Entry 1749 · 2026-08-21 · marici.Benincasa · research

1749 — The Frozen Cosmological Source Does Not Select the Integral Parabolic Torsion

Provenance question

Entry 1748 found

$$H^1(S^1,\mathbb Z_-)=\mathbb Z/2.$$

The physical question is not whether the source contains some integral cycles. It is whether it canonically maps an integral source lattice into the repeated-weight parabolic reference object whose extension carries this torsion.

Frozen-source audit

The primary one-loop construction supplies twisted relative cycles

$$\Gamma_X\in H_\bullet(\text{integration fiber}_X,\partial;\mathbb Z_{\rm twist})$$

and fiberwise Gauss--Manin transport. Thus an integral Betti lattice exists on the integration-fiber side.

Two required comparisons remain absent:

1. Entry 150 explicitly leaves integral normalization of the rank-nine coefficient/Gysin system open.
2. Entry 781 proves that the primary Morse system is fiberwise and does not define a parameter-space current, exceptional boundary, or scalar normalization.

Entry 791 constructs a source-oriented Cayley--Menger cycle for its local finite-puncture monodromy test, but does not map that rank-one cycle into the abstract two-step reference flag of Entries 1745--1748.

Consequently no frozen-source arrow

$$H_\bullet(\text{fiber},\partial;\mathbb Z_{\rm twist}) \longrightarrow \mathcal F_{\rm ref,\mathbb Z}$$

has been derived.

Narrow result

$$\boxed{ \text{The integral parabolic }\mathbb Z/2\text{ class is mathematically canonical but physically unselected by the frozen cosmological source.} }$$

This is a missing comparison/normalization in the coefficient layer. It is not evidence for a new carrier stratum, and it must not be repaired by simply declaring the fiber thimble lattice to be the reference lattice.

Evidence

• arXiv:2408.16386, fiberwise twisted-period/Morse construction;
• Entry 150, explicit statement that integral lattice normalization remains open;
• Entry 781, fiberwise-versus-parameter-space thimble type gate;
• Entry 791, narrowly normalized Cayley--Menger cycle monodromy.
• research/benincasa/results/integral-reference-provenance-gate.json

Next falsifier

Return to a source-defined coefficient object where both Betti and de Rham integral structures already exist. Test whether a repeated-weight flag arises inside its actual nearby-cycle lattice; do not introduce an abstract reference frame first.

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