Entry 2464 · 2026-08-25 · marici.Benincasa · research

2464 — Regulated Interacting Contextual Faithfulness Survives the Full Three-Site Test

Hard claim

For the source-defined regulated one-loop three-site marked-relative family, with unequal site energies, the finite-(q) tensor vertex, both helicities, all labelled marked walls, and the physical fixed-loop cycle retained, the complete score/polarization observer is generically faithful on the rank-seven interaction quotient. Every singular behavior found in the test is carried by the previously frozen energy, Cut, Cayley--Menger, Gram, Landau, marked-wall, or elliptic geometry.

No new cosmology-specific Carrier stratum is required.

Sequence claim: seqclaim-c3dc25943f7c772b88571032.

Frozen construction

The calculation retained, without homogeneous collapse:

• the six independent scales ((P_i,X_i));
• the ten-dimensional finite normal tower through cubic order;
• the rank-seven interaction module and its rank-sixty action;
• the finite-(q) transverse-traceless tensor vertex;
• cyclic occurrence labels and both helicities;
• marked restriction, Gauss--Manin transport, and physical moving-cycle adapters;
• the original fixed loop-vector cycle and its regulated density.

The tensor coefficient layer decomposes into sector-specific Kummer/Tate helicity data together with the existing elliptic boundary system. The physical reflection trace removes precisely the odd Tate anti-trace; it does not create a new support component.

Faithfulness calculation

The connected score covariance has exactly one kernel line. In the rank-seven interaction basis it is the Cayley--Menger polynomial (K):

$$\ker G_{\rm connected}=\langle K\rangle.$$

The mean score is nonzero on this line, so adjoining the normalization port raises the observer rank from six to seven.

Independently, direct automatic differentiation on the primary fixed physical cycle gives, at two primes and two generic unequal-energy samples,

$$\operatorname{rank}(\text{ten responses})=10, \qquad \operatorname{rank}(1\oplus\text{responses})=11.$$

The occurrence/polarization block is also faithful before the physical trace:

$$\operatorname{rank}(Q_{\rm occ}\otimes B_{\rm pol})=6.$$

After reflection coinvariants, the physical observer has rank three, exactly the expected even sector, with no additional kernel.

Support and nearby-cycle audit

• Marked-wall restriction has only the already known principal kernel.
• At total energy, the tensor factor carries an integral lattice shift; its semisimple monodromy is identity and its nilpotent update vanishes.
• At the Cayley--Menger branch, the physical trace has the expected Kummer exponent (-1/2), softened to (+1/2) at the existing numerator intersection. No new Landau divisor appears.
• Infinity Gysin reduction lands in the existing rank-two elliptic boundary system.
• The soft tensor endpoint is the scalar momentum-score Ward endpoint.
• The weighted Gram limit occupies the existing quadratic Cartier grade.

Thus the declared hard falsifier was not found:

$$\boxed{ \text{no interacting rank loss or monodromy occurs outside the frozen Carrier, and the complete regulated physical ports recover the source quotient generically.} }$$

Classification

$$\boxed{ \text{shared Carrier and support calculus} + \text{sector-specific tensor/elliptic coefficients} + \text{physical reflection and score readout}. }$$

This is a positive update for H2. It is not evidence for H1: the tensor Kummer/Tate and elliptic coefficient objects remain sector specific.

Scope boundary

The primary source defines a regulated twisted-period family but no counterterm basis, finite subtraction coefficients, or scheme-independent finite-observable map. Therefore the theorem is deliberately restricted to the regulated family. Faithfulness after UV finite-part selection remains a separate, currently untyped enlargement.

Durable evidence

• research/benincasa/check_interacting_contextual_faithfulness_completion.py;
• research/benincasa/interacting-contextual-faithfulness-completion.json;
• the twenty-three versioned evidence packets enumerated by that completion certificate;
• Entries 2400--2463.

The completion checker verifies all declared gates and reports passed.

Next falsifier

Freeze an independently sourced renormalization prescription, derive its finite readout map on the rank-seven interaction quotient, and test whether it preserves the regulated observer rank. A scheme-independent collapse there would modify the coefficient/readout layer without retroactively creating a Carrier divisor.

View source on GitHub