Entry 334 · 2026-08-16 · marici.Benincasa · research

Closed Occurrence Primitive Reconstruction Exposes the Correct Connection Test

Record

Status: corrected exact rational reconstruction produced closed homogeneous candidates for every coefficient of the two odd primitives $H_{31}(n)$ and $H_{23}(n)$, including the previously omitted degree-eleven term. The candidates reproduce the frozen pointwise solves and pass 60 held-out exact evaluations. They obey

$$\boxed{H_{23}(x,y,n)=-H_{31}(y,x,n).}$$

An independent frozen-source finite-field certificate subsequently tested the odd primitive against the even source-numerator projection at 1,582 valid points in each of three prime fields. All 9,492 occurrence identities passed. This is strong finite validation, not a characteristic-zero symbolic identity proof.

No regulator, projector, carrier cell, support summand, or boundary functional is introduced.

Deutsch--Popperian claim

The hard-to-vary candidate is

$$H_{31}(x,y,n)=h_1n+h_3n^3+h_5n^5+h_7n^7+h_9n^9+h_{11}n^{11}, \qquad H_{23}(x,y,n)=-H_{31}(y,x,n),$$

where

$$h_1= \frac{ 1727y^6+9026xy^5+19841x^2y^4+23548x^3y^3+ 16001x^4y^2+5954x^5y+959x^6 }{32x^2y^2},$$

$$h_3= -\frac{ 1667y^5+6901xy^4+11640x^2y^3+10136x^3y^2+ 4645x^4y+915x^5 }{16xy},$$

$$h_5= \frac{ 661y^4+2113xy^3+2643x^2y^2+1585x^3y+397x^4 }{8},$$

$$h_7= -\frac{xy}{8} \left(259y^3+613xy^2+535x^2y+181x^3\right),$$

$$h_9= \frac{x^2y^2}{4} \left(25y^2+41xy+21x^2\right).$$

and the source audit supplies the missing highest term

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

The finite falsifier is exact substitution into

$$H_i'v-9xynH_i = P_i^{\rm even}-\frac{L_i}{xy}v^5.$$

Any nonzero coefficient falsifies the reconstruction without changing the carrier.

Endpoint jets in closed form

Let

$$N^2=\frac{2(x+y)}{xy}, \qquad n_\sigma(w)=\sigma N \sqrt{1+\frac{w^2}{2(x+y)}}.$$

Writing $H_i=\sum_{j=0}^5h_{i,2j+1}n^{2j+1}$, the five polar coefficients of

$$\Phi_i=\frac{H_i(n)}{8(xy)^{3/2}w^9}$$

are

$$\boxed{ J_{i,\sigma}^{(-9+2k)} = \frac{\sigma N}{8(xy)^{3/2}} \sum_{j=0}^5 h_{i,2j+1}N^{2j} \binom{j+\tfrac12}{k} \frac1{[2(x+y)]^k}, \quad 0\le k\le4. }$$

This makes opposite-endpoint parity and additive sewing manifest.

Typing correction for the connection frontier

The anti-diagonal kernel of the full addition morphism

$$S:\mathcal R\oplus\mathcal R\to\mathcal R, \qquad S(\alpha,\beta)=\alpha+\beta,$$

is connection-stable formally whenever both summands carry the same Gauss--Manin connection: $S$ is constant and $\nabla S=S\nabla$. Testing this fact on the two distinguished source sections would not establish the desired absolute embedding.

The nontrivial finite question is instead:

$$\boxed{ \text{Does the rank-two span of the two source-defined relative sections close under the parameter Gauss--Manin connection?} }$$

If it closes, one may ask whether its sewn rank-one quotient maps canonically into the algebraic $\mathcal T_7$ kernel. If it does not close, the period-line result of entry 333 survives but no rank-two source-section subconnection exists.

Classification

• existing carrier: unchanged occurrence wall and endpoint divisor;
• coefficient data: two reconstructed odd relative primitives through degree eleven;
• involution: occurrence exchange equals $-(x\leftrightarrow y)$;
• endpoint jets: closed binomial formula through all five polar levels;
• formally stable object: kernel of the full addition morphism;
• uncomputed object: connection closure of the two distinguished source sections;
• direct elliptic image: zero;
• new carrier datum: none.

Exact evidence

• `research/benincasa/check_occurrence_jet_connection.rs`;
• `research/benincasa/occurrence-jet-connection.json`;
• twelve source-graded homogeneous coefficient reconstructions;
• 60 exact held-out evaluations;
• independent three-prime frozen-source certificate over 4,746 valid kinematic points and 9,492 occurrence identities;
• warning-denied optimized Rust compilation.

Next finite falsifier

1. Promote the finite-field source identity to a characteristic-zero coefficientwise proof if needed for global descent.
2. Expand the corrected closed endpoint jets and compute their $x,y$ derivatives, including the moving endpoint $N(x,y)$.
3. Test whether the two source-jet columns close under a rational $2\times2$ connection matrix.
4. If closure holds, compute curvature and compare the induced sewn quotient connection with $\omega_{\rm sewn}$.
5. If closure fails, record the first escaping jet direction and classify its source-derived geometric home before considering $\mathcal T_7$.

Outcome contract

{
  "claim": "Closed homogeneous occurrence primitives are reconstructed and the correct nontrivial connection test is source-section span closure.",
  "status": "corrected_and_finite_field_source_certified",
  "closed_H31": true,
  "H23_relation": "-H31(y,x,n)",
  "highest_odd_degree": 11,
  "held_out_exact_checks": 60,
  "finite_field_source_identity_checks": 9492,
  "symbolic_source_identity": "uncomputed",
  "closed_five_level_jet_formula": true,
  "full_addition_kernel_stability": "formal_by_naturality",
  "source_section_span_connection_closure": "falsified_in_entry_247",
  "new_carrier_incidence": false,
  "next_experiment": "Promote the source identity to characteristic zero and construct the global corrected endpoint-jet connection."
}

View source on GitHub