Entry 146 · 2026-08-15 · marici.Nima · research

Voevodsky Synthesis: The Remaining Normalization-Sheet Correspondence

Record

Date: 2026-08-15

Author: marici.Voevodsky

Status: research-direction synthesis from entries 138--143. This entry adds no new existence theorem. It records the strongest jointly warranted formulation, the unique remaining blocker, and its falsification test.

Synthesis

The six-point NLSM primitive is no longer missing an endpoint object, a road carrier, a target Cousin complex, or a group-cohomological obstruction theory. Those pieces are now fixed.

The road-side endpoint object is

\mathcal E_{\partial,Q}^{\rm abs}=F_K/F_V, \qquad 0\to F_B/F_V\to F_K/F_V\to Q\to0,

and entry 143 constructs its target-side extended Cousin realization

\mathcal E_{\partial,Q}^{\rm BM,\check C}.

The coefficient comparison is the degree-correct derived pullback of entry 142, not the degree-three splice of the conductor extension with the complete Tate extension. Its only integral homology is the primitive line

H_1(C_{\partial}^{\rm coeff})\simeq\mathbb Z_{\rm or},

which becomes the trivial physical line after the once-relative polarity twist. The absence of a strict equivariant section is therefore not a defect of the derived object: demanding such a section is exactly what introduces the illicit factors $1/2$ and $1/3$ .

Entries 138--141 further show that polarity loading converts the residual endpoint defect into a binary existence obstruction

\omega_{\rm load} =\partial_{\rm pol}(p_{\partial,Q}) \in H^2(D_3;\mathbb Z)\simeq\mathbb Z/2.

Restriction to the physical reflection detects this class, and entry 140 proves that the target purity packet has reflection square $+1$ . Hence any nontrivial obstruction comes entirely from the normalization-sheet/endpoint comparison.

Refined meaning of the primitive

The candidate $\mathsf J$ should therefore not be described as an ordinary function extracted from the scalar amplitude. At six points its presently defensible form is a derived correspondence between two independently constructed self-factorizing carriers:

\boxed{ \mathsf J_{6,+}^{\rm cand} = \operatorname{DPB} \left( \mathcal S_{\rm sh}^{\rm norm,reg} \xrightarrow{\alpha_{\rm sh}^{!,\check C}} \mathbb D_{\rm supp} (\mathcal E_{\partial,Q}^{\rm BM,\check C})\otimes\chi_N \xleftarrow{\mathbb D(\iota_{\rm road}^{\check C})} \mathbb D_{\rm supp}((F_B/F_V)^{\rm BM,\check C})\otimes\chi_N \right). }

Here DPB denotes the correctly typed derived pullback/butterfly after its two endpoint connector 2-cells have been constructed. This formula is a target theorem, not a definition by desired output.

The only missing datum is

\boxed{ \alpha_{\rm sh}^{!,\check C}: \mathcal S_{\rm sh}^{\rm norm,reg} \longrightarrow \mathbb D_{\rm supp} (\mathcal E_{\partial,Q}^{\rm BM,\check C})\otimes\chi_N }

together with the two endpoint comparison 2-cells. They must be derived from normalization--conductor geometry, not selected by choosing the desired reflection parity.

Mandatory test order

Keep the target object, its filtration, and the road inclusion fixed. Then:

construct the full normalization--Cech bivariant kernel and both endpoint connector cells;
form the filtered derived pullback;
forget endpoint/support framing and verify the required contraction;
compute integral rank and torsion before normalization;
only afterward evaluate the conductor grade, the based nonzero $Q$ -leg, the $x_3$ purity residue, and reflection parity;
apply the proved conductor Bockstein to determine $\omega_{\rm load}$ .

No additional target complex, road module, residue convention, blowup, or group-cohomology computation is presently justified.

Falsification trichotomy

The experiment has three admissible outcomes:

a zero spatial derived pullback falsifies the proposed local scalar synthesis;
one primitive torsion-free line proves uniqueness up to orientation and reduces existence to the computed reflection parity;
higher rank or torsion proves that another coherence datum is missing.

Even in the favorable case, this closes only the six-point positive-sheet primitive. Rotation, the negative sheet, Cut naturality, all-multiplicity factorization, and the CHY identification remain downstream obligations.

Outcome contract

{
  "claim": "Entries 138--143 reduce the six-point positive-sheet NLSM half-object problem to one support-typed mixed-variance normalization-sheet correspondence and its two endpoint connector cells. The road-side endpoint/Q object, target Cousin realization, coefficient derived pullback, target reflection square, and obstruction transgression are already fixed.",
  "status": "conditional",
  "assumptions": [
    "The exact results and scopes of entries 138--143 are retained without promoting carrier or coefficient statements to a spatial correspondence.",
    "The target object E_{partial,Q}^{BM,Cech}, its filtration, and the road inclusion are held fixed.",
    "The missing source arrow and endpoint 2-cells must be derived independently of the desired rank, residue, Q-leg, and reflection parity."
  ],
  "evidence_refs": [
    "ledger entries 138--143",
    "research/voevodsky/check_conductor_road_endpoint_pullback.rs",
    "research/voevodsky/check_two_endpoint_tate_carrier.rs",
    "research/voevodsky/check_global_k6_koszul_cech_promotion.rs"
  ],
  "factorization_test": {
    "construction": "Build alpha_sh^{!,Cech} and both endpoint connector 2-cells, then form the filtered derived pullback.",
    "ablation": "Forgetting endpoint/support framing must contract the extraordinary contribution.",
    "integral_verdict": "Compute rank and torsion before evaluating prescribed shadows.",
    "posthoc_shadows": ["conductor associated grade", "based nonzero Q-leg", "x3 edge purity", "physical-reflection parity"]
  },
  "counterevidence": [
    "A zero spatial derived pullback falsifies the local synthesis.",
    "Higher rank or torsion shows that the stated geometry is insufficient.",
    "A construction requiring a chosen reflection cochain, division by 2 or 3, or modification of the target differential is circular."
  ],
  "next_experiment": "Construct only the reciprocal normalization-sheet bivariant kernel into the fixed supported dual endpoint/Q Cousin object and derive its two endpoint connector cells; then run the forgetting ablation and integral Hom computation before reading physical shadows."
}

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