Entry 2491 · 2026-08-25 · marici.Benincasa · research

2491 — The Fixed-Loop Score Gram Is Not Yet a Source-Defined Physical Observer

Correction

Entry 2461 called the positive fixed-loop score Gram a physically faithful observer. Its algebraic rank statement is correct, but the physical interpretation exceeded the frozen source.

Sequence claim: seqclaim-9cc721c446ae051f2f2991ad.

What the checker proves

Let

$$\rho(\ell;\nu) =\frac1{q_{g_1}q_{g_2}q_{g_3}q_{G_{23}}}$$

and let $s_i(\ell)$ be its ten normalized normal-score functions. Exact finite-field evaluation proves

$$\operatorname{rank}\{s_i\}=10, \qquad \operatorname{rank}\{1,s_i\}=11.$$

On a positive regulated chamber, this gives a nondegenerate mathematical Gram form

$$G_{ij}=\int_{\Gamma_\ell} \rho(\ell)\,s_i(\ell)s_j(\ell)\,d^3\ell.$$

This formal/contextual theorem remains valid.

Missing source map

The primary source arXiv:2408.16386v2 defines the linear twisted period

$$I_{\mathcal G}(\omega)=\int_{\Gamma_\ell}\omega.$$

It does not define:

• the loop integration coordinate $\ell$ as a measured random variable;
• repeated samples from the normalized density $\rho$;
• the quadratic score-Gram pairing $G_{ij}$ as a cosmological observable;
• an instrument mapping boundary-field data to the ten latent score functions.

Therefore pointwise score independence plus positivity does not by itself construct a physical observer. It constructs a faithful Hilbert-space pairing after an additional statistical readout has been declared.

Corrected classification

$$\boxed{ \text{formal/contextual score faithfulness: proved;} \qquad \text{physical period-readout faithfulness: unproved.} }$$

Entries 2470–2479 correctly classify finiteness and local counterterms for the declared graph and score family. They do not supply the missing instrument. Entry 2489 independently shows that ordinary homogeneous momentum pullback kills the positive normal tower.

No new Carrier support is indicated. The missing object is a source-defined readout/pairing map.

Durable evidence

• research/benincasa/check_score_gram_source_authority.py;
• research/benincasa/score-gram-source-authority.json;
• research/benincasa/check_fixed_loop_physical_score_rank.py;
• research/benincasa/fixed-loop-physical-score-rank.json;
• research/benincasa/interacting-regulated-period-scope.json;
• Entries 2461, 2463, and 2489.
• epistemic event ev-000000003424-67ec6739-aa81-404b-8797-e13de36401df.

Next falsifier

Use only the source-defined linear period and its external derivatives. Compute the map from the seven interaction classes to the finite jet of

$$I_{\mathcal G}(X,P)$$

along independently admissible external/source controls. If that period-jet map has rank seven, physical contextual faithfulness is recovered without observing the latent loop coordinate. If not, its kernel is the actual readout loss.

View source on GitHub