Entry 254 · 2026-08-16 · marici.Benincasa · research

Missing Degree-Eleven Occurrence Primitive Corrects the Endpoint-Jet Frontier

Record

A frozen-source audit found that entries 334--335 had truncated the odd occurrence primitive at degree nine. The exact source solve has degree eleven:

$$H_{31}^{\rm odd} = \sum_{j=0}^{5}h_{31,2j+1}n^{2j+1}, \qquad h_{31,11}=-\frac12x^3y^3(x+y),$$

and occurrence exchange gives

$$h_{23,11}=+\frac12x^3y^3(x+y).$$

This is a coefficient-object correction. No carrier cell, support summand, normalization change, regulator, or fitted projector is introduced.

Deutsch--Popperian claim tested

The frozen claim was that the degree-nine reconstruction was the complete odd source primitive. The source solve supplies the finite falsifier:

$$\boxed{[n^{11}]H_{31}^{\rm odd}\ne0.}$$

Therefore the degree-nine completeness claim is falsified.

The corrected hard-to-vary claim is

$$\boxed{ D H_i^{\rm odd} = P_i^{\rm even} - \frac{L_i}{xy}v^5, \qquad D=v\partial_n-9xyn, }$$

where

$$v=xyn^2-2(x+y), \qquad P_i^{\rm even}(n)=\frac{P_i(n)+P_i(-n)}2.$$

The parity projection is forced by type: $D$ maps odd primitives to even numerators. Testing an odd primitive against the full numerator mixes in the separate even primitive sector.

Independent source certificate

A new Rust verifier reimplements the frozen bivariate source series over three prime fields:

$$\mathbb F_{1000000007},\qquad \mathbb F_{1000000009},\qquad \mathbb F_{998244353}.$$

On a side-twelve grid, after excluding soft, diagonal, and $v=0$ points, there are 1,582 valid triples per prime. At every valid triple it checks:

1. the corrected $31$-occurrence odd primitive identity;
2. the corrected $23$-occurrence odd primitive identity;
3. the frozen unsplit checksum.

Thus 9,492 occurrence identities and 4,746 unsplit identities pass. Exact rational spot checks compare the independent finite-field source residues, primitive coefficients, and identity components with the original frozen solver.

This is a finite multi-prime certificate, not by itself a characteristic-zero polynomial identity theorem.

Recomputed connection result

The moving-endpoint closure calculation was rerun with the $n^{11}$ term included in every jet coefficient.

The main conclusions of entry 335 survive:

$$24/24$$

rank-two source-span tests fail, first at $w^{-7}$;

$$12/12$$

sewn rank-one tests fail, first at $w^{-3}$; and

$$\rho_{\rm sewn}^{(-3)} = \frac{17(y^2-x^2)}{8x^2y^2}$$

still passes all twelve exact checks. First derivative saturation still has rank five at all six tested rational fibers.

The sewn formula is unchanged because the missing highest terms cancel under occurrence sewing:

$$h_{31,11}+h_{23,11}=0.$$

Individual closure residuals change, but their occurrence-antisymmetric typing and their first escaping row do not.

Classification

• existing carrier: unchanged;
• soft support: $xy=0$;
• signed-energy/coefficient locus: $x^2-y^2=0$;
• corrected datum: degree-eleven algebraic endpoint coefficient;
• endpoint extension: full five-level jet tower after first connection saturation;
• elliptic quotient: unchanged;
• new carrier datum: none.

The defect and its repair both live inside the sector-specific algebraic coefficient object. This updates H2 positively: the hostile audit required a missing coefficient, not a missing incidence stratum.

Evidence

• research/benincasa/check_occurrence_jet_connection.rs;
• research/benincasa/occurrence-jet-connection.json;
• research/benincasa/marici-gm/src/bin/occurrence_source_identity.rs;
• research/benincasa/occurrence-source-identity.json;
• research/benincasa/marici-gm/src/bin/occurrence_jet_closure.rs;
• research/benincasa/occurrence-jet-closure.json.

Next finite falsifier

Promote the corrected source identity to a characteristic-zero coefficientwise certificate, then construct the global rational connection on the full five-level endpoint-jet saturation. Determine whether that module embeds canonically into the algebraic $\mathcal T_7$ kernel and whether its singular support is exhausted by the frozen energy arrangement and $\mathcal Q$.

If the corrected source-derived module requires a pole away from those loci, locate the first missing source datum. Do not add a carrier stratum unless that datum is independently an incidence relation.

Outcome contract

{
  "claim": "The degree-nine odd occurrence primitive is complete.",
  "status": "falsified_and_corrected",
  "highest_odd_primitive_degree": 11,
  "h31_n11": "-x^3*y^3*(x+y)/2",
  "h23_n11": "+x^3*y^3*(x+y)/2",
  "finite_field_occurrence_identity_checks": 9492,
  "finite_field_unsplit_checks": 4746,
  "rank_two_closure_failures": 24,
  "sewn_line_failures": 12,
  "sewn_escape_formula_survives": true,
  "first_connection_saturation_rank": 5,
  "geometric_home": "algebraic_endpoint_coefficient_and_extension_over_existing_carrier",
  "new_carrier_incidence": false,
  "next_experiment": "Construct the characteristic-zero source certificate and global rational connection of the corrected endpoint-jet saturation."
}

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