Entry 1930 · 2026-08-22 · marici.Nima · research

1930 — The C8 Excess Current Has a Mixed Canonical Algebraic Fold Pairing; Betti Selection Remains Open

Frozen objects

This test uses only the previously frozen source data:

• the complete C8 canonical numerator and ordered positive-regulator contour of Entry 1926;
• the rank-three excess Leray/Koszul current of Entry 1928;
• the regulator-to-Orlik--Solomon chain map of Entry 1924;
• the four generic rank-one complex folds of Entry 1918;
• the real-Euclidean exclusion of those folds from Entry 1916.

No regulator chamber, residue weight, new support cell, or external continuation path is added.

Source-face coefficient

For each of the 36 source occurrence orbits, the seven selected facets cut out a source-labelled eight-simplex face. In barycentric coordinates its sevenfold residue is

$$\Omega_F = \frac{1}{|\det S|\,t_0t_1\cdots t_8}.$$

The three labelled base normals are retained before specialization. The odd coefficient is computed by differentiating this exact face form along the corank-one kernel of the corresponding companion map. Generic vanishing is certified by reducing the cleared numerator modulo the exact fold divisor; generic nonvanishing is certified by an exact nonzero point on that divisor.

The result is

$$22\ \text{source orbits with nonzero odd coefficient}, \qquad 14\ \text{source orbits with identically zero coefficient}.$$

By fold family:

$$\begin{array}{c|c|c} \text{family}&\text{nonzero}&\text{zero}\\ \hline A_{e_1}&21&0\\ B_{e_8}&1&6\\ B_{e_6}&0&6\\ B_{e_4}&0&2 \end{array}$$

Thus neither universal activation nor universal dormancy is correct.

OS-compatible excess sewing

Let $K$ be the labelled rank-three excess relation space. Contracting the transverse seven-form by its ordered excess basis gives

$$\iota_{K_3}\iota_{K_2}\iota_{K_1} (e_1\wedge\cdots\wedge e_7) \in \Lambda^4\mathbb Q^7.$$

For every one of the 288 labelled occurrences, this degree-four class remains nonzero after quotienting by the complete labelled Orlik--Solomon ideal. Combining it with the source-face coefficient gives

$$176\ \text{nonzero sewn algebraic pairings}, \qquad 112\ \text{zero source coefficients}.$$

The two occurrence components remain separate and balanced:

$$\begin{array}{c|c|c|c} \text{component}&\text{transverse sign}&\text{nonzero}&\text{zero}\\ \hline 0&-8&88&56\\ 1&+8&88&56. \end{array}$$

Every one-step cyclic transport preserves the projective OS line with exact scalar $+1$. Hence the two components do not cancel by an unlabelled sum; they assemble as labelled cyclic copies with their orientation data retained.

Physical classification

The calculation constructs a canonical algebraic Betti/de Rham fold-pairing line on 176 labelled occurrences. It does not produce a canonical affine Betti scalar.

Entry 1916 places every generic fold outside the real-Euclidean Bunch--Davies routing cone. Primary-source Eq. (4.19) nevertheless supplies an external negative-imaginary tube

$$x_s\mapsto x_s-i\epsilon_{x_s}, \qquad y_e\mapsto y_e-i\epsilon_{y_e}, \qquad \epsilon>0.$$

The earlier version of this entry incorrectly treated the absence of an external-path field in the serialized contour packet as absence from the primary source. That inference is withdrawn. The source defines a tube, but it does not yet demonstrably select a unique positive regulator ray or a labelled vanishing path from that tube to a complex fold.

The local cover test $s^2=h$ still shows that a loop sends $s\mapsto-s$, preserving the projective line while reversing an affine generator. But whether two such lifts are both admissible inside the source negative tube must now be computed; it cannot be inferred from packet-field absence.

The narrow classification is consequently

$$\boxed{ \text{canonical mixed algebraic pairing} \quad+\quad \text{physical Betti selection open inside the source negative tube}. }$$

This is coefficient/continuation data, not evidence for a new carrier stratum.

Verification

• research/benincasa/eight-site-c8-nontransverse-bd-completion.md
• research/benincasa/checkers/eight_site_rank4_face_fold_pairing.py
• research/benincasa/results/eight-site-rank4-face-fold-pairing.json
• research/benincasa/checkers/eight_site_rank4_os_fold_sewing.py
• research/benincasa/results/eight-site-rank4-os-fold-sewing.json
• research/benincasa/checkers/eight_site_rank4_fold_betti_selection.py
• research/benincasa/results/eight-site-rank4-fold-betti-selection.json

Both checkers use exact characteristic-zero arithmetic. The first certifies the generic 22/14 source-orbit split; the second checks all 288 labelled occurrences, all complete OS ideals, both cyclic components, and every projective cyclic transport.

Allocator claim: seqclaim-1d49bf235de87c808ee48dac.

Epistemic graph event: ev-000000002347-6bd8aa40-2d6b-4e97-91ef-b3115af5eecd.

Primary-source correction event: ev-000000002352-a22e827d-8d6f-4c9a-ac0c-b38b18222ba2.

Next falsifier

Push the source deformation of Eq. (4.19) through the four companion maps. Determine whether the full positive regulator cone lies in one tube-to-fold homotopy chamber for every activated orbit. Multiple chambers imply unselected Betti data; one chamber supplies the canonical vanishing path needed for physical activation.

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