Common-anchor action audit

Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Action and kinetic-normalization tests locate the unresolved fixed-reference bridge.

NOT A PHYSICS MODULE. This note records the action-level attempt to derive or falsify the V2 common-anchor law as it proceeds.

1. Target

The relation under test is

$$\frac{(M_d^\star)^2}{g_{dd}}=\frac{m_e^2}{g_{66}},$$

or, in frequency language,

$$\frac{\omega_d^\star}{\omega_e}=\sqrt{\frac{g_{dd}}{g_{66}}}.$$

The task is to determine whether this follows from the written V2 projected dynamics rather than being an additional scale ansatz.

2. First source-level result

The self-contact module explicitly starts from a sector-diagonal scalar interaction

$$\mathcal L_{\rm sc}=-\frac{\kappa_d}{2}J_d^2$$

and derives the projected scalar coefficient

$$\mu_d=\kappa_dR_dB_d.$$

The finite-band contact theorem then gives

$$M_{n,d}=\mu_dS(n,d).$$

Therefore the written operator ordering is

$$\boxed{\mu_d\;\longrightarrow\;\mu_dS(n,d)}.$$

The sector coefficient is formed first; the simplex/contact eigenvalue multiplies it afterward.

Section 9.1 of the self-contact module introduces the common-anchor equation separately as a projected fixed-point condition and explicitly states that the master equation of motion must supply that equation. No equation in the inspected self-contact passage identifies $\kappa_dR_dB_d$ with $g_{dd}$ or $\sqrt{g_{dd}}$.

3. Cross-depth source audit

The bound/free module writes reciprocal cross-depth interactions schematically as

$$S_{ab}=g_{ab}\int dt\,I_{ab},$$

whose variations give $g_{ab}$ times the other sector's marginal/contact density.

The higher-dimensional-polarization module writes a schematic current/gauge interaction

$$S_{\rm int}=g_{ab}\int_{Q_a}\langle A_a,(\pi_{ba})_*J_b\rangle.$$

These establish incidence, reciprocity, representation filtering, and the existence of coupling coefficients. In the inspected passages they do not identify the scalar self-contact coefficient $\kappa_d$, the projected scalar coefficient $\mu_d$, or an occupied reference frequency with $g_{dd}$ or $\sqrt{g_{dd}}$.

The polarization source in fact says that a nonzero invariant vertex leaves its coupling coefficient to the projected dynamics.

4. Exact placement test for the rank-one proposal

Let

$$v_d=\sqrt{g_{dd}},\qquad G=vv^T,$$

and for the proposed depth-3, depth-4, and electron reference bands define

$$D_\star=\operatorname{diag}(1,15,18564).$$

If the rank-one response acts on primitive pre-contact coefficients,

$$\sigma=GJ=v(v^TJ),$$

then the written contact projection gives

$$M^\star=D_\star\sigma=(D_\star v)(v^TJ).$$

Hence

$$\boxed{ \frac{M_d^\star}{M_6^\star} =\frac{D_d^\star\sqrt{g_{dd}}}{D_6^\star\sqrt{g_{66}}} }.$$

This is Case B, not the desired common-anchor law.

If instead a deeper master equation acts directly on the already-projected occupied reference coefficients,

$$M^\star=vC,$$

then

$$\boxed{ \frac{M_d^\star}{M_6^\star}=\sqrt{\frac{g_{dd}}{g_{66}}} }.$$

This is Case A.

The algebraic way to begin before contact multiplication and still end in Case A is to use

$$\widetilde G=D_\star^{-1}vv^TD_\star^{-1},$$

so that $D_\star\widetilde GJ$ lies along $v$. But this changes the primitive diagonal couplings to

$$\widetilde g_{dd}=\frac{g_{dd}}{(D_d^\star)^2}.$$

That is not a harmless relabeling of the quoted V2 couplings unless an independent projection or kinetic-normalization theorem derives exactly this weighting.

5. Interim verdict

The naive rank-one source-to-self-energy mechanism does not derive the common-anchor law when attached directly to the scalar coefficient already written in V2. With the explicit self-contact ordering it produces Case B.

Case A remains possible only if the deeper projected master dynamics supplies additional structure, for example:

1. a rank-one equation acting directly on occupied reference frequencies;
2. an independently derived two-sided reference-count weighting before contact projection;
3. sector-dependent projected overlap or kinetic factors that cancel the reference counts exactly; or
4. a nonlinear fixed-point equation in which the reference-band factors enter before the final ratio is formed.

None of those mechanisms has yet been found in the inspected source passages.

The identity

$$g_{34}^2=g_{33}g_{44}$$

is therefore insufficient by itself. It establishes rank one for a coupling-coefficient matrix, but does not determine the operator layer on which that matrix acts. Since $D_\star v$ is not proportional to $v$, the simplex projection and the proposed rank-one image do not commute in the way required by the anchor law.

The remaining problem is an operator-ordering and normalization problem, not an unknown-amplitude problem.

6. Next proof obligation

Trace the actual derivation and mathematical role of

$$g_{33}=8\sqrt7,\qquad g_{44}=12/\sqrt7,\qquad g_{34}=4\sqrt6,\qquad g_{66}=1/4.$$

For each value, determine whether it is defined as a scalar self-interaction coefficient, a polarization/gauge normalization, a stiffness, a kernel eigenvalue, or another object. Only then can the use of these same numbers in the scalar reference-frequency fixed-point equation be classified as derived, merely compatible, or category-invalid.

No V1 material is used as evidence here.

7. Historical V2 action supplies a real candidate mechanism - but not yet a current-V2 derivation

The git history contains an earlier V2, not V1, formulation from August 28, 2026. It is useful provenance because it shows what the common-anchor statement was intended to mean before the present projective-contact modules were written.

That historical V2 wrote one quartic master interaction

$$S_{\rm int} =\frac12\sum_{d,e}g_{de} \int K_{de}(\xi_d,\xi_e)\,\rho_d\rho_e,$$

with

$$g_{de}=v_dv_e, \qquad g_{dd}=v_d^2.$$

For the displayed kernel,

$$K_{de}=(\xi_d\cdot\xi_e)^2.$$

The same historical text then states the lightest-mode fixed point

$$\frac{(M_d^\star)^2}{g_{dd}} =\frac{m_e^2}{g_{66}},$$

which is exactly the present common-anchor condition.

This matters because the rank-one-response idea was not invented during the present audit: an earlier V2 master action already contained a rank-one coupling matrix and explicitly described one collective interaction channel.

7.1 Exact variation of the historical rank-one action

Let

$$\rho_d=\bar\Psi P_d\Psi.$$

For a symmetric kernel and symmetric coupling matrix, variation of the quadratic density functional with respect to $\rho_d$ gives

$$\Sigma_d =\sum_e g_{de}K_{de}\rho_e.$$

Using $g_{de}=v_dv_e$,

$$\boxed{ \Sigma_d =v_d\sum_e v_eK_{de}\rho_e. }$$

This is an important refinement of the simpler toy calculation $\Sigma=v(v^TJ)$.

Rank one of the coefficient matrix alone does not yet imply

$$\Sigma_d=v_dC$$

with one sector-independent scalar $C$, because the geometric kernel still carries the external sector label $d$ through $K_{de}$.

The required additional condition is

$$\boxed{ \sum_e v_eK_{de}\rho_e=C \quad\text{independent of }d }$$

on the relevant fixed-point branch, or an equivalent projector/normalization theorem that reduces the full kernel to a common scalar functional.

The historical prose asserts precisely such a common condensate, but the displayed action by itself does not prove that the $K_{de}$ dependence collapses this way.

7.2 Why this still does not automatically solve the present projective-contact problem

The historical V2 and current V2 place the spectral count at different conceptual stages.

The current self-contact theorem derives

$$M_d^\star=D_d^\star\mu_d,$$

where $\mu_d$ is the sector-normalized scalar contact coefficient before multiplication by the occupied-band dimension.

Therefore, if the historical rank-one response $\Sigma_d$ is identified with the current primitive coefficient $\mu_d$, then

$$M_d^\star =D_d^\star v_dC,$$

which is Case B:

$$\frac{M_d^\star}{M_6^\star} =\frac{D_d^\star v_d}{D_6^\star v_6}.$$

To recover Case A in the current architecture, one must establish one of two precise identifications:

1. post-projection response: the historical $\Sigma_d$ corresponds directly to the occupied reference eigenvalue $M_d^\star$, not to $\mu_d$; or
2. count-compensated primitive response: projection of the historical kernel produces $$\mu_d\propto\frac{v_d}{D_d^\star},$$ so that the subsequent contact multiplier restores $M_d^\star\propto v_d$.

The historical source does not perform the present finite-band projective-contact calculation, so it cannot decide between these two identifications.

7.3 A second incompatibility that must be resolved

The historical V2 derives a self-consistent harmonic stiffness

$$\lambda_d=\left(\frac{g_{dd}}2\right)^{2/3}$$

from the quartic kernel, and then separately states that the physical mass spectrum is

$$m(n,d)=m_{{\rm scale},d}S(n,d).$$

But the ordinary isotropic harmonic-oscillator eigenvalues scale linearly with excitation number and with $\sqrt{\lambda_d}$; their degeneracy count is $S(n,d)$ only after cumulative counting. The historical source does not derive why the eigenfrequency itself equals the cumulative degeneracy rather than the usual oscillator eigenvalue.

The current projective-contact theorem was written precisely to supply a genuine operator whose occupied eigenvalue is $S(n,d)$. Consequently, importing the historical harmonic-well fixed point wholesale would mix two distinct spectral mechanisms.

7.4 Result of the historical-source test

Survives as provenance and a candidate mechanism:

• V2 originally intended $g_{de}$ to be one rank-one quartic self-interaction matrix;
• V2 originally intended the electron-relative scale law to be a kernel-vacuum fixed point;
• the square-root coupling ratios therefore have a coherent action-level motivation, not merely a numerical coincidence.

Does not yet survive as a derivation in the current V2 operator architecture:

• rank one of $g_{de}$ does not by itself remove the sector dependence of the geometric kernel $K_{de}$;
• the historical action does not derive the inverse reference-count factor required if its response is pre-contact;
• the historical harmonic-well spectrum and the current projective-contact spectrum are not the same operator construction.

The common anchor is therefore better motivated than the current files alone made visible, but still unproved in present V2.

8. Sharpened next calculation

The next decisive calculation is now concrete. Project the historical-style rank-one master kernel through the current finite-band projectors and ask whether its occupied-line matrix element has the form

$$\langle z_d;N_d|P_d\mathcal K_\infty P_d|z_d;N_d\rangle =C\sqrt{g_{dd}}$$

or

$$=C\sqrt{g_{dd}}\,D_d^\star,$$

or instead acquires a compensating factor $1/D_d^\star$ before the contact eigenvalue is applied.

Equivalently, define the projected source functional

$$F_d[\rho] =\sum_e v_e\,\langle z_d;N_d|P_dK_{de}[\rho_e]P_d|z_d;N_d\rangle.$$

Case A requires

$$D_d^\star F_d[\rho] \propto \text{constant in }d,$$

when the primitive scalar coefficient is used, whereas a direct post-projection formulation requires simply

$$F_d[\rho]\propto\text{constant in }d.$$

This is now the exact normalization theorem to prove or falsify.

9. Fixed reference normalization is compatible with band blindness

The current self-contact theorem states that, for the ordinary family law,

$$M_{n,d}=\mu_dS(n,d),$$

with all explicit band dependence carried by the reproducing-kernel diagonal $S(n,d)$. Its action-level domain explicitly requires the scalar contact strength $\mu_d$ to be band-blind.

This sharpens the Case A/Case B fork.

Suppose the historical rank-one master response is attached to the current primitive scalar coefficient:

$$\mu_d=v_dC, \qquad v_d=\sqrt{g_{dd}}.$$

Then the present contact theorem gives

$$M_d^\star=D_d^\star v_dC,$$

and therefore Case B.

The only primitive-level algebraic rescue previously identified was

$$\mu_d\propto\frac{v_d}{D_d^\star}.$$

Here $D_d^\star$ is fixed once per sector. Dependence on that fixed reference does not imply dependence on the varying occupied band $n$ or $N=n-1$. Thus $\mu_d=Cv_d/D_d^\star$ is band-blind and gives $M_{n,d}=Cv_dD_{n,d}/D_d^\star$, with $M_d^\star=Cv_d$. The earlier claim that band blindness alone excludes this normalization is withdrawn. Replacing $D_d^\star$ by the running $D_{n,d}$ would violate band blindness, but that is a different prescription.

Therefore:

$$\boxed{ \mu_d=Cv_d\text{ forces Case B; fixed }\mu_d=Cv_d/D_d^\star\text{ gives Case A.} }$$

A fixed inverse-reference normalization changes the primitive-response identification, not the band-blind contact theorem. Its physical origin remains unproved; it cannot be silently inserted as a result of the historical kernel.

This rules out only the simultaneous unweighted primitive response $\mu_d=Cv_d$ and the desired reference ratios with unequal reference counts.

Case A can follow from a post-contact / occupied-reference fixed point, or from a derived fixed sector normalization at the primitive layer. Neither route has been derived from the current master action.

10. Current V2 explicitly records the missing coordinate-kernel to phase-ray bridge

Section 8.6 of the present self-contact module distinguishes two operators:

$$P_{N,d}|q|^2P_{N,d}$$

from the coordinate harmonic kernel, and

$$D_N|z;N\rangle\langle z;N|$$

from phase-ray point contact.

It then states that their relation still requires either:

1. a master interaction written locally on the homogenized phase-ray bundle; or
2. a derivation showing that the coordinate-space nonlinear kernel transforms into occupied-ray contact after oscillator and sector projection.

This is exactly the missing bridge between the historical V2 master kernel

$$(\xi_d\cdot\xi_e)^2$$

and the present projective-contact mass operator.

So the source audit now yields a stronger documentary conclusion:

$$\boxed{ \text{the current V2 source itself identifies the historical-to-current operator bridge as unfinished.} }$$

Consequently, the historical statement that the kernel vacuum supplies the common anchor cannot presently be imported as a completed derivation. It points to the right location in the dynamics, but the required projection theorem has not yet been written.

11. Revised status after the bridge audit

Established in current V2:

• occupied phase-ray point contact gives the exact simplex eigenvalue $D_N=S(n,d)$;
• the ordinary family law has a band-blind primitive coefficient $\mu_d$;
• therefore a primitive response proportional to $\sqrt{g_{dd}}$ produces Case B.

Established as historical V2 provenance:

• an earlier V2 master action used a rank-one coupling matrix $g_{de}=v_dv_e$;
• it intended one collective self-interaction channel;
• it stated the electron-relative common-anchor law as a kernel-vacuum fixed point.

Still missing:

• the projection of that coordinate-space rank-one kernel into the current occupied phase-ray scalar-contact operator;
• a proof that the resulting complete occupied reference eigenvalue, rather than the primitive coefficient, lies along $v_d$;
• or any other current-V2 equation that yields $$M_d^\star\propto\sqrt{g_{dd}}$$ with a derived fixed sector normalization or occupied-reference fixed point.

The common-anchor law is not disproved. The unweighted primitive identification is excluded; a fixed inverse-reference normalization remains algebraically allowed but physically underived.

12. Kinetic-normalization audit: no present rescue of the primitive rank-one anchor

The remaining obvious place for a compensating factor was canonical or kinetic normalization. The current V2 mass module does not provide one.

Its operator ordering is explicit:

1. the spatial Dirac/Laplace operator constructs the mode eigenspace;
2. the localized contact operator maps the rank of that eigenspace to the rest-energy eigenvalue;
3. the sector coefficient is $$\mu_d=\kappa_dR_dB_d,$$ and is independent of the band label under the ordinary mass theorem.

No reference-band wavefunction renormalization or kinetic factor proportional to

$$\frac1{D_d^\star}$$

appears in that derivation.

A separate gauge/polarization module does say that propagation laws and kinetic normalizations of the projected gauge connection remain to be obtained from the master action. But that statement concerns projected connection components and cross-sector propagation; no current source identifies such a normalization with the scalar rest-energy contact coefficient, and no formula supplies the reference-band dependence required here.

The cross-depth contact theorem also preserves complete normalized marginal weight:

$$\int \bar\rho_{b\to a}=1.$$

Thus full marginalization does not furnish an inverse simplex factor either.

Combining these facts gives the following present-V2 no-go statement.

12.1 Primitive-rank-one incompatibility theorem

Assume simultaneously:

1. the ordinary current V2 contact law $$M_{n,d}=\mu_dD_{n,d}, \qquad D_{n,d}=S(n,d);$$
2. band blindness of the primitive scalar contact coefficient $\mu_d$;
3. normalized cross-depth marginalization with no band-dependent loss of weight;
4. identification of the historical rank-one response with that primitive coefficient, $$\mu_d=C\sqrt{g_{dd}}$$ for one sector-independent $C$.

Then for any chosen reference bands,

$$\boxed{ \frac{M_d^\star}{M_6^\star} = \frac{D_d^\star}{D_6^\star} \sqrt{\frac{g_{dd}}{g_{66}}} }$$

and not

$$\sqrt{\frac{g_{dd}}{g_{66}}}$$

unless accidentally $D_d^\star=D_6^\star$.

For the adopted references,

$$D_3^\star=S(1,3)=1, \qquad D_4^\star=S(3,4)=15, \qquad D_6^\star=S(13,6)=18564,$$

so the accidental-equality condition fails drastically.

Therefore the four assumptions above are mutually incompatible with the common-anchor law.

At least one must change. Since assumptions 1--3 are explicit parts of the current V2 contact construction, the natural conclusion is:

$$\boxed{ \text{the desired anchor is incompatible with the unweighted identification }\mu_d=Cv_d. }$$

It can instead act on the complete occupied reference eigenvalues, or arise from a derived fixed sector weighting such as $\mu_d=Cv_d/D_d^\star$. The latter preserves assumptions 1--3 but replaces assumption 4; it is not excluded merely because it acts at the primitive layer.

12.2 Numerical size of the contradiction

The difference is not perturbative.

The desired Case A factors relative to the electron reference are

$$\sqrt{g_{33}/g_{66}}\approx 9.2013065, \qquad \sqrt{g_{44}/g_{66}}\approx 4.2593773.$$

A primitive rank-one response followed by the present contact theorem instead gives

$$\frac{1}{18564}\sqrt{g_{33}/g_{66}} \approx 4.9565\times10^{-4},$$

and

$$\frac{15}{18564}\sqrt{g_{44}/g_{66}} \approx 3.4416\times10^{-3}.$$

These differ from the required reference-frequency ratios by factors exactly equal to the missing count ratios. This cannot be repaired by unit conventions.

13. Status of the common-anchor attack

The first attempted action-level derivation has therefore failed in a precise way.

It did not fail because rank-one coupling is useless. Historical V2 shows why rank one is a serious clue. It failed because the current exact simplex theorem fixes where the count enters, and a primitive rank-one response then places that count on the wrong side of the proposed fixed point.

The surviving proof target is now narrower:

Derive a fixed-point equation for the complete occupied reference eigenvalue $M_d^\star$, or derive the fixed sector normalization of the primitive coefficient (for example $1/D_d^\star$) from the master projection.

Any future derivation that merely writes

$$\mu_d\propto\sqrt{g_{dd}}$$

has already been ruled out for the present V2 mass operator.

Related

• Research status — current proof boundaries and correction guidance.