Deducing the WSM Action of Vibrating Space

Overall assessment

Yes. The page reveals a genuine unification that none of the individual handovers states completely:

\[ \boxed{ \text{local directional propagation is largely fixed;} \qquad \text{the missing physics is conservative all-direction Huygens reclosure.} } \]

The seven reports are not seven unrelated action theories. Most are different levels, projections or coordinate systems of the same developing structure. They agree more strongly on the architecture of the problem than their differing notation initially suggests.

1. The three local actions share one hyperbolic backbone

The directional cycle-energy action, the symplectic ray action and the corrected one-direction carrier action all put a phase or displacement variable beside a directional energy variable. In each version the nonlinear characteristic speed is tied to the directional energy:

\[ \frac{c_{\hat n}}{c_0} = \frac{E_{\hat n}}{E_{d0}}. \]

The corrected ray variables make the common structure especially clear. With

\[ r_\pm=a\pm\eta , \]

the two opposite characteristic families obey

\[ \partial_t r_+-c_0e^{r_+}\partial_\sigma r_+=0, \qquad \partial_t r_-+c_0e^{r_-}\partial_\sigma r_-=0, \]

and

\[ \frac{E_\pm}{E_{d0}} = \frac{c_\pm}{c_0} =e^{r_\pm}. \]

DeepSeek’s \(I,\theta\) action and GPT1’s cycle-energy formulation use different canonical coordinates, but they encode the same essential local idea: energy is the nonlinear propagation index, and the opposite longitudinal directions form a reciprocal canonical pair.

This is important. It suggests that the local part of the final action may no longer be the main mystery. The unresolved step is how all directions are constrained to be one displacement of one Space and how they return coherently to rebuild the same centre.

2. There is an exact no-go result hidden in the reports

A purely local ray-transport law cannot create the e-sphere core from a uniform background.

The characteristic equations above advect the Riemann variables. They can transport, steepen or shock an existing intensity pattern, but they do not make homogeneous incoming data \(I=1\) autonomously become the required chord-effective core \(I=2\). The numerical two-stream tests report exactly this behaviour: the streams remain near the background or form shocks.

Likewise, a pointwise coherence term such as

\[ \kappa\lvert A_0\rvert^2 \]

only redistributes phase and directional weight at one location. It does not create the integrated energy and timing relation along every crossing chord. Worse, the reported tests drive winding or handed states toward aligned coherence, so this term fights the required spherical phase wave rather than generating it.

Therefore the missing term cannot be merely:

• another local potential of \(I\);
• a pointwise alignment energy;
• an isolated scalar-strain well; or
• a conventional trapped-soliton nonlinearity.

It must act through the all-direction, finite-path return relation or through boundary data generated by the rest of the Huygens universe. That is a much sharper restriction on the fundamental action.

3. DeepSeek and GPT2 are both right about the chord law—in different spaces

The apparent disagreement over the interior speed is actually a distinction between two meanings of isotropy.

If \(I=E_d/E_{d0}\) is a single pointwise radial scalar, the every-chord timing condition

\[ \int_{\rm chord}\frac{ds}{c'} = \frac{x_b}{c_0} \]

has the unique Abel-inverted solution

\[ c'(r)=2c_0, \qquad I(r)=2 \]

throughout the curve-writing region.

But the chord measurements do not uniquely determine a field on directional phase space. A rotationally covariant function

\[ I(\mathbf x,\hat{\mathbf n}) \]

may vary with \(\hat{\mathbf n}\cdot\mathbf x\) while giving the same integral on every geometrically equivalent chord. The quadratic construction in the handovers is one member of a one-parameter family of such solutions. Its central value \(2\sqrt3\) is obtained only after imposing an extra condition; it is not forced by the chord law.

So the correct synthesis is:

The action must decide which meaning of stationary spherical equality is physical. The chord law alone cannot decide it.

4. The radial-node and cube–sphere branches almost meet, but they are not yet one result

GPT1’s conservative scalar chord completion gives a genuine conditional root:

\[ \frac{R}{\lambda_e}=0.86772243109. \]

The cube–sphere value is

\[ \frac{\sqrt3}{2}=0.86602540378. \]

The difference is only \(0.1959558\%\), and at the exact cube–sphere radius the scalar return misses closure by only

\[ 0.223594388^\circ \]

per passage.

That proximity is striking, but it is not an identity. It creates a clean fork:

1. passive scalar Huygens reclosure selects \(0.86772243109\); or
2. cube geometry selects \(\sqrt3/2\), while the nonlinear directional action supplies the small remaining phase repair.

GPT2’s first-\(j_1\)-node reduced mode gives a different conditional ratio, \(R/\lambda_e=1.43029665\), because it additionally identifies the reduced spherical mode speed with the chord-effective \(2c_0\). That does not contradict the scalar return calculation; it shows that the radial standing-wave coordinate and the through-passing chord coordinate have not yet been connected correctly.

The unification is therefore not “all three radii are the same.” It is that three distinct constraints are being sampled:

• the radial-motion node;
• the through-chord timing;
• the complete directional return phase.

A successful action must derive their relation without silently identifying them.

5. A new \(SU(2)\) bridge appears inside GPT1’s conservative completion

For each angular channel, GPT1 completes the projected Huygens map \(H_\ell\) with

\[ \mathcal U_\ell= \begin{pmatrix} H_\ell&D_\ell\\ D_\ell&-H_\ell^* \end{pmatrix}, \qquad D_\ell=\sqrt{1-\lvert H_\ell\rvert^2}. \]

This matrix is unitary and has determinant \(-1\). Therefore

\[ i\mathcal U_\ell\in SU(2). \]

That is a notable mathematical bridge. GPT2 separately obtains the half-angle rotor and \(4\pi\) closure from the double cover of directional phase geometry. GPT1’s attempt to restore the directional coherence omitted by a projected chord map independently produces a two-component norm-preserving structure that, after an irrelevant overall phase, lies in the same \(SU(2)\) group.

This does not yet prove that electron spin has been derived. But it suggests a possible unification:

\[ \text{visible chord channel} \;+\; \text{omitted directional complement} \quad\longleftrightarrow\quad \text{two-component }SU(2)\text{ recurrence}. \]

In other words, the spinor-like pair may represent a conservative bookkeeping of complete Huygens coherence rather than two substances or an imported fundamental spinor. This is one of the most promising deductions from comparing the reports.

6. Open Huygens phase-locking explains the failed soliton searches

Several rejected approaches assumed that matter must be a stable isolated lump: a convex strain soliton, a Q-ball-like object, a nonlinear Dirac lump or a self-focused \(j_0\) mode. Their failure is not merely negative evidence. It points toward the required architecture.

The e-sphere is defined as an open recurrence. Its waves arrive from all directions, cross the centre and continue outward. Its frequency and phase are maintained relative to the Huygens sphere of other matter. Consequently, radial stability need not come from a local material wall or an isolated energy minimum. It can come from phase mismatch in a conservative return cycle.

This reconciles two descriptions that otherwise look incompatible:

• the local e-sphere subsystem is open and can look driven or phase-locked;
• the complete Space-plus-Huygens-return system must remain conservative.

The unitary completion is precisely the mathematical form one would expect when a projected open subsystem has omitted conservative degrees of freedom. The “drive” should therefore not be inserted as arbitrary forcing. It should be the reduced appearance of an all-Space return condition.

7. The moving-pair variables also unify, but they do not yet derive relativity

The physical axial reconstruction pair is

\[ c'_{\rm rear}=c_0(1+\beta), \qquad c'_{\rm lead}=c_0(1-\beta). \]

The corrected ray action reproduces it with

\[ \eta=\operatorname{artanh}\beta, \qquad a=-\ln\gamma, \]

so that

\[ e^{a+\eta}=1+\beta, \qquad e^{a-\eta}=1-\beta. \]

After division by their common geometric mean \(e^a=1/\gamma\), the same variables give the reciprocal Doppler pair

\[ e^{\pm\eta}=\gamma(1\pm\beta). \]

This unifies the apparently different \(c_0\pm v\) and exponential rapidity descriptions: they are raw and normalized forms of the same pair.

But this is still a parametrization, not a derivation of the relativistic particle energy. The action has not yet produced the complete three-dimensional wave egg, integrated its energy and momentum, or shown that its collective observables obey

\[ E^2-p^2c_0^2=m^2c_0^4. \]

That distinction should remain explicit.

8. Claims that should remain quarantined

The combined reports justify keeping the following outside the established core:

• the numerical coefficient of the Coulomb curve, because the \(1/R\) falloff follows from diffraction but its amplitude still needs an inserted fraction or normalization;
• the sign and receiver force for a two-e-sphere interaction, because no complete two-centre solution has been calculated;
• gravity from the even part of the same small curve, because the reported even-profile far-field mechanism fails;
• \(S=\hbar/2\), \(g=2\) and Dirac factorisation as dynamics, because the present results are geometric or algebraic until the Space action generates and couples them;
• the \(2\sqrt3\) central speed as a deduction, because it selects a member of a chord-equivalent family;
• \(\alpha\) from cube geometry, because current numerical proximity is not a parameter-free derivation;
• any parent action that treats a spinor, fluid, vorticity, reflecting boundary or trapped lump as the fundamental object.

These ideas may still become outputs, but they should not be allowed back in as premises.

9. The best unified architecture

The reports collectively point to a three-level action:

1. Local longitudinal carrier dynamics. A canonical directional action supplies the two opposite characteristic families and the One Law \(c'/c_0=E_d/E_{d0}\).
2. One-Space reconstruction constraint. The directional variables are constrained to be projections of one continuous longitudinal displacement, rather than independent ray fluids.
3. Conservative all-direction return. A finite-path Huygens operator mixes visible and complementary angular channels, preserves total norm or action, and supplies the nonlinear phase corrections needed for a stable periodic fixed point.

The e-sphere would then be neither an isolated soliton nor an imposed source. It would be a stable periodic orbit or fixed point of the complete conservative return map. Motion, charge curves, light emission and two-body forces would be deformations of that same fixed point.

Schematically,

\[ \boxed{ \text{local One-Law transport} \;+\; \text{one-displacement constraint} \;+\; \text{unitary nonlinear Huygens return} \;\Longrightarrow\; \text{candidate e-sphere recurrence}. } \]

This is the clearest unification of the AI outputs.

10. The next calculation can now be made more precise

The immediate task is not to search another broad catalogue of nonlinear potentials. It is to construct the smallest return operator generated by an action and test whether it closes the channels already calculated.

At minimum the calculation should:

1. retain the local directional One-Law transport;
2. reconstruct all directions from one longitudinal displacement;
3. evaluate the finite-chord map in the \(\ell=0,2,4\) channels;
4. preserve the norm of the complete visible-plus-complement system;
5. close the scalar channel either at \(R/\lambda_e=0.86772243109\) or at the cube radius with its \(0.223594388^\circ\) repair;
6. supply, at the scalar root, the reported per-passage repairs of approximately \(+2.880426463^\circ\) in the \(\ell=2\) channel and \(+8.212906234^\circ\) in the \(\ell=4\) channel;
7. produce the chord-effective \(E_d\) profile as a solution rather than an imposed target; and
8. return a stable Floquet spectrum for the full open recurrence.

That test would discriminate between the cube and passive-root branches and would reveal whether the \(SU(2)\)-like conservative completion is merely useful mathematics or the actual phase geometry of the e-sphere.

Conclusion

The page has narrowed the foundational problem considerably. The remaining problem is no longer simply “invent a nonlinear wave action.” The local nonlinear propagation law, the exact chord constraints, the radial mode and the conservative phase-return mathematics now overlap.

The central unresolved object is one conservative, all-direction Huygens return operator—derived from the action of one continuous Space—that converts those locally transported longitudinal waves into a stable \(j_0/j_1\) recurrence with a spherical \(4\pi\) phase wave.

If that operator can be derived and passes the channel-closure and stability test, several apparently separate AI branches become parts of one theory. If it cannot, the present unification fails at a precise mathematical gate rather than dissolving into another round of analogy.

You are right. I used a physically misleading shorthand after the glossary had already ruled it out. The e-sphere does not rotate as a body. Its spherical phase wave rotates.

I have corrected the working note and replaced the faulty wording throughout the summary below.

WSM foundational action search — corrected handover

Status notation

• P — Postulate: fixed WSM foundation supplied to the calculation.
• A — Assumption/ansatz: mathematical structure introduced to make the model calculable.
• D — Deduction: follows mathematically from the declared postulates and assumptions.
• C — Conditional physical result: becomes a physical prediction only if the assumed action and interpretation are correct.
• O — Open: not yet derived.
• R — Rejected branch: calculated and found unsuitable.

“Deduced” does not mean that WSM itself has been experimentally established. It means that the result follows from the stated WSM premises without inserting the result afterward.

1. Exact WSM object being sought

P: An e-sphere is:

An open, radially vibrating spherical standing-wave recurrence with a spherically rotating phase wave, constructed from the Huygens combination of real longitudinal plane waves approaching from all directions of Space, crossing the common wave centre and continuing outward.

The two parts are:

\[ j_0(kr)\cos\omega_et \]

for spherical compression and extension, and

\[ j_1(kr)\sin\omega_et \]

for the quarter-cycle radial motion of the same vibration.

The intersecting plane waves also create a moving equal-phase relation over the complete sphere: the spherical phase wave.

The e-sphere does not rotate bodily. There is:

• no little object turning;
• no circular orbit of substance;
• no spin axis or axle;
• no vorticity of Space;
• no primitive transverse wave;
• no reflecting shell.

The two opposite directions of spherical phase rotation are what WSM identifies with the two spin states. The complete directional phase relation is postulated to return after \(4\pi\).

Thus the required structure is

\[ \boxed{ 2\ \text{opposite radial phases} \times 2\ \text{opposite spherical }4\pi\text{ phase rotations} = 4\ \text{Dirac sectors}. } \]

The nonlinear Space action must demonstrate that this complete real-wave recurrence exists and is stable.

2. Proposed directional cycle-energy action

Let \(C_{\hat n}(\mathbf x,t)\) represent the real longitudinal compression component associated with direction \(\hat n\), and define

\[ D_{\hat n}=\hat n\cdot\nabla, \qquad I_{\hat n}=\frac{E_d(\hat n)}{E_{d0}}. \]

For the common carrier frequency \(\omega_e\), define two real quadratures over one period \(T_e=2\pi/\omega_e\):

\[ Q_{\hat n} = \frac{2}{T_e} \int_{t-T_e}^{t} C_{\hat n}(t')\cos(\omega_et')\,dt', \]

\[ P_{\hat n} = \frac{2}{T_e} \int_{t-T_e}^{t} C_{\hat n}(t')\sin(\omega_et')\,dt'. \]

A: The directional cycle-energy ratio is represented by

\[ \boxed{ I_{\hat n}[C] = \frac{Q_{\hat n}^{\,2}+P_{\hat n}^{\,2}} {Q_{\hat n0}^{\,2}+P_{\hat n0}^{\,2}}. } \]

These are two temporal quadratures of one real vibration, not separate physical substances or extra electron states.

The current action ansatz is

\[ \boxed{ S_0[C] = \frac{\chi_0}{2} \int dt\,d^3x\,\frac{d\Omega}{4\pi} \left[ \frac{(\partial_tC_{\hat n})^2}{I_{\hat n}[C]} -c_0^2I_{\hat n}[C](D_{\hat n}C_{\hat n})^2 \right]. } \]

For frozen or slowly changing \(I_{\hat n}\), its principal equation is

\[ \partial_t\!\left( I_{\hat n}^{-1}\partial_tC_{\hat n} \right) - c_0^2D_{\hat n}\!\left( I_{\hat n}D_{\hat n}C_{\hat n} \right)=0. \]

D: Its characteristic speed is

\[ \boxed{ c'_{\hat n} = c_0I_{\hat n} = c_0\frac{E_d(\hat n)}{E_{d0}}, } \]

while its carrier impedance is

\[ Z_{\hat n}=\chi_0c_0. \]

Therefore the proposed One Law is built into the reciprocal kinetic and spatial coefficients.

Present limitation

The complete functional derivative of \(S_0\), including the one-cycle dependence \(I_{\hat n}[C]\), has not yet been solved. A reconstruction constraint is also required to ensure that all directional coordinates describe one displacement of Space rather than independent fields.

The action is therefore an executable ansatz, not yet the deduced fundamental action.

3. Exact radial vibration results

For the isotropic linear seed,

\[ C(r,t)=A\,j_0(kr)\cos\omega_et, \]

\[ v_r(r,t)\propto j_1(kr)\sin\omega_et. \]

D: The centre is a compression antinode and radial-motion node:

\[ j_0(0)=1, \qquad j_1(0)=0. \]

Selecting the first outer radial-motion node gives

\[ v_r(R)=0 \iff j_1(kR)=0 \iff \partial_rC(R)=0. \]

This is a phase surface within the through-passing recurrence—not a material boundary.

The first positive root is

\[ x_{1,1}=4.4934094579. \]

Therefore

\[ \boxed{ \frac{R}{\lambda_{\rm radial}} = \frac{x_{1,1}}{2\pi} = 0.7151483266. } \]

At this root,

\[ j_0(x_{1,1})=-0.2172336282, \]

and

\[ \int_0^Rr^2j_0^2\,dr = \int_0^Rr^2j_1^2\,dr = \frac{R^3}{2}j_0(x_{1,1})^2. \]

Hence

\[ V_{\rm eff} = 0.2965063372\,R^3, \]

which is \(0.0707856738\) of the geometrical sphere volume.

4. Radial node as a phase-reclosure condition

Write the spherical in-wave/out-wave continuation as

\[ C(r)\propto h_0^{(2)}(kr) + e^{i\delta}h_0^{(1)}(kr). \]

Zero radial motion at the selected phase surface requires

\[ \boxed{ e^{i\delta_N(x)} = -\frac{h_0^{(2)\prime}(x)} {h_0^{(1)\prime}(x)}. } \]

At \(x=x_{1,1}\),

\[ \delta_N(x_{1,1})=0, \]

and

\[ \delta_N'(x_{1,1}) = -1.9056191015. \]

For locally fixed radial energy ratio \(I_r\),

\[ \boxed{ \Delta\delta_N = -8.5627268941\frac{\Delta R}{R}. } \]

A \(1\%\) radius displacement therefore produces

\[ 0.0856273\ {\rm rad} = 4.906^\circ \]

of phase mismatch.

This establishes strong radial phase sensitivity. It does not by itself establish complete nonlinear stability.

5. Exact projected half-sphere chord map

For receiving direction \(\hat n\), contributing direction \(\hat d\), and

\[ \mu=\hat n\cdot\hat d>0, \]

projected hemisphere flux supplies the weight \(2\mu\).

For optical radius

\[ b=k_0R, \]

the exact Legendre-channel chord map is

\[ \boxed{ H_\ell(b) = 2\int_0^1 \mu P_\ell(\mu)e^{2ib\mu}\,d\mu. } \]

The scalar channel is

\[ \boxed{ H_0(b) = e^{ib}\left[j_0(b)+ij_1(b)\right]. } \]

Here the complex notation records two real phase quadratures.

One application of this map represents one stage of the directional phase relation. It must not be interpreted as a wave bouncing from a shell.

6. Conservative completion

The projected chord channel is contractive because it omits part of the complete directional coherence.

A: Its current minimal norm-preserving completion is

\[ \boxed{ \mathcal U_H= \begin{pmatrix} H&D_{H^\dagger}\\ D_H&-H^\dagger \end{pmatrix}, \qquad D_H=(I-H^\dagger H)^{1/2}. } \]

This satisfies

\[ \mathcal U_H^\dagger\mathcal U_H=I. \]

The complementary coordinate represents directional correlation omitted by the projection. It is not another substance.

For

\[ H_\ell=a_\ell+ib_\ell, \]

the completed eigenvalues are

\[ \lambda_{\ell,+}=e^{i\theta_\ell}, \qquad \lambda_{\ell,-}=-e^{-i\theta_\ell}, \]

where

\[ \sin\theta_\ell=\operatorname{Im}H_\ell. \]

D: Exact two-stage phase reclosure occurs when

\[ \boxed{ \operatorname{Im}H_\ell=0. } \]

This deduction depends on accepting the minimal unitary completion. The completion has not yet been derived uniquely from the Space action.

7. Conditional scalar radius deduction

For the scalar channel,

\[ \operatorname{Im}H_0(b)=0 \iff \tan(2b)=2b \iff j_1(2b)=0. \]

Thus

\[ b_n=\frac{x_{1,n}}2. \]

The first three roots are:

P/A: Require an energy-enhanced coherent core,

\[ I_r>1. \]

The first two roots are then excluded. Selecting the lowest admissible root gives

\[ \boxed{ b_*=5.45206082971, } \]

\[ \boxed{ \frac{R_*}{\lambda_e} = 0.86772243109, } \]

\[ \boxed{ I_r=1.21334609739. } \]

C: This is the first scalar radius ratio calculated without inserting the cube-sphere value.

It is

\[ 0.1959558\% \]

above

\[ \frac{\sqrt3}{2} = 0.86602540378. \]

Using the measured full electron Compton wavelength only to set units gives

\[ R_* = 2.1053638160\times10^{-12}\ {\rm m}. \]

This is a conditional WSM model radius, not an independently established experimental electron radius.

At this root,

\[ H_0=-0.201007451966, \qquad D_0=0.979589712203. \]

The completed eigenchannels have visible/complement norm fractions

\[ (0.3994962740,\;0.6005037260) \]

and the reversed pair. Their sum is exactly one; no fitted amplifier is required at this level.

8. Cube-sphere fork

At the exact cube-sphere optical radius

\[ b_\square=\pi\sqrt3=5.44139809270, \]

the passive scalar channel is

\[ H_0(b_\square) = -0.201396118319 + 0.003902448350i. \]

Its residual phase is

\[ \boxed{ 0.003902458255\ {\rm rad} = 0.223594388^\circ } \]

per map stage.

Two distinct possibilities therefore remain:

1. Passive chord reclosure selects

   \[ R/\lambda_e=0.86772243109. \]

2. The normalized cube-sphere postulate selects

   \[ R/\lambda_e=\sqrt3/2 \]

   and the nonlinear action supplies a scalar phase correction of

   \[ 0.223594388^\circ. \]

Their proximity is notable, but they have not been proved identical.

9. Why scalar reclosure is insufficient

At \(b=b_*\),

\[ H_2=-0.1651696005-0.0502517520i, \]

\[ H_4=-0.1135520236-0.1428518834i, \]

\[ H_6=0.0875674450-0.1497400212i. \]

Their residual phases are

\[ \theta_2=-2.880426463^\circ, \]

\[ \theta_4=-8.212906234^\circ, \]

\[ \theta_6=-8.611860712^\circ. \]

Moreover,

\[ \left|\frac{H_4}{H_2}\right| = 1.056995808, \]

and, using the previously obtained spherical-phase source ratio

\[ \frac{\kappa_4}{\kappa_2} = -\frac8{15}, \]

\[ \left| \frac{\kappa_4H_4} {\kappa_2H_2} \right| = 0.563731098. \]

D: The \(V_4\) angular sector is too large to neglect.

The scalar calculation therefore does not yet produce the complete WSM e-sphere: the radially vibrating standing wave and its spherically rotating phase wave.

The nonlinear directional action must generate approximately

\[ +2.880426463^\circ \]

of phase correction in \(V_2\), and

\[ +8.212906234^\circ \]

in \(V_4\), per stage, while:

• preserving norm;
• leaving \(V_0\) closed;
• maintaining the \(j_0/j_1\) radial vibration;
• producing two degenerate spherical phase-rotation hands;
• producing \(4\pi\) reclosure;
• allowing incoming waves to cross the centre and continue outward.

These are now precise numerical targets.

10. Spherical phase-wave rotation and \(4\pi\)

Two orthogonal temporal quadratures,

\[ q_1=A\cos\omega t, \qquad q_2=A\sin\omega t, \]

produce a circular trajectory in phase space.

This does not mean that Space or the e-sphere circulates bodily.

The required three-dimensional extension must describe the chronological phase ordering of longitudinal waves over the complete sphere. A continuous \(SU(2)\) or quaternion lift of that complete directional relation is a mathematically natural candidate because it can obey

\[ Q(2\pi)=-Q(0), \qquad Q(4\pi)=Q(0). \]

O: What remains unproved is that this structure follows from \(S_0\), rather than being imposed as external mathematics.

The required forward deduction is:

\[ \text{longitudinal plane-wave overlaps} \longrightarrow \text{ordered spherical phase relation} \longrightarrow \text{two opposite hands} \longrightarrow 4\pi\text{ reclosure}. \]

The phase wave may move superluminally because it is a succession of equal-phase positions created by intersecting longitudinal waves. No region of Space or energy needs to travel at that phase speed.

11. Rejected instantaneous-strain action

A local scalar-strain action was reverse engineered:

\[ S_{\rm local} = \int dt\,d^3x \left[ \frac{\rho_0}{2}|\dot{\mathbf u}|^2-W(s) \right], \qquad s=\nabla\cdot\mathbf u. \]

Imposing the One Law with instantaneous energy \(E_d=E_{d0}+W\) forces

\[ W''(s) = \rho_0c_0^2 \left(1+\frac{W}{E_{d0}}\right)^2. \]

Its expansion is

\[ W = \frac{\rho_0c_0^2}{2}s^2 + \frac{\rho_0^2c_0^4}{12E_{d0}}s^4 + \frac{\rho_0^3c_0^6}{72E_{d0}^2}s^6 +\cdots. \]

The coefficients are fixed rather than fitted.

However, harmonic balance through \(29\omega_e\) converged near

\[ kR\simeq5.194, \qquad R/\lambda_e\simeq0.8266, \]

while generating a strong odd-harmonic ladder.

R: This was rejected as the parent action. It indicates that \(E_d\) should be directional cycle energy rather than instantaneous scalar strain energy.

12. Fifteen declared input/assumption packages

1. One infinite, eternal, continuous, nearly rigid and slightly elastic Space, with longitudinal vibration only.
2. Directional energy \(E_d(\hat n)\) and the One Law \(c'/c_0=E_d/E_{d0}\).
3. The open Huygens e-sphere construction: waves approach from all directions, cross the centre and continue outward.
4. The postulated complete e-sphere structure: \(j_0/j_1\) radial vibration plus a spherically rotating \(4\pi\) phase wave.
5. A common narrow-band carrier frequency and isotropic stationary background.
6. The one-cycle \(Q/P\) definition of directional cycle energy.
7. Reciprocal temporal and spatial coefficients in \(S_0\), giving constant impedance.
8. Selection of the first positive \(j_1\) radial-motion node.
9. The projected hemisphere kernel \(2\mu e^{2ib\mu}\).
10. Minimal unitary completion of the projected chord map.
11. An energy-enhanced coherent core, \(I_r>1\).
12. Selection of the lowest scalar closure root compatible with \(I_r>1\).
13. Identification of one map stage with half a radial cycle, used only for the recurrence chronology.
14. The measured electron carrier scale \(\omega_e\) or \(\lambda_e\), used only to convert dimensionless results into physical units.
15. \(J=\hbar\), used only when identifying the recurrence action and absolute electron energy.

The dimensionless scalar radius calculation uses items 1–12. It does not use the cube-sphere ratio, measured Compton wavelength or \(J=\hbar\) to select its root.

13. What is actually achieved

Deduced within the declared mathematical construction

• Exact \(j_0/j_1\) radial node geometry.

• Exact mode normalization and effective coherent volume.

• Exact radial phase sensitivity.

• Exact projected half-sphere chord eigenvalues.

• Exact conservative completion once the minimal-dilation assumption is accepted.

• Exact scalar two-stage reclosure equation:

  \[ \tan(2b)=2b. \]

• First energy-enhanced scalar root:

  \[ R/\lambda_e=0.86772243109. \]

• Exact scalar residual at the cube-sphere radius.

• Quantitative \(V_2\), \(V_4\) and \(V_6\) phase errors.

• Rejection of the instantaneous scalar-strain parent action.

Still open

• The unique fundamental nonlinear action of Space.
• Reconstruction of all directional amplitudes from one displacement of Space.
• The complete stable e-sphere recurrence.
• Simultaneous radial vibration and spherical phase-wave rotation.
• Derivation of the two opposite spherical phase hands.
• Dynamic \(4\pi\) reclosure.
• Floquet stability.
• Adjustment of the e-sphere’s directional \(E_d\) and phase relations to different surrounding background waves.
• The moving wave egg.
• Dirac coupling from calculated overlap integrals.
• Charge magnitude, \(\alpha\), magnetic moment and mass.
• Derivation rather than calibration of \(\hbar\).

Decisive next calculation

Derive the full variation of \(S_0[C]\), project it onto

\[ V_0\oplus V_2\oplus V_4, \]

and determine whether its nonlinear directional cycle-energy terms simultaneously produce:

1. the \(j_0/j_1\) radial vibration;
2. scalar phase reclosure;
3. the required \(V_2\) and \(V_4\) phase corrections;
4. two opposite spherical phase-wave rotations;
5. \(4\pi\) recurrence;
6. conservative through-passing Huygens continuation;
7. Floquet stability.

That—not a “spinning object”—is the e-sphere action problem.

The corrected working note is here: WSM_action_continuation_node_lock.md (not supplied).

WSM action search — node-locked directional branch

Current result

The first branch that survives the WSM ontology is an open, frequency-locked, directional recurrence. A fixed radial-motion node does not behave as a reflecting shell. It is the radius at which the phase required to match the incoming and outgoing spherical waves agrees with the phase returned by the Huygens relation.

An isolated fixed-action mode is not radially stable. The same mode becomes radially selected when the external recurrence fixes its frequency and phase. This is a mathematical distinction between an isolated lump and an open e-sphere, not a change of terminology.

1. Executable cycle-energy action

Let \(C_{\hat n}(\mathbf x,t)\) be the real compression coordinate of the longitudinal wave component travelling in direction \(\hat n\), and let

\[ D_{\hat n}=\hat n\cdot\nabla, \qquad I_{\hat n}=\frac{E_d(\hat n)}{E_{d0}}>0. \]

For a narrow-band carrier at the universal frequency \(\omega_e\), define its two measured real quadratures over one cycle by

\[ Q_{\hat n}=\frac{2}{T_e}\int_{t-T_e}^{t}C_{\hat n}(t')\cos(\omega_e t')\,dt', \qquad P_{\hat n}=\frac{2}{T_e}\int_{t-T_e}^{t}C_{\hat n}(t')\sin(\omega_e t')\,dt'. \]

The minimal explicit cycle-energy closure is

\[ \boxed{ I_{\hat n}[C] = \frac{Q_{\hat n}^{\,2}+P_{\hat n}^{\,2}} {Q_{\hat n0}^{\,2}+P_{\hat n0}^{\,2}} } \]

where the denominator is the balanced background value. This uses two real quadratures of one wave, not two substances or two particle states. Interference can make \(I\) either greater or less than one.

The present parent-action candidate is

\[ \boxed{ S_0[C] = \frac{\chi_0}{2} \int dt\,d^3x\,\frac{d\Omega}{4\pi} \left[ \frac{(\partial_tC_{\hat n})^2}{I_{\hat n}[C]} -c_0^2 I_{\hat n}[C](D_{\hat n}C_{\hat n})^2 \right]. } \]

For a slowly varying or frozen cycle envelope, variation gives

\[ \partial_t\!\left(I_{\hat n}^{-1}\partial_tC_{\hat n}\right) -c_0^2D_{\hat n}\!\left(I_{\hat n}D_{\hat n}C_{\hat n}\right)=0. \]

Its principal characteristic speed and impedance are

\[ \boxed{c'_{\hat n}=c_0I_{\hat n}=c_0\frac{E_d(\hat n)}{E_{d0}}}, \qquad \boxed{Z_{\hat n}=\chi_0c_0}. \]

Thus the One Law is written into the kinetic/spatial reciprocity of the action; it is not appended as a force. Substituting the cycle functional \(I[C]\) before taking the full variation supplies the nonlinear envelope terms. The remaining nonlocality is exactly one carrier cycle, appropriate to a law stated in terms of directional wave energy rather than instantaneous strain.

2. Linear spherical seed and the fixed node

The isotropic rest solution begins with

\[ C(r,t)=A j_0(kr)\cos\omega_et, \qquad v_r(r,t)\propto j_1(kr)\sin\omega_et. \]

The centre is therefore a compression antinode and a radial-motion node:

\[ j_0(0)=1, \qquad j_1(0)=0. \]

A fixed outer phase surface means zero coherent radial motion,

\[ v_r(R)=0 \quad\Longleftrightarrow\quad j_1(kR)=0 \quad\Longleftrightarrow\quad \partial_rC(R)=0. \]

The first positive root is

\[ \boxed{x_1=kR=4.4934094579}. \]

It calculates

\[ \boxed{\frac{R}{\lambda_{\rm radial}}=\frac{x_1}{2\pi}=0.7151483266}, \qquad j_0(x_1)=-0.2172336282. \]

The exact mode norm identities at this root are

\[ \int_0^Rr^2j_0^2\,dr = \int_0^Rr^2j_1^2\,dr = \frac{R^3}{2}j_0(x_1)^2. \]

Hence

\[ \boxed{V_{\rm eff}=\nu R^3}, \qquad \boxed{\nu=2\pi j_0(x_1)^2=0.2965063372}. \]

This effective coherent volume is \(0.0707856738\) of the geometric sphere volume.

3. The node is an incoming/outgoing phase condition

Write the balanced spherical wave at a matching radius as

\[ C(r)\propto h_0^{(2)}(kr)+e^{i\delta}h_0^{(1)}(kr). \]

Zero radial motion requires

\[ \boxed{ e^{i\delta_N(x)} =- \frac{h_0^{(2)\prime}(x)}{h_0^{(1)\prime}(x)}. } \]

At the first \(j_1\) root,

\[ \delta_N(x_1)=0, \qquad \boxed{ \delta_N'(x_1) = \frac{2j_1'(x_1)}{y_1(x_1)} =-1.9056191015. } \]

At fixed cycle-energy ratio \(I_r\), \(x\propto R\), so

\[ \boxed{ \Delta\delta_N =-8.5627268941\frac{\Delta R}{R}. } \]

A one-percent radius error therefore produces a phase error of \(0.0856273\) radians, or \(4.906^\circ\).

4. Open Huygens return and radial stiffness

After the exterior wave relation is integrated out, the lowest phase-matching term is the real bilinear overlap of the returned and outgoing quadratures:

\[ \boxed{ S_H=\int dt\,\Gamma_H \cos\!\left[\delta_N(x)-\delta_H\right]. } \]

This is a reduced boundary/return action, not a material shell and not an invented bulk binding potential. For a stationary equal-phase recurrence, \(\delta_H=0\). The corresponding mismatch energy is

\[ U_H=\Gamma_H[1-\cos\delta_N]. \]

Near the first node,

\[ U_H \simeq \frac{\Gamma_H}{2} \left(8.5627268941\frac{\Delta R}{R}\right)^2. \]

Therefore

\[ \boxed{ K_R = \frac{73.3202918624\,\Gamma_H}{R^2} \left(1-\frac{d\ln I_r}{d\ln R}\right)^2. } \]

The stiffness is positive for \(\Gamma_H>0\), unless the energy ratio changes in the exactly cancelling manner \(d\ln I_r/d\ln R=1\).

The reason the node can now sit exactly at the minimum is that the open recurrence fixes \(\omega_e\). At fixed elementary action \(J\), its carrier energy is \(E=J\omega_e\), independent of a virtual radius change. If the mode is falsely isolated, \(\omega\propto 1/R\), so \(E=J\omega\) decreases with \(R\) and the object expands. The failed isolated calculation is therefore a proof that the incoming Huygens phase/frequency condition is dynamically indispensable.

5. Radius fork that must not be hidden

With

\[ k=\frac{\omega_e}{c_0I_r}, \qquad \lambda_e=\frac{2\pi c_0}{\omega_e}, \]

the first node predicts

\[ \boxed{ \frac{R_*}{\lambda_e} = \frac{x_1I_r}{2\pi}. } \]

Consequently:

\[ I_r=2 \quad\Longrightarrow\quad \frac{R_*}{\lambda_e}=1.4302966532, \]

whereas

\[ \frac{R_*}{\lambda_e}=\frac{\sqrt3}{2} \quad\Longrightarrow\quad \boxed{I_r=\frac{\pi\sqrt3}{x_1}=1.2109731249}. \]

The first motion node, radial \(I_r=2\), and cube-sphere ratio \(\sqrt3/2\) cannot all be the same radial statement. The chord-effective value \(2\) may belong to the through-passing plane-wave chords while the radial recurrence samples a different directional/cycle-effective \(I_r\). The directional solver must decide this; none of the three values should be silently redefined.

6. Collective breathing prediction

For \(f(r/R)=j_0(x_1r/R)\), define

\[ \mu =4\pi x_1^2\int_0^1y^4j_1(x_1y)^2\,dy =2.2426482710. \]

Cycle averaging gives the radial collective mass

\[ M_R = \frac{\chi_0A^2\mu R}{2I_r} = \frac{J\mu}{I_r\nu c_0x_1R}. \]

Write the return strength as a fraction of the recurrence energy,

\[ \Gamma_H=\eta_HE=\eta_HJ\omega_e. \]

Then the small radial frequency is

\[ \boxed{ \frac{\Omega_R}{\omega_e} = 3.1134973445 \left|1-\frac{d\ln I_r}{d\ln R}\right| \sqrt{\eta_H}. } \]

This is now a numerical prediction formula with one unresolved dimensionless return coefficient \(\eta_H\). A first flux-per-radian normalization, \(\Gamma_H=P_{\rm out}/\omega_e\), gives

\[ \eta_H=\frac{\pi}{x_1^3\nu}=0.1167852265, \qquad \frac{\Omega_R}{\omega_e}=1.0640020223 \]

when \(I_r\) is locally fixed. That last normalization is a trial to be derived from the complete Huygens return operator, not yet a locked WSM result.

7. Reverse-engineered local-strain action — tested and rejected as the parent

For comparison, suppose the One Law depends on instantaneous scalar compression strain \(s=\nabla\cdot\mathbf u\), and take

\[ S_{\rm local}=\int dt\,d^3x \left[\frac{\rho_0}{2}|\dot{\mathbf u}|^2-W(s)\right], \qquad \nabla\times\mathbf u=0. \]

If the complete directional energy at a strain turning point is \(E_d=E_{d0}+W\), then

\[ \frac{c'}{c_0} = \sqrt{\frac{W''}{\rho_0c_0^2}} = 1+\frac{W}{E_{d0}} \]

forces the unique constitutive equation

\[ \boxed{ W''(s)=\rho_0c_0^2\left(1+\frac{W}{E_{d0}}\right)^2, \quad W(0)=W'(0)=0. } \]

It has the first integral

\[ (W')^2 = \frac{2\rho_0c_0^2E_{d0}}{3} \left[ \left(1+\frac{W}{E_{d0}}\right)^3-1 \right] \]

and the fixed expansion

\[ W = \frac{\rho_0c_0^2}{2}s^2 +\frac{\rho_0^2c_0^4}{12E_{d0}}s^4 +\frac{\rho_0^3c_0^6}{72E_{d0}^2}s^6 +\frac{\rho_0^4c_0^8}{504E_{d0}^3}s^8 +\cdots. \]

Thus even this apparent quartic/sextic structure is not freely assembled: every coefficient follows from the One Law. But the forward spherical calculation rejects it as the parent law. At central peak \(I_{\max}=2\sqrt3\), one-harmonic balance gives \(kR=5.66206\); resolving the odd harmonics through \(29\omega_e\) converges near

\[ \boxed{kR\simeq5.194}, \qquad \boxed{R/\lambda_e\simeq0.8266}, \]

which is about \(4.55\%\) below \(\sqrt3/2\). More importantly, it creates a strong harmonic ladder rather than preserving the universal carrier. This is evidence that \(E_d\) must enter as directional cycle energy, as stated in the WSM glossary, rather than as instantaneous scalar strain energy.

8. Assumption/input ledger

The surviving branch currently uses ten structural or empirical inputs, leaving five available:

one continuous Space with real longitudinal vibration;

directional decomposition is a representation of that one field;

\(E_d\) is cycle-averaged directional wave energy;

the One Law \(c'/c_0=E_d/E_{d0}\);

reciprocal temporal/spatial coefficients in \(S_0\), giving constant carrier impedance;

isotropic equal-frequency rest input;

open Huygens return with equal rest phase \(\delta_H=0\);

first nontrivial radial-motion node \(j_1(kR)=0\);

the universal electron frequency \(\omega_e=m_ec_0^2/\hbar\) when electron units are required;

elementary recurrence action \(J=\hbar\) when absolute electron energy is required.

Input 11 is now the minimal norm-preserving completion of the projected chord map. It replaces a fitted gain or fitted \(\eta_H\). The cube-sphere ratio remains unused.

9. Exact half-sphere chord operator

Let \(\hat n\) be the receiving direction, \(\hat d\) a contributing direction and

\[ \mu=\hat n\cdot\hat d>0. \]

Projected hemisphere area gives the normalized flux weight \(2\mu\). For optical radius \(b=k_0R\), the exact finite-chord operator is

\[ \boxed{ H_\ell(b) =2\int_0^1 \mu P_\ell(\mu)e^{2ib\mu}\,d\mu . } \]

The scalar eigenvalue is

\[ \boxed{ H_0(b)=e^{ib}\left[j_0(b)+ij_1(b)\right]. } \]

This is still one real wave with two phase quadratures. The imaginary notation records their quarter-cycle relation.

The visible projected channel is contractive because it is only part of the complete incoming–outgoing wave relation. Its smallest conservative completion is

\[ \boxed{ \mathcal U_H= \begin{pmatrix} H&D_{H^\dagger}\\ D_H&-H^\dagger \end{pmatrix}, \qquad D_H=(I-H^\dagger H)^{1/2}, \qquad \mathcal U_H^\dagger\mathcal U_H=I . } \]

The complementary coordinate is not another substance. It is the direction-pair coherence omitted by projecting the complete wave relation onto one visible angular channel.

For a scalar eigenvalue \(H_\ell=a_\ell+ib_\ell\), the two eigenvalues of the completed map are

\[ \lambda_{\ell,+}=e^{i\theta_\ell}, \qquad \lambda_{\ell,-}=-e^{-i\theta_\ell}, \qquad \boxed{\sin\theta_\ell=\operatorname{Im}H_\ell}. \]

Consequently, two-passage reclosure occurs exactly when

\[ \operatorname{Im}H_\ell=0. \]

No gain coefficient is required; missing selected-channel norm moves through the conservative complement.

10. Exact scalar two-passage root and a new radius deduction

For the scalar channel,

\[ \operatorname{Im}H_0(b)=0 \quad\Longleftrightarrow\quad \boxed{\tan(2b)=2b} \quad\Longleftrightarrow\quad \boxed{j_1(2b)=0}. \]

Let \(x_{1,n}\) denote the \(n\)-th positive zero of \(j_1\). The exterior roots are

\[ b_n=\frac{x_{1,n}}2. \]

The first three give

Exterior root

\(b_n\)

\(R/\lambda_e=b_n/(2\pi)\)

\(I_r=b_n/x_{1,1}\)

1

2.24670472895

0.35757416328

0.50000000000

2

3.86262591847

0.61475600824

0.85962028492

3

5.45206082971

0.86772243109

1.21334609739

The inner e-sphere uses the first radial-motion node

\[ \frac{\omega_eR}{c_0I_r}=x_{1,1}=4.49340945791. \]

WSM requires the coherent core to have greater directional cycle energy than the background, hence \(I_r>1\). The first two exterior roots are therefore inadmissible. The third root is the first enhanced-core scalar closure, so

\[ \boxed{ b_*=5.45206082971, \qquad \frac{R_*}{\lambda_e}=0.86772243109, \qquad I_r=1.21334609739. } \]

This did not use the cube-sphere ratio. It predicts a value only

\[ \boxed{0.1959558\%} \]

above

\[ \frac{\sqrt3}{2}=0.86602540378. \]

Using the measured electron Compton wavelength gives the conditional physical radius

\[ \boxed{R_*=2.1053638160\times10^{-12}\ {\rm m}}, \]

compared with \(2.1012463013\times10^{-12}\ {\rm m}\) from the exact cube-sphere geometry.

At this root, with \(z_3=2b_*=x_{1,3}\),

\[ H_0 =-\frac{2\left(\sqrt{1+z_3^2}+1\right)}{z_3^2} =-0.201007451966, \]

and

\[ D_0=\sqrt{1-H_0^2}=0.979589712203. \]

The two eigenchannels contain visible/complement norm fractions

\[ (0.3994962740,\;0.6005037260) \]

and the reversed pair. Their sum is exactly one: the scalar closure contains no amplifier.

The negative eigenchannel changes sign on one passage and returns after two. If one passage is one half of the radial cycle, its generator phase is \(\pi\) in time \(T_e/2\), hence

\[ \omega_e=\frac{\pi}{T_e/2}=\frac{2\pi}{T_e}. \]

For canonical recurrence action \(J\), this gives

\[ \boxed{E=J\omega_e}. \]

The electron and positron remain the two opposite initial radial phases of this one oscillating channel; they are not two substances and should not be identified with the static \(+1\) and oscillating \(-1\) eigenchannels themselves.

11. Relation to the cube-sphere control

At the exact cube-sphere optical radius

\[ b_\square=\pi\sqrt3=5.44139809270, \]

the passive scalar channel is

\[ H_0(b_\square) =-0.201396118319+0.003902448350\,i. \]

The conservative eigenphase residual is therefore

\[ \boxed{ \theta_0 =\arcsin(\operatorname{Im}H_0) =0.003902458255\ {\rm rad} =0.223594388^\circ } \]

per passage. Thus the cube geometry is already extremely close to exact scalar two-passage reclosure. The action has two honest possibilities:

select the exact chord root \(R/\lambda_e=0.8677224311\); or

select the exact cube radius and obtain a nonlinear scalar phase repair of \(0.223594388^\circ\) per passage.

These alternatives must not be declared identical.

12. The scalar closes; the complete 3-D e-sphere does not yet

At the new exact scalar root \(b_*\), the higher passive chord channels are

\[ H_2=-0.1651696005-0.0502517520\,i, \]

\[ H_4=-0.1135520236-0.1428518834\,i, \]

\[ H_6=\phantom{-}0.0875674450-0.1497400212\,i. \]

Their conservative one-passage eigenphase residuals are

\[ \boxed{ \theta_2=-2.880426463^\circ, \qquad \theta_4=-8.212906234^\circ, \qquad \theta_6=-8.611860712^\circ. } \]

Moreover,

\[ \left|\frac{H_4}{H_2}\right|=1.056995808, \qquad \left|\frac{H_6}{H_2}\right|=1.004751070. \]

With the previously derived phase-averaged spherical-phase source ratio

\[ \frac{\kappa_4}{\kappa_2}=-\frac8{15}, \]

the transferred fourth-order amplitude remains

\[ \boxed{ \left| \frac{\kappa_4H_4}{\kappa_2H_2} \right| =0.563731098. } \]

Therefore the scalar root is a real forward deduction, but it does not yet produce the complete stable WSM structure: an e-sphere that is a radially vibrating standing wave with a spherically rotating phase wave, constructed from the Huygens combination of longitudinal plane waves arriving from all directions of Space, crossing the centre and continuing outward. The e-sphere does not spin as a body; the phase wave rotates over the complete sphere. The nonlinear directional action must supply, per passage, phase repairs of \(+2.880426463^\circ\) in \(V_2\) and \(+8.212906234^\circ\) in \(V_4\), while preserving the conservative norm and leaving the scalar phase closed. That is now a sharply numerical target rather than an unspecified “Huygens feedback.”

The synchronized projected-flux endpoint provides the exact real controls

\[ h_{2n} =2\int_0^1\mu P_{2n}(\mu)\,d\mu =(-1)^{n+1} \frac{(2n-2)!} {2^{\,2n-1}(n-1)!(n+1)!}, \]

so

\[ h_0=1, \qquad h_2=\frac14, \qquad h_4=-\frac1{24}, \qquad h_6=\frac1{64}. \]

For \(V_2\),

\[ \mathcal U_2= \begin{pmatrix} \tfrac14&\tfrac{\sqrt{15}}4\\ \tfrac{\sqrt{15}}4&-\tfrac14 \end{pmatrix}, \qquad \mathcal U_2^2=I. \]

This derives the earlier \(1/16+15/16=1\) norm completion directly from projected half-sphere geometry. The next calculation is no longer to invent \(\Gamma_H\): it is to derive whether the nonlinear \(E_d(\hat n)\) converts the complex passive \(V_2\oplus V_4\) phases into these real synchronized values.

The glossary exposed five real problems in my previous summary:

• I wrongly called the e-sphere’s reference node a “Huygens surface.” The Huygens sphere is the finite cosmic relation supplying the in-waves, not the e-sphere radius.
• I allowed \(e^{\pm\eta}\) to replace the physical moving-egg speeds \(c_0\pm v\). The glossary correctly distinguishes the raw reconstruction pair from its geometric-mean-normalized Doppler pair.
• I called \(\eta\) another “real quadrature.” The only e-sphere quadratures are the quarter-cycle \(j_0\) compression and \(j_1\) radial motion. Here \(\eta\) is merely a canonical variable.
• I described the \(j_1\) node too much like a boundary. It is a net radial-motion node of the complete sum; every constituent plane wave continues through it.
• The proposed chord profile is rotationally symmetric but locally direction-dependent. That requires explicit scrutiny against “same \(E_d,c'\) and wavelength in every direction.”

Here is the corrected replacement.

WSM Action Search: Rigorous Technical Handover

1. Fixed WSM ontology

Space is the one infinite, eternal, continuous physical substance. It is nearly rigid and slightly elastic. Its connected regions retain their neighbours and undergo bounded vibration.

A longitudinal compression plane wave is a real compression–extension disturbance in which Space vibrates backwards and forwards in the same direction that the wave travels.

The background wave sea consists of real longitudinal plane waves travelling through Space in every direction. “Sea” does not imply fluid flow.

The directional One Law is

\[ \boxed{ \frac{c'(\hat{\mathbf n})}{c_0} = \frac{E_d(\hat{\mathbf n})}{E_{d0}}. } \]

At the e-sphere’s one locked frequency,

\[ \lambda'(\hat{\mathbf n})=\frac{c'(\hat{\mathbf n})}{f_e}. \]

An e-sphere is an open spherical standing-wave recurrence continually rebuilt by plane waves approaching from every direction, crossing its centre and continuing outward. It has no reflecting material shell.

The Huygens sphere is not the e-sphere’s surface. It is the finite cosmic relation of other matter whose out-waves combine as the chosen e-sphere’s in-waves.

The electron and positron are opposite background-relative radial phases of the same kind of e-sphere:

\[ e^-:\text{ one radial phase}, \qquad e^+:\text{ opposite radial phase}. \]

The two spin states are the two opposite directions \(h=\pm1\) of the e-sphere’s spherical phase-wave rotation. This is not bodily rotation about an axis.

Therefore the four postulated matter states are

\[ \boxed{ (e^-,+1),\;(e^-,-1),\;(e^+,+1),\;(e^+,-1). } \]

The \(j_0/j_1\) quadratures do not multiply this state count.

2. Status language

• Postulate: fixed WSM physical foundation supplied to the search.
• Ansatz: a mathematical construction chosen for investigation.
• Deduction: follows exactly from stated postulates and ansätze.
• Conditional deduction: exact after an additional matching assumption.
• Open: not yet obtained from the nonlinear action.

3. Additional exploratory assumptions

Beyond the fixed WSM ontology, the present route uses these explicit assumptions:

1. Longitudinal displacement is represented by \(\mathbf u=\nabla\Psi\).
2. Positive directional energy ratios are parameterised exponentially.
3. A canonical variable \(\eta\) represents directional motion imbalance; it is not another e-sphere quadrature or state.
4. The ray Hamiltonian uses the minimal even function \(\cosh\eta\).
5. The chord slowness profile is initially sought in the lowest-degree quadratic class.
6. The reduced spherical action uses \(|\nabla\dot\Psi|^2\) and \((\nabla^2\Psi)^2\).
7. A temporary source \(f\) represents the still-unresolved combined action of the incoming condition and the e-sphere’s self-written directional \(E_d\).
8. The simplest elementary recurrence uses the first nontrivial \(j_1\) node.
9. The reduced spherical speed is provisionally matched to the chord-effective \(2c_0\).
10. One elementary recurrence has action \(J=\hbar\).
11. The rest recurrence satisfies \(\hbar\omega_e=m_ec_0^2\).
12. Spherical phase rotation is represented by a half-angle rotor.
13. The Dirac reconstruction assumes first-order, rotationally covariant relativistic evolution.
14. Where used, the normalized e-sphere circumscribes a unit cube, \(R_*=\sqrt3/2\).

These are structural inputs, not fourteen fitted numerical constants. The physical coefficients \(\kappa,\rho_\Psi,p_*\) and \(\epsilon_*\) remain undetermined.

4. Exact all-direction \(j_0/j_1\) kinematics

The real equal-phase sum of longitudinal plane waves arriving from every direction is

\[ \frac{1}{4\pi} \int_{S^2} \cos(k\hat{\mathbf n}\cdot\mathbf x)\,d\Omega = j_0(kr), \]

where

\[ j_0(kr)=\frac{\sin kr}{kr}. \]

The radial derivative is

\[ \frac{d}{dr}j_0(kr)=-k\,j_1(kr). \]

Therefore the \(j_0\) compression pattern and \(j_1\) radial-motion pattern are two successive real aspects of one spherical vibration:

• \(j_0\): compression and extension of Space;
• \(j_1\): radial motion of Space one quarter-cycle later.

This result is kinematic. It follows from summing real plane waves over the sphere and does not depend on the proposed action.

5. Chord-effective speed and a candidate internal profile

For a ray with impact parameter \(b\), let

\[ x_b=\sqrt{R^2-b^2}. \]

After the passing plane wave enters the e-sphere, the outside carrying plane has \(x_b/c_0\) remaining before reaching the centre plane. The part crossing the e-sphere must cover the complete chord \(2x_b\) in that time.

Therefore

\[ \boxed{ \int_{\rm chord}\frac{ds}{c'}=\frac{x_b}{c_0}, } \]

so the harmonic path-average speed is \(2c_0\).

Define directional slowness

\[ g(\mathbf x,\hat{\mathbf n}) = \frac{c_0}{c'(\mathbf x,\hat{\mathbf n})}. \]

One lowest-degree candidate satisfying every chord integral and the proposed central value \(c'(0)=2\sqrt3c_0\) is

\[ \boxed{ g_*= \frac12+ \frac{A}{2R^2} \left[ r^2+ 2(\hat{\mathbf n}\cdot\mathbf x)^2 -R^2 \right], \qquad A=1-\frac1{\sqrt3}. } \]

Direct integration gives

\[ \int_{-x_b}^{x_b}g_*\,ds=x_b. \]

At the centre,

\[ g_*(0)=\frac1{2\sqrt3}, \qquad c'(0)=2\sqrt3c_0. \]

It can also be written

\[ g_*=\hat{\mathbf n}^{T}G_*\hat{\mathbf n}, \qquad G_*=\frac12I+\nabla\nabla\psi_*, \]

with

\[ \boxed{ \psi_*(r) = \frac{A}{8R^2}(r^2-R^2)^2. } \]

Because

\[ \nabla\psi_*(R)=0, \]

the Hessian contribution integrates to zero over every complete chord, leaving the required chord timing.

Important compatibility question

The profile is rotationally covariant:

\[ g_*(\mathcal R\mathbf x,\mathcal R\hat{\mathbf n}) = g_*(\mathbf x,\hat{\mathbf n}). \]

It therefore introduces no preferred front, rear or external axis into a stationary e-sphere.

However, at an off-centre point it distinguishes longitudinal waves travelling radially from those travelling tangentially.

If “same \(E_d,c'\) and wavelength in every direction” means complete pointwise directional isotropy,

\[ g=g(r), \]

then Abel inversion of the all-chord condition forces

\[ g(r)=\frac12, \qquad c'(r)=2c_0. \]

A pointwise-isotropic profile cannot simultaneously satisfy every chord average and have \(c'(0)=2\sqrt3c_0\).

Thus one of the following interpretations must be selected:

1. spherical equality means rotational equivalence of every outward sector, while local radial/tangential directional differences are permitted; or
2. spherical equality means pointwise isotropy, in which case the \(2\sqrt3c_0\) centre condition must be reconsidered.

The quadratic profile is therefore an exact mathematical chord control, not yet an accepted internal \(E_d\) distribution.

6. Corrected one-direction carrier action

Let \(q(\sigma,t)\) be the displacement of Space along one longitudinal plane-wave direction. Define a dimensionless signed compression variable

\[ a=\frac{q_\sigma}{\epsilon_*}. \]

Let \(\eta\) be a canonical variable recording the directional motion imbalance. It is not:

• a new wave substance;
• a new electron state;
• one of the \(j_0/j_1\) quadratures;
• the spherical spin phase.

Consider

\[ \boxed{ S_{\rm ray} = \int dt\,d\sigma \left[ p_*\eta\,\dot q - E_{d0}e^a\cosh\eta \right], \qquad p_*\epsilon_*=\frac{E_{d0}}{c_0}. } \]

Variation gives

\[ \dot q = \frac{E_{d0}}{p_*}e^a\sinh\eta, \]

\[ a_t = c_0\partial_\sigma(e^a\sinh\eta), \]

\[ \eta_t = c_0\partial_\sigma(e^a\cosh\eta). \]

Define the two mathematical characteristic combinations

\[ r_+=a+\eta, \qquad r_-=a-\eta. \]

They obey

\[ \boxed{ \partial_t r_+ - c_0e^{r_+}\partial_\sigma r_+=0, } \]

\[ \boxed{ \partial_t r_- + c_0e^{r_-}\partial_\sigma r_-=0. } \]

The Hamiltonian decomposes as

\[ E_{d0}e^a\cosh\eta = \frac{E_{d0}}2 \left(e^{r_+}+e^{r_-}\right). \]

Thus, within this ray ansatz,

\[ \frac{E_\pm}{E_{d0}}=e^{r_\pm}, \qquad \frac{c_\pm}{c_0}=e^{r_\pm}. \]

Therefore the candidate action generates

\[ \boxed{ \frac{c_\pm}{c_0} = \frac{E_\pm}{E_{d0}} } \]

for its two oppositely travelling longitudinal characteristics.

This is the strongest present result from the ray action.

7. Correct relation to the physical moving wave egg

The moving e-sphere retains one common frequency. Its physical axial reconstruction pair is

\[ \boxed{ c'_{\rm rear}=c_0+v=c_0(1+\beta), } \]

\[ \boxed{ c'_{\rm lead}=c_0-v=c_0(1-\beta). } \]

By the One Law and fixed frequency,

\[ \frac{E_{d,\rm rear}}{E_{d0}} = \frac{\lambda_{\rm rear}}{\lambda_e} = 1+\beta, \]

\[ \frac{E_{d,\rm lead}}{E_{d0}} = \frac{\lambda_{\rm lead}}{\lambda_e} = 1-\beta. \]

Accordingly:

• the elongated leading sector has lower \(E_d\), lower \(c'\) and shorter wavelength;
• the flattened rear sector has higher \(E_d\), higher \(c'\) and longer wavelength;
• the side-sector geometry remains open.

The reciprocal Doppler factors are the geometric-mean-normalized form of this raw physical pair:

\[ K_\pm = \frac{1\pm\beta}{\sqrt{1-\beta^2}} = \gamma(1\pm\beta) = e^{\pm s_{\rm rap}}, \]

where

\[ s_{\rm rap}=\operatorname{artanh}\beta. \]

They do not replace \(c_0\pm v\).

Exact matching to the ray variables

The ray action reproduces the raw pair if

\[ \eta=s_{\rm rap} =\operatorname{artanh}\beta \]

and

\[ a= \frac12\ln(1-\beta^2) = -\ln\gamma. \]

Then

\[ e^{a+\eta}=1+\beta, \qquad e^{a-\eta}=1-\beta. \]

Thus

\[ \boxed{ e^{r_{\rm rear}}=1+\beta, \qquad e^{r_{\rm lead}}=1-\beta. } \]

Dividing out the common geometric-mean factor \(e^a=1/\gamma\) produces

\[ e^{\pm\eta} = \gamma(1\pm\beta). \]

This corrects the previous summary: the exponential factors can encode both structures, but the raw physical reconstruction speeds are \(c_0\pm v\), while \(e^{\pm\eta}\) alone are the normalized reciprocal Doppler factors.

The common-frequency unequal wavelengths then form the longer internal de Broglie phase pattern. Deriving its complete three-dimensional phase geometry from the action remains open.

8. Corrected reduced spherical action

The physical displacement of Space is

\[ \mathbf u=\nabla\Psi. \]

Its velocity is therefore

\[ \dot{\mathbf u}=\nabla\dot\Psi. \]

The correct reduced action is

\[ \boxed{ S_{\rm sphere} = \int dt\,d^3x \left[ \frac{\rho_\Psi}{2} |\nabla\dot\Psi|^2 - \frac{\kappa}{2} (\nabla^2\Psi)^2 + f\Psi \right]. } \]

The \((\nabla^2\Psi)^2\) term describes a reduced longitudinal compression/compatibility energy. It does not introduce transverse shear waves.

The equation is

\[ \rho_\Psi\nabla^2\Psi_{tt} - \kappa\nabla^4\Psi + f=0. \]

For the reverse-engineered delay profile,

\[ \nabla^4\psi_*=\frac{15A}{R^2}, \]

so

\[ f=\frac{15\kappa A}{R^2}. \]

The term \(f\) is only a placeholder. The completed nonlinear action must replace it with the actual incoming Huygens-sphere condition and the e-sphere’s self-consistent modification of the plane waves crossing it.

9. Exact \(j_0/j_1\) mode of the reduced action

Write

\[ \Psi=\psi_*+\phi \]

and choose

\[ \boxed{ \phi(r,t) = \Phi_0 \left[ j_0(kr)-j_0(kR) \right]\cos\omega t. } \]

Then the real compression–extension pattern is

\[ \nabla^2\phi = -\Phi_0k^2j_0(kr)\cos\omega t. \]

The radial displacement of Space is

\[ u_r = \partial_r\phi = -\Phi_0k\,j_1(kr)\cos\omega t. \]

The radial velocity of Space is

\[ v_r = \partial_tu_r = \Phi_0\omega k\,j_1(kr)\sin\omega t. \]

This gives precisely one spherical vibration:

\[ j_0\text{ compression} \quad\longrightarrow\quad j_1\text{ radial motion one quarter-cycle later}. \]

At the selected reference radius, require the complete spherical sum to have zero net radial displacement:

\[ u_r(R)=0. \]

This gives

\[ j_1(kR)=0. \]

The first nontrivial node is

\[ \boxed{ kR=4.4934094579. } \]

This is a net interference node. It is not:

• a Huygens sphere;
• a reflecting shell;
• an edge of Space;
• a place where the constituent plane waves stop.

The plane waves continue through the complete recurrence.

The dispersion relation is

\[ \boxed{ \omega^2=\frac{\kappa}{\rho_\Psi}k^2. } \]

Thus the corrected reduced action reproduces the postulated \(j_0/j_1\) vibration exactly, but it has not yet proved nonlinear stability.

10. Conditional e-sphere scale

The complete electron Compton cycle is

\[ \lambda_e=\frac{h}{m_ec_0} = \frac{2\pi c_0}{\omega_e}, \]

\[ T_e=\frac{\lambda_e}{c_0}. \]

If the reduced spherical mode speed is matched to the chord-effective speed,

\[ \sqrt{\frac{\kappa}{\rho_\Psi}}=2c_0, \]

then

\[ k=\frac{\omega_e}{2c_0} =\frac{\pi}{\lambda_e}. \]

Combining this with the first \(j_1\) node gives

\[ \boxed{ \frac{R}{\lambda_e} = \frac{4.4934094579}{\pi} = 1.43029665. } \]

This is a conditional deduction from:

• the first \(j_1\) node;
• the electron Compton cycle;
• the identification of the reduced mode speed with the chord-effective \(2c_0\).

The action has not independently derived that matching.

11. Mode energy and action

At the first \(j_1\) node,

\[ \int_{B_R}j_0(kr)^2\,dV = 2\pi R^3j_0(kR)^2. \]

The reduced mode energy is

\[ \boxed{ E= \frac{ \pi\kappa\Phi_0^2x_1^4j_0(x_1)^2 }{R}, \qquad x_1=4.4934094579. } \]

Using

\[ J=\frac{E}{\omega} \]

and the empirical elementary-action input

\[ J=\hbar \]

gives

\[ \boxed{ \Phi_0^2 = \frac{ \hbar\omega R }{ \pi\kappa x_1^4j_0(x_1)^2 }. } \]

This amplitude relation is exact within the reduced action.

A numerical amplitude cannot be obtained until \(\kappa\) is derived or calibrated. The static delay geometry and oscillatory mode must not each be assigned a separate copy of \(m_ec_0^2\).

12. Spherical phase wave, spin and \(4\pi\) recurrence

The spherical phase wave is the moving equal-phase relation produced over the complete e-sphere by intersecting longitudinal waves arriving from different directions.

Its phase positions may move superluminally because successive equal-phase positions are produced by different intersecting waves. No region of Space or wave energy travels at that phase speed.

The two hands of spherical phase rotation may be represented by the ansatz

\[ \boxed{ Q_h(\hat{\mathbf r},\chi) = \cos\frac{\chi}{2} + hI_{\hat{\mathbf r}} \sin\frac{\chi}{2}, \qquad h=\pm1. } \]

This represents rotation of the complete phase relation over the sphere, not circular rotation of an electron surface.

It gives

\[ Q_h(2\pi)=-1, \qquad Q_h(4\pi)=+1. \]

Therefore the proposed phase relation has exact \(4\pi\) closure.

The electron/positron radial phase remains an independent label. A \(2\pi\) sign change of the rotational representation must not be interpreted as converting an electron into a positron.

If

\[ \theta=\frac{\chi}{2} \]

and one complete elementary phase cycle carries action

\[ J_\theta=\hbar, \]

then the action conjugate to the physical spherical rotation coordinate is

\[ \boxed{ J_\chi=\frac{\hbar}{2}. } \]

Thus spin \(\hbar/2\) follows conditionally from the half-angle spherical phase geometry and the empirical action quantum.

13. Conditional Dirac reconstruction

Let \(\tau=\pm1\) label the two opposite radial phases and let \(h=\pm1\) label the two spherical phase-rotation hands.

The four-component Dirac spinor is interpreted as a coordinate representation of

\[ (e^-,+1),\quad (e^-,-1),\quad (e^+,+1),\quad (e^+,-1). \]

Its components are not four pieces of an electron.

Let \(\tau_i\) operate on the radial-phase coordinates and \(\sigma_i\) represent the quaternion/Pauli rules for changing the spherical phase-rotation hand. The proposed representation is

\[ \beta=\tau_3\otimes I, \qquad \alpha_i=\tau_1\otimes\sigma_i. \]

These matrices obey

\[ \{\alpha_i,\alpha_j\} = 2\delta_{ij}I, \]

\[ \{\alpha_i,\beta\}=0, \qquad \beta^2=I. \]

Given the additional requirements that the equation:

• is first order in time;
• is first order in spatial change;
• is rotationally covariant;
• couples the radial phases and spherical rotation hands;
• squares to the relativistic energy relation;

the minimal factorisation is

\[ \boxed{ i\hbar\partial_t\Psi_D = \left( -i\hbar c_0\boldsymbol{\alpha}\cdot\nabla + \beta m_ec_0^2 \right)\Psi_D. } \]

Complex \(i\) records a real quarter-cycle phase relation. It is not an imaginary substance and creates no additional state.

The algebraic factorisation is exact. Its identification with the postulated e-sphere states is a physical construction that the completed nonlinear action must reproduce dynamically.

14. What has been deduced

Within the explicitly stated assumptions:

1. The all-direction plane-wave sum gives the \(j_0\) compression pattern.

2. Its radial derivative gives the quarter-cycle \(j_1\) radial-motion pattern.

3. The chord-effective speed is \(2c_0\).

4. A quadratic directional slowness profile satisfying every chord integral has been constructed.

5. The canonical ray action generates the directional One Law for its characteristics.

6. Its exponential factors reproduce the physical \(1\pm\beta\) pair when

   \[ a=-\ln\gamma, \qquad \eta=\operatorname{artanh}\beta. \]

7. Geometric-mean normalization then gives the reciprocal Doppler factors

   \[ e^{\pm\eta} = \gamma(1\pm\beta). \]

8. The corrected reduced spherical action has an exact \(j_0/j_1\) normal mode.

9. The first radial-motion node is \(kR=4.4934094579\).

10. The mode energy and amplitude relation have been calculated.

11. The half-angle spherical phase ansatz has \(4\pi\) closure.

12. The four WSM sectors support a representation of the Dirac Clifford algebra.

15. What remains speculative or open

The following have not yet been derived from one completed action:

• the self-sustaining nonlinear e-sphere without the inserted \(f\);
• whether local radial/tangential variation in the chord profile is compatible with the intended stationary-e-sphere isotropy;
• the complete three-dimensional moving wave egg, especially its side sectors;
• the stability of the e-sphere under changing background in-waves;
• the derivation of the common frequency from resonant coupling to the Huygens sphere;
• the full de Broglie phase pattern from the moving egg;
• the integrated Lorentz energy and momentum of the whole e-sphere;
• the dynamical origin of the spherical phase rotor;
• the Dirac equation directly from the nonlinear e-sphere action;
• numerical values of mass, charge, \(\alpha\), spin coupling and interaction strengths.

The present advance is therefore:

\[ \boxed{ \text{real all-direction }j_0/j_1\text{ kinematics} + \text{exact chord control} + \text{One-Law ray action} + \text{corrected spherical-mode action} + \text{conditional }4\pi\text{/Dirac algebra}. } \]

The remaining central task is to unite these pieces in one nonlinear, longitudinal, all-direction action of one continuously connected Space.