FROM THE REAL E-SPHERE TO DIRAC, FEYNMAN AND QED

Can the causal response of one finite e-sphere wave-centre core, continually rebuilt by an open all-direction wave relation, become the successful momentum-space calculus of Dirac and QED—and calculate \(\alpha\), \(g\), and the anomalous magnetic moment without turning the mathematics into a second reality?

Picture the proposed physics before reading a symbol. Space is not an empty stage containing waves. Space is the one infinite, continuously connected elastic wave medium, already vibrating in every direction. A lepton is a persistent spherical ordering of those real waves. Calm plane waves arrive from infinite Space as in-waves. They flow through the living e-sphere and its wave centre; beyond that centre the same continuing carriers are out-waves, travelling onward toward infinity. Nothing hits a material surface and nothing turns around. Their phases close into a breathing, globally handed spherical pattern: local circulation planes are distributed over the whole sphere, with no rigid spin axis hidden inside it.

Now perturb it. A bound transition changes one stable closure into another; an acceleration changes the directional wave egg—the complete organisation of \(E_d\), local wave speed, wavelength, phase, curvature and convergence around the moving centre. The waves still flow through. They leave carrying a changed curvature and coherence pattern. When such a changed train later flows through another bound e-sphere, its curved in-wave fronts do not strike a point-particle. They reorganise a real extended resonator. If frequency, direction, phase ordering, polarisation and curve geometry match an allowed deformation, thousands of small coherent changes add: the coherent centre can shift, the directional shape can change, and the receiver can close into a new stable pattern.

The outgoing waves do not “reshape Space” as though Space were a container separate from them. While they propagate, their changed curvature, phase relation and directional \(E_d\) are changes of vibrating Space itself. Fresh in-waves already travelling inward can cross that moving changed organisation and acquire a new phase relation through the One Law. The out-wave continues outward; the in-wave continues inward. This causal cross-wave reconditioning is the physical event whose reduced mathematics WSM is asking to connect with QED sources, propagators, vertices, momenta and amplitudes.

Current answer

QED is already a magnificent compression of electromagnetic response. WSM now asks a narrower physical question: can the four Dirac coordinates, the two radiative light modes, the conserved electric current and their precision corrections all be recovered from one finite e-sphere core and the real longitudinal waves that continually cross it?

P fixed WSM postulate.

Part I — The Physical Question

1. What QED calculates—and what WSM is asking

Quantum electrodynamics is one of the most successful calculational frameworks ever built. Given measured masses, charges and a state prescription, it predicts scattering amplitudes, atomic shifts and magnetic moments with extraordinary accuracy. Its propagators, spinors, vertices, symmetry identities and perturbative corrections are not casual metaphors. They are an exceptionally constrained mathematical response calculus.

The WSM question is therefore not whether those calculations work. They do. The question is what physical motion they are compressing.

Nothing in that dictionary licenses an easy equivalence. A retarded mechanical Green function is not automatically a Feynman propagator. A \(4\pi\) orientation closure is not automatically a Dirac equation. A finite radius does not automatically remove every ultraviolet problem. The point is narrower and stronger: a localised periodic wave structure naturally produces Floquet sidebands, poles, residues, two-momentum kernels, higher-variation vertices and internal Fourier convolutions. QED proves the precision of its response calculus once its field content, spinor representation, gauge structure, masses, coupling, state prescription and renormalisation conditions are supplied. WSM asks whether those successful inputs can be recovered as outputs of one real wave system.

1.1 The standard WSM must meet

A replacement foundation must recover, not dismiss:

• the observed poles, residues, thresholds and cross sections;
• the electron and positron spectra and charge-conjugate response;
• the Ward–Takahashi identity and the optical theorem;
• exactly two freely radiative electromagnetic polarisations;
• the running electromagnetic coupling;
• the measured \(F_1(q^2)\) and \(F_2(q^2)\);
• the electron and muon anomalous magnetic moments;
• the absence of visible e-sphere sidebands or compositeness at tested momentum transfer.

WSM asks for more—an intelligible physical cause—but it receives no discount on the measured mathematics.

1.2 What the Quantum Physics page hands to QED

This page does not begin from nothing. The preceding Quantum Physics essay has already isolated a real-wave skeleton: an open periodic e-sphere with linked \(j_0/j_1\) quadratures; two opposite spherical phase-wave hands with \(4\pi\) closure; two radial phases times two hands; a first-harmonic Huygens photon sector with two helicities; phase-reference redundancy; a directional conserved-current target; quadratic receiver work; and a nonlinear reclosure programme for one completed event. It also shows why Bell correlations cannot be produced by locally prewritten setting-independent instructions: an entangled event must be a nonfactorisable boundary reclosure of one connected train.

2. One reality, two descriptions

Position space is where the proposed event happens. A source’s real bound pattern changes. The change writes a directional train on waves passing through it. Those waves cross actual intervening Space. A receiving e-sphere deforms and absorbs work. Momentum-space QED is a transform of that whole relation.

The transform does not make \(\mathbf k\) a second arena. It labels how much of each spatial oscillation is present in the real pattern. Nor does an intermediate momentum label prove that a tiny “virtual particle” followed a hidden trajectory. It labels an internal convolution channel in the transformed response.

This distinction also prevents the reverse mistake. Fourier coordinates are not unreal or optional; they are exact information about the real wave configuration. WSM must reproduce their algebra, including the analytic structure that QED obtains through complex time ordering. Ontology and mathematics should be separated without insulting either.

Part II — The WSM Object

3. P1–P3 and the directional causal chain

Immediate deduction from P1 As the one substance, Space cannot be bounded, created or interrupted by another substance. It is infinite, eternal and continuous.

P1–P3 are the fundamental postulates; additional working assumptions and approximations are stated where used. The complete recurrence, its two spherical hands, stability, mass, charge, light coupling, action scale and all precision physics must follow from the one direction-resolved WSM Action.

The Action is therefore written schematically as \(\mathcal A_{\rm WSM}[Z]\), where \(Z\) is one direction-resolved state of Space. Its coordinates may be transformed, projected or compressed, but no transformed coordinate becomes a second substance. P3 fixes the physical target before variation: the Action must return the finite core at the stated radius, not scan the radius until a desired number appears.

4. The open e-sphere and its spherical phase wave

The e-sphere is finite at its core and open in its support. Longitudinal plane waves approach from every direction, cross the common centre and continue. The outgoing wave is not a reflected incoming wave; “in” and “out” only name the direction of the same travelling relation relative to the chosen centre.

A · all-direction geometry The equal angular sum of the calm plane waves gives

The linked longitudinal motion is the regular radial \(j_1\) quadrature:

Compression and radial motion are one recurrence a quarter-cycle apart. Writing that real temporal quarter-turn as \(J\), with \(J^2=-1\), gives the physical meaning compressed by the usual symbol \(i\).

4.1 Spin is a spherical phase wave, not a rotating shell

The scalar \(j_0/j_1\) breathing pattern does not yet contain two spin states. The physical hand appears when the equal-phase intersections of waves arriving from different directions move around the sphere in a coherent order. Each local displacement remains longitudinal along its own plane-wave direction, yet the linked equal-phase relation travels around the whole spherical organisation. Reverse that angular order and the spherical phase wave has the opposite hand.

A · exact carrier representation Quaternion notation packages the radial carrier as

This is an exact solver identity. The physical hands are supplied by the non-scalar angular organisations of the arriving directions; a bare reversal of the radial \(j_1\) quadrature is time reversal and does not manufacture a second spin state.

4.2 The four WSM configurations

Each complex Dirac coordinate records two real quadratures. The four complex components therefore do not imply eight physical species, and the propagation basis introduced later is only another coordinate system on the same four-state space.

A one-e-sphere \(4\pi\) closure gives the spinorial rotation law. Exchange statistics, Pauli exclusion and the closed-fermion-loop sign remain properties of the complete many-e-sphere state and its action measure; they cannot be obtained by counting one-body turns.

5. Charge is a persistent relation; light is its change

Let a source e-sphere carry phase class \(q_{\phi,s}\) and a receiver carry \(q_{\phi,r}\). The observable interaction sign is relational:

A global reversal of every phase label changes no interaction. “Positive” and “negative” are names for the two classes; attraction and repulsion depend on their relation.

For two real harmonic contributions, the elementary interference identity is exact:

It proves that self-terms are phase-even and the cross-term reverses with the relative phase. The complete WSM constitutive response is more general, so the honest directional statement is

where the coupled action must determine the kernel \(\mathcal K\), its sign and normalisation. In real-wave language, the source e-sphere changes the portion of a longitudinal plane-wave front that crosses its finite wave volume. One relative radial phase raises the directional \(E_d\), raises \(c'\) and initially bows that portion ahead of the undisturbed carrying plane; the opposite phase lowers \(E_d\), lowers \(c'\) and initially leaves the changed portion behind it. The two signed writes later alter a receiver's reclosure in opposite directions. They are real advanced and delayed wavefront portions—not unexplained “apart” and “together” orientations.

5.1 Two oppositely written wavefront curves: signed source write, common propagation delay

Let \(\sigma=+1\) denote the portion initially advanced ahead of the carrying plane and \(\sigma=-1\) the portion initially delayed behind it. A compact reciprocal-slowness control writes their opposite source signs and their later common delay as

C · reciprocal-slowness control At the source exit surface take \(g(0)=0\), and separately posit that the exact hemispherical half-weight sets the signed fractional-slowness amplitude \(\delta(0)=1/2\). The chord-average result does not derive this identification; it only motivates a sharply normalised constitutive trial. Given these two stated premises, the following arithmetic is exact:

Thus the numbers \(2c_0\) and \(2c_0/3\) are not algebraic errors. Their harmonic mean restores \(c_0\). Their physical status is C because the living action must decide whether a source-written hemispherical deformation really enters the complete directional \(E_d\) through this reciprocal-slowness rule. This statement fixes the status once; the real-wave argument below can then proceed without repeating it.

A pure sky-phase dipole has an exact all-amplitude read. For constant \(\mathbf a\),

The whole regular carrier translates; no small-amplitude expansion is involved. The physical reads must nevertheless remain separate:

Here \(\mathbf T\) is only the momentum-flux ledger of the same vibrating Space, not a second substance. In a homogeneous linear control a curved phase front need not have a lower local characteristic speed; it may lag axially because of geometry while every local ray still travels at \(c_0\). WSM makes a stronger causal proposal: where the complete overlap and coherence actually lower \(E_d\), the One Law lowers \(c'\). The living action must calculate that \(E_d\) change rather than infer it from curvature alone.

The reciprocal factors \(W\pm P=e^{\pm s}\) remain exact in the constitutive and transfer ledgers; they are not substituted for these two physical writing speeds. Likewise, a common gravitational delay is not created by merely squaring or exponentiating the odd charge write. It is the phase-even residue formed during propagation and reclosure.

Both oppositely written wavefront portions gain area as they spread. At fixed transported action their directional modulation density falls; through the One Law their later speed and phase accumulation change. They widen, flatten and can fall behind the flatter carrying plane even though one portion began ahead of it. The nonlinear transport calculation determines the common \(g(D)\), its range and its strength; it must not be counted twice as both “area delay” and a separate writing term.

5.1a Four real-wave relations that must not be merged

For the proposed response in which the receiver is pushed away from the source, the source-facing near side becomes the trailing side and the far side becomes the leading side. The Relativity-page closure candidate is visually precise but still constitutive:

Shape alone does not assign wavelength. The last column follows only on the invariant-carrier branch \(c'=f_e\lambda'\); the coupled moving e-sphere must establish that branch.

5.2 A stable pattern is not continual radiation

A stable e-sphere continually transforms the waves flowing through it, but after each complete cycle the same excess organisation recurs. Its mean radiative excess flux is zero:

Radiation appears when a bound standing-wave mode changes, or when an accelerating, scattering or externally driven e-sphere changes its deformation non-adiabatically. Atomic and molecular emission are the clean discrete bound-to-bound cases. Separate the endpoint contrast, the time-dependent source and the radiative solution:

Ahead of the causal change, remote Space still carries the old relation; behind it, the new relation. The finite radiative train is the time-dependent outgoing part generated while the source changes. It is not automatically equal to the static difference of the endpoints.

Define \(\mathcal P_{\rm adv}\) and \(\mathcal P_{\rm del}\) as projections onto the portions initially written ahead of and behind the carrying plane, and \(\mathcal P_{\rm rad}^{\rm EM}\) as the part that couples to freely propagating electromagnetic receiver modes. WSM's present candidate is that ordinary electron light is dominated by the initially advanced wavefront relation, with no freely radiative residue in the oppositely written delayed channel:

That is a sharp Tier-C calculation, not a definition. The solved train must decide among three real possibilities: genuinely one-signed advanced curvature; an advanced-dominant pulse with compensating shoulders; or alternating advanced and delayed curvature whose net phase memory vanishes while its action still travels outward. Outward energy flux, curve polarity and zero-frequency memory are different ledgers. Positronium annihilation is a particularly severe test: the final inward overlap must generate the same observed electromagnetic radiation through the changed source geometry, without inventing a different kind of light.

5.3 Why a changed curve can carry momentum without being a massive object

Imagine a sequence of plane-wave fronts approaching a receiving e-sphere. A flat train preserves the existing balanced pattern. An axis-centred bowed train brings its phase change first near the axis and later off-axis. The receiver’s real susceptibility therefore receives a directed sequence of coherent changes. Translation cannot be an unchanged isotropic sphere merely carrying an assigned velocity: for an isotropic directional wave weighting \(\mathcal J_p\),

A nonzero translational state therefore requires a directional \(V_1\) component somewhere in its phase, wavenumber, propagation or coherence weighting. The literal hemispherical sag is \(R|\mu|\), so its bare signed advance-minus-delay profile is already the pure dipole

It contains no bare \(V_3\). The older profile below is a distinct, exact once-weighted response: it applies only if the action supplies one additional projected-area, crossing-flux or receiver factor \(w(\mu)=|\mu|\):

Use this weighted profile with ordinary \(d\mu\), or use the bare profile with the corresponding weighted measure, but do not apply the same factor twice. In the ordinary angular norm the \(V_1\) fraction of this declared weighted response is \(15/16\); after recentering the leading \(V_3\) term carries \(35/36\) of its remaining odd norm, and \(V_1+V_3\) captures \(575/576\). Under the Huygens comparison norm \(2|\mu|d\mu\), the \(V_1\) fraction is \(24/25\). These are geometric projection fractions, not energy fractions and not bare-hemisphere facts. The receiving susceptibility decides whether the extra weight exists and what physical contour results; an incident Legendre coefficient is not automatically an e-sphere shape amplitude. If a \(P_3\) deformation is present, its exact quadratic angular product is

The receiving e-sphere’s centre moves because the real waves flowing through it have been reorganised. Momentum is visible as the directed redistribution of that same wave motion: the incoming curve overlaps the translational collective coordinate while simultaneously driving internal deformation. The train transfers momentum without being a massive pellet. For one free canonical Fourier mode, the general action ledger is

and hence

This is the precise content of “nonzero energy and momentum, zero rest mass” for one on-shell free Fourier mode: it has no rest frame. If completed source–receiver closure selects the universal value \(\mathscr J_{\mathbf k}=J_*\), these relations become \(E=J_*\omega\) and \(\mathbf p=J_*\mathbf k\); only the further blind result \(J_*=\hbar\) gives their measured conventional form. A finite train containing non-collinear on-shell components can have nonzero composite invariant mass without becoming a localised matter closure; its peak can also evolve subluminally while its small-amplitude components and causal front remain luminal.

Light is therefore not a substance called energy. It is a real changed wave configuration whose curvature and coherence carry positive background-relative Hamiltonian and action. Define that energy without double counting:

The tangent subtraction removes the linear ledger of an active background; near a stable constrained sea, \(E_{\rm rel}=\tfrac12\langle\delta Z,H''[Z_{\rm bg}]\delta Z\rangle+\cdots\). It is mandatory for a long tail. If a constitutive perturbation falls as \(r^{-p}\), the naive linear volume excess requires \(p>3\), whereas its tangent-subtracted quadratic remainder converges for \(p>3/2\). A spherical \(r^{-2}\) tail can therefore have finite \(E_{\rm rel}\) even though \(\int(\mathcal H-\mathcal H_{\rm bg})\,d^3x\) does not. The non-overlapping conservation statement is

Only when the sea settles back to the same background and all released excess leaves in one radiative train may an isolated downward transition \(a\to b\), \(E_a>E_b\), be reduced to \(E_a-E_b=\int dt\oint_{\partial B}\mathbf S_{\rm train}\cdot d\mathbf A\).

Uniform displacement or tilt has no paired restoring excess; a localised bowed wavefront portion has edges, gradients and curvature, and therefore can carry positive excess Hamiltonian.

5.4 Why light has no separate antiphoton

Any real travelling wave satisfies

The conjugate Fourier component is part of the same real wave train, not a second charged species. WSM proposes that matter–antimatter labels arise from opposite closed relative-phase classes of bound structures. A neutral propagating change need not inherit that binary. In WSM language, there is no missing antiphoton exception: the “conjugate” is already required by the reality of the wave.

Pair creation is the inverse formation test. The available invariant energy must reach \(2m_ec_0^2\), and one freely propagating luminal train cannot create an electron–positron pair in the calm homogeneous background with no second participating excitation while conserving four-momentum. A second train, nucleus or other recoiling structure is required. The coupled action must then show how one incoming real-wave relation closes into two opposite \(q_\phi\) e-sphere branches while conserving the complete Space-wave energy, momentum and orientation ledger.

5.5 The finite-train memory test

A finite one-signed phase advance cannot simply begin and end without accounting for its zero-frequency content. Define the permanent phase memory by

Either compensating shoulders make this zero, or a genuine permanent phase displacement remains. Solving this waveform is one of the cleanest ways to distinguish a finite light train from a loose drawing of repeated displaced wavefronts.

Part III — Action and Response

6. One physical state and its useful representations

A direction-resolved wave state may be expressed in real-space fields, Huygens directions, Fourier amplitudes, canonical quadratures or collective coordinates. These are views of one motion. The transformation is acceptable only when its measure, antipodal relation, constraints and inverse map are explicit.

A finite bound change writes a real train on successive plane waves. Use the complete canonical ledger

Here \(\zeta\) records front displacement, \(\Pi_\zeta\) its conjugate wave momentum, \(\Gamma\) the directional phase/orientation organisation and \(\Pi_\Gamma\) its conjugate motion. Dropping either conjugate variable turns a moving wave relation into a static drawing.

7. The slow timing mode and the \(1/R\) exterior

The raw overlap of two rapid carriers oscillates as \(j_0(k_eR)\); it cannot be Coulomb's monotone exterior. The long-range candidate lives instead in the relative timing of complete recurrences. A periodic e-sphere may be shifted around its cycle without changing its isolated energy, so the phase tangent \(Z_\vartheta=\partial_{\theta_c}Z_e\) is the natural slow coordinate.

C · slow recurrence mode

In real wave language, a source e-sphere changes the crossing time of particular portions of the longitudinal waves flowing through it. Those changed portions continue outward. When they join the in-wave sky of another e-sphere, their slowly varying timing relation shifts where and when that receiver can reclose.

The physical charge requires three equal readings of this one relation: the conserved generator of the source, the far flux it writes and the response of a calibrated receiver. The force is not the curve alone; it is the complete incoming-minus-outgoing stress integrated over the receiver.

8. The e-sphere Hessian: the real object behind the electron propagator

The present one-Space action search uses one primary material motion \(X\). Its Huygens image \(\Xi\), matter texture \(\Gamma\), lifted rotor \(Q\), signed recurrence coordinate \(\zeta\), and reconstruction multipliers are constrained descriptions of that same motion. Write their aggregate as \(Z\), with constraint surface \(\mathcal C[Z]=0\). Schematically, the frozen direction-resolved WSM Action has the form

This is an action architecture, not yet Nature's frozen law. Its purpose is precise: forbid a Huygens or quaternion representation from becoming a second substance, and make every response pole an excitation of one constrained Space state.

Suppose the solved action produces a stable periodic e-sphere \(Z_e(t)\). Let \(P_{\mathcal C}\) map independent physical variations into the tangent space of the linearised constraints. The response operator is the reduced Hessian

Constraint, multiplier and duplicated reconstruction directions must be eliminated before poles, norms or particle counts are read. A zero or negative direction of the unreduced block matrix is not automatically a physical mode or ghost; the decisive objects are the reduced symplectic form, reduced Hamiltonian and residues on the physical quotient.

If the Hessian depends only on observable functionals, \(T_{\mathcal L}\) may equal \(T_{\rm obs}\). If a permitted coupling retains the unsquared lifted sign, its block period may equal \(T_{\rm lift}\). Both are candidate periods until the solved temporal phase and orientation path are synchronized. The action—not notation—must decide.

The exact causal response satisfies

Action–response guardrail. The constrained action is reciprocal and conservative. Its ordinary single-history Hessian is not itself a purely retarded kernel. Retardation is selected after variation by the physically prepared past and open outgoing boundary condition; equivalently, unresolved real-wave coordinates may be eliminated using their retarded solution when constructing the driven response. Eliminating them at the stationary-action level instead generally produces a time-symmetric reduced functional.

This is initially a Floquet Green function, not the propagator of a time-independent point. The background is periodic, so a disturbance can enter with one harmonic and leave with another.

Choosing the e-sphere centre selects one member of a continuously translated family of solutions. The centred representative does not itself display translation invariance, although the complete homogeneous action retains it. Response about that pinned representative is therefore generally a two-momentum kernel:

Before inversion, collective neutral directions must be separated:

• translation of the e-sphere centre;
• displacement along its periodic orbit, or clock phase;
• global orientation and any genuine internal symmetry.

For a correctly solved e-sphere in homogeneous Space, translation is exact:

Let \(P_\perp\) remove the genuine collective neutral directions. The deformation Green response is then defined on the complementary subspace by

These modes are not rubbish to delete. They become collective coordinates. Restoring the translational coordinate of the whole source–electron–receiver system is what eventually restores total momentum conservation and its \(\delta\)-function.

8.1 Vertices are higher variations, not drawings

Once the modes of \(\mathcal L_e\) are known, the interaction tensors are obtained from higher variations of the same action:

projected on the solved source, radiative and receiver modes. A Feynman vertex can then be read as compressed bookkeeping for one real place and time where those response patterns overlap. The diagram is useful; it is not the ontology.

9. From real positions and times to momentum-space QED

Write the linearised e-sphere operator as

Whenever the inverse exists on the non-collective subspace, its causal response has the Neumann expansion

with the signs fixed by the definition of \(V\). In position space every product means integration over real intermediate places and times. Schematically,

Fourier transformation converts that convolution into multiplication and internal momentum integration:

The loop momentum \(\ell\) is therefore naturally read as the Fourier coordinate of an unobserved intermediate response. This is the precise, defensible bridge. It does not mean that a QED loop is one carrier making a closed trip. The numerators, state prescription, symmetry factors, exchange signs, crossing relations and counterterm structure must emerge when the unresolved real-wave response is projected into the effective QED calculus.

9.1 What Floquet harmonics mean visually

A static object scatters one incoming sinusoidal pattern into spatial harmonics. A periodic object does more: during the encounter its own internal phase advances. A train arriving when the sphere is breathing outward does not see exactly the same geometry as one arriving a quarter-cycle later when its oriented spherical-flow quadrature is largest. The response therefore contains sidebands

In the transform, the integer \(n\) records exchange with the periodic background. In real Space it is simply the number of internal e-sphere cycles by which the perturbation’s phase bookkeeping differs. Whether any such sideband is freely observable is an empirical question. The physical electron must explain why scattering sees one clean electron pole rather than a conspicuous ladder of compositeness resonances.

If the internal clock phase is not prepared, average over it:

A response fact/C electron reduction Averaging the unprepared cycle origin restores dependence on \(t-t'\) and removes coherence between different choices of the clock origin. It does not generally erase the diagonal Floquet spectrum: poles at quasienergies shifted by \(n\Omega_{\mathcal L}\) can survive in the averaged spectral response. One clean observed electron pole therefore requires vanishing residues, exact selection rules, inaccessible energies or a collective reduction that makes the extra harmonics unobservable. Phase averaging alone is not the explanation; the solver must return every surviving residue.

9.2 How one momentum is recovered

The two-momentum kernel does not conflict with momentum conservation. It describes a sphere pinned at a chosen origin. Promote that origin to a collective coordinate \(\mathbf X(t)\), include the recoil of source and receiver, and transform the complete translation-invariant system. The common translation then yields

while the internal form factor retains the finite response of the e-sphere. Total momentum conservation is a symmetry of the whole real wave relation, not a property one may demand of a deliberately pinned subsystem.

10. From retarded response to the Feynman propagator

Four response objects must be kept separate:

A retarded kernel vanishes before a disturbance can arrive. The symmetric \(G_H\) records state fluctuations but is not causal response. A Feynman kernel orders positive- and negative-frequency parts differently so that asymptotic scattering amplitudes have the required analytic structure. Obtaining any one of these objects does not obtain the others. The familiar free targets are, schematically,

WSM must derive the spectral density and state rule that turn its real causal response into the corresponding time-ordered correlator. It must also derive why fermionic exchange contributes a minus sign. Declaring “retarded equals Feynman” would merely rename the missing calculation.

10.1 Absolute wave-time and time ordering

WSM assumes the vibrating medium has an absolute phase evolution: the waves of Space oscillate, and that oscillation orders events. The e-sphere’s own clock is a collective phase measured relative to that background and modified by motion and local \(c'\). This does not make every coordinate-time label observable, nor does it license superluminal signalling.

In-waves and out-waves describe direction relative to a centre. They move forward in absolute wave-time. Mathematically time-symmetric kernels can be useful for stationary boundary-value problems, but the physical radiation condition must be selected by the actual prepared past and open exterior. A positron is not required to be a little electron literally travelling into yesterday.

10.2 The state-construction gate

The minimum derivation must show how a stationary statistical state of the real wave sea supplies a two-point function with:

• the correct positive spectral measure;
• the correct \(i0\) boundary prescription;
• causal commutator or response support;
• the observed fluctuation–response relation;
• the correct asymptotic in/out reduction.

Only then may the WSM response be called the physical origin of a QED propagator.

C Three action ledgers are kept separate throughout the reduction:

The target entries are not permission to insert \(\hbar\). If recurrence and every narrow completed transition share one calculated scale, define \(J_*=J_{\rm cl}=J_{\rm tr}\) and use

Only after the solved action returns \(J_*=\hbar\) may one write \(\omega_e=m_ec_0^2/\hbar\). The sea value \(J_{\rm sea}=J_*/2\) is a further state calculation, not an automatic half of the same result.

Part IV — Dirac and QED Gates

11. Two radial phases × two spherical hands: the four-mode Dirac closure

The free Dirac equation supplies the first-order structure to recover from the four WSM e-sphere configurations. The reduced law below has an explicit conserved action current; its physical identification requires the modes and coupling to be derived from the underlying WSM dynamics. The configuration basis is

A Hadamard turn on the radial factor gives the equivalent propagation/chiral basis. Its coordinates \(\psi_+\) and \(\psi_-\) still carry the same two spherical hands; they are not extra physical grades.

The opposite derivative signs record reciprocal propagation coordinates. Real in-waves and out-waves occur within every \((s_q,h)\) state. Applying the reciprocal operator twice gives

The same four-mode law has the positive conserved action density and current

This is the action norm of the reduced four-mode law. Electric charge is a further signed current derived from the interacting recurrence.

11.1 The familiar matrix equation is the compressed form

The matrices do not cause the motion. They preserve the multiplication table of the real quarter-cycle carrier and its two spherical hands while the whole e-sphere translates. The Action must calculate \(J_*=\hbar\), \(\Omega_D=\omega_e\) and the collective spatial derivative; relabelling the internal carrier wavenumber as the de Broglie wavenumber would only imitate that projection.

11.2 The finite e-sphere projection

Let \(\mathcal E_a\) be the four tangents of the solved e-sphere. Projection of the quadratic wave evolution returns the Hermitian action metric \(G\), translation matrices \(C_i\) and rest-recurrence matrix \(R\). Define

Two convergence numbers make this a finite calculation rather than a resemblance:

Passing these tests means that the Dirac equation is the calculated envelope law of the real e-sphere. It also exposes failure cleanly: wrong coefficients, a non-positive action norm or leakage into an extra low mode cannot be hidden by changing matrix notation.

11.3 Antimatter and statistics remain distinct

Electron and positron are the opposite background-relative radial phases fixed by P3. Multiplying the complete state by \(-1\) changes no quadratic read and is not charge conjugation. Either radial phase may carry either spherical hand.

The one-body half-angle law does not itself derive antisymmetric exchange, Pauli exclusion or the closed-loop minus sign. Those belong to the topology, measure and reclosure of the complete many-e-sphere state.

12. Electric current, the phase connection, Ward identity and unitarity

Charge is not a coloured label attached to an e-sphere. It is the signed ability of one recurrence to write a long-range timing relation and of another recurrence to read it. One physical electric charge must appear identically in three calculations:

12.1 The recurrence-phase connection

Suppose the solved source waves write a path-dependent timing one-form \(\mathcal C_\mu dx^\mu\) on the receiving recurrence. Transport of its real quadrature pair is then compressed by

In real wave language, \(\mathcal C_\mu\) records how source-changed plane waves alter recurrence timing along neighbouring paths. Its curvature records the net timing change around a closed comparison. It is a response ledger of the same vibrating Space, not a second substance.

12.2 Ward–Takahashi

A long-wavelength change of the source-written phase connection must equal the corresponding change of the e-sphere's inverse response. If the two sides differ, the proposed vertex is not the conserved motion of the same Space.

12.3 Optical theorem and the two light modes

Response removed from one resolved outgoing pattern must reappear in the complete set of other real-wave patterns. Together with positive spectral weight, the reduced radiation sector must contain exactly two physical optical residues, one gauge null/Gauss constraint and no scalar optical residue.

12.4 Receiver work and completed events

A matched receiver accumulates coherent deformation through its physical action metric \(G\). Quadratic work naturally produces a squared response amplitude, but the square alone does not select one exclusive result or normalize all competing outcomes. Those require the complete source–apparatus–receiver boundary reclosure and the many-channel action measure.

13. Repeated causal response and why QED draws loops

A concise nonlinear WSM response equation has the form

Iterating it produces a hierarchy:

Each term is repeated causal interaction: real changes cross connected regions of Space and alter later encounters. No carrier reverses direction. Fourier transformation turns unobserved intermediate positions into momentum integrals. A closed QED loop appears when the response closes as the trace below.

The closed real-position-space target is a trace:

The closure in this expression is algebraic. A QED loop appears when unresolved intermediate fluctuations are contracted into a closed trace; it is not a picture of one real wave leaving an electron, travelling around a circle and coming home. The mathematical trace may close while every underlying physical propagation leg remains forward-causal flow-through. The conventional Gaussian effective-action target is schematically \(\Delta\mathcal A_{\rm eff}\sim(i\hbar/2)\operatorname{Tr}\ln\mathcal L_{\rm bos}\) for bosonic fluctuations of the real e-sphere; WSM must obtain the state measure, \(\hbar\) normalisation and distinct fermionic sign from its own reduction before identifying the two calculi.

A QED loop is therefore more than repeated Huygens coherence or repeated flow-through interaction. A complete derivation must generate:

• the spinor and tensor numerators;
• fermionic minus signs;
• time-ordering and \(i0\) prescriptions;
• symmetry factors and crossing;
• the Ward identities at every order;
• the observed logarithmic scale dependence;
• the optical theorem’s allowed cuts.

There is no theorem that the response terminates after four layers. A finite number of geometric sectors does not imply a nilpotent nonlinear response operator. If a termination exists, the action must demonstrate the selection rule.

13.1 All paths without tiny travellers

The path integral can also be read without imagining a point particle secretly testing every road. A real extended e-sphere writes phase into the connected wave routes allowed by the medium. Stationary-phase families reinforce; rapidly varying neighbouring phases cancel. The integral over histories is the standard formal amplitude representation. The WSM task is to derive its measure, configuration space and phase \(e^{iS/\hbar}\) from the one wave action—not merely to rename it “Huygens.”

13.2 Renormalisation and the spectral contract

A finite e-sphere does not automatically solve renormalisation. QED relates regulator-dependent intermediate quantities to physical pole locations, residues and inclusive observables through symmetry-preserving conditions. A WSM replacement must either reproduce that structure or calculate the same observables by a finite response with no hidden cutoff dependence.

Equivalently, the reduced two-point functions need a positive physical spectral measure: one electron pole or controlled resonance, two photon massless residues, the correct multiparticle continua and no negative-norm contribution. This is the real-wave counterpart of the Källén–Lehmann and unitarity audit. Extended structure may soften ultraviolet response, leave conventional running intact over the tested range, or fail; only the solved spectrum decides.

14. The rank-two electromagnetic radiation target

The free paired action is generous: every unconstrained coefficient \(g_A\) becomes a gapless luminal mode. If \(V_0\oplus V_2\oplus V_4\) were treated as independent, that would give fifteen real coefficients before quadratures—far more than the two electromagnetic polarisations observed.

This is a linear-order falsifier, not a distant detail. The coupled action must show which combinations are:

• constraints rather than independent modes;
• slaved into the e-sphere’s internal closure;
• bound or non-radiative;
• absent from the long-range source residue;
• or quantitatively below observational bounds.

The electromagnetic target is a rank-2 transverse projector for a spin-1 radiative response, with exactly two freely propagating receiver residues. For propagation direction \(\hat{\mathbf k}\),

The latest WSM construction supplies a concrete candidate for this rank rather than merely demanding it. For propagation along \(\hat{\mathbf k}\), parameterise the orthogonal Huygens great circle by

A source-written curve/phase disturbance has angular expansion \(\xi(\varphi)=\sum_m\xi_m e^{im\varphi}\). Its first harmonic gives

Therefore \(\hat{\mathbf k}\cdot\mathbf A_\perp=0\) identically, and rotation by \(\theta\) about the propagation axis gives phases \(e^{\pm i\theta}\): exactly two helicity-\(1\) hands with \(2\pi\) closure. Each constituent Space wave remains longitudinal along its own \(\hat{\mathbf n}\); transversality belongs to the collective first angular moment. This is exact geometry once the first-harmonic sector is selected.

D spectral gate The foundational angular Hessian must still show that \(m=\pm1\) are the unique massless poles, that \(m=0\) is constrained or lifted, and that \(|m|\ge2\) modes are gapped or non-radiative. A reverse-engineered operator such as \((\partial_\varphi^2+1)^2\) has the desired kernel; only the Space action can establish that Nature uses it.

Continuity guardrail. WSM has therefore already constructed the geometrical two-helicity candidate; it has not “forgotten” photon helicity. What remains is the harder spectral theorem: the reduced constrained Hessian must make this pair—and only this pair—the healthy massless radiative residue, with the gauge null direction and Gauss constraint but no scalar photon. The scalar-jet matter texture cannot fill this role: its infinitesimal tangent is longitudinal, \(\delta\Gamma\propto\boldsymbol\sigma\!\cdot\!\mathbf k\), whereas the photon residue is transverse to \(\mathbf k\).

“Transverse” here describes the two differential response patterns seen by matter. It need not mean that a second transverse substance has been added to longitudinal Space. WSM must calculate how directional longitudinal phase relations project into those two receiver quadratures.

A constrained static or near-field longitudinal response is not a third freely radiative polarisation. The decisive count is the number of independent on-shell residues that carry energy to distant receivers.

Do not import the gravitational \(+\) and \(\times\) spin-2 tensors into this section. Electromagnetic linear polarisations are spin-1 response quadratures; their circular combinations are the two helicity hands. The representations and source residues are different even if both are carried by the same underlying waves.

Part V — Constants, Form Factors and Magnetic Moments

15. Fine structure as one source–receiver stress ratio

The fine-structure constant is not merely a beautiful number. It measures how strongly one charged recurrence changes another compared with the receiver's inertia. Enclose receiver B, integrate the incoming-minus-outgoing Noether stress through one complete recurrence and define

Every term has a visible wave meaning. \(s_{q,A}s_{q,B}\) is the relative radial-phase sign; \(\Delta\mathbf P_B\) is momentum delivered by the complete real-wave stress; and \(M_B\) is the action cost of changing the maintained receiver wave egg. Separate write and read amplitudes depend on coordinate normalization; this complete ratio does not.

15.1 The fixed geometric skeleton

A · exact arithmetic/C · electromagnetic identification

The factors have a clean reading: \(4\pi\) is the all-direction solid angle and \(2k_0R\) is the diametral phase across P3's fixed e-sphere. The value lies about \(0.203413\%\) below the CODATA inverse coupling \(137.035999177(21)\). That near equality is a powerful skeleton target, not permission to move the radius.

D · decisive normalization The same solved e-sphere and two-centre stress must return the measured value without measured \(\alpha\) in the Action, boundary or initial seed. If the missing factor divides symmetrically between source writing and receiver reading, the conditional per-leg diagnostic is \(\sqrt{\alpha/\alpha_0}=0.9989824174\ldots\); the stress calculation must decide whether that factor exists.

15.2 Where \(4\pi\) belongs

Unit conventions can move \(4\pi\) between charge, flux and field definitions. They cannot create the physical coupling. The derivation is complete only when the same convention-independent source flux, receiver response and inertial normalization give \(\alpha\), \(F_1(0)=1\) and the measured force.

16. The electromagnetic vertex, form factors and the pointlikeness wall

Continuing the declared conventional target convention \(\hbar=c_0=1\), define the electromagnetic vertex and current matrix element of an on-shell charged spin-\(\tfrac12\) mode by

The physical sign is carried by \(Q_{\rm em}\); \(\mathcal V^\mu_{\rm em}\) is the charge-stripped vertex, and its dimensionless form factors are normalised by

In WSM these functions must be residues of the solved, finite, periodic current—not inserted phenomenological decorations.

16.1 What \(F_1\) means in real waves

An external train does not push every part of the e-sphere in the same phase when its wavelength becomes comparable to the structure’s coherence support. The integral of the local current therefore contains cancellations. \(F_1(q^2)\) records how the complete charge-current pattern looks when probed by wave curvature of spatial scale \(1/|\mathbf q|\).

A large coherence support need not equal a large charge radius if positive and negative current contributions cancel almost perfectly. But that cancellation must be demonstrated. For a conventional spacelike variable \(Q^2=-q_{\rm tr}^2>0\), the small-\(Q\) slope defines the Dirac current radius through

This is the Dirac current radius. If an electric Sachs radius is quoted, use instead

in the declared natural-unit target convention, with the appropriate \(\hbar c\) conversion when \(Q\) is expressed in energy units.

The structural WSM form factor means the background- and soft-radiation-subtracted current response of the solved e-sphere. Comparison with scattering must use an inclusive infrared-safe observable in the same convention; the intrinsic current profile, its correlation support and an infrared-dressed experimental slope are not automatically the same radius.

16.2 The experimental wall

If the phase-count support is calibrated with the full electron Compton wavelength, then

A model-dependent LEP2 Bhabha-scattering analysis gave an illustrative direct form-factor bound

a ratio

Equivalently,

This is not automatically fatal because coherence support, mass support and electromagnetic current radius need not coincide. It is nevertheless a severe and constructive symmetry target. If an unsuppressed current slope were naturally of order \(R_{\rm coh}^2\), the source/read projection would have to cancel or suppress it by more than about one part in \(5.6\times10^{13}\). “Almost spherical” is not enough; the solved current needs an exact symmetry, orthogonality or source–receiver identity. That is precisely how a large real coherence support could remain electromagnetically pointlike. The page must not call the extended e-sphere successful until its calculated \(F_1(q^2)\) passes this test.

16.3 The Floquet sideband wall

A periodic extended structure generically offers inelastic sidebands \(q^0+n\Omega_e\). The physical e-sphere must explain why low-energy experiments see one stable electron mass and no obvious internal excitation spectrum. Possible outcomes are constrained selection rules, sidebands outside accessible energy, vanishing residues, or a genuinely rigid collective sector. “The harmonics are virtual particles” is not an explanation.

Zero total cycle-averaged excess flux is weaker: a periodic structure can absorb in one open channel and emit in another while conserving total action. The solver must therefore report every open harmonic residue separately. Closed, bound and off-shell response may still dress the current and contribute to \(F_2\); the kill condition forbids only unobserved freely outgoing components. Unitarity is not identical to non-radiation.

16.4 Do not confuse \(F_1\) with vacuum polarisation

\(F_1(q^2)\) is the electron’s current form factor. The running coupling is governed by the background/radiative two-point response, conventionally written through a vacuum-polarisation function \(\Pi_{\rm vac}(q^2)\):

Here \(\Pi_{{\rm vac},{\rm ren}}\) is distinct from canonical momentum \(\Pi\). WSM may interpret it as polarisation of the common wave background and its bound structures, but it must reproduce the measured logarithms, thresholds and screening sign. Above one Dirac-lepton threshold, the leading target is

The same response must give the correct mass thresholds and the imaginary part associated with real production. A finite electron current and sea polarisation are different calculations.

17. \(g\), \(F_2\), and the anomalous magnetic moment

The magnetic moment is the cycle-averaged moment of the e-sphere's background-relative current about its collective centre:

The angular momentum must be cycle-averaged about the same centre and derived from the same background-relative Noether momentum/action ledger of the real waves; its conventional reduced description is the stress-energy/current ledger. Their relation defines \(g\):

For the standard on-shell vertex convention,

when \(F_1(0)=1\). These equations define the required output. They are not obtained by counting \(2\pi\) and \(4\pi\), geometric-algebra grades, quadratures or Huygens eigenvalues.

17.1 What \(F_2\) physically measures

The anomaly is not total egg area, total deformation energy or an unsigned rotation rate. In the standard light-front current representation it is exactly the left–right dipole of the transverse current density:

This exact all-orders identity and the finite-transfer Pauli profile are derived by Hoyer and Kurki.

The two sides are read relative to a prepared spin direction. Their difference changes sign when the spherical hand reverses. In WSM language, the magnetic perturbation makes one side of the real flow-through current rebuild slightly more strongly than the other.

The minimal parity algebra is equally sharp. Split source and read into hand-even and hand-odd parts,

For an even Hessian \(H\), the signed response is

A magnetic anomaly therefore needs even–odd interference. A square such as \(\varpi^2\) can shape the magnitude, but cannot carry the hand sign alone.

Local circular-motion lemma

Purely longitudinal waves can create the required local circular displacement when many directions meet in quarter-phase coherence:

Every constituent displacement is parallel to its own propagation direction. At the centre,

Its spatial quadratures remain gradients:

They are curl-free separately. Their cross-direction, quarter-phase sum creates local circular motion without inserting a fundamental transverse Space wave.

A This proves the local longitudinal quadrature mechanism. B §4.1 joins the local planes by a frame-free pair correlation and the sphere-varying operator \(h\mathsf J_{\hat r}\).

17.2 The complete one-loop Pauli shape

Use spacelike transfer \(Q\) and its Breit half-rapidity:

For a real wave direction \(\hat n\) and transfer direction \(\hat d\), let

If \(\mu=\cos\theta\), the same parameter is the spinor half-angle share:

The before/after real-wave Doppler factors in the symmetric Breit relation are

The same transverse mismatch is the noncommutativity of two spatial orientation reads. With \(P(\hat n)=\tfrac12(\mathbf1+\boldsymbol\sigma\!\cdot\!\hat n)\),

The matrices compact the real orientation algebra; the waves themselves remain real and longitudinal.

What “one loop” means here. In QED it names the first term of a controlled expansion in the measured coupling. WSM may use the same order label only after its action derives the small parameter, canonical power counting and current normalisation. Write the required weak-response expansion as

The conventional QED target for the first coefficient is

The normalized one-loop QED Pauli form factor is exactly the isotropic directional compliance

Straighten the direction coordinate with \(y=\operatorname{artanh}\mu=\tfrac12\ln[x/(1-x)]\). Uniform \(x\) becomes the normalized real profile

The before and after directional profiles are the same real shape shifted by \(\pm\eta_Q\). Their coherent overlap is the complete one-loop Pauli shape: motion separates the profiles, and spherical reclosure measures what still overlaps.

The transferred momentum changes the relative orientation of the two real wave legs. Directions parallel to \(\hat d\) remain matched; transverse directions slip most. The sphere rebuilds from the coherent all-direction average.

The exact one-loop QED target that the WSM response must generate is therefore

A benchmark The complete shape is exact, and \(J_{\rm em}/J_{\rm loop}=\alpha_{\rm WSM}/(2\pi)\) is an exact identity under the declared action definitions. C physical connection The living current must derive the expansion, select that ratio in the magnetic read and excite the correctly normalised spherical-hand response. This preserves the strong result without calling a supplied QED integral an action-derived WSM loop.

It also obeys the blind differential test

17.3 A minimal real closure model

Two real quadratures already contain the required denominator. Let \(\mathsf J^T=-\mathsf J\), \(\mathsf J^2=-I\), and

Minimise the directional mismatch cost

The settled state is exact:

Writing \(\tan\chi=p\) gives the vivid form

The returned direction is the equal standing-wave mean of the original quadrature and its twice-rotated partner. If their canonical weights are \(w\) and \(1-w\), the component along the original direction is

The QED kernel \((1+p^2)^{-1}\) therefore selects \(w=1/2\): equal canonical in/out weight. C This is a blind test of the living action. The trial scale \(\Lambda_{\rm tr}\) remains distinct from the q-phase stiffness \(\Lambda_q\).

The minimal norm-preserving completion makes the wave motion visible:

C Motion does not erase the spherical action in this completion; it turns a direction-dependent share into the wave egg. The living action must make this complementary channel the actual Lorentz/de Broglie deformation.

The coefficient \(F_2\) is hand-even; the physical magnetic response is \(hF_2\). The odd quadrature \(p/(1+p^2)\) describes motion-odd phase leakage and vanishes at \(Q=0\); it is not the anomaly itself.

17.4 The action metric selects the directional weight

Let \(C_h\) be the physical hand tangent pulled back from the living e-sphere and \(\mathcal G_{\rm action}\) its kinetic/action metric:

Canonical normalisation must make the mean source/read weight flat on every equal-\(\upsilon\) latitude:

This is the exact uniqueness condition for recovering \(\mathcal G_P(X)\) for every spacelike \(X\). Hand and azimuthal structure may remain; only their latitude average must be constant.

The source–dimension theorem makes the result quantitative. With effective weight \(\upsilon^r\) in \(d\) dimensions,

Its slope is

The Pauli slope \(1/6\) and its ultraviolet logarithm both select \(d_{\rm eff}=3\). Since physical Huygens Space is already three-dimensional, the effective magnetic weight must have \(r=0\).

The raw transverse norm is a decisive negative control:

It is the wrong \(d_{\rm eff}=5\) response; changing \(\alpha\) cannot repair its shape. Explicit division by \(\sqrt{\upsilon}\) is one possible angular normalisation, while full action whitening is the physical operation the solver must perform.

An axisymmetric sea perturbation \(w(\mu)=1+\epsilon_2P_2+\epsilon_4P_4+\cdots\) leaves \(\mathcal G_w(0)=1\) but changes the slope to

Background anisotropy can change the finite-transfer shape; it cannot create the universal zero-transfer prefactor. That scale belongs to the intrinsic signed hand source and phase stiffness.

17.5 Finite-direction benchmark quadrature and the \(2m_e\) edge

The exact angular probability measure is

In the adapted Jacobi basis, multiplication by \(\upsilon\) is tridiagonal with

An \(N\)-rung benchmark quadrature is exactly \(N\) antipodal Gauss–Legendre direction pairs:

where \(\mu_j>0\) are the positive zeros of \(P_{2N}\). Each node is a latitude ring after azimuthal integration—not a physical e-sphere axis. The chain is zonal:

The Jacobi-matrix/Gaussian-quadrature correspondence and endpoint-zero asymptotics are standard results collected in the NIST DLMF orthogonal-polynomial chapter and its Jacobi-zero section.

The exact value is \(\mathcal G_P(2)=0.760345996300946\ldots\). For \(X>0\), Gauss and Gauss–Radau give certified brackets. The first two are

At \(X=2\), \(0.7602339181\le0.7603459963\le0.7603603604\). The closest pole approaches the continuum edge as

Resolving a distance \(\delta\) from threshold needs roughly \(N\gtrsim\pi/(2\sqrt\delta)\). At fixed \(N\), the ultraviolet tail is \(C_N/X\), not \(2\ln X/X\); uniform high-transfer resolution needs \(N=O(\sqrt X)\).

Quadrature direction \(\ne\) physical e-sphere axis; quadrature pole \(\ne\) particle.

17.6 One angular measure, distinct QED reads—and complementary GDH consistency

Let \(\mathsf M_\upsilon\) multiply a directional state by \(\upsilon\), and let \(e_0\) be the isotropic state. Then

Define the master angular logarithm

The same angular measure gives the renormalized one-loop vacuum-polarisation benchmark

The apparent singularity at \(X=0\) cancels: \(\widehat\Pi(0)=0\), \(\widehat\Pi=X/15+O(X^2)\), and \(\widehat\Pi(2)=1/9\).

At the common timelike edge, \(\beta=\sqrt{1-4/t}\), the two receivers weight the same directions oppositely:

The Pauli read enhances the balanced in/out branch merger; the screening read inserts its squared imbalance and vanishes there. The exact edge follows from \(\upsilon_{\max}=1/4\Rightarrow t_{\rm edge}=4\). A physical \(2m_e\) pair mode is the corresponding Hessian and nonlinear two-branch test, with conserved energy, phase and exchange structure.

Shared measure is not shared ontology. The Pauli and vacuum-polarisation benchmarks use the same one-loop angular parameter and threshold geometry, but their source operators, receiver functionals, tensor structure and normalisations differ. This exact mathematical unity is a valuable clue for one underlying Space; it does not prove that one scalar WSM mode produces both observables.

Complementary real-scattering test of the Schwinger prefactor

In natural units, with a fixed helicity-difference convention, the linearised GDH sum rule reads

The tree-level derivative route to the leading anomaly is established by Pascalutsa, Holstein and Vanderhaeghen and in their expanded analysis.

Use \(\varepsilon=\hbar\omega/(m_ec_0^2)\). Before differentiating, the complete tree spectrum must obey the zeroth-order cancellation

The differentiated target is

The negative tail is essential: the real response must reverse phase or cancel; an all-positive weakening cascade cannot pass this test.

The spacelike Pauli kernel is a positive Stieltjes response, whereas the GDH helicity-difference spectrum necessarily changes sign. They therefore cannot both be produced by one positive scalar weakening channel with one unsigned read. At minimum the living e-sphere requires a hand-resolved matrix response—or two distinct projections of a multi-channel causal operator. This strengthens the scalar-dressing no-go without turning Pauli and GDH into separate substances.

D The direct signed-current result and this helicity-scattering result must agree. Their agreement tests the hand mode, current, action normalisation, causal state, Ward identity and optical theorem at once.

17.7 The flow-through-dressed magnetic response

The AMM programme has one exact response reduction. Split the projected perturbations into magnetic/current modes \(M\) and every coupled deformation/Space-response mode \(D\), with genuine collective zero modes separated before inversion:

Eliminating the driven deformation sector gives the retarded magnetic operator

This Schur complement is mathematical compression of an enlarged causal system, not the physical mechanism and not the path taken by one carrier. The mechanism is the real shell/cross-wave recurrence displayed in §§17.8–17.9: a weak source-generated magnetic configuration changes the living e-sphere; changed out-waves cross fresh in-waves; the One Law alters what those in-waves later carry back through the centre. Eliminating the unresolved deformation and surrounding Space-wave coordinates produces the effective magnetic operator above. Projecting the corrected current vertex gives the target \(F_2(0)\). The solved symmetry—not a prior story—must decide whether \(D\) contains \(V_2,V_3,V_4,V_6\), bilocal coherence or other coupled sectors.

The local paired kernels provide a useful scale check, not a licence to truncate. Their even angular eigenvalues include

so a double insertion in a purely local Schur-complement response would weight \(V_4\) relative to \(V_2\) by \((h_4/h_2)^2=2.78\%\) in the kinetic kernel and \((v_4/v_2)^2=1.5625\%\) in the stiffness kernel. But the complete finite-radius chord flow-through operator is different: at \(k_0R=\pi\sqrt3\), the Action-page calculation finds \(|H_4/H_2|\approx1.06\). The factor \(e^{2ib\mu}\) in \(H_\ell(b)=2\int_0^1\mu P_\ell(\mu)e^{2ib\mu}d\mu\) records phase accumulated across a complete chord; it never requires reflection. The local percentages are therefore diagnostic only. They do not show that the full e-sphere’s \(V_4\) response is small; the complete coupled operator must retain it.

17.8 What the AMM calculation looks like in real Space

1. Solve the unperturbed open periodic e-sphere \(Z_e(t)\).
2. Derive its conserved electric current, mass, collective angular momentum and magnetic moment.
3. Let the real external apparatus establish an infinitesimal, slowly varying source-generated magnetic/phase configuration.
4. Remove the translational, orbit-phase and orientation collective zero modes correctly.
5. Solve the driven Floquet deformation, retaining every symmetry-allowed sector.
6. Follow each affected in-wave as it flows through the wave centre and continues outward carrying the changed curvature and phase written by the deformed e-sphere.
7. Where a changed out-wave crosses fresh waves already travelling inward, calculate the resulting change of the complete directional \(E_d\), hence of their local speed, phase accumulation and curvature.
8. Propagate those fresh changed in-waves onward to the e-sphere and solve the next, smaller local correction.
9. Recompute the current and extract the term linear in the magnetic perturbation.
10. Project the resulting vertex onto \(F_1\) and \(F_2\).

The image is concrete. A weak magnetic configuration does not twist an abstract intrinsic arrow. It changes the relative timing of real waves flowing into different directions of the spherically rotating phase system. The wave egg changes: directional \(E_d\), phase, curvature, convergence, circulation and current all adjust together. The carriers still flow through. What changes is the geometry they carry onward and therefore the conditions encountered by later in-waves.

The One Law at the crossing

For a locally fixed carrier frequency, the One Law gives the direct phase ledger

If the perturbing \(E_d\) profile were static, this would give the familiar weak phase integral \(\delta\phi\simeq-k_0\int(\delta E_d/E_{d0})ds\). But the relevant profile is itself moving. In the simplest one-dimensional antiparallel control, let \(\delta E_d/E_{d0}=f(r-c_0t)\) move outward while the fresh in-wave follows \(r=r_1-c_0t\). Along that in-wave the argument changes twice as fast, so

The factor \(1/2\) belongs to this antiparallel equal-speed control, not to every three-dimensional crossing. In the full e-sphere the relative direction, local \(c'\), curvature and angular coherence all enter the kernel. That angular generalisation is part of the calculation rather than a number to insert by hand.

Causal memory without a returning wave

Every possible crossing radius carries its own delay. If the changed out-wave reaches radius \(r\) and modifies an inward carrier there, the causal delay to the later e-sphere response is

in the calm-background control. At the phase-count radius \(R/\lambda_0=\sqrt3/2\), this particular ledger gives \(\tau(R)\simeq\sqrt3\,T_0\); it is a characteristic \(r=R\) delay, not a reflecting boundary or a universal minimum.

A reduced linear description should therefore be a distributed causal-memory equation rather than one literal out-and-back cycle:

Here \(K_\perp\) is a cross-wave response kernel: changed outward propagation, local state-dependent interaction with a different inward carrier, and that carrier's subsequent inward propagation, all projected away from the exact collective zero modes. It is not a reflection kernel. Fourier transforming this continuous causal memory produces the familiar resolvent form

Finite spherical shell encounters

Let an outward shell \(m\) and a fresh inward shell \(n\) carry direction-dependent longitudinal shifts \(\zeta_m^+(\Omega)\) and \(\zeta_n^-(\Omega)\). Their actual crossing event is

The memory state must retain harmonic content, hand, phase, quadrature and both directions:

The open retarded cascade \(M_{\rm ret}\) carries disturbances away through fresh waves; the closed periodic operator \(M_{\rm cyc}\) tests e-sphere reclosure. Stability requires \(r(M_{\rm ret})<1\), while non-normal transients are measured separately by

At an ordered two-direction crossing, keep the symmetric and handed invariants distinct:

For two independently isotropic directions,

A One exact number is now read three ways: the isotropic handed crossing of two directions, the variance of the real \(\operatorname{sech}\) rapidity profile, and the selected three-dimensional e-sphere geometry. The oriented numerator dies as two directions become parallel, leaving a finite Pauli-like channel. Shell generation, \(X_Q\)-moment and QED loop order remain different ledgers.

This also separates two experiments that should never be conflated. After a finite impulse, non-collective deformation of a stable open e-sphere should decay locally as changed wave action is carried away to infinity. Under a persistent weak magnetic drive, however, the electron approaches a new driven periodic flow-through state. The AMM is not a leftover after the external perturbation disappears. It is the \(B\to0\) magnetic/current derivative of that converged state, projected onto the Pauli form factor \(F_2(0)\) with the same current normalisation that gives \(F_1(0)=1\).

Do not force a scalar \(V_3\) term merely because the moving wave egg contains odd directional structure. A quasi-static magnetic perturbation has definite symmetry. \(V_2\) and \(V_4\) are natural even deformation sectors; \(V_3\) enters only if the solved source, orientation and boundary relation permit it. The wave egg itself is the complete directional state, not one Legendre coefficient.

17.9 \(g=2\) and the Schwinger action ratio

\(g=2\) is the baseline of the derived minimal Dirac current; \(g-2\) is the finite flow-through response.

The reduced Dirac current gives \(g=2\) conditionally when the solved collective mode has the positive Clifford reduction of §11, every grade carries the same minimal q-odd phase connection, and no independent Pauli term is inserted. Then the exact algebraic square is

so the nonrelativistic current contains \(-Q_{\rm em}\mathbf S\cdot\mathbf B_{\rm eff}/m_e\), fixing the baseline \(g_0=2\). Here \(\mathbf B_{\rm eff}\) is the curvature of the reduced phase-transport connection made by the same real waves, not an added magnetic substance. The \(S^3\) half-angle versus \(S^2\) full-angle geometry explains visually why the factor two is natural; it does not normalise \(\boldsymbol\mu/\mathbf S\). Finite e-sphere deformation and wave-sea feedback then add the independent transverse coefficient \(F_2\).

QED’s leading target is

The WSM action gives this coefficient a transparent normalization ledger without treating circumference as proof:

The second equality follows only if the same solved action first gives \(J_{\rm em}/J_{\rm cl}=\alpha_{\rm WSM}\). It is therefore a normalization target, not an independent derivation of Schwinger’s coefficient. One complete internal quadrature cycle encloses \(2\pi J_{\rm cl}\) in phase space while the associated spherical orientation closes over \(4\pi\); neither circumference fixes the magnetic current. The direct Pauli response and the GDH integral ask the same blind question in different ways: does the derived current, with \(F_1(0)=1\), return the signed ratio \(F_2(0)=\alpha/(2\pi)\)?

17.10 Why one scalar dressing fails

The first QED coefficients provide a useful structural test of any proposed WSM response before the full calculation is attempted. With \(u=\alpha/\pi\), the mass-independent electron series begins

Suppose, only as a control, that the entire finite-e-sphere correction were one scalar geometric screening channel,

Matching the two-loop coefficient forces \(\lambda=0.6569579312\ldots\), which then predicts a three-loop coefficient \(\lambda^2/2\simeq0.2158\), nowhere near \(1.18124\ldots\). Thus a one-pole scalar geometric recurrence cannot be the AMM mechanism.

17.11 The nonlinear cascade beyond one loop

The quadratic Hessian gives the linear response and the first retarded sideband mixing. Further turns of the physical cascade necessarily sample higher variations of the same frozen Action:

In Space, one perturbation changes the e-sphere; the changed e-sphere writes altered out-waves; their crossings change later in-waves; those in-waves rebuild the e-sphere again. At each turn the response can mix new radial, angular and temporal harmonics. The algebra above is the compact perturbative ledger of that living sequence.

Euler–Heisenberg: the quartic wave tribunal

For two emergent optical waves the response must vanish for one ideal null wave, vanish for co-propagating null waves and grow with the squared angular crossing factor. The exact symmetric angular selection rule is

Thus two first-harmonic optical modes can drive scalar breathing and quadrupolar internal response. If their parity-even quartic invariants are \(X=-F_{\mu\nu}F^{\mu\nu}/4\) and \(Y=-F_{\mu\nu}\widetilde F^{\mu\nu}/4\), QED requires the Euler–Heisenberg ratio

The same Action derivatives that produce the AMM must produce this \(4{:}7\) optical response, its signs and its positive spectrum.

17.12 The muon is the decisive repeat calculation

The muon calculation must use the same action, current definition and projection with a long-lived metastable periodic/Floquet lepton resonance. It cannot be obtained by taking the electron cascade and adding a hand-chosen proton-overlap term. The electromagnetic calculation must determine the muon mass branch, support, current and \(F_2^\mu(0)\). Its lifetime is a downstream weak-transition calculation of the unified action, not a QED output.

A successful WSM calculation should produce electron and muon masses and \(F_2(0)\) from the same electromagnetic response law, then allow the weak-sector calculation to determine the muon lifetime. If each lepton requires a new geometric correction chosen from its measured anomaly, no unification has occurred.

18. Present numerical benchmarks

The direct electron number comes from the 2023 one-electron Penning-trap measurement. The CODATA value is a recommended adjusted constant and should not be presented as though it were a second direct experiment. A clean AMM comparison should also show the independent atom-recoil determinations. Using their quoted standard uncertainties, the Rb and Cs inverse-\(\alpha\) values differ by about \(5.49\sigma\); provenance therefore matters.

A Penning trap does not expose two little classical rotations for direct viewing. It resolves quantum transitions and reconstructs the anomaly from frequency relations after cavity, relativistic and trap corrections. The WSM calculation must predict that same operational result.

The final Fermilab report distinguishes its \(127\)-ppb combined result from the \(124\)-ppb experimental world average. Against the 2025 Theory Initiative value, the world-average difference is approximately \(38.5(63.7)\times10^{-11}\), so there is no present significant experiment–theory discrepancy. Tensions among data-driven dispersive inputs and between evaluation methods remain active. This page therefore does not advertise a settled “muon anomaly” as proof for or against WSM.

Part VI — The Calculation That Can Close or Break the Programme

19. Frozen e-sphere and QED response protocol

The next advance is one executable wave calculation. Freeze the foundation and the Action before comparing a precision number; then let the real wave state answer.

1. Freeze one direction-resolved Action. Publish every field, coefficient, constraint, boundary term, symmetry and normalization.
2. Derive the wave mechanics. Obtain characteristics, canonical form, Noether current and complete stress directly from that Action.
3. Solve the calm sea. Establish its directional spectrum, positive physical modes and absence of hidden duplicate carriers.
4. Solve the living P3 e-sphere. Use \(R=\sqrt3\lambda_0/2\), an open exterior and no reflecting wall, point source or fitted electron mass.
5. Measure finite relative energy. Subtract the sea value and tangent; derive \(M=E_{\rm rel}/c_0^2\) and the recurrence action \(J_*\).
6. Prove stability. Report Floquet quartets, Krein signatures, continuum resonances, convergence and the fate of every collective zero mode.
7. Construct the two spherical hands. Exhibit their direction weights, opposite phase rotation, spherical quadratic reads and \(4\pi\) closure.
8. Move the e-sphere. Recover the complete wave egg and distinguish the raw unequal-speed ledger from the reciprocal Lorentz–de Broglie representation.
9. Derive charge three times. Calculate the Noether generator, active far write and passive receiver read; require their equality and \(F_1(0)=1\).
10. Project Dirac. Calculate \(G,C_i,R\), \(\epsilon_D\), \(\epsilon_{\rm leak}\), the positive norm and \(\Omega_D=\omega_e\).
11. Derive the optical sector. Obtain the recurrence-phase connection, Ward–Takahashi identity, optical theorem, two radiative residues and no scalar photon.
12. Follow the AMM cascade. Perturb the e-sphere, propagate its changed out-waves, calculate their crossings with fresh in-waves and sum the convergent retarded rebuilding of the current.
13. Calculate the precision packet. Return \(\alpha\), \(F_1(Q^2)\), \(F_2(Q^2)\), the Pauli edge, electron and muon AMM, running and light-by-light response from the unchanged vertices.
14. Publish before comparison. Hash the input, numerical resolution and output packet before unblinding against measured constants.

20. Direct failure conditions

• No finite, open, stable e-sphere occurs at P3's fixed radius.
• The physical quadratic action metric is indefinite, or unstable modes survive refinement.
• The two non-scalar spherical hands cannot be constructed from the direction-resolved longitudinal state.
• The four-mode projection fails positivity, Clifford closure, \(\Omega_D=\omega_e\) or \(\epsilon_{\rm leak}\to0\).
• Noether, active and passive charge disagree, or \(F_1(0)\ne1\).
• The reduced optical spectrum has other than two physical radiative residues, or contains a scalar optical ghost.
• Ward–Takahashi, the optical theorem or positive spectral weight fails.
• The fixed source–receiver stress does not return the measured \(\alpha\).
• The same current does not return both the static AMM and its absorptive/GDH representation.
• The normalized Pauli response misses its spacelike shape, \(2m_e\) edge, discontinuity or positive spectral form.
• Electron and muon require different hand-chosen corrections or new fitted geometry.
• The retarded AMM cascade grows without bound, double-counts modes or depends on a reflecting exterior wall.

21. Quarantine: routes not to re-import

• Q A determinant-only volume law used as the direction-resolved WSM foundation.
• Q A paired/exponential representation added as a second physical carrier.
• Q Vorticity, ordered strain or matrix holonomy renamed as the physical spherical phase wave.
• Q Scalar-jet or borrowed soliton topology identified with the electron or electric charge.
• Q Invented reciprocal reconstruction grades used to double the Dirac state count.
• Q A single wave said to reverse direction and return to its source.
• Q P3's radius, \(\alpha\) or an AMM coefficient tuned after comparison.
• Q One-body \(4\pi\) closure treated as a derivation of many-body fermion statistics.
• Q A response trace, Schur complement or four-wave transfer called a QED fermion loop before physical mode projection and the state measure are derived.
• Q Quadratic receiver work alone treated as a normalized, exclusive Born rule.

22. The living AMM cascade, compressed for AI

This is the physical idea that must remain alive through every mathematical reduction. A weak magnetic arrangement changes the timing and directional balance of the real waves entering an e-sphere. The e-sphere does not acquire a little abstract arrow; its complete wave egg changes—its phase intersections, curvature, convergence and circulating current.

The changed out-wave in the middle of this chain continues outward. At a surrounding crossing region it overlaps a different plane wave already travelling inward. Because the complete directional state there has changed, P2 changes the inward wave's local propagation speed and accumulated phase. That inward wave continues to the centre and participates in the next reconstruction. The next changed e-sphere writes a smaller correction onto later out-waves, and the causal sequence continues through progressively more distant crossings.

After eliminating the resolved exterior coordinates, the same history becomes a retarded recurrence:

The periodic base state converts this spatial history into Floquet sidebands. A static probe enters the zero external harmonic, mixes through \(n\Omega_e\), and returns to the zero harmonic as a changed cycle-averaged current:

This is the WSM candidate for anomalous magnetism: finite reciprocal reconstruction of the current through real crossed waves. The electron and muon calculations must use the same Action, current and causal summation. The direct magnetic moment and the absorption spectrum are two readings of this one cascade.

23. From Feynman’s calculus back to a physical picture

This page began from a dissatisfaction that has lasted roughly thirty years. Richard Feynman’s QED: The Strange Theory of Light and Matter displayed astonishing predictive power while insisting that the physical picture would remain strange. Geoffrey Haselhurst’s response was not to deny the numbers, but to keep asking what real motion could make those numbers inevitable.

The proposed answer is visually simple even where the mathematics is difficult: one vibrating medium; spherical standing-wave matter; advanced or delayed curved portions written onto continuing longitudinal plane waves when bound patterns change; receiving e-spheres displaced and reshaped by coherent incoming curvature; and a causal cascade in one connected Space. An in-wave flows through a wave centre and continues outward. If the e-sphere has changed, that continuing out-wave carries changed geometry. Farther out it crosses fresh in-waves already travelling toward the centre; through the state-dependent One Law the crossing changes their local phase and curvature. Those fresh in-waves later reach the e-sphere and help determine its next state. No carrier turns around. The feedback is the evolving relation among different waves in the same Space.

The tier labels now carry the scientific status once, allowing the argument itself to stay physical and direct. Exact geometry remains exact; solver hypotheses remain testable; historical reverse engineering stays visible in the folded archive.

Human physical imagination and AI algebraic endurance have made this page possible. Neither should dominate the other. A vivid picture without a solved response can drift into story; mathematics without a picture can become detached bookkeeping. The aim is their reconciliation in one object that can be computed, measured and seen in the mind.

23.1 Primary numerical and mathematical sources

• Fan, Myers, Sukra & Gabrielse, Measurement of the Electron Magnetic Moment, Phys. Rev. Lett. 130, 071801 (2023).
• CODATA 2022 adjustment of the fundamental constants, Rev. Mod. Phys. 97, 025002.
• Morel, Yao, Cladé & Guellati-Khélifa, independent rubidium-recoil determination of \(\alpha\) (2020).
• Parker, Yu, Zhong, Estey & Müller, independent caesium-recoil determination of \(\alpha\) (2018).
• Muon \(g-2\) Collaboration, 2025 127-ppb Fermilab measurement.
• Muon \(g-2\) Collaboration, final Fermilab report and world average (2026).
• Muon \(g-2\) Theory Initiative, 2025 Standard-Model update.
• Bourilkov, LEP2 Bhabha-scattering compositeness analysis, Phys. Rev. D 62, 076005 (2000).
• Schwinger, On Quantum-Electrodynamics and the Magnetic Moment of the Electron (1948).
• Hoyer & Kurki, finite-transfer Pauli form factor and exact transverse-current dipole.
• Pascalutsa, Holstein & Vanderhaeghen, linearised GDH derivation of the leading anomalous moment; see also the expanded analysis.
• NIST DLMF, orthogonal polynomials and Jacobi matrices; Jacobi zeros and endpoint asymptotics.
• Yuan et al., experimental elastic spin from real displacement quadratures.
• Dunne, Euler–Heisenberg effective action review.
• Henriksson, McPeak, Russo & Vichi, Rigorous Bounds on Light-by-Light Scattering (2021).
• Akhilesh et al., Bargmann invariants and global geometric phase in finite-dimensional SU(2) systems.
• Battey-Pratt & Racey, spherical rotation geometry; neutron \(4\pi\) interferometry analysis.
• Laporta & Remiddi, analytical three-loop electron \(g-2\) coefficient, Phys. Lett. B 379, 283–291 (1996).
• Dirac, The Quantum Theory of the Electron (1928).

23.2 WSM corpus handoff

• Action of Vibrating Space: the fixed P1–P3 foundation, direction-resolved Action problem, e-sphere closure, Huygens geometry and mode gates.
• Quantum Theory: discrete bound changes, receiver work, Born-rule gate, Bell constraints and many-body relations.
• Relativity: moving e-sphere, reciprocal Doppler structure, clock phase, gravity and luminal response.

QED remains the precision target. The e-sphere remains the physical hypothesis. The bridge now has two complementary, non-identical calculations of the same anomaly: direct signed current and dispersive GDH scattering.

Real waves enter. A real spherical rhythm receives them. The e-sphere writes curved portions onto the plane waves that continue outward. Those waves cross fresh inward-travelling waves, changing the real wave relation from which the next sphere is rebuilt. The equations above are the disciplined attempt to make that visible motion equal the mathematics already measured so precisely.

Geoffrey Haselhurst, in Human–AI scientific collaboration. Reconstructed in current WSM real-wave language, 11 September 2026.