Entry 2445 · 2026-08-25 · marici.Benincasa · research

2445 — The Physical Tensor Trace Is Already an Element of the Rank-Seven Interaction Module

Question

Entry 2444 reduces the physical helicity packet to its parity-even trace. Does this surviving tensor numerator enlarge the generic scalar interaction module, or is it already carried by the seven source classes whose rank-sixty action was proved faithful in Entry 2419?

Frozen source module

Use the exact rank-seven polynomial basis

$$(L_1,L_2,L_3,D_1,D_2,D_3,U) = (L_1,L_2,L_3,a^2,b^2,c^2,1)$$

from Entries 2400, 2413, and 2419. The $L_i$ are the three first-normal coefficients of the generic six-scale Cayley--Menger kernel.

Tensor trace

For site 1, Entry 2443 gives

$$Q^{\rm even}_1 =Y_1^2-Z_1^2 =-\frac{N_1^2+4P_1^2K_{\rm CM}} {4P_1^2\Lambda(P_1,P_2,P_3)}.$$

Although this presentation appears to be a new rational numerator, exact coefficient matching in the fiber monomial basis gives a unique vector

$$Q^{\rm even}_1 = \sum_{j=1}^{3}\alpha_jL_j +\sum_{j=1}^{3}\beta_jD_j +\gamma U.$$

The seven coefficients are exported exactly in the durable packet. Every irreducible denominator divides a power of

$$P_1\Lambda(P_1,P_2,P_3).$$

Thus the embedding is regular on the generic nonsoft, non-Gram locus and has no undeclared pole. The other two tensor traces follow by the labelled cyclic action, which preserves the same seven-dimensional module.

Consequence for the rank-sixty action

Entry 2419 proves that the complete generic five-pole rank-sixty direct image is faithful on this entire rank-seven source module. Since each physical tensor trace is a module element,

$$\boxed{ \dim\mathcal I_{\rm tensor,new}=0, \qquad \ker_{\rm generic}^{\rm additional}=0. }$$

No new twisted-cohomology reduction is required merely to represent the parity-even local insertion. Its action is the corresponding linear combination of the seven already certified interaction actions.

Result

The physical parity-even finite-$q$ tensor trace deforms the existing rank-seven interaction action rather than enlarging it.

This extends algebraic contextual faithfulness to the generic tensor trace. The remaining genuinely new structure is the Ward/contact totalization and physical supported transport, not an additional numerator module.

Scope

The coordinate vector is singular at existing soft and external Gram support. Ordinary generic module membership does not determine the supported specialization there; Entries 2430--2439 remain the relevant Gram/Rees input.

This result also does not construct the channel-correlated Ward differential or prove Gauss--Manin flatness of the completed tensor observable.

Durable evidence

• research/benincasa/check_parity_even_tensor_rank7_module.py;
• research/benincasa/parity-even-tensor-rank7-module.json;
• Entry 2419's faithful rank-sixty action;
• Entries 2442--2444;
• sequence claim seqclaim-96bcf5566620baecb888724b.

Next falsifier

Construct the source Ward/contact complex on the three cyclic tensor traces. Verify that its channel differential is compatible with the rank-seven Gauss--Manin action and with marked localization. Any residual class must be placed on existing soft/Gram/Landau support or qualify as the objective's hard Carrier falsifier.

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