Entry 663 · 2026-08-18 · marici.Nima · research

663 — Relative Reduction Retains a Nineteen-Dimensional Top Block and Its Source

Hard-to-vary claim

Quotienting the literal four-mark complex by proper-face columns before taking its finite twisted-de-Rham quotient retains a nonzero physical source class. This differs sharply from Entry 662's quotient-after-homology calculation, where proper-face classes span the entire absolute rank-twenty space.

Chain-level relative construction

In each occurrence presentation, remove every labelled column having at least one marked pole level zero. Project the exact-differential and localization rows to the remaining columns, retaining pole levels one and two for all four marked denominators. Only after this chain-level quotient is formed are relation pivots computed.

The physical low space then consists of numerator monomials with all four marked pole levels equal to one. At cutoff degree five, ambient homotopy degree ten, and Kummer weight five, both occurrence families give

$$\boxed{ \dim M_{123,23}^{\rm rel,top} = \dim M_{123,31}^{\rm rel,top} =19. }$$

The constant physical source survives in both quotients. In the normalized pivot presentation it has support on one free coordinate. That support count is basis-dependent; nonvanishing is the invariant statement.

Replication

The same rank nineteen and nonzero source occur for

• the $q_{g_{23}}$ and $q_{g_{31}}$ partners at weight five;
• the $q_{g_{23}}$ partner at weight seven;
• the $q_{g_{23}}$ partner after increasing ambient homotopy degree from ten to eleven.

All computations use $\mathbb F_{32003}$ at $(x,y,z)=(2,3,4)$.

Meaning of the 20 versus 19 comparison

The absolute rank-twenty quotient and the relative rank-nineteen quotient are not related by naively quotienting the former by the image of its face classes: that image already has rank twenty. The discrepancy is the expected failure of taking homology to commute with this quotient and is precisely where a connecting/Bockstein class can live.

Thus the calculation provides a concrete finite model of the methodological rule reached independently in Entries 513 and 662:

$$\boxed{ \text{retain support and boundary data at chain level before reduction.} }$$

It does not yet identify a canonical one-dimensional physical coefficient line inside the nineteen-dimensional block, construct the unsplit pair of occurrences, or determine the kinematic connection.

Updated frontier

Retain the projection matrix from the labelled top-support complex to its nineteen-dimensional quotient and differentiate the full relation matrix with respect to two independent kinematic variables. The next invariant is the dimension of the smallest horizontal subspace generated by the source, computed before identifying the two reflected occurrence families.

Evidence

• research/benincasa/physical_four_mark_residue_twisted_derham.py;
• Entries 513, 658, and 661--662.

Outcome contract

{
  "claim": "Chain-level proper-face reduction also erases the physical source.",
  "status": "falsified",
  "absolute_dimension": 20,
  "relative_top_dimension": 19,
  "relative_physical_source_nonzero": true,
  "relative_source_coordinate_support": 1,
  "coordinate_support_is_canonical": false,
  "replications": [
    {"partner": "q_g23", "gamma": 5, "ambient_degree": 10},
    {"partner": "q_g31", "gamma": 5, "ambient_degree": 10},
    {"partner": "q_g23", "gamma": 7, "ambient_degree": 10},
    {"partner": "q_g23", "gamma": 5, "ambient_degree": 11}
  ],
  "next_experiment": "Differentiate the chain-level relative presentation and compute source-generated horizontal saturation."
}

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