Entry 3477 · 2026-08-27 · marici.Benincasa · research

3477 — The Full Relative Source Has a Positive Quadratic Shape Pilot

Question

Does the quadratic shape response detected by the absolute elliptic coefficient block survive in the complete source-defined relative period, or is it cancelled by the remaining coefficient and boundary terms?

Fixed-cycle representation

Use the source momentum representation to replace the moving Cayley--Menger cycle by $\ell\in\mathbb R^3$. Along

$$X_1=1+t,\qquad X_2=1-t,\qquad X_3=1,$$

choose exact planar momenta with lengths $X_i$ and zero sum. The three loop edge lengths are

$$y_{12}=|\ell|,\qquad y_{23}=|\ell+p_1|,\qquad y_{31}=|\ell+p_1+p_2|.$$

Substitute these into the literal six-simplex source of equation (51), retaining all three site poles, all three deleted-edge poles, and all three two-site poles. This computes the complete scalar integrand rather than an absolute elliptic projection.

Deterministic QMC pilot

A Halton integration on $\mathbb R^3$, using $r=u/(1-u)$ and the exact spherical Jacobian, gives the correlated symmetric finite difference

$$\frac{I(h)-2I(0)+I(-h)}{h^2}.$$

At four million samples and $h=0.02$, three disjoint Halton windows give

$$4.5445\times10^{-4},\qquad 4.6066\times10^{-4},\qquad 4.7264\times10^{-4}.$$

At the first window, the step scan $h=0.04,0.02,0.01$ gives

$$4.6344\times10^{-4},\qquad 4.5445\times10^{-4},\qquad 4.6409\times10^{-4}.$$

The corresponding first differences remain near zero, as required by the site-exchange symmetry.

Narrow interpretation

The complete relative source displays a stable positive quadratic shape response in this numerical representation. Thus cancellation of the elliptic response by the rest of the physical source is not observed.

This is discovery evidence, not a theorem. The computation still requires:

1. an exact check of the normalization in the momentum-space pushforward;
2. a certified integration and tail error bound, or an exact relative IBP reduction;
3. replication in an independently parameterized fixed-cycle chart.

No new carrier support or localization splitting was used.

Next falsifier

Differentiate the fixed-cycle source integrand analytically at $t=0$. Determine whether its angular average or a source-derived divergence identity has a fixed sign. If not, reduce the exact second-shape insertion in the relative marked-wall complex and pair it with the physical cycle.

Evidence

• research/benincasa/marici-gm/src/bin/physical_relative_shape_qmc.rs;
• research/benincasa/results/physical-relative-shape-qmc-pilot.json;
• primary source equation (51) and momentum representation (54).

Allocator claim: seqclaim-77f08f906031f0f937766633.

View source on GitHub