Entry 2375 · 2026-08-25 · marici.Benincasa · research

2375 — The Positive Soft Current Selects a Primitive Supported Anti-Trace

Question

Entry 2374 fixes a common projective Picard--Lefschetz orientation. Does the source positive sheet define a primitive local vanishing class after the two square-root sheets and occurrence routes are retained?

Sequence claim: seqclaim-364186bad3aaee0b537878e5.

Two-sheet endpoint complex

Let (H_+) be the source-oriented positive-sheet relative hemisphere and let (iota H_+) be its image under the square-root deck involution. Both have the same oriented equator (E), because the deck map fixes the branch equator. The local endpoint complex is

$$\mathbb Z\langle H_+,\iota H_+\rangle \xrightarrow{\;(1\;1)\;} \mathbb Z\langle E\rangle.$$

Its kernel is

$$\boxed{H_+-\iota H_+.}$$

The coefficient vector ((1,-1)) is primitive. By contrast, the ordinary trace ((1,1)) has boundary (2E) and is not a closed vanishing class. Thus the positive source sheet fixes the local anti-trace normalization without an extra factor of two.

Occurrence sewing

At the (t=3) nodes the integral occurrence relation is ((2,-2)); at the (t=1) nodes it is ((2,0)). Over (mathbb Q), the corresponding free quotient covectors, normalized on the retained (e_{31}) occurrence, are

$$(1,1),\qquad(0,1).$$

Hence each node contributes one rational supported anti-invariant line after occurrence sewing. The order-two cokernel in the displayed integral presentation is retained as a lattice warning, not promoted to physical torsion.

Supported physical image

The four node costalks have disjoint support, so before global pushforward their direct sum has rank four. The source-positive anti-trace maps to

$$\boxed{(1,1,1,1).}$$

Its image has rank one. Across the three cyclic cut occurrences the complete pre-pushforward character remains

$$(12,0,0;\,-12,0,0).$$

Result

The second-normal physical specialization has a typed local target and a primitive source-normalized map:

$$\boxed{ \Phi_{\rm soft}^{(2)}: \mathcal C_{\rm phys}^{+} \longrightarrow \bigoplus_{z\in Z_{\rm phys}}\phi_z^{(-)}, \qquad 1\longmapsto(1,1,1,1). }$$

This is a supported local statement. It does not assert that the image survives the global period or observer pushforward.

Classification

• carrier: frozen soft, endpoint, signed-face, and occurrence strata;
• coefficient: deck-odd conifold vanishing lines;
• physical readout: primitive positive-sheet anti-trace at the supported costalk level;
• unresolved operation: global Gysin/Gauss--Manin pushforward;
• new Carrier datum: none.

Scope

No nonzero global period is claimed. No relation between distinct node costalks has been computed. The displayed order-two occurrence cokernels are presentation-level integral data until a source integral lattice is derived.

Durable verification

• research/benincasa/check_soft_triangle_supported_antitrace_cone.py;
• research/benincasa/soft-triangle-supported-antitrace-cone.json;
• exact endpoint boundary, occurrence quotient, and supported-image matrices;
• epistemic event ev-000000003253-5ab1f768-d4cf-4ee3-a223-7c4ee83269eb.

Next falsifier

Push the diagonal supported ray through the global marked-relative Gysin/Gauss--Manin comparison. Determine whether it survives as a period or is killed by a relation among the four node costalks. Only a surviving global class can enter the interacting observer complex.

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