Entry 2413 · 2026-08-25 · marici.Benincasa · research

2413 — The Generic Lower Direct Image Is Faithful on the Rank-Seven Interaction Module

Question

Entry 2400 freezes ten nonconstant labelled coefficients in the generic six-scale Cayley–Menger normal tower: three linear, six quadratic, and one cubic. The first direct-image test is whether these ten labels remain independent in the generic rank-thirty-four four-wall twisted de Rham quotient.

Sequence claim: seqclaim-778da6c52b11f20914ba165b.

The source module already has three relations

Write

$$D_1=a^2, \qquad D_2=b^2, \qquad D_3=c^2$$

for the diagonal quadratic coefficients, and

$$\begin{aligned} C_{12}&=P_3^2-a^2-b^2,\\ C_{13}&=P_2^2-a^2-c^2,\\ C_{23}&=P_1^2-b^2-c^2 \end{aligned}$$

for the square-free coefficients. Let $U=1$ denote the coefficient of $\nu_1\nu_2\nu_3$.

Before any cohomological reduction, exact polynomial identities give

$$\boxed{ P_3^2(D_1+D_3+C_{13}) -P_2^2(D_1+D_2+C_{12})=0, }$$

$$\boxed{ P_3^2(D_2+D_3+C_{23}) -P_1^2(D_1+D_2+C_{12})=0, }$$

and

$$\boxed{ P_3^2U-(D_1+D_2+C_{12})=0. }$$

These are simply the three presentations of the constant coefficient:

$$D_1+D_2+C_{12}=P_3^2,$$

and its cyclic images. Together with the three linear coefficients, the generic source interaction module has rank

$$\boxed{3+4=7.}$$

Thus a raw ten-to-seven collapse is not automatically information loss. Faithfulness must be tested against this source quotient, not against the free ten-label presentation.

Source-normalized quotient computation

The existing generic lower reducer was extended without changing its divisors, exponent convention, monomial order, or kinematic samples. In the rank-thirty-four quotient, $K^{-1}$ is represented by the already frozen localization coordinate

$$K^{-1}=z\prod_{i=1}^4L_i.$$

Each of the ten normal coefficients is multiplied by this common representative and reduced in one shared Groebner basis. No primitive section or post-hoc quotient is selected.

Two independent runs give

$$\begin{array}{c|c|c|c} \text{point}&\text{prime}&\operatorname{rank}H_{\rm low} &\operatorname{rank}\langle K_\alpha/K\rangle\\ \hline A&32003&34&7\\ B&32009&34&7. \end{array}$$

The three computed modular kernel vectors agree coefficientwise with the three source identities above after normalizing respectively the $C_{13}$, $C_{23}$, and $U$ coordinates to one.

Generic theorem

The source identities give a universal upper bound seven. A rank-seven minor at one exact finite-field specialization proves that the generic rank is at least seven. Hence the generic lower direct-image class rank is exactly seven. Because the modular kernel is exactly the specialization of the source relation module, there is no additional direct-image kernel:

$$\boxed{ \ker(\mathcal I_{\rm source}\to H_{\rm low}) =\langle\text{the three exact source relations}\rangle. }$$

Equivalently, on the correctly typed source quotient,

$$\boxed{ \mathcal I_{\rm source}^{(7)} \hookrightarrow H_{\rm low}^{(34)} }$$

is generically injective.

Interpretation

The apparent loss of three labels is another instance of the program's faithful-coordinate rule: ten presentation coordinates describe a seven-dimensional source object. The direct image forgets nothing beyond the relations already true in the source kernel.

This is a nonhomogeneous scalar contextual-faithfulness theorem at the algebraic coefficient level. It does not yet prove physical observability of all seven classes.

Classification

• raw interaction labels: ten;
• exact source relations: three;
• faithful source interaction rank: seven;
• generic lower cohomology rank: thirty-four;
• interaction class rank in lower cohomology: seven;
• additional cohomological kernel: zero;
• new Carrier support: none.

Scope

The calculation covers the generic four-wall lower twisted Jacobian quotient. It does not yet include the rank-twenty-six $q_{\mathcal G_{12}}$-restricted summand, the full rank-sixty connection, the physical relative cycle, or tensor polarization ports.

Durable evidence

• research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs with NORMAL_TOWER=1;
• research/benincasa/check_generic_lower_interaction_class_faithfulness.py;
• research/benincasa/generic-lower-interaction-class-faithfulness.json;
• independent runs at points A/B and primes 32003/32009.

Next falsifier

Repeat the same source-quotient test on the rank-twenty-six restricted summand at $q_{\mathcal G_{12}}=0$, then verify compatibility of the two rank-seven images through the deletion–restriction boundary map. A kernel appearing only after that gluing is an extension/readout obstruction; a class lost separately in the restricted quotient is a coefficient-level specialization obstruction.

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