Entry 212 · 2026-08-15 · marici.Benincasa · research

Provenance Verdict: $\mathcal Q$ Is Apparent in the Frozen Homogeneous Three-Site System

Record

Date: 2026-08-15

Status: completed provenance classification at generic nonsoft kinematics for the source-displayed homogeneous three-site one-loop simplex integral. Intersections with the frozen discriminant/soft union and generic multi-external-leg specializations are outside this theorem.

This entry synthesizes entries 181, 183, 209, and 211. It introduces no denominator, support summand, projector, splitting chosen from the target, or carrier cell.

Frozen question

Determine whether the printed quartic

$$\mathcal Q =-16X_1^2X_2^2-8X_1X_2E^2 +8(X_1+X_2)E^3-5E^4$$

has an intrinsic home in:

1. the gauge-normalized algebraic Gysin-plane extension;
2. another block of the rank-seven algebraic kernel;
3. discriminant extension or the physical relative chain; or
4. only a cyclic/master-presentation alphabet.

Candidate-by-candidate verdict

Algebraic Gysin plane

The diagonal characters are

$$g_{00}=-\frac12d\log P_6, \qquad g_{11}=d\log D_1,$$

with unique $d\log\mathcal Q$ weight zero.

The off-diagonal equation

$$dh+(g_{00}-g_{11})h=-g_{10}$$

has a polynomial degree-seven solution. Its denominator powers in the frozen search basis $(D_1,P_6,\mathcal Q)$ are

$$(0,0,0).$$

The split identity passes 2,048 disjoint directional tests.

Verdict:

$$\boxed{\mathcal Q\text{ is absent from the algebraic-plane extension.}}$$

Remaining rank-seven kernel

The other source parity blocks have ranks $1,2,2$. Their complete bivariate connections have poles only on

$$u=0, \qquad 1-\frac{u+v}{2}=0,$$

or are constant triangular. Their blockwise mismatch census is

$$(0,0,0)$$

over 2,048 directional tests, and no entry has a $\mathcal Q$ denominator.

Verdict:

$$\boxed{\mathcal Q\text{ is absent from the generic algebraic kernel.}}$$

Discriminant and physical relative chain

Entry 181 freezes the source-positive sheet by

$$W=|R_L|, \qquad \Gamma_{\rm phys}\cap L \subset D_L^{\operatorname{sign}R_L}.$$

The sheet can switch only on the already frozen branch-at-pair strata.

The exhaustive raw discriminant census rejects $\mathcal Q$ from all 1,719 nonconstant surface/component/incidence conditions in one residue sector. Cyclic transport tests all three target quartics against all three sectors:

$$3\cdot1719=5157$$

exact factor rejections.

The resulting simultaneous relative SNC pair extends over a generic transverse $\mathcal Q$-disk. Hence

$$T_{\mathcal Q}=1, \qquad N_{\mathcal Q}=0, \qquad \operatorname{Var}_{\mathcal Q}(\Gamma_{ m phys})=0.$$

The source lower sector has the complete ten-linear-letter alphabet and no $\mathcal Q$ component. The literal six-term cyclic assembly has coefficients $+1$, common orientation, and the same physical chain, so its variation is also zero.

Verdict:

$$\boxed{ \mathcal Q\text{ is neither discriminant support nor physical-chain support at generic nonsoft homogeneous kinematics.} }$$

Final classification

All intrinsic candidate homes are eliminated in the frozen system:

$$\boxed{ \mathcal Q \text{ is apparent cyclic/master-presentation alphabet data.} }$$

More precisely:

• existing carrier support: no $\mathcal Q$ component;
• coefficient support: none generically;
• algebraic extension support: none;
• physical relative-cycle support: none;
• cross-sector extension: none detected by the literal source period;
• genuinely new carrier datum: none.

The occurrence of $\sqrt{\mathcal Q}$ in a printed algebraic letter is therefore not a singular-support statement. A singular master-basis presentation may retain apparent $\mathcal Q$ poles without changing the period or its monodromy.

Scope boundary

This verdict is generic along $\mathcal Q=0$ for the displayed homogeneous three-site simplex integral. It excludes intersections with:

• soft support;
• the ten lower-sector linear letters;
• the frozen residue/coefficient discriminant union.

It does not classify generic multi-external-leg algebraic letters, integral lattice extension, or an arbitrary nonhomogeneous specialization.

Evidence

• entry 181 and research/benincasa/q_sheet_resolution_result.json;
• entry 183 and research/benincasa/cyclic_q_assembly_result.json;
• entry 211 and research/benincasa/marici-gm/q-algebraic-kernel-certificate.json;
• primary source temp/arxiv-2408.16386-source/sections/applications.tex, SHA-256 3e92460fe2e34dc21a537c784dab3b2fbcd9b7cfee9e7372f06971b50d8b6f9b.

Outcome contract

{
  "objective": "Determine the intrinsic home of Q in the frozen homogeneous three-site one-loop coefficient system.",
  "status": "complete_generic_nonsoft_homogeneous_system",
  "algebraic_plane_Q_support": false,
  "other_rank7_blocks_Q_support": false,
  "discriminant_Q_support": false,
  "physical_chain_Q_support": false,
  "cross_sector_Q_extension": false,
  "T_Q": "identity",
  "N_Q": 0,
  "Var_Q_physical": 0,
  "classification": "apparent_cyclic_master_presentation_alphabet_data",
  "existing_carrier_sufficient": true,
  "new_carrier_datum": false,
  "scope": "generic nonsoft locus of the source-displayed homogeneous three-site simplex integral"
}

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