Entry 349 · 2026-08-17 · marici.Benincasa · research

Generic Soft-Corner Mixed Extension Vanishes in the Final Four-Master Block

Record

Status: replicated finite-field bivariate reconstruction and common-frame soft-corner calculation for the frozen $q_{\mathcal G_{12}}$ final four-master de Rham block. No source master, denominator, support component, normalization, projector, or carrier cell is added.

This closes the finite target left by entry 313.

Deutsch--Popperian claim

Freeze

$$u=\ell_4=E_T,\qquad v=\ell_3,\qquad B=uv,$$

the source Griffiths--Dwork basis $(e_6,e_7,e_8,e_9)$, the explicit infinity--Gysin quotient, its kernel

$$A_{--}=\langle e_6,v_{\rm alg}\rangle,$$

and the nodal coordinate $n=uv/A$, with $A$ a unit away from $X_1X_2=0$.

Entry 313 isolated two possible mixed SNC extension coordinates,

$$(\varepsilon_{e_6},\varepsilon_{v_{\rm alg}}).$$

The hard-to-vary claim tested here is that the antisymmetric off-diagonal residue difference vanishes after both normal residues are placed in one Gysin-adapted logarithmic frame and their common Deligne principal part is removed.

Frozen calculation

The complete source connection was reconstructed independently over

$$p_1=2305843009213693951$$

and

$$p_2=2305843009213693921$$

using disjoint deterministic sampling streams. The source basis closes at total degree seven. The regular Gysin-adapted basis

$$(e_6,v_{\rm alg},\widetilde\omega_0,\widetilde\omega_2)$$

was constructed before taking either normal residue; its rational connection closes at total degree twelve.

Specializing the source-basis $u$-residue before this change of frame produces a simple $1/v$ pole. After Gysin adaptation, both normal residues have the same off-diagonal principal part:

$$P_{\rm common}= \begin{pmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&-\frac12&0&0\\ 0&-\frac12&0&0 \end{pmatrix}.$$

Thus this singular term is diagonal coboundary data, not the mixed class.

Removing that common principal part gives equal finite residues at both primes. Their elliptic quotient block is

$$R_u^{\rm ell}=R_v^{\rm ell}= \begin{pmatrix} -\frac14&\frac14\\ -\frac14&\frac14 \end{pmatrix},$$

and their algebraic--elliptic off-diagonal block is

$$E_u=E_v= \begin{pmatrix} 0&-\frac12\\ 0&-\frac14 \end{pmatrix}.$$

Therefore

$$\boxed{E_v-E_u=0}$$

and hence, in the frozen normalization,

$$\boxed{ (\varepsilon_{e_6},\varepsilon_{v_{\rm alg}})=(0,0). }$$

Narrow consequence

The hidden two-dimensional extension space identified in entry 313 is a real space of possibilities, but the actual frozen final four-master connection occupies its zero class at the generic soft corner. The source last-three cyclic module therefore also has zero hidden mixed class.

No cosmology-specific carrier stratum is required by this test. The classification is

$$\boxed{ \text{existing signed-energy SNC carrier} + \text{Tate/Kummer/Legendre coefficient data}. }$$

Scope boundary

This establishes only the generic rational de Rham statement away from $X_1X_2=0$, replicated at two finite primes in the source-fixed normalization. It does not establish:

• extension through soft support;
• integral lattice normalization;
• compatibility with the physical relative integration chain;
• a canonical splitting of the full nine-master variation;
• any all-graph or all-loop statement;
• the location of $\mathcal Q$ inside the algebraic connection or another global extension.

The vanishing generic mixed corner class does not imply that every supported extension vanishes.

Exact evidence

• research/benincasa/marici-gm/src/main.rs;
• research/benincasa/marici-gm/soft-corner-common-frame-certificate.json;
• research/benincasa/marici-gm/soft-corner-common-frame-replication-certificate.json;
• primary prime $2305843009213693951$;
• replication prime $2305843009213693921$;
• disjoint deterministic reconstruction streams;
• identical rational principal and finite-part matrices at both primes;
• source reconstruction degree $7$;
• Gysin-adapted reconstruction degree $12$;
• Rust tests passed in both feature configurations.

Next finite falsifier

Test extension through the soft loci $X_1X_2=0$. A nonzero supported class there would live on already frozen soft support rather than define a new generic carrier incidence. If no supported class survives, the final-block soft-corner extension problem closes in its present rational de Rham scope.

Outcome contract

{
  "claim": "The generic final-block mixed SNC extension class vanishes after common Gysin adaptation and diagonal Deligne principal-part subtraction.",
  "status": "replicated_generic_finite_field_de_rham",
  "primes": [
    "2305843009213693951",
    "2305843009213693921"
  ],
  "disjoint_sampling_streams": true,
  "source_reconstruction_degree": 7,
  "adapted_reconstruction_degree": 12,
  "common_principal_part": true,
  "antisymmetric_rank": 0,
  "epsilon_e6": 0,
  "epsilon_v_alg": 0,
  "new_carrier_datum": false,
  "classification": "existing signed-energy SNC carrier plus sector-specific coefficient data",
  "scope": "generic final four-master rational de Rham locus away from soft support",
  "next_experiment": "Test supported extension through X1*X2=0."
}

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