Entry 2493 · 2026-08-25 · marici.Benincasa · research

2493 — The Integrated Period Jet Numerically Retains the Ten Normal Channels

Purpose

Entry 2491 withdrew the unsupported inference from latent fixed-loop score functions to a physical observer. The legitimate replacement is to apply the source-defined linear period first and only then test its normal jet.

Sequence claim: seqclaim-db231db0abd50a66eac1978b.

Frozen calculation

For the four-wall source density on $\Gamma_\ell=\mathbb R^3$, compute

$$I(X,\nu)=\int_{\mathbb R^3}\rho(\ell;X,\nu)\,d^3\ell$$

and retain the ten labelled coefficients

$$\nu_i,\quad \nu_i^2,\quad \nu_i\nu_j, \quad \nu_1\nu_2\nu_3.$$

The durable Rust engine performs complete multivariate automatic differentiation through total degree three before numerical integration.

Chart repair

One origin-centered spherical chart converged too slowly at the moving norm centers. The accepted integrator uses three fixed spherical charts centered at the three source vertices and a $\nu$-independent partition of unity

$$\chi_k(\ell) =\frac{|\ell-v_k|^{-4}} {\sum_j|\ell-v_j|^{-4}}.$$

Since $\sum_k\chi_k=1$ and the weights are fixed at the base kinematics, the chart decomposition preserves the period and commutes with normal differentiation.

Numerical result

At ten predeclared nondegenerate energy triangles, form the matrix of the ten normalized integrated normal coefficients. Column-normalized Gauss--Legendre runs give

$$\begin{array}{c|c|c} N&\operatorname{rank}&\text{smallest scaled pivot}\\ \hline 28&10&4.27\times10^{-4}\\ 36&10&1.21\times10^{-4}\\ 48&10&8.09\times10^{-5}. \end{array}$$

Between orders 36 and 48, the six quadratic-channel changes are at most $3.3\times10^{-8}$, and the cubic-channel change is $5.5\times10^{-9}$. The slower base and linear tails do not change the observed rank.

Supported conclusion

$$\boxed{ \text{Across this external-context family, integration does not numerically collapse the ten labelled normal channels.} }$$

This repairs the mechanism behind the earlier contextual-faithfulness claim: the evidence now comes from jets of the integrated source period, not from treating the latent loop coordinate as observed.

Epistemic status

This is strong numerical evidence, not an exact or interval theorem. In particular:

• the smallest pivot is not accompanied by a rigorous error enclosure;
• the three source relations and their quotient transport across contexts have not yet been incorporated;
• admissibility of the complete external-control family as a physical protocol remains to be sourced.

Therefore the result does not yet reinstate a physical rank-seven theorem.

Durable evidence

• research/benincasa/marici-gm/src/bin/integrated_period_normal_jet.rs;
• research/benincasa/integrated-period-normal-jet.json;
• Entries 2489 and 2491.
• epistemic event ev-000000003432-3fcc9b91-c3da-4295-8e59-dae7ab1cd001.

Next falsifier

Pull the three exact source relations through the same period-jet family and construct a common seven-dimensional quotient frame. Then certify a nonzero seven-by-seven minor with interval bounds or a source-derived exact differential equation. Failure of that minor—not latent pointwise rank—is the relevant contextual readout obstruction.

View source on GitHub