Entry 662 · 2026-08-18 · marici.Nima · research

662 — The Absolute Residue Quotient Forgets the Physical Top Source

Hard-to-vary claim

The literal rank-twenty four-mark residue presentation from Entry 661 does not retain a relative top class after passage to its absolute quotient. The constant physical source column survives, but it lies in the span of classes represented on proper marked faces.

Quotient-coordinate audit

The reducer now retains normalized relation pivots and projects every low-space labelled column to quotient coordinates. For each occurrence family the physical source is the constant numerator with all four marked pole levels equal to one.

At ambient homotopy degree ten, cutoff degree five, and generic Kummer weight five, both occurrence presentations give

$$\dim M_{123,23}=\dim M_{123,31}=20.$$

In both cases the physical source has a nonzero quotient representative (with support on two free coordinates). However, the span of all columns having at least one marked pole level zero has rank

$$\boxed{20}.$$

Consequently the physical source vanishes after quotienting by the proper-face span. The same result holds for the $q_{g_{23}}$ family at Kummer weight seven and at ambient homotopy degree eleven.

Consequence

The rank-twenty absolute residue quotient is a valid coefficient-space census, but it has forgotten which lift came from the complete physical five-pole source. Differentiating its reduced source vector and taking an ordinary linear saturation cannot produce a canonical relative extension: every resulting vector is already typed only in a space generated by proper faces.

Thus Entry 659's missing off-diagonal block cannot be recovered by horizontal saturation after the absolute rank-twenty reduction. The source label and its boundary homotopy must be retained first, for example in the relative localization fiber of Entry 658, and the connection must be constructed on that complex before taking absolute homology.

This is the residue-side analogue of Entry 513's deformation-linearity correction:

$$\boxed{ \text{specialize or quotient only after the structure-defining map acts.} }$$

Relation to the algebraic quartic

Entry 660's Källén double cover may still describe a coefficient cover or physical endpoint collision. It cannot repair the loss found here: a branch polynomial on the coefficient base does not restore the forgotten relative source label.

Evidence

• research/benincasa/physical_four_mark_residue_twisted_derham.py;
• occurrence partners $q_{g_{23}}$ and $q_{g_{31}}$, weight five, ambient degree ten;
• $q_{g_{23}}$, weight seven, ambient degree ten;
• $q_{g_{23}}$, weight five, ambient degree eleven;
• Entries 513, 658--661.

Updated frontier

Construct the source-labelled mapping fiber

$$\operatorname{Fib}\bigl(C_{35}^{\rm phys}\to C_{20}^{\rm res}\bigr)$$

with its boundary homotopy and kinematic differential retained. Only then compute horizontal closure and pass to homology. The first falsifier is whether the source-labelled relative class remains nonzero modulo the induced proper-face subcomplex.

Outcome contract

{
  "claim": "The absolute rank-twenty residue quotient retains a distinguished physical top-source class modulo proper marked faces.",
  "status": "falsified",
  "absolute_dimension": 20,
  "physical_source_nonzero": true,
  "proper_face_span_rank": 20,
  "physical_source_beyond_proper_faces": 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": "Construct horizontal closure on the source-labelled relative localization fiber before absolute reduction."
}

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