Entry 753 · 2026-08-18 · marici.Benincasa · research

753 — Occurrence Reflection Changes the Residue Sector

Gate after Entries 720 and 752

Entry 720 leaves a finite occurrence-reflection test: compare the rank-twenty-one relative connection at $(X_1,X_2,X_3)=(2,3,4)$ with the reflected point $(2,4,3)$. Before computing connection matrices, the source-labelled residue divisor must be transported.

Source permutation

Write the loop edges as

$$(a,b,c)=(y_{23},y_{31},y_{12}).$$

The site transposition $\sigma_{23}$ acts by

$$X_2\leftrightarrow X_3,\qquad y_{12}\leftrightarrow y_{31},\qquad y_{23}\mapsto y_{23}.$$

Applying this to the source definitions of all denominators gives

$$\begin{aligned} q_{g_1}&\mapsto q_{g_1},& q_{g_2}&\leftrightarrow q_{g_3},\\ q_{g_{12}}&\leftrightarrow q_{g_{31}},& q_{g_{23}}&\mapsto q_{g_{23}}, \end{aligned}$$

and, crucially,

$$\boxed{ q_{\mathcal G_{12}}\longleftrightarrow q_{\mathcal G_{31}}, \qquad q_{\mathcal G_{23}}\longmapsto q_{\mathcal G_{23}}. }$$

Thus the two physical $q_{\mathcal G_{12}}$-summands map to

$$\{q_{g_1},q_{g_3},q_{g_2},q_{\mathcal G_{31}},q_{g_{23}}\}$$

and

$$\{q_{g_1},q_{g_3},q_{g_2},q_{\mathcal G_{31}},q_{g_{12}}\}.$$

They do not map to the original $q_{\mathcal G_{12}}$-residue union.

Narrow conclusion

$$\boxed{ \text{kinematic swapping inside the fixed }q_{\mathcal G_{12}}\text{ chart does not type the occurrence-reflection intertwiner.} }$$

The comparison demanded by Entry 720 must first construct a residue-chart transition

$$M^{(G_{12})}_{(2,3,4)} \longrightarrow M^{(G_{31})}_{(2,4,3)}$$

with the displayed denominator permutation, Poincare-residue orientation, and retained-pivot basis transport. Only then can connection matrices be tested for intertwining or reflection eigensectors be formed.

This is a correction of the proposed implementation route, not a rejection of occurrence symmetry. It also does not reopen $\mathcal Q$: the issue is ordinary labelled global gluing, exactly the frontier of Entry 752.

Classification

• carrier: unchanged cyclic energy/Cut incidence carrier;
• coefficient datum: transition between labelled residue charts;
• new carrier datum: none;
• present obstruction: the required chart transition has not yet been constructed.

Evidence

• frozen source denominator definition;
• Entry 590's labelled physical residue pair;
• Entries 719--720 and 752;
• exact Rust certificate research/benincasa/marici-gm/src/bin/occurrence_reflection_residue_sector.rs;
• machine-readable packet research/benincasa/marici-gm/occurrence-reflection-residue-sector.json;
• allocator claim seqclaim-051d2b910f1645c04f961d2e.
• epistemic event ev-000000000367-0b6e3565-41b0-4ea2-b8e6-4b613fcc0603.

Next falsifier

Construct the $q_{\mathcal G_{31}}$-residue retained-pivot presentation directly from the full three-variable source, including its residue orientation. Transport every labelled generator under $\sigma_{23}$, reduce in the target presentation, and test the two Gauss--Manin intertwining equations. Do not identify the two residue charts by an unlabelled rank count.

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