Stationary E-Sphere and Moving Wave Egg

Physical foundation

WSM Postulates

WSM Wave Action: the mathematics already in hand

Abstract / Summary

Real plane waves arrive from every direction, overlap, pass through the centre and continue. At rest, an isotropic coherent combination produces a spherical vibration at one fixed location. In the moving construction, the rear-entering contribution advances faster and the front-entering contribution advances more slowly, so their recurring meeting point advances through Space.

This essay calculates that picture using declared assumptions. The axial assignment \(c'_r=c_0+v\), \(c'_f=c_0-v\) supplies a front–rear asymmetry. For a stable, uniformly translating, single-period recurrence represented by two opposed coherent background waves, absence of cumulative phase slip requires phase matching at the moving centre. This fixes their frequency ratio. The separate assumption that their geometric-mean frequency is preserved fixes the remaining scale and gives the reciprocal Doppler factors, the Lorentz factor, the translating carrier and the de Broglie phase modulation exactly. The phase rate followed at the moving centre is \(\omega_0/\gamma\); its complete recurrence therefore lasts \(\gamma T_0\) in background-Space time.

A separate, explicitly assumed surface model spreads fixed assigned wave-layer energy over a flattened rear cap and an elongated front cap. It gives \(S_{\rm egg}/S_0=\gamma^2\). Choosing joined half-spheroids makes their heights, volumes and geometrical centroids calculable. The rear hemisphere-to-disk limit is consistent with \(|v|<c_0\). The same input factors \(1\pm\beta\) generate both the frequency and area appearances of \(\gamma\); these are connected consequences, not independent confirmations.

Essential WSM glossary

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

Claim-status key

1. Postulates, real-wave language and assumptions

WSM treats matter as recurrent wave organisation of Space. Space vibrates locally; wave patterns propagate. The centre of an e-sphere is the location at which its complete wave relation is continually reconstructed. An “egg” here is a proposed contour of that organisation, not a material shell that reflects the waves.

These postulates supply the WSM physical foundation. This note retains the corpus's infinite, eternal, continuous Space. The numerical choice of units can set \(c_0=E_{d0}=\lambda_0=1\); dimensional factors remain visible here to prevent different lengths and periods from being identified accidentally.

One universal intrinsic standard; several distinct phase measurements

Retained WSM cosmic-clock requirement. Stationary and moving e-spheres share the intrinsic standard \(f_0\). Electron and positron remain opposite radial phases relative to their reconstructing background wave sea, with no cumulative phase slip against that relation. This physical requirement is not removed when motion changes the coordinate readings of phase. Throughout this page, including its corpus summaries, “common frequency” means the common intrinsic recurrence unless a fixed-position frequency is explicitly specified. Section 6.4 distinguishes this background-relative lock from imposing the spatially uniform phase \(\omega_0t\) on every moving centre.

The coordinate \(t\) counts change using the balanced background-Space reference. For a local plane-wave phase, \(c'=f_{\rm fixed}\lambda'\) relates speed, crest spacing and frequency measured in the same coordinates. The intrinsic standard alone does not identify \(f_{\rm fixed}\) inside a moving egg.

Dependency notation. P denotes a postulate; H an auxiliary assumption; A exact mathematics within stated premises; B a deduction from named premises; C a proposed physical identification; D an unfinished calculation; Q a rejected identification. H-S labels stationary assumptions, H-M motion assumptions and H-G geometry assumptions. Each theorem states the assumptions it actually uses.

2. The stationary spherical vibration

Real waves first. Equal coherent contributions arrive from every direction. Opposed contributions balance, while their sum repeatedly compresses and releases the same central region. Space at that region vibrates; the centre of the recurring pattern stays in one place.

H-S1 — Isotropic coherent background. Directional weights, reference wavenumber and phase alignment are equal over the full sphere.

H-S2 — Linear reference kinematics. The reference waves obey \(\omega_0=c_0k_0\), may be superposed linearly, and have compressive strain \(\chi=-\nabla\cdot\mathbf s\). This is a linear reference construction, not an exact finite-deformation continuity law.

H-S3 — Open recurrence. The proposed e-sphere is maintained by phase-related waves that enter, cross and continue through its central region. Its physical continuation is assumed to reclose periodically. This assumption does not itself establish nonlinear stability.

2.1 The all-direction sum

Let \(A\) be a dimensionless compression amplitude. The real angular sum is

Choose the polar axis along \(\mathbf r\), write \(\mu=\widehat{\mathbf n}\cdot\widehat{\mathbf r}\), and integrate. The sine part is odd in \(\mu\), so

The limit \(j_0(0)=1\) makes the compression regular at the centre. This exact angular identity uses H-S1–H-S2; P3's radius is not needed to derive it.

2.2 Displacement and velocity follow from the compression

Within H-S2, a radial displacement satisfies

Integration therefore gives

Regularity at \(r=0\) excludes \(C(t)/r^2\). The remaining field has \(s_r\sim-Ar\cos\omega_0t/3\) near the centre. Consequently,

Compression and displacement share cosine time dependence with the displayed opposite sign and different radial factors. Vibrational velocity has sine time dependence: it is in quarter-cycle quadrature. At an extremum of compression the vibrational velocity vanishes. These are three readings of one longitudinal vibration.

2.3 Why the reference centre stays fixed

The vanishing directional first moment expresses the balance. More directly, the derived field depends on \(r=|\mathbf r|\) about one fixed origin and has no translating spatial argument. Under the stated isotropic phase alignment it reconstructs about that same origin. A vanishing first moment by itself, without these phase assumptions, would not prove stationarity of every possible wave state.

2.4 The P3 reference geometry

Result and scope. H-S1–H-S2 give the exact linear spherical reference; P3 supplies its designated core scale. A nonlinear variable-speed core can require corrections to the reference profiles. The core's finite radius does not truncate the extended wave or by itself prove finite energy relative to the background. H-S3, nonlinear stability and the two angular spin hands require additional physical construction.

3. Real curves on passing plane waves

Real waves first. If a portion of a plane wave crosses a region faster than the surrounding portion, it emerges advanced. The front has acquired a curve. The phase advance does not by itself mean that the stationary e-sphere has supplied extra energy to every passing wave.

A phase-screen description is

The corresponding complex phase multiplier \(e^{-ik_0\zeta}\) has magnitude one. At the screen it preserves the amplitude norm and changes phase. That limited identity neither adds energy nor proves every aspect of the complete wave-energy distribution: transverse gradients, amplitudes and subsequent propagation must be treated consistently. It also does not establish a general absence of source–receiver energy exchange.

A stationary hemisphere imprint is a separate speed-profile control

For straight parallel paths through a sphere, let \(b\) be the impact parameter and \(L(b)=2\sqrt{R^2-b^2}\) the chord. If the interior path speed is a constant \(c_s\), then after exit the advance relative to the unmodified plane is

The choice \(c_s=2c_0\) gives \(\zeta(b)=\sqrt{R^2-b^2}\), an exact hemispherical profile under those path assumptions. Curved rays or a varying speed require the actual travel-time integral.

This chord-speed control and the linear \(j_0/j_1\) reference are distinct reduced calculations. Likewise, the representative axial rates and cap-average densities used below do not specify an entire core profile. Identifying the stationary hemisphere imprint and moving egg as solutions of one local speed field is an explicit further requirement. In particular, the reduced axial rate tends to \(c_0\) at rest, whereas this uniform-chord control uses \(2c_0\); they cannot be assigned to the same path average without an additional physical relation.

4. The moving axial construction

Take motion along \(+x\), let \(v\ge0\), and define \(\beta=v/c_0\). “Rear” means the side behind the advancing centre; “front” means the leading side. We have not yet imposed \(\beta<1\).

4.1 Moving-recurrence premises and reciprocal closure

H-M1 — Axial reconstruction rates. Over a stated interval \(\Delta t\) of background-Space time, the rear-entering contribution is assumed to cover the balanced reference distance \(c_0\Delta t\) plus the centre's advance \(v\Delta t\); the front-entering contribution covers that reference distance minus the advance. These distances per background time are identified with representative physical axial characteristic rates. They are not a point-by-point solution for \(c'(\mathbf x,\widehat{\mathbf n},t)\).

H-M2 — Stable single-period background-wave representation. The same uniformly translating recurrence is assumed to admit two opposed coherent background waves at speed \(c_0\), representing its axial fundamental mode. The moving e-sphere reconstructs the same state one-to-one each period, with a fixed compatible phase offset and no cumulative phase slip. In real-wave language, corresponding crests, zero crossings and opposite crests keep arriving together at the centre. Equality of their encountered phase rates follows as a necessary resonance condition in Section 6.1; it is not a further independent assumption. Matching this background representation through the variable-speed core is assumed, not calculated here.

H-M3 — Reciprocal frequency closure. The two positive background frequencies preserve their geometric mean:

H-M3 supplies one scalar normalization not fixed by phase matching. Equal coherent amplitudes are an additional state choice used for the clean cosine product; the frequency and phase relations do not require equal amplitudes.

4.2 Speeds, densities and travel distances

H-M1 states a specific distance rule. Dividing by the same background-time interval gives

The signed propagation velocities are \(+c'_r\) for the rear-entering wave and \(-c'_f\) for the front-entering wave. Applying P2 to the representative sectors gives

The rear sector is denser and faster; the front is less dense and slower. Over the reference interval \(T_0\),

These are travel distances and a centre displacement. They do not determine simultaneous crest spacings unless the relevant fixed-position frequencies are supplied.

Conditional internal wavelengths. If the common intrinsic e-sphere frequency \(f_e=f_0\) is additionally identified with the internal crest-passage frequency in the coordinates used for \(c'\), then

This is an explicit additional internal assignment. It neither identifies these spacings with the calm-Space Fourier wavelengths derived in Section 6, nor establishes their physical connection. The numerical tables retain their travel-distance labels.

4.3 The two-sided recurrence has a speed boundary

Under H-M1, both directional characteristic rates must remain positive for the assumed two-sided inward recurrence. At \(v=c_0\), the front-entering rate \(c_0-v\) vanishes. Above \(c_0\), it reverses sign and contradicts that inward-rate assumption. Therefore

Reversing the motion exchanges front and rear, giving \(|v|<c_0\). This is a conditional limit of H-M1, not a proof from P1–P3 alone. It makes no claim that the front-entering wave has the longer journey: under H-M1 the rear contribution covers the longer reference distance.

5. The equal fixed-position frequency diagnostic

Real waves first. Assigning different propagation speeds to two waves does not yet say how often successive crests pass a fixed point. This diagnostic tests equal frequencies at fixed positions in background Space. It does not reject the universal cosmic clock or one common intrinsic recurrence.

For this diagnostic only, give the two rates \(c_0(1+\beta)\) and \(c_0(1-\beta)\) the same fixed-position frequency \(\omega_0\). Then

The sum factorizes exactly:

The first spatial factor stays fixed. It is not the translating carrier derived below. Thus the physical bridge cannot be justified merely by assigning a common fixed-position frequency and unequal wavelengths. H-M2 and H-M3 state the moving background representation and its normalization; mapping the internal waves to that representation remains required. This diagnostic uses \(1/(1-\beta^2)\) directly; the Lorentz factor is derived next.

6. Lorentz–de Broglie phase deduction

The opposed-wave construction continues the standing-wave approach discussed by Milo Wolff. His displayed moving-wave factorization inserts the relativistic Doppler factors. Here the phase-lock condition and the additional geometric-mean assumption are written separately, so the normalization and the consequences it enables remain visible.

6.1 The unique positive 1:1 phase-lock branch

Real waves first. The centre chases the rear-entering, forward-going wave and meets the front-entering, opposite-going wave head-on. For the assumed single-period e-sphere to reconstruct the same state every cycle, both waves must present the same phase recurrence rate to that centre. Equal phase-arrival rates therefore require unequal fixed-position frequencies.

At \(x=vt\), the signed phase rates are \(-\omega_+(1-\beta)\) and \(+\omega_-(1+\beta)\). Since cosine is even, the two oscillations must have equal rate magnitudes to maintain the assumed 1:1 reconstruction. For \(|\beta|<1\), the accumulated difference between their encountered phases, after allowing for their opposite propagation signs and an initially compatible offset, is

If the bracket is nonzero, the waves continually slip out of their original phase relationship at the moving centre. The same constructive meeting would not reconstruct the assumed state each cycle. Absence of this secular phase drift therefore requires phase matching. The rate-magnitude alternatives are

For \(|\beta|<1\) and positive frequencies, the second equation is impossible. Thus the positive phase-matching branch is the unique necessary condition within H-M2's single-period, 1:1 recurrence:

General periodicity could admit other harmonic relationships; H-M2 specifically assumes the 1:1 recurrence. This necessary resonance condition does not prove the full recurrence's nonlinear stability. Nor does it fix how fast the entire frequency pair runs. The most general positive matched pair can be written

Every positive common scale \(\bar\omega\) satisfies phase matching. Equivalently, write

6.2 Geometric-mean closure fixes the scale

Real waves first. H-M3 says that motion redistributes the stationary frequency scale reciprocally between the opposed directions: the fixed-position frequency increases on one side and decreases on the other by equal multiplicative amounts. The underlying rest scale is not collectively rescaled. This sets \(\bar\omega=\omega_0\); stable 1:1 phase matching alone does not require that choice.

H-M3 therefore gives

The raw crossing factors have geometric mean \(\sqrt{1-\beta^2}\). Dividing by that mean makes the pair reciprocal. With \(\beta=\tanh\eta\),

The normalization is exact after H-M3 is stated. Algebraic normalization alone does not demonstrate that a physical three-dimensional egg enforces H-M3.

6.3 Two real waves: translating carrier and de Broglie modulation

Use the cosine addition identity and \(\omega_0=c_0k_0\):

The first factor's equal-phase positions advance with the centre. Its full spatial period is \(\lambda_{\rm carrier}=\lambda_0/\gamma\). The second factor records the progressing phase alignment of the same real waves:

For \(0<v<c_0\), the phase modulation has speed

The moving phase alignment is not a piece of Space moving at \(c_0^2/v\), and its speed alone does not define a signal or energy-transport speed. The constituents of this background representation propagate at \(c_0\). At rest \(K_{\rm dB}=0\): the modulation is spatially uniform, so no finite phase speed is assigned.

“Carrier contraction” here means the spatial period of this cosine factor. Interpreting it as contraction of every physical ruler requires a bound-matter construction.

6.4 The centre phase clock and the full moving recurrence

Real waves first. Count phase while following the advancing meeting point. The spatial change of phase subtracts from the fixed-position temporal rate. The larger fixed-position modulation frequency and slower moving-centre recurrence are consistent readings of the same phase.

Define a centre phase-clock parameter by \(d\tau_{\rm ph}=(q/\omega_0)dt\). Then

C — Physical clock identification. Identifying \(\tau_{\rm ph}\) with measured proper time or with the reading of every physical clock is an additional physical requirement. The axial algebra establishes this centre phase clock.

Opposite radial phases remain locked. For opposite-phase versions of the same recurrence, with the same motion and background conditions, reverse the complete radial vibration: when one compresses, the other stretches. At corresponding points,

At \(\gamma=2\), both centre phases advance at \(\omega_0/2\) in background time and remain exactly half a cycle apart. This establishes preservation of the opposition within that matched pair; it does not by itself establish the cosmic lock between centres undergoing different motions.

Locking to real background waves. A moving centre encounters those waves at successive places. A fixed phase offset to its local reconstructing wave relation need not mean a fixed offset to \(\omega_0t\). If \(\Theta_{\rm in}(\mathbf x,t)\) denotes that actual incoming phase relation, a fixed offset requires

This is a wave-sampling identity, not a solution for the cosmic wave sea. The incoming phases for different motions must come from that same sea, not be chosen independently to fit the desired rates. Their opposite background-relative offsets are a retained WSM requirement. A spatially uniform imposed phase \(\Theta_{\rm in}=\omega_0t\) would instead require \(\dot\phi=\omega_0\) for every motion, which cannot also equal \(\omega_0/\gamma\) when \(v\ne0\). If that stricter simultaneous synchrony is required, the identification of this calculated phase with the physical radial clock must be revised. H-M3 remains a separate assumption in either case.

The internal travel distances and external wavelengths have reciprocal expressions. They are different physical measurements. A local internal crest spacing remains an output of the matched internal phase field.

6.5 Dispersion and measured energy–momentum relations

The derivative is the group-velocity expression for a narrow wave packet constructed from this dispersion branch. The single equal-amplitude pair is itself spatially extended; it is not a localized packet.

H-Q1 — Physical identifications and calibration. To express the phase relations in measured energy and momentum, adopt

Here \(J_*\) is the action-per-radian conversion scale. These maps and its calibration are additional physical inputs. De Broglie's historical wave-mechanical account also makes explicit the energy–frequency and momentum–wavelength connections; see his 1929 Nobel lecture.

B — Dependency statement. The stable single-period, two-wave representation in H-M2 requires phase matching as a necessary resonance condition and fixes the frequency ratio. H-M3 supplies the independent geometric-mean normalization. Together with the background dispersion they give the positive reciprocal pair and phase relations; equal amplitudes give the displayed product. H-M1 supplies the proposed internal interpretation and its speed boundary. P3 is needed when lengths are expressed in the specified e-sphere radius, not for the trigonometric identity. H-Q1 supplies the measured energy–momentum interpretation.

7. A calculable surface model of the wave egg

Real waves first. Assign a fixed amount of existing wave energy to each corresponding thin cap layer. Compressing that layer onto a smaller rear surface raises its average density; spreading it over a larger front surface lowers its average density. To calculate an actual contour, choose a definite surface family and keep its assumptions visible.

7.1 Four geometric assumptions

H-G1 — Fixed rim. The front and rear caps join on a circle of radius \(R\), equal to the resting reference radius.

H-G2 — Joined half-spheroids. Choose an oblate rear half-spheroid of height \(a_r\) and a prolate front half-spheroid of height \(a_f\). Their tangent planes agree at the rim, but their curvatures generally differ there. This chosen contour need not be the exact physical egg.

H-G3 — Fixed-thickness, fixed-energy layers. Corresponding thin layers have the same physical thickness \(\delta\) and the same assigned energy as their stationary reference layers. With cap-average density \(\overline E_d\), adopt the thin-layer relation

For a physical layer of nonzero thickness this is the stated thin-layer approximation. It is not a relation for the entire enclosed volume, and it does not follow solely from the absence of net energy donation. Following a fixed fraction of a wavelength would be a different thickness rule.

H-G4 — Cap-average to axial-density identification. Identify the average density of each cap with the corresponding representative axial density from H-M1 and P2:

This is a substantive approximation. An integrated cap energy would otherwise determine only an average, not the density encountered by the relevant axial wave. The two fixed assigned layer energies are not identified with the complete moving-state energy \(\gamma m_ec_0^2\).

7.2 Exact area identities within the model

The rear area decreases, while the front area increases. Direct addition, subtraction and multiplication give

These identities require H-G3–H-G4 and the reference cap area. They do not require the half-spheroid shape H-G2. Thus changing the contour family can preserve these area relations while changing heights and volumes.

The equality to the phase quantities also uses H-M2–H-M3. Both calculations inherit \(1\pm\beta\); taking their geometric means is the shared mathematical operation. The area identities do not independently prove frequency-product preservation.

7.3 Unique heights in the chosen family

Let \(q_f=a_f/R\), \(q_r=a_r/R\). In coordinates \(\xi\) measured from the joining plane, each half has \(\xi^2/a^2+\rho^2/R^2=1\). Its curved area can be written without a branch singularity as

For \(q>0\),

The function grows without bound as \(q\) grows. Therefore, for \(0<\beta<1\), each equation has one solution in its stated branch:

Equivalent closed expressions, useful for computation, are

The spherical value is obtained by continuity at \(q=1\). Once H-G2 is chosen, no subsequent fitting of these heights is used.

7.4 Volumes, expansion and a compact approximation

Expanding the area integral about \(q=1\) gives \(I(q)=1+\tfrac23(q-1)+\tfrac1{15}(q-1)^2+O((q-1)^3)\). Inverting the two area equations yields

The sum is even in \(\beta\), since reversing motion exchanges the caps. Near the limiting speed,

A compact numerical approximation is

On the 1,999-point grid \(\beta=0.0005,0.0010,\ldots,0.9995\), the largest computed \(|H_{\rm app}/H-1|\) is 0.15913%, approximately \(0.1592\%\), at \(\beta=0.8985\). This is a grid-tested approximation, not a proved error bound on the entire interval. Its coefficient \(4/\pi\) is close to \(51/40\) and reproduces the leading divergence; those facts motivate the approximation but do not prove its intermediate accuracy.

7.5 Hemisphere to disk: the radius-independent boundary

For the chosen single rear cap spanning the fixed circular rim, its area is at least the disk area. The same bound applies to any regular spanning surface whose projection covers that disk:

Under P2 and H-G4, the representative rear speed is therefore at most \(2c_0\). With H-M1,

Equality flattens the rear to a disk and sends the front area to infinity; in the chosen half-spheroid family its front height and volume also diverge. A finite nondegenerate egg has \(v<c_0\). The \(R=1\) areas are \(2\pi\) and \(\pi\); at \(R=\sqrt3\lambda_0/2\) they are \(3\pi\lambda_0^2/2\) and \(3\pi\lambda_0^2/4\). The factor of two is unchanged.

This is a geometric consistency explanation of the same conditional speed boundary. It inherits the axial premise and is not an independent experimental confirmation.

8. Where the axial waves meet

Real waves first. The egg's joining plane, its midpoint, its volume centroid and the meeting point of specified waves are different locations. Follow the translating outline while calculating the arrivals.

H-G5 — Simultaneous entry and constant path rates. One opposed pair enters simultaneously from the two tips at background time \(t=0\) and retains the H-M1 speeds throughout its approach. The joining plane is at \(x=0\) initially and advances as \(x=vt\). This is an additional axial timing model.

In coordinates \(\xi=x-vt\) relative to the moving outline, these become \(\xi_r=-a_r+c_0t\) and \(\xi_f=a_f-c_0t\). Equating the two positions gives

The meeting point lies inside the elongated front, measured from the joining plane. Its distance from each contemporaneous tip is the same, \(RH\). In background coordinates it is \(x_{\rm meet}=\xi_{\rm axial}+vT_{\rm approach}\). Distances travelled since entry need not equal distances to the moving tips at arrival.

For the chosen enclosed solid with uniform volume weighting, direct integration gives

This is a geometrical centroid, not a derived energy centre. H-G5 establishes one axial pair's meeting. Identifying it with the complete e-sphere centre requires oblique and transverse waves to reclose there with the appropriate phases as well.

8.1 Compare the two durations using the same reference

These durations already differ at rest: simultaneous tip-to-centre approach takes \(R/c_0\), whereas a complete reference phase cycle takes \(\lambda_0/c_0\). Comparing speed-dependent ratios to each quantity's own rest value gives \(H\) versus \(\gamma\); comparing the durations themselves in units of \(T_0\) requires the factor \(R/\lambda_0\).

The series \(H=1+51\beta^2/40+\cdots\) and \(\gamma=1+\beta^2/2+\cdots\) differ as well. Thus even rescaling the approach time to agree with the phase period at rest does not make their moving behaviour identical. No identification of the cap-approach time with the phase clock is adopted.

9. Numerical eggs and the speed sequence

9.1 The example \(v=\sqrt3c_0/2\), \(\gamma=2\)

With \(\lambda_0=1\) for the numerical length unit and \(R=\sqrt3/2\), the calculation gives:

The phase and travel relations are particularly simple:

During the shorter reference interval \(T_0\), the centre advances only \(\sqrt3\lambda_0/2=R\). The external component wavelengths are \(\lambda_+=(2-\sqrt3)\lambda_0\) and \(\lambda_-=(2+\sqrt3)\lambda_0\). These reciprocal values must not be relabelled as the internal travel distances of the same component.

9.2 One fixed model over the speed sequence

All rows below solve the same two area equations. Heights are measured in \(R\); the time columns use the common reference \(T_0\) with the P3 radius. No row-specific shape parameter is fitted.

The sequence shows a regular rear flattening and front elongation. It also shows why an area identity, a volume ratio and a clock rate must not be interchanged: they are distinct functions or use distinct reference measures.

10. Common rules and transverse geometry

The compact connection between the phase and area modules is

Each equality has named premises above. H-M3 fixes the phase normalization, while H-G3–H-G4 fix the cap-area normalization. Reusing the same \(1\pm\beta\) structure explains the common formulas; it does not remove either physical assumption.

The transverse radius is an assumption of the surface model

The axial phase theorem contains no transverse radius. The constant rim enters through H-G1. Replacing it by \(b(\beta)\) would introduce a new function unless another physical condition determines it. If the same absolute cap areas were retained, the shape equations would become

A small percentage error in an approximation to \(H\) is not evidence for a transverse-radius change by that percentage. A radius change must be calculated through these equations and a stated additional constraint.

An optional first-order directional continuation is \(c'(\widehat{\mathbf n})/c_0=1+\beta\mu+O(\beta^2)\), where \(\mu=\widehat{\mathbf n}\cdot\widehat{\mathbf v}\). It leaves orthogonal directions unchanged at first order. It does not determine the second-order profile or prove a particular three-dimensional contour. The one-dimensional cosine carrier likewise fixes an axial phase spacing, not the entire egg surface or universal transverse ruler behaviour.

The carrier's shortened axial period and the cap model's elongated front describe different proposed features. One complete wave solution must specify how both arise and how its oblique waves meet. Treating them as already the same contour would reintroduce the contradiction this note removes.

11. Minimum Description Length: count every independent choice

MDL evaluates the description of a model together with the data left to encode under that model. A familiar two-part expression is

In this setting, the model description includes laws, auxiliary assumptions, state and boundary choices, free functions, calibrated constants and the precision with which they are supplied. The conditional data description requires a specified observation and error model. Information introduced by selecting or adjusting a model after seeing data belongs in its full model-selection accounting; it cannot be treated as a free deduction. See Peter Grünwald's tutorial on MDL.

Minimum assumptions for the phase result. Admit a stable, uniformly translating, single-period recurrence represented by two opposed coherent background waves (H-M2), and preservation of their geometric-mean rest frequency (H-M3). Phase matching then follows as the necessary no-slip resonance condition; it is not counted again as an independent physical assumption. The condition fixes the ratio and H-M3 fixes the remaining scale. H-M1 is the separate proposed internal interpretation. H-G1–H-G5 are not needed for that phase theorem. Counting labels is not a unique count of independent scientific inputs: H-Q1 alone contains several physical identifications.

Special relativity is commonly summarized by two fundamental postulates, but counting headline postulates is not an MDL comparison. The complete background assumptions, bridge identifications, empirical parameters and residual fit must be encoded consistently for both descriptions. No comparative MDL ranking is claimed here. The Stanford discussion of symmetry and relativity also distinguishes the headline postulates from other assumptions entering the transformations.

The present achievement is structural reuse: one assumed asymmetry and one phase normalization generate a connected collection of formulas. H-M3 was selected to recover the known Lorentz target; the cap family was chosen to make a wave geometry calculable. Their economy makes them useful candidates to test. It does not make the recovered target an independent prediction.

MDL can guide work now: freeze a small candidate model, calculate its consequences, charge for any added freedom, and retain failures. A complete Action could later replace auxiliary assumptions by deductions, reducing the independently supplied description. Quantitative candidate calculations need not wait for that final construction.

12. Decisive checks and present result

The assumptions allow bounded checks. For a proposed moving solution, extract its positive background pair and calculate

The present phase theorem requires both residuals to vanish, within the accuracy of the calculation. A reliable nonzero phase-lock residual rules out the assumed stable 1:1 two-wave representation for that solution; a nonzero geometric-mean residual rejects or modifies H-M3.

The computed speed sequence, area identities and phase factorization are quantitative results of the declared construction. They are not new experimental measurements. The half-spheroid crossing time fails the proposed identification with the Lorentz phase period; that failed identification is retained as a constraint on future models.

References and scope

The uploaded WSM essays and successive stationary/moving-wave drafts supply the premises and candidate model developed here. This new note reconciles the stated phase, time, distance and surface calculations; it does not revise the other corpus pages or certify their unrelated claims.

1. Geoffrey Haselhurst, Wave Structure of Matter: Truth and Reality — the WSM foundation and corpus map.
2. Geoffrey Haselhurst and human–AI collaboration, Quantum Theory and Wave Mechanics — the e-sphere, real curves and moving-wave account.
3. Relativity and Gravity — the real-wave interpretation of moving matter and phase clocks.
4. WSM Mathematical Physics: Full Derivations — the shared mathematical ledger, reference wave identities and physical boundaries.
5. Classical Action and Quantum Wave — the nonlinear construction programme and retained failures.
6. Louis de Broglie, The Wave Nature of the Electron, Nobel Lecture, 12 December 1929 — the historical phase, energy and momentum connections.
7. Milo Wolff, The Wave Structure of Matter, mathematical appendix — the opposed-wave Doppler factorization.
8. Peter D. Grünwald, A Tutorial Introduction to the Minimum Description Length Principle, 2004.
9. Katherine Brading and Elena Castellani, Symmetry and Symmetry Breaking, Stanford Encyclopedia of Philosophy, Fall 2019 edition, §2.1.1.
10. WSM Simplicity and Inputs and Experimental Tests and Predictions — the broader assumption and test ledgers.

Numerical method. The cap heights were obtained by bracketed root solution of their monotone area functions and checked against independent numerical surface integrals. The displayed values are rounded. The approximation error is a discrete-grid result. Every time comparison on this page states its reference period.

Prepared 7 September 2026 · Revised 11 September 2026 · Working conditional derivation · Canonical address: quantum-physics-lorentz-debroglie-wave.html

WHY THIS CORPUS EXISTS

Geoffrey Haselhurst · Natural Philosopher · Human–AI Collaboration

Geoffrey Haselhurst is an Australian natural philosopher, inventor, ecological restorer, former international hockey player and ocean sailor who has pursued a physically intelligible account of reality for nearly thirty years. The 2026 WSM corpus joins his persistent picture of real waves in one continuous elastic Space to intensive collaboration with artificial intelligence. This history proves no equation. It explains the origin, continuity, working method and human purpose of the programme—and why physics, philosophy, ecology, evolution, mind and civilisation appear here as connected parts of one inquiry.