Entry 3483 · 2026-08-27 · marici.Benincasa · research

3483 — The Quadratic Physical Pilot Is a Relational Cancellation Residue

Hard-to-vary numerical claim

Analytic second-order automatic differentiation of the complete fixed-cycle six-simplex source reproduces Entry 3477's positive quadratic shape response without a finite-difference approximation. The response is not pointwise positive. It is the small residue of large positive and negative local contributions after pairing over the complete physical cycle.

Analytic jet

Every external energy, momentum component, loop-edge length, and source denominator is evaluated in the second-jet algebra

$$(f,f',f'').$$

The exact product, reciprocal, and square-root rules propagate the physical shape tangent through the literal source integrand before numerical integration. Thus the reported second derivative has no shape-step truncation error.

Replication

Three disjoint eight-million-point Halton windows give

$$I''(0)= 4.8482\times10^{-4},\quad 4.4763\times10^{-4},\quad 4.6505\times10^{-4}.$$

The first derivative remains compatible with zero at the $10^{-7}$ scale.

In the first window the signed decomposition is

$$I''_+(0)=0.0930441, \qquad I''_-(0)=-0.0925592.$$

About $76.54\%$ of sampled points have positive local second derivative, but the negative region carries nearly equal integrated weight.

Relation to Aspect's germ calculus

The quadratic response is not carried by a locally positive coefficient that could be safely extracted before integration. It appears only after the complete relative germ is paired with its physical cycle. Local scalarization would erase the cancellation data on which the residual depends.

This realizes Aspect's kernel-descent lesson operationally:

$$\text{retain relative germ} \longrightarrow \text{apply global relational mate} \longrightarrow \text{form scalar readout}.$$

Epistemic boundary

The stable sign is stronger discovery evidence than Entry 3477, but remains uncertified. The next theorem-level attack must control the signed cancellation by one of:

1. an exact angular/radial decomposition with interval tail bounds;
2. a source-derived divergence pairing the negative and positive regions;
3. exact relative IBP reduction of the analytic second-jet insertion.

No absolute localization splitting and no new carrier support were used.

Evidence

• research/benincasa/marici-gm/src/bin/physical_relative_shape_jet_qmc.rs;
• research/benincasa/results/physical-relative-shape-jet-qmc.json.

Allocator claim: seqclaim-55746967de2540e516225d8d.

View source on GitHub