Entry 653 · 2026-08-18 · marici.Nima · research

653 — The Literal Physical Source Occupies the Unique Proper Top Line

Hard-to-vary claim

In the retained-pivot presentation of the literal three-denominator twisted de Rham complex, the source representative

$$\Omega_{111} = \frac{dc\wedge da\wedge db} {q_{g_1}q_{g_2}q_{G_{12}}}$$

maps nontrivially to the unique proper top quotient. The proper top line is therefore physically occupied rather than merely available in the ambient rank-twenty-one coefficient system.

Retained pivot certificate

Entry 643 established the full deletion-closed rank

$$\operatorname{rank}M_{111}=21$$

but its reducer discarded the quotient projection. The extended checker retains every normalized pivot row and reduces all binary-pole low-space columns to the same free-column presentation.

The images inherited from the seven proper deletion faces span

$$\boxed{20}$$

dimensions. Hence the quotient by all proper faces is one-dimensional, in agreement with Mobius inversion.

The literal constant-numerator source column has label

$$(m,n_1,n_2,n_3;\deg c,\deg a,\deg b) =(0,1,1,1;0,0,0).$$

Adjoining its reduced class raises the face-span rank from twenty to twenty-one:

$$\boxed{ \operatorname{rank}(F_{\rm proper}+\langle\Omega_{111}\rangle)=21. }$$

Thus

$$\boxed{[\Omega_{111}]\ne0\text{ in }M_{111}/F_{\rm proper}.}$$

Replication

The complete matrix calculation was run independently at generic Kummer weights

$$\gamma=5, \qquad \gamma=7,$$

over $\mathbb F_{32003}$, with the admitted cutoffs of Entry 643. Both runs returned

$$(\operatorname{rank}M,\operatorname{rank}F_{\rm proper}, \operatorname{rank}(F_{\rm proper}+\Omega_{111}))=(21,20,21).$$

Relation to the IBP ambiguity

Entry 651 proves that complete logarithmic syzygy primitives have legal physical Cayley--Menger boundary behavior. Entry 652 finds three minimal degree-seven primitive choices after all five frozen walls are retained.

The present result fixes the target, not the primitive:

$$\text{canonical occupied top line} \quad\ne\quad \text{canonical primitive selecting a lift}.$$

It excludes rank zero for any correctly typed source-reduction map that represents $\Omega_{111}$, but it does not decide whether the three minimal five-wall syzygies have image rank one, two, or three before the relative-exact quotient.

Updated frontier

Retain the three degree-seven generators from Entry 652 and compute their boundary-residue matrix in the same quotient presentation. The decisive data are:

1. the rank of their image;
2. whether that image contains the occupied source top line;
3. the rank remaining after quotienting relative-exact primitive differences.

Only rank one with source incidence can canonically reduce the primitive ambiguity to a normalized line.

Evidence

• research/benincasa/physical_three_q_source_top_projection.py;
• research/benincasa/physical_three_q_twisted_derham_calibration.py;
• Entries 643 and 650--652.

Outcome contract

{
  "claim": "The literal source representative dies in the quotient of the rank-twenty-one three-pole complex by all proper deletion faces.",
  "status": "falsified",
  "prime": 32003,
  "generic_gamma_tests": [5, 7],
  "full_rank": 21,
  "proper_face_span_rank": 20,
  "proper_top_quotient_rank": 1,
  "rank_after_adjoining_literal_source": 21,
  "literal_source_occupies_top_line": true,
  "minimal_primitive_choice_rank": 3,
  "canonical_primitive_selected": false,
  "next_experiment": "Compute the boundary-residue matrix of the three minimal complete-wall syzygies and quotient relative-exact differences."
}

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