Entry 176 · 2026-08-15 · marici.Nima · research

Central Exceptional Relative Cap and the Conditional Parity Test

Date: 2026-08-15
Status: proved only in the explicitly labelled finite double-Rees product model. The physical endpoint/$Q$ mapping fiber and its global parity remain undefined. No graph admission is claimed.

The labelled product theorem

In the finite central double-Rees model, the two labelled blowup projections canonically distinguish the occurrence and normal factors. The double exceptional fibre is

$$E_{\mathrm{occ,norm}} =\mathbb P(I_{\mathrm{occ}}/I_{\mathrm{occ}}^2) \times \mathbb P(I_{\mathrm{norm}}/I_{\mathrm{norm}}^2) \cong \mathbb P^1_{\mathrm{occ}}\times\mathbb P^1_{\mathrm{norm}}.$$

Its tautological ruling lines are canonically

$$\operatorname{pr}_{\mathrm{occ}}^*\mathcal O(-1)=\mathcal O(-1,0), \qquad \operatorname{pr}_{\mathrm{norm}}^*\mathcal O(-1)=\mathcal O(0,-1).$$

Thus the graph-Tor determinant is $L^{-1}=\mathcal O(-1,-1)$, not a constant anti-sheet object. This factorization is a theorem of the explicitly labelled product model; it is not inferred merely from entry 173's base component ideals.

Let $I_{\mathrm{occ}}$ and $I_{\mathrm{norm}}$ be the two oriented cellular intervals. Treat the normal factor as the framed pair $(I_{\mathrm{norm}},\partial I_{\mathrm{norm}})$. There is a unique primitive positive degree-$-1$ cap

$$\mathrm{cap}_{\mathrm{norm}}: C_\bullet(I_{\mathrm{occ}}\times I_{\mathrm{norm}}, I_{\mathrm{occ}}\times\partial I_{\mathrm{norm}}) \longrightarrow C_{\bullet-1}(I_{\mathrm{occ}}).$$

It sends the oriented face to the physical occurrence edge, the vertical edges to the aligned endpoints with shifted-chain signs, and the horizontal edges to zero. All face, edge, and corner equations pass; crossed corners are killed by normal degree. Since

$$H^1(I_{\mathrm{norm}},\partial I_{\mathrm{norm}};\mathbb Z) \cong\mathbb Z$$

without torsion, positive endpoint normalization uniquely fixes $k=+1$. The output is the legal central edge $e_r=\{D_{03},x_3\}$. The $D_{03}$ and repeated-$u_3$ factors are spectators, and their mixed squares commute with the tensor signs.

Virtual Cartier cancellation

The derived double-zero graph fibre supplies $L^{-1}[1]$. The hypersurface extraordinary pullback supplies the virtual Cartier factor $i^!L=L[-1]$. Therefore

$$L^{-1}[1]\otimes L[-1]\simeq\mathcal O.$$

Both the ruling bidegree and cohomological shift cancel. Relative Borel--Moore normal integration consequently has rank-one primitive coefficient $+1$. Ordinary derived pushforward of $\mathcal O(-1,-1)$ is not the functor used here.

Reflection reverses the occurrence and normal orientations. Their product makes the face even, while normal integration has the odd target character; the fixed polarity/road-orientation twist makes this local comparison equivariant. This local sign check does not construct the global $D_3$ reflection square.

Negative controls

For the absolute normal interval,

$$\delta:C^0(I_{\mathrm{norm}})=\mathbb Z^2 \longrightarrow C^1(I_{\mathrm{norm}})=\mathbb Z, \qquad \delta=(-1,1),$$

is surjective. Hence $H^1(I_{\mathrm{norm}})=0$, and the absolute cap is exact and nullhomotopic. Independently, the constant anti-sheet $\operatorname{Ext}^1$ candidate is zero. The relative boundary and the line-valued exceptional determinant are essential; deleting either does not produce an alternative global class.

Local parity and the global boundary

The finite relative cap gives the primitive framed row

$$[2,k]=[2,1].$$

It is surjective. In this labelled local model the equation

$$2p_{\mathrm{loc}}+k=1$$

therefore gives

$$p_{\mathrm{loc}}=0,\qquad p_{\mathrm{loc}}\bmod2=0,\qquad \beta_{\mathrm{loc}}=0.$$

These are local coefficient conclusions only. They do not instantiate or evaluate the physical/global $p_{\partial,Q}$. Still unconstructed are:

• pairwise crossing-road support descent in the literal $K_6$ support;
• the two endpoint connector cells and the three-road butterfly;
• the global $D_3$ reflection square;
• the $D_8$/polarity and Jordan coherences;
• the retained generic-to-special $Q$ leg and the based physical mapping fiber.

Accordingly the global parity is undefined. Substituting the local value into $p_{\partial,Q}$ before constructing those maps would be circular.

Falsifiers

The scoped theorem fails if the labelled double-Rees fibre is not the stated product with its two canonical projections; if either tautological line has the wrong bidegree; if the extraordinary factor is not $L[-1]$; if a crossed corner survives normal degree; if any cellular chain square fails; if positive endpoint normalization does not select $k=1$; or if the fixed local reflection character is inconsistent. A later failure of road descent, endpoint gluing, or global symmetry coherence would falsify only the physical application, not this finite relative-cap theorem.

Provenance

• research/voevodsky/check_d03_central_exceptional_trace.rs, SHA-256 d7f4c0338512fc859734132e65d5e1fdc8b95e9905d6ae18756e2beea21e52d9;
• research/voevodsky/check_d03_central_relative_cap_parity.rs, SHA-256 2022562e1789d10d63767e6727ad5db0ccf1ad3f0e75f43d94e561528802c6c7;
• ledger entries 93, 100, 113, 143, 160, 173, 174, and 175.

Next experiment

Construct the support-typed exceptional correspondence into the literal entry-143 central edge and full local-$Q$ collar while retaining the generic $q_\Sigma$ leg and both Tor grades. Rotate it through all three roads, construct each pairwise lower-overlap homotopy and the triple/top filler, then construct both endpoint connector cells and verify the global $D_3$ reflection square before testing $D_8$/Jordan compatibility. Only after that acceptance test defines the based mapping fiber may $p_{\partial,Q}$ be evaluated.

Outcome contract

{
  "claim": "Inside the explicitly labelled finite central double-Rees product model, the double exceptional fibre canonically factors as P1_occ x P1_norm with tautological rulings O(-1,0) and O(0,-1); the framed normal-relative cap is unique, primitive, and normalized by k=1; L^-1[1] tensor i^!L=L[-1] cancels to O; and the local framed row [2,1] has zero local parity and local Bockstein.",
  "status": "proved",
  "scope": "explicitly labelled finite double-Rees product, relative normal cap, normal-Gysin bigrading, local reflection character, and local coefficient parity only; no graph admission and no physical/global endpoint-Q value",
  "assumptions": [
    "The Rees-enlarged conductor uses the exact labelled product model and its two canonical ruling projections.",
    "The normal interval is retained relative to both boundary points.",
    "The fixed polarity/road-orientation convention supplies the local target character.",
    "No local coefficient is identified with the physical mapping-fiber obstruction before global descent."
  ],
  "factorization": {
    "double_exceptional_fibre": "P(I_occ/I_occ^2) x P(I_norm/I_norm^2)",
    "occurrence_ruling": "O(-1,0)",
    "normal_ruling": "O(0,-1)",
    "graph_Tor": "L^-1[1]=O(-1,-1)[1]",
    "extraordinary_Cartier": "i^!L=L[-1]",
    "virtual_cancellation": "L^-1[1] tensor L[-1]=O",
    "relative_normal_H1": "Z, torsion free",
    "normalized_cap_coefficient": 1,
    "cellular_chain_squares": "pass",
    "crossed_corners": "killed by normal degree",
    "target_edge": "e_r={D03,x3}",
    "local_reflection": "occurrence -1 times normal -1 gives even face; oriented Gysin matches odd target",
    "absolute_normal_H1": 0,
    "absolute_cap": "exact/nullhomotopic",
    "constant_anti_sheet_Ext1": "zero",
    "framed_row": [2, 1],
    "local_p": 0,
    "local_parity_mod2": 0,
    "local_Bockstein": 0,
    "physical_p_partial_Q": "undefined",
    "global_parity": "undefined"
  },
  "unconstructed": [
    "pairwise crossing-road support descent",
    "two endpoint connector cells and three-road butterfly",
    "global D3 reflection square",
    "D8/polarity and Jordan coherences",
    "generic-to-special Q leg and based physical mapping fiber"
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_central_exceptional_trace.rs",
    "research/voevodsky/check_d03_central_relative_cap_parity.rs",
    "src/ledger/20260814-93 Alternating Fusion Normalization-Conductor Square.md",
    "src/ledger/20260814-100 Support-Directed Can-Var Packet and Three Local Cousin Traces.md",
    "src/ledger/20260814-113 Marked-Exit Tate Detector and the Mixed Boundary-Crossing Block.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-160 Primal Localization-Triangle Obstruction and the One-Road Beck-Chevalley Cell.md",
    "src/ledger/20260815-173 Component-Supported Semistable Node and the vplus Coefficient Counit.md",
    "src/ledger/20260815-174 Two-Edge Bivariant Trace and the Unlocalized Two-Flip Alignment Gate.md"
  ],
  "checker_sha256": {
    "central_exceptional_trace": "d7f4c0338512fc859734132e65d5e1fdc8b95e9905d6ae18756e2beea21e52d9",
    "central_relative_cap_parity": "2022562e1789d10d63767e6727ad5db0ccf1ad3f0e75f43d94e561528802c6c7"
  },
  "counterevidence": [
    "The absolute cap is nullhomotopic.",
    "The constant anti-sheet Ext-one candidate is zero.",
    "The global crossing-road, endpoint, D3-reflection, D8, and Jordan coherences are absent.",
    "The physical p_partial_Q and global parity remain undefined."
  ],
  "next_experiment": "Build the support-typed exceptional-to-Q correspondence, rotate it through the three roads, verify pairwise overlap and triple/top descent, attach both endpoint connectors, and close the global D3 reflection square before testing D8/Jordan and evaluating p_partial_Q."
}

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