Audit of existing V2 candidates for the dimensionless lock coefficient

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: Research record, section 59: Audit of existing V2 candidates for the dimensionless lock coefficient.

Provenance

Preserved from interaction notebook section 59, recorded 2026-09-05 and corrected 2026-09-06. Original sequence contents. This is an exploratory research record, not an independently established V2 result. Read the related later audits before using an intermediate conclusion.

59. Audit of existing V2 candidates for the dimensionless lock coefficient

The remaining unknown after sections 57--58 is the dimensionless interaction coefficient

$$\widehat\kappa, \qquad \frac{\omega_{\rm normal}}{\omega_e}=\sqrt{8\widehat\kappa}$$

under the canonical local sigma-model reduction. Several already existing V2 quantities were tested as possible sources of $\widehat\kappa$.

59.1 Candidate: the depth-six diagonal coupling $g_{66}=1/4$

Algebraically one could set

$$\widehat\kappa=g_{66}=\frac14,$$

which would give

$$\frac{\omega_{\rm normal}}{\omega_e}=\sqrt2.$$

Nothing is mathematically inconsistent about that number. However, the written V2 modules do not contain an operator identity equating the frame-lock Hessian with the diagonal charged-lepton coupling $g_{66}$.

The fixed-point relation involving $g_{66}$ concerns sector reference frequencies/masses. It does not by itself identify $g_{66}$ with the coefficient of a cross-depth orientation potential.

Therefore

$$\boxed{ \widehat\kappa=g_{66} }$$

is presently an unsupported identification, not a derivation.

59.2 Candidate: the electron contact eigenvalue $S(13,6)$

The exact scalar contact theorem gives

$$S(13,6)=\binom{18}{6}=18564.$$

This is a genuine eigenvalue, but of the finite-band occupied-ray scalar self-contact operator.

The frame lock is instead a cross-depth orientation problem. No theorem in the current modules maps the scalar occupied-ray eigenvalue into the Hessian of the $3+3$ frame-link potential.

Using

$$\widehat\kappa=S(13,6)$$

would therefore double-purpose an unrelated eigenvalue without an operator map. It is rejected as unjustified.

59.3 Candidate: invariant-vertex multiplicity

The higher-dimensional-polarization module defines the allowed local vertices through an invariant space

$$\mathcal V_{abc} = \operatorname{Inv} \left(F_a\otimes R_b^*\otimes R_c\right).$$

If this space is one-dimensional, then the tensor structure of the allowed vertex is unique up to scale.

But one-dimensionality does not fix the coefficient multiplying that tensor. Multiplying the unique invariant tensor by any real constant leaves it in the same one-dimensional invariant space.

Therefore multiplicity one can prove

$$\boxed{ \text{unique interaction channel} }$$

but not

$$\boxed{ \text{unique interaction strength}. }$$

59.4 Candidate: the mixed spin-connection normalization

The higher-dimensional spin connection contains the mixed/extrinsic term

$$\frac12\omega_\mu^{\alpha i}\Gamma_{\alpha i}.$$

For distinct Euclidean Clifford directions, the generator $\Gamma_{\alpha i}$ has fixed algebraic norm. Consequently the explicit factor $1/2$ fixes a relative quadratic normalization proportional to

$$\frac14|\omega_{\rm mix}|^2.$$

This is promising because it shows that once a mixed connection is derived, its contribution to a quadratic orientation energy has a fixed Clifford normalization.

However, rescaling

$$\omega_{\rm mix}\mapsto c\,\omega_{\rm mix}$$

leaves the representation content and Clifford algebra unchanged while changing the quadratic strength by

$$c^2.$$

Thus Clifford normalization fixes the coefficient relative to the mixed-connection magnitude, but does not derive that magnitude.

59.5 Candidate: normalized Procrustes curvature

The unit frame-mismatch functional

$$3-\operatorname{tr}(QQ_{\rm scat}^T)$$

has isotropic tangent Hessian

$$2I_3$$

at its minimum.

This is an exact geometric normalization. It fixes the shape and degeneracy of the three lock modes. But the physical interaction is

$$\widehat\kappa \left[3-\operatorname{tr}(QQ_{\rm scat}^T)\right],$$

so the overall spectral coefficient remains $\widehat\kappa$.

59.6 Result of the coefficient audit

The tested existing V2 quantities separate cleanly into two classes:

1. quantities that fix geometry or representation structure:
   • the unique invariant channel, when its multiplicity is one;
   • the Clifford factor $1/2$ in the mixed spin connection;
   • the unit Procrustes Hessian;
   • the scattering-selected frame $Q_{\rm scat}$;
2. quantities that are genuine eigenvalues but belong to a different operator:
   • $g_{66}$ in the sector fixed-point relation;
   • $S(13,6)$ in the scalar occupied-ray self-contact operator.

None of these presently supplies an operator theorem fixing

$$\boxed{\widehat\kappa}.$$

The closest route is the mixed/extrinsic connection. If the projected master geometry derives

$$\omega_\mu^{\alpha i}$$

from the scattering state, then Clifford normalization and the Procrustes geometry together would determine the resulting frame-lock Hessian with no additional arbitrary tensor structure.

The remaining unknown would then be reduced to the norm/eigenvalue of a concrete mixed-connection block rather than an abstract phenomenological constant.

59.7 Sharpened next target

The next calculation should therefore not guess a numerical value for $\widehat\kappa$. It should construct the mixed/extrinsic connection induced by the scattering-selected split and ask whether its normalized curvature has a dimensionless eigenvalue fixed by the existing sector geometry.

Schematically the desired closure is

$$\boxed{ (p,b) \to Q_{\rm scat} \to \omega^{\alpha i}_{\rm mix} \to F_{\rm mix} \to \widehat\kappa \to \omega_{\rm lock}/\omega_e. }$$

At this stage the orientation problem is solved, while the coefficient problem has been reduced to a specific mixed-connection spectral calculation.

Related

• Research status — current proof boundaries and correction guidance.
• Frequency normalization requires the correct lock coordinate and kinetic metric — preceding section in the research sequence.
• Maurer--Cartan test: the scattering frame alone cannot generate its own restoring curvature — following section in the research sequence.