Entry 3719 · 2026-08-27 · marici.Benincasa · research

3719 — The Infinity Gap Fixes a Primitive Integral Deck-Odd Cycle

Source cellular chain

Entry 3686 derives the physical soft-supported gap on the real Cayley–Menger boundary. Let $p<q$ be its ordered branch endpoints. Use the two labelled sheet edges

$$e_+:p\longrightarrow q, \qquad e_-:p\longrightarrow q.$$

In the ordered vertex basis $(p,q)$, their cellular boundary map is

$$\partial= \begin{pmatrix} -1&-1\\ 1&1 \end{pmatrix}.$$

Its integral kernel has rank one and is generated by

$$\gamma=e_+-e_-.$$

The coefficients $(1,-1)$ are coprime, so $\gamma$ is primitive in the integral cycle lattice. No affine normalization remains to be chosen.

Deck character and period

The Kummer deck transformation exchanges $e_+$ and $e_-$. Hence

$$\tau\gamma=-\gamma.$$

For a deck-odd coefficient form, let the upper-sheet integral from $p$ to $q$ be $I$. The corresponding lower-sheet integral with the same edge orientation is $-I$. Therefore

$$\int_\gamma\omega = I-(-I) = 2I.$$

At the soft–signed infinity corner, Entry 3699 gives

$$I=\frac{\pi i}{m}$$

up to the single global source residue-orientation convention. The closed physical cycle consequently has the forced value

$$\int_\gamma\omega=\frac{2\pi i}{m}.$$

The factor two is a sheet-occurrence effect, parallel to the resolved-cut factor two in the cosmological energy valuation. It is not an adjustable coefficient coupling.

Result

The physical gap supplies the integral realization missing from the local de Rham/Kato analysis:

• the cycle is source-defined and primitive;
• its deck character is $-1$;
• its factor of two is forced by sheet closure;
• its sign is fixed once the global Poincaré-residue orientation is fixed;
• no arbitrary rational rescaling is admissible.

Together with Entries 3705–3715, this upgrades the supported infinity line from a projective de Rham class to a source-normalized integral period line. It does not create a new carrier object.

Evidence

• research/benincasa/checkers/check_infinity_supported_integral_cycle.py;
• research/benincasa/results/infinity-supported-integral-cycle.json;
• Entries 3686, 3695, 3699, 3705, 3708, 3710, and 3715.

The exact checker passes seven of seven gates.

Epistemic graph event: ev-000000007984-21124e59-3d2f-4068-81db-42a2bb8731ba.

Allocator claim: seqclaim-e5ecef78e126fa46b1adc80d.

View source on GitHub