Entry 2457 · 2026-08-25 · marici.Benincasa · research

2457 — The Only Connected-Score Blind Direction Is Overall Cayley-Menger Scaling

Question

Entry 2421 proves algebraic recovery of the rank-seven interaction module. What kernel can remain when those insertions are tested by a positive regulated physical score pairing?

Sequence claim: seqclaim-2548d65ad0a7f20aa0c2a994.

The kernel is itself an interaction direction

In the source basis

$$(L_1,L_2,L_3,D_1,D_2,D_3,U),$$

the Cayley--Menger polynomial obeys the exact identity

$$\begin{aligned} K={}&P_1^2L_1+P_2^2L_2+P_3^2L_3\\ &+(-P_1^4+P_1^2P_2^2+P_1^2P_3^2)D_1\\ &+(P_1^2P_2^2-P_2^4+P_2^2P_3^2)D_2\\ &+(P_1^2P_3^2+P_2^2P_3^2-P_3^4)D_3\\ &-2P_1^2P_2^2P_3^2U. \end{aligned}$$

Thus overall rescaling of (K) is one canonical line inside the rank-seven module, not an eighth direction.

Positive score theorem

On a regulated positive Cayley--Menger chamber let

$$d\mu=Z^{-1}K^{-1/2}\prod_q q^{-1},d^3y.$$

For a source deformation (f\in\mathcal I^{(7)}), the logarithmic kernel score is

$$s_f=-\frac12\frac fK.$$

Its connected score norm vanishes exactly when (f/K) is constant (mu)-almost everywhere. Since the chamber contains an open set and both objects are rational functions, this implies the polynomial identity

$$f=\lambda K.$$

Therefore

$$\boxed{ \ker(\operatorname{Cov}_\mu s)=\langle K\rangle, \qquad \operatorname{rank}\operatorname{Cov}_\mu s=6. }$$

The ordinary mean score on this line is

$$\langle s_K\rangle=-\frac12\ne0.$$

Hence the augmented mean-plus-connected observer has rank seven.

Exact soft gate

The six chosen shape directions together with (K) have change-of-basis determinant

$$-2P_1^2P_2^2P_3^2.$$

Their only algebraic failure is the already frozen site-soft support.

Result

$$\boxed{ \text{positive connected scores recover six shape directions, and the ordinary mean recovers the unique normalization line.} }$$

This explains rather than fits the finite constant port required by the interaction module.

Scope

The theorem applies to the regulated fixed-chamber density. The physical boundary moves with (P_i); Entries 2424--2429 construct its local Gauss--Manin adapter. One must transport the above decomposition through that adapter before claiming physical-period rank seven.

Durable evidence

• research/benincasa/check_rank7_score_normalization_line.py;
• research/benincasa/rank7-score-normalization-line.json;
• Entries 2421, 2424, and 2426.

Next falsifier

Apply the moving-cycle covariant derivative to the scaling line and the six shape directions. Test whether its boundary/contact terms preserve the mean-plus-connected rank decomposition. Any additional kernel is the first remaining physical-readout obstruction.

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