Entry 3724 · 2026-08-27 · marici.Benincasa · research

3724 — The Physical Infinity Cycle Defines a Canonical Leray Covector on the Source Masters

Inputs

Entries 3678–3722 derive a source-normalized physical infinity cycle whose period covector on the elliptic basis $(\omega_0,\omega_2)$ is

$$\ell_{\rm phys}=\begin{pmatrix}1&1\end{pmatrix},$$

up to the single global Poincaré-residue orientation.

The explicit infinity-Gysin matrix on the final three simple-pole masters is

$$R_\infty= \begin{pmatrix} 1&\dfrac{E^2+y^2}{2}&\dfrac{E^2+x^2}{2}\\[2mm] 0&-\dfrac{E^2+x^2}{2}& -\dfrac{x^2(E^2+y^2)}{2y^2} \end{pmatrix}$$

in the source basis $(e_7,e_8,e_9)$.

Canonical composition

Composing the physical period with the Gysin map gives

$$\ell_{\rm src}=\ell_{\rm phys}R_\infty = \begin{pmatrix} 1& \dfrac{y^2-x^2}{2}& \dfrac{E^2(y^2-x^2)}{2y^2} \end{pmatrix}.$$

Including the double-pole master, the final-block covector is

$$\begin{pmatrix} 0& 1& \dfrac{y^2-x^2}{2}& \dfrac{E^2(y^2-x^2)}{2y^2} \end{pmatrix}$$

on $(e_6,e_7,e_8,e_9)$.

This is a composition of two independently derived maps. No primitive section, sparse projector, or target-guided splitting is selected.

Descent through the algebraic kernel

The double-pole master obeys

$$R_\infty(e_6)=0.$$

Direct exact substitution also gives

$$\ell_{\rm src}(v_{\rm alg})=0.$$

Hence the covector annihilates the complete two-dimensional algebraic kernel

$$\langle e_6,v_{\rm alg}\rangle.$$

It therefore descends canonically to the elliptic Gysin quotient. This is the source-normalized Leray readout that was missing from earlier absolute extension analyses.

Soft–signed specialization

At the supported corner $x=y$,

$$\ell_{\rm src}\big|_{x=y} = \begin{pmatrix}1&0&0\end{pmatrix}.$$

Thus the physical supported line selects precisely the source master $e_7$. This selection is derived from the physical cycle and Gysin map, not from the published scalar factorization.

The only denominator in the generic covector is supported on $y=0$, an existing soft coordinate wall. The quartic $\mathcal Q$ is absent.

Result

The final four-master block now has a canonical source-to-physical readout:

$$\mathcal M_{4,--} \xrightarrow{R_\infty} H^1(D_\infty)(-1) \xrightarrow{\ell_{\rm phys}} \mathbb C.$$

It is:

• independent of algebraic-kernel representatives;
• source-normalized integrally;
• supported only on existing soft/signed-energy geometry;
• free of generic $\mathcal Q$-support.

This does not determine the complete rank-twelve physical readout. It does close the physically activated final-block Leray covector.

Evidence

• research/benincasa/checkers/check_infinity_physical_leray_covector.py;
• research/benincasa/results/infinity-physical-leray-covector.json;
• Entries 3678–3722;
• the explicit infinity-Gysin projection recorded in the cosmology ledger.

The exact checker passes eight of eight gates.

Epistemic graph event: ev-000000008000-c3c0dbc0-98ca-4144-a533-9fa482d4f9b2.

Allocator claim: seqclaim-2f9219f394db2e25308bc8b5.

View source on GitHub