690).
• James Clerk Maxwell, On Physical Lines of Force (1861–1862).
• Albert Einstein and Leopold Infeld, The Evolution of Physics (1938).
• Albert Einstein, “Physics and Reality” (1936).
• Albert Einstein, Ether and the Theory of Relativity (1920).
• Max Planck, The Genesis and Present State of Development of the Quantum Theory.
• Albert Einstein, On a Heuristic Point of View Concerning the Production and Transformation of Light (1905).
• Albert Einstein, Considerations Concerning the Fundaments of Theoretical Physics (1940).
• Albert Einstein, “Max Planck in Memoriam” (1948), reprinted in Ideas and Opinions.
• Albert Einstein to Michele Besso, 12 December 1951, “What are light quanta?”.
• Albert Einstein, “On the Method of Theoretical Physics” (Herbert Spencer Lecture, 1933), reprinted in Ideas and Opinions; note dated 9 June 1952 in Relativity: The Special and the General Theory.
• Niels Bohr, On the Constitution of Atoms and Molecules (1913).
• Gustav Hertz, The Results of the Electron-Impact Tests in the Light of Bohr’s Theory of Atoms, Nobel lecture.
• Arthur Compton, A Quantum Theory of the Scattering of X-rays by Light Elements (1923).
• Louis de Broglie, Nobel lecture (1929).
• Satyendra Nath Bose, Plancks Gesetz und Lichtquantenhypothese (1924).
• Werner Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen (1925), and Nobel lecture.
• Erwin Schrödinger, Nobel lecture and Are There Quantum Jumps?.
• Erwin Schrödinger, The Interpretation of Quantum Mechanics: Dublin Seminars (1949–1955) and Other Unpublished Essays.
• Wolfgang Pauli, Nobel lecture.
• Enrico Fermi, On the Quantization of the Monoatomic Ideal Gas, English translation of the 1926 paper.
• P. A. M. Dirac, On the Theory of Quantum Mechanics (1926).
• Max Born, Nobel lecture.
• John von Neumann, Mathematical Foundations of Quantum Mechanics, English edition.
• P. A. M. Dirac, The Quantum Theory of the Emission and Absorption of Radiation (1927), and The Quantum Theory of the Electron (1928).
• P. A. M. Dirac, Quantised Singularities in the Electromagnetic Field (1931).
• Carl D. Anderson, The Positive Electron (1933).
• Robert S. Van Dyck Jr., Paul B. Schwinberg and Hans G. Dehmelt, New High-Precision Comparison of Electron and Positron g Factors (1987).
• J. Bailey et al., Measurements of Relativistic Time Dilatation for Positive and Negative Muons in a Circular Orbit (1977).
• M. J. Borchert et al., A 16-Parts-per-Trillion Measurement of the Antiproton-to-Proton Charge–Mass Ratio (2022).
• E. P. Battey-Pratt and T. J. Racey, Geometric Model for Fundamental Particles (1980).
• Albert Einstein, Boris Podolsky and Nathan Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? (1935).
• Erwin Schrödinger, Discussion of Probability Relations between Separated Systems (1935).
• David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables I and II (1952).
• David Bohm, Wholeness and the Implicate Order (1980).
• John S. Bell, On the Einstein Podolsky Rosen Paradox (1964).
• Simon Kochen and Ernst Specker, “The Problem of Hidden Variables in Quantum Mechanics,” Journal of Mathematics and Mechanics 17 (1967), 59–87.
• John S. Bell, On the Impossible Pilot Wave (1982), and Six Possible Worlds of Quantum Mechanics.
• John Clauser, Michael Horne, Abner Shimony and Richard Holt, CHSH inequality (1970).
• Stuart Freedman and John Clauser, first Bell test of this form (1972).
• Alain Aspect, Jean Dalibard and Gérard Roger, time-varying-analyser Bell test (1982).
• Bas Hensen et al., loophole-free Bell inequality violation using electron spins (2015).
• Marissa Giustina et al., loophole-free photon Bell test (2015).
• Lynden Shalm et al., loophole-free photon Bell test (2015).
• Juan Yin et al., satellite-based entanglement distribution and Bell test (2017).
• Paul Kwiat et al., high-intensity polarisation-entangled photon source (1995).
• Hou Shun Poh et al., Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion (2007).
• H. J. Kimble, M. Dagenais and L. Mandel, Photon Antibunching in Resonance Fluorescence (1977).
• C. K. Hong, Z. Y. Ou and L. Mandel, two-photon interference (1987).
• Eric Palmerduca and Hong Qin, Photon topology (2024).
• Johan Henriksson, Brian McPeak, Francesco Russo and Alessandro Vichi, Rigorous Bounds on Light-by-Light Scattering (2021).
• William Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits (1997).
• Ryszard Horodecki, Paweł Horodecki and Michał Horodecki, Violating Bell Inequality by Mixed Spin-½ States: Necessary and Sufficient Condition (1995).
• Wojciech Zurek, Decoherence, Einselection, and the Quantum Origins of the Classical (2003).
• John Wheeler and Richard Feynman, Interaction with the Absorber as the Mechanism of Radiation (1945).
• Richard Feynman, Nobel lecture on QED.
• Sin-Itiro Tomonaga, Nobel lecture on QED.
• Paul Dirac, Directions in Physics (1978); Richard Feynman, QED: The Strange Theory of Light and Matter (1985).

Further sources for the causal and probability tests

• P. Grangier, G. Roger and A. Aspect, Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences, Europhysics Letters 1, 173–179 (1986). See also Grangier’s author review of single-photon experiments.
• D. Dürr, S. Goldstein, T. Norsen, W. Struyve and N. Zanghì, Can Bohmian mechanics be made relativistic?, Proceedings of the Royal Society A 470, 20130699 (2014).
• C. Sanner et al., Optical clock comparison for Lorentz symmetry testing, Nature 567, 204–208 (2019).
• Dilleys Ferreira, Romain Bachelard, William Guerin, Robin Kaiser and Mathilde Fouché, Connecting field and intensity correlations: the Siegert relation and how to test it (2020); author preprint.
• M. Schultze et al., Delay in Photoemission, Science 328, 1658–1662 (2010).
• D. Dürr, S. Goldstein and N. Zanghì, Quantum Equilibrium and the Origin of Absolute Uncertainty (1992; arXiv version 2003), and Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory (2004).
• A. Valentini, Signal-Locality in Hidden-Variables Theories (2002).
• D. Salart et al., Testing the speed of ‘spooky action at a distance’, Nature 454, 861–864 (2008).
• J. Yin et al., Bounding the speed of ‘spooky action at a distance’, Physical Review Letters 110, 260407 (2013).
• M. J. W. Hall, Local Deterministic Model of Singlet State Correlations Based on Relaxing Measurement Independence, Physical Review Letters 105, 250404 (2010).
• T. H. Boyer, Random electrodynamics: The theory of classical electrodynamics with classical electromagnetic zero-point radiation, Physical Review D 11, 790–808 (1975).
• E. Nelson, Derivation of the Schrödinger Equation from Newtonian Mechanics, Physical Review 150, 1079–1085 (1966).
• T. C. Wallstrom, Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations, Physical Review A 49, 1613–1617 (1994).
• T. M. Nieuwenhuizen and M. T. P. Liska, Simulation of the hydrogen ground state in Stochastic Electrodynamics-2: Inclusion of relativistic corrections (2015).
• J. L. Friar and I. Sick, Muonic Hydrogen and the Third Zemach Moment, Physical Review A 72, 040502(R) (2005).
• A. Valentini and H. Westman, Dynamical Origin of Quantum Probabilities, Proceedings of the Royal Society A 461, 253–272 (2005).
• M. Bader et al., Efficient coupling to an optical resonator by exploiting time-reversal symmetry, New Journal of Physics 15, 123008 (2013).
• T. Guerreiro et al., Single-photon space-like antibunching (2012).
• J.-D. Bancal et al., Quantum nonlocality based on finite-speed causal influences leads to superluminal signaling, Nature Physics 8, 867 (2012).
• B. F. Toner and D. Bacon, The Communication Cost of Simulating Bell Correlations, Physical Review Letters 91, 187904 (2003).
• M. F. Pusey, J. Barrett and T. Rudolph, On the reality of the quantum state, Nature Physics 8, 475 (2012).

Continuation

Next essay: The Electron: Dirac, Feynman, QED, \(\alpha\) and the Anomalous Magnetic Moment.