Entry 194 · 2026-08-15 · marici.Nima · research

Endpoint Equivariant Splitting No-Go and the Odd Counit Gate

Date: 2026-08-15
Status: scoped integral endpoint-splitting obstruction proved. The minimal endpoint-supported odd-counit enlargement is characterized algebraically but not geometrically constructed. No global extraordinary-kernel no-go and no graph admission are claimed.

Finite endpoint complex

Work only in the explicitly labelled integral endpoint coefficient model. The homogeneous polarity-equivariance differential on the defect line is

d_{\mathrm{eq}}=[2]:\mathbb Z\longrightarrow\mathbb Z.

Its Smith normal form is $[2]$ . Consequently

H^0=0, \qquad H^1\cong\mathbb Z/2.

A normalized endpoint splitting has the form

s_n(1)=(n+1,n).

Polarity exchanges the two endpoint entries, so equivariance would require

n=-n-1, \qquad\text{equivalently}\qquad 2n+1=0.

This equation has no integral solution. Its obstruction is the nonzero class

[1]\in\operatorname{coker}[2]\cong\mathbb Z/2.

This is an intrinsic no-go for an integral polarity-equivariant normalized splitting inside the stated finite endpoint complex.

Why the existing special Tor class does not repair it

Adjoin a special $\operatorname{Tor}_1$ variable in degree $-1$ with zero differential. The resulting local complex has the form

\mathbb Z[-1]\xrightarrow{0}\mathbb Z\xrightarrow{2}\mathbb Z.

It adds

H^{-1}\cong\mathbb Z

but leaves

H^1\cong\mathbb Z/2.

Thus a closed odd class whose endpoint boundary is zero cannot absorb the splitting obstruction. In particular, this finite calculation does not turn the entry-176 central exceptional Tor class or the entry-192 DNC $\operatorname{Tor}_1$ symbol into an endpoint correction. Their required support comparisons are absent, and a zero endpoint restriction would leave the same cokernel.

The checker takes the literal central-support restriction in this finite packet to be zero as a declared scoped input. It does not derive that restriction from a six-functor, purity, deformation-to-the-normal-cone, or ringed-support theorem.

Minimal endpoint-supported odd counit

Let a new framed variable $\epsilon_{v_+}$ have boundary

d\epsilon_{v_+}=m\,e_{\mathrm{def}},

where $e_{\mathrm{def}}$ generates the endpoint equivariance defect line. The presentation becomes

\left[2\;m\right]:\mathbb Z^2\longrightarrow\mathbb Z,

with

\operatorname{coker}\left[2\;m\right] \cong \mathbb Z/\gcd(2,m).

Therefore the obstruction dies exactly when $m$ is odd. The smallest normalized choice is

m=\pm1.

Accordingly the minimal algebraic repair is one odd, endpoint-supported counit whose boundary evaluates primitively on the defect line. Merely adding a closed odd generator, a degree-zero polarity line, or an even endpoint evaluation does not suffice.

This statement characterizes the required enlargement; it does not construct $\epsilon_{v_+}$ as a geometric class. A geometric realization must type it on the selected endpoint support and prove that its boundary is odd after the literal endpoint restriction.

Scope boundary and dependencies

The theorem depends only on the finite integral endpoint matrix, normalized splittings, polarity exchange, and the declared zero central-support restriction. It is compatible with, but does not instantiate, the following repository data:

entry 138 supplies the loaded polarity character, not an endpoint-supported odd boundary;
entry 158 fixes the local Cartier/coorientation degree budget and leaves the global primal trace and endpoint comparison unconstructed;
entry 176 constructs a degree-zero relative cap after cancellation of its exceptional $[1]$ and extraordinary $[-1]$ shifts;
entry 190 isolates the missing branch-selected logarithmic Beck--Chevalley comparison;
entry 192 constructs a line-valued DNC Bockstein but not its restriction to the literal endpoint packet.

The minimal missing geometric arrow is consequently an endpoint-supported extraordinary or logarithmic counit

\gamma_{v_+}^{\mathrm{odd}}: \mathcal W_{v_+} \longrightarrow E_{v_+}^{\mathrm{BM},\check C}

together with a proved boundary equation

d\gamma_{v_+}^{\mathrm{odd}} =\pm e_{\mathrm{def}} \pmod{2}.

Its source support $\mathcal W_{v_+}$ , branch/conormal framing, literal entry-131 purity comparison, and entry-143 Boolean/costalk realization remain unconstructed. No assertion here rules out an extraordinary, proper, DNC, nearby-cycle, or logarithmic Gysin kernel that supplies such a map.

Until that map, its reflected endpoint mate, the generic $Q$ leg, and the global endpoint connector coherences are constructed, the physical endpoint/ $Q$ mapping fiber is uninstantiated. Hence $p_{\partial,Q}$ , its parity, and its Bockstein remain undefined.

Falsifiers

The scoped no-go is falsified if the endpoint equivariance matrix is not $[2]$ , if its Smith diagonal is not $[2]$ , if an integer solves $2n+1=0$ , or if adjoining a zero-boundary degree- $-1$ generator changes the cokernel of $[2]$ .

The minimal-repair criterion is falsified if $\operatorname{coker}\left[2\;m\right]$ is not $\mathbb Z/\gcd(2,m)$ , or if the obstruction vanishes for even $m$ or survives for odd $m$ .

A future geometrically typed odd counit does not falsify the theorem: it realizes precisely the enlargement absent from the scoped endpoint complex.

Provenance and validation

Exact certificate:

research/voevodsky/check_d03_endpoint_equivariant_splitting_obstruction.rs, SHA-256 cc84e4baa08e47e69d559d8dbdbeb18d4d426c0b81b2fba759e0d3f7ac4b99f5.

The checker certifies the matrix, Smith form, affine parity obstruction, zero-boundary Tor negative control, and odd-boundary repair criterion. Its central-support vanishing is explicitly a declared input. It does not certify a spatial comparison or a global nonexistence theorem.

Relevant ledger inputs are entries 131, 138, 143, 158, 176, 190, and 192.

Next experiment

Construct a branch-selected endpoint support object $\mathcal W_{v_+}$ from the logarithmic DNC/conductor geometry and a support-typed counit $\gamma_{v_+}^{\mathrm{odd}}$ . Verify, without inverting a branch or normal parameter, that its literal entry-143 endpoint boundary is primitive and odd. Then construct the reflected endpoint counit and connector square before attaching the generic $Q$ leg or evaluating any physical parity.

Outcome contract

{
  "claim": "In the finite integral endpoint equivariant splitting complex the homogeneous differential is [2], so H0=0, H1=Z/2 and the normalized affine equation 2n+1=0 has no integral solution. A zero-boundary Tor1 variable adds Hminus1=Z but leaves the obstruction. Enlarging the presentation to [2 m] kills the obstruction exactly when m is odd.",
  "status": "proved_scoped_integral_obstruction",
  "scope": "finite integral endpoint equivariant splitting complex and algebraic odd-boundary repair criterion only; central-support restriction is a declared input; no global extraordinary-kernel no-go, spatial comparison, physical mapping fiber, or graph admission",
  "factorization": {
    "equivariance_matrix": [[2]],
    "smith_diagonal": [2],
    "H0": "0",
    "H1": "Z/2",
    "normalized_splitting": "s_n(1)=(n+1,n)",
    "affine_obstruction": "2n+1=0 has no integral solution",
    "zero_boundary_Tor": "adds Hminus1=Z and leaves H1=Z/2",
    "enlarged_presentation": "[2 m]",
    "enlarged_cokernel": "Z/gcd(2,m)",
    "repair_condition": "m odd",
    "minimal_normalized_evaluation": "m=+/-1",
    "central_support_restriction": "zero is a scoped declared input",
    "endpoint_supported_odd_counit": "algebraically characterized, geometrically unconstructed",
    "physical_mapping_fiber": "unconstructed",
    "physical_p_partial_Q": "undefined"
  },
  "checker_sha256": "cc84e4baa08e47e69d559d8dbdbeb18d4d426c0b81b2fba759e0d3f7ac4b99f5",
  "evidence_refs": [
    "research/voevodsky/check_d03_endpoint_equivariant_splitting_obstruction.rs",
    "src/ledger/20260814-131 D03 Cartier Edge Purity and the Scoped PC Promotion.md",
    "src/ledger/20260815-138 Physical Polarity Loading and the Shifted Butterfly Obstruction.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-158 Local Gysin Sufficiency No-Go and the Global Mapping-Fiber Definition Gate.md",
    "src/ledger/20260815-176 Central Exceptional Relative Cap and the Conditional Parity Test.md",
    "src/ledger/20260815-190 Rees-Line Cancellation and the Log Branch-Selected Beck-Chevalley Gate.md",
    "src/ledger/20260815-192 Flat DNC Log-Node Bockstein and the Toric Framing Gate.md"
  ],
  "unconstructed": [
    "geometric endpoint-supported odd counit with primitive odd boundary",
    "branch/conormal support and logarithmic or extraordinary comparison",
    "literal entry131 purity and entry143 endpoint realization",
    "reflected endpoint mate and endpoint connector coherences",
    "generic Q leg and physical endpoint mapping fiber",
    "physical p, parity, and Bockstein"
  ],
  "counterevidence": [
    "A closed Tor1 variable has zero boundary and leaves the Z/2 obstruction.",
    "The entry176 exceptional shift is canceled by its extraordinary Cartier shift and is not a spare odd endpoint class.",
    "The declared central-support restriction is not derived from spatial geometry.",
    "No present arrow identifies the DNC or exceptional support with the literal endpoint defect line."
  ],
  "minimal_repair": "Construct one endpoint-supported odd counit whose boundary evaluates by an odd integer, minimally +/-1, on the endpoint equivariance defect line.",
  "next_experiment": "Build the branch-selected endpoint support object and odd counit, prove its literal endpoint boundary is primitive without inversion, and attach its reflected mate and connector square before the generic Q leg."
}

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