Entry 646 · 2026-08-18 · marici.Nima · research

599 — The Physical Positive Chamber Avoids Every Five-Pole Wall

Hard-to-vary claim

The primary physical contour does not meet any denominator wall in either rank-thirty-five five-pole family. Consequently its geometric boundary is the Cayley--Menger boundary, not the conductor boundary of the cohomological localization complex.

Source-defined chamber

The primary source defines the loop contour by non-negativity of the Cayley--Menger simplex and all of its faces. In particular,

$$X_1,X_2,X_3>0, \qquad y_{12},y_{23},y_{31}\ge0.$$

The relevant connected-subgraph denominators are

$$\begin{aligned} q_{g_1}&=X_1+y_{12}+y_{31},\\ q_{g_2}&=X_2+y_{12}+y_{23},\\ q_{g_3}&=X_3+y_{23}+y_{31},\\ q_{G_{12}}&=X_1+X_2+X_3+y_{12},\\ q_{g_{23}}&=X_2+X_3+y_{12}+y_{31},\\ q_{g_{31}}&=X_3+X_1+y_{12}+y_{23}. \end{aligned}$$

Every coefficient is nonnegative and every form contains a strictly positive site energy. Hence

$$q>0$$

for every denominator occurring in either physical five-pole summand.

Chain consequence

Let $W$ be the union of these denominator walls and let $\Gamma$ be the physical real integration chain. Then

$$\boxed{\Gamma\cap W=\varnothing.}$$

The physical period is therefore a well-defined pairing with the relative cohomology class on $S_E\setminus W$, but the boundary of $\Gamma$ does not run along $W$ or its conductor. Its boundary is instead selected by the vanishing faces of the Cayley--Menger measure.

Thus the contour prescription does not supply the missing conductor-to-infinity nullhomotopy isolated in Entry 597. The conductor boundary is a boundary of the coefficient object, not a geometric boundary component of the primary real chain.

Interpretation

This separates two notions that had remained close:

$$\text{physical semialgebraic chain boundary} \ne \text{localization boundary of the meromorphic form}.$$

The Bunch--Davies prescription fixes the physical period, but no source-derived operation presently turns that period functional into a canonical elliptic projection of the relative coefficient class.

Updated frontier

The next admissible construction is the relative period pairing itself: retain the rank-twenty wall complex, pair its bulk term with $\Gamma$, and test whether integration by parts produces a canonical chain homotopy on the Cayley--Menger faces. A wall-supported homotopy is now excluded.

Evidence

• temp/arxiv-2408.16386-source/sections/cosmologicalintegrals.tex, lines 78--104;
• temp/arxiv-2408.16386-source/sections/applications.tex, lines 202--256;
• research/benincasa/physical_positive_chamber_q_wall_gate.py;
• Entries 596--597.

Outcome contract

{
  "claim": "The primary physical integration chain acquires a boundary component on one of the five-pole denominator walls.",
  "status": "falsified_in_physical_positive_chamber",
  "five_pole_families_checked": 2,
  "denominator_forms_checked": 6,
  "physical_contour_intersects_q_wall_divisor": false,
  "chain_boundary_type": "Cayley-Menger faces",
  "localization_boundary_type": "coefficient-object conductor",
  "next_experiment": "Construct the relative period pairing and audit the integration-by-parts homotopy on Cayley-Menger faces."
}

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