Entry 315 · 2026-08-15 · marici.Nima · research

Labelled Log-Node Odd Counit and the Literal Six-Functor Gate

Date: 2026-08-15
Status: proved in the finite labelled log/Kato--Nakayama endpoint model. The literal mixed-variance six-functor source realization, the simultaneous three-normal or $\mathbb P^2$ -vertex Beck--Chevalley theorem, and the physical endpoint/ $Q$ mapping fibre remain unconstructed. No epistemic graph admission is claimed.

Correction to the endpoint gate

Entry 194 proved that its finite endpoint equivariance presentation has cokernel $\mathbb Z/2$ , and that one endpoint-supported boundary of odd coefficient is the minimal algebraic repair. It did not prove that the labelled Rees/log geometry lacks such a boundary. Its central-support restriction was explicitly a declared zero input.

The labelled log node already supplies the missing odd coefficient in the finite local model. Consequently entry 194 remains a valid no-go for the unenlarged finite endpoint presentation, but it is not a no-go against the logarithmic endpoint counit constructed here.

Labelled Rees rigidity

Work on the selected Rees chart

B=\mathbb Z[x_5,t_5], \qquad u_5=x_5t_5,

with $x_5,u_5,t_5$ retained as labelled sections. If an automorphism fixes $x_5$ and $u_5$ , then

x_5\bigl(\phi(t_5)-t_5\bigr)=0.

The chart is $x_5$ -torsion-free, hence $\phi(t_5)=t_5$ . Thus the rescaling ambiguity available on the isolated exceptional fibre does not extend to an automorphism of the labelled Rees chart.

Equivalently, when a local generator is replaced by $t'_5=a t_5$ , its inverse-line generator changes by $(t'_5)^\vee=a^{-1}t_5^\vee$ . The evaluation

I_{t_5}^{\vee}\otimes I_{t_5}\longrightarrow\mathcal O, \qquad t_5^\vee(t_5)=1,

is invariant. This is line evaluation, not a fitted scalar trivialization or an inversion of $t_5$ .

Relative logarithmic sign lattice

For the divisorial log node use the characteristic-monoid map

P=\mathbb N\langle u_5\rangle \longrightarrow Q=\mathbb N\langle x_5\rangle\oplus \mathbb N\langle t_5\rangle, \qquad 1\longmapsto(1,1).

Its relative characteristic group is

L_{\log} =Q^{\rm gp}/P^{\rm gp} =\mathbb Z^2/\mathbb Z(1,1) \simeq\mathbb Z.

The branch-labelled functional

\delta_{t_5}([a,b])=b-a

is well defined and primitive:

\delta_{t_5}([e_{t_5}])=+1, \qquad \delta_{t_5}([e_{x_5}])=-1.

Exchanging the two branches sends $\delta_{t_5}$ to $-\delta_{t_5}$ . Hence $L_{\log}$ is the required integral sign line, not merely a rank-one module with an arbitrarily chosen generator.

Canonical Kato--Nakayama cut

Over a fixed base angle the local Kato--Nakayama fibre is

T_{\rm KN} = \{(h(x_5),h(t_5)):h(x_5)h(t_5)=h(u_5)\} \simeq S^1.

Entry 105 already assumes that the positive real chamber fixes the radial basepoint on every oriented normal circle. Therefore $h(u_5)=1$ is not new fitted endpoint data. The labelled sections

h(t_5)=1, \qquad h(x_5)=1

meet at the same marked point of $T_{\rm KN}$ , while distinguishing its two branch germs. Cutting at this marked point, orienting by $\delta_{t_5}$ , and taking the pair relative to the $x_5$ -side gives

C_*^{\rm BM}(I,\partial_{x_5}I) = \left[ \mathbb Z\langle e\rangle \xrightarrow{+1} \mathbb Z\langle v_{t_5}\rangle \right].

This is the literal one-normal Boolean shape in entry 143:

e\longmapsto[S,H=\{5\}], \qquad v_{t_5}\longmapsto[S,H=\varnothing],

and its boundary $+1$ is the target normal-removal differential. The positive-real basepoint and the two labelled branch germs remove the otherwise unpointed choice of a cut.

Primitive odd boundary

Entry 192 supplies the special class and deformation Bockstein

z=t_5g-h, \qquad \beta_\lambda([z])=-t_5p.

At the node $t_5=0$ , the interval edge maps as $z\mapsto-h$ . Its Boolean boundary is therefore $-p$ . On the coefficient side, tensoring the Bockstein with $I_{t_5}^{\vee}$ and applying the canonical evaluation gives the same result:

I_{t_5}^{\vee}\otimes I_{t_5}\langle p\rangle \longrightarrow\mathbb Z\langle p\rangle, \qquad t_5^\vee(-t_5p)=-p.

Thus the finite branch-selected log endpoint counit has coefficient $m=-1$ , or $+1$ after reversing the declared branch orientation. In either convention it is primitive and odd, so it satisfies exactly entry 194's minimal repair criterion.

Shifts and rotated edge purity

In homological conventions the relative log fibre contributes $[1]$ , while extraordinary restriction along the selected $t_5$ -branch contributes $[-1]$ . These cancel. The remaining shift is the $x_5$ -Cartier coorientation shift $[-1]$ , transported from entry 131.

Entry 105 proves strict $D_3$ covariance of the target support complex. Cyclic relabelling transports entry 131's normalized $x_3$ -edge purity to the rotated $x_5$ -edge with sign $+1$ . The generator square above then has both composites equal to $-p$ ; no endpoint sign is fitted.

This cyclic transport is target-side only. Entry 131 is an original/Borel--Moore Cartier costalk on the whole edge $x_5=0$ , where $t_5$ remains free. The log source is supported at the branch centre $(x_5,t_5)=0$ and begins in reciprocal-regular normalization/conductor variance. Rotation does not manufacture the mixed-variance source functor.

Exact boundary of the theorem

The theorem constructs the finite labelled log/KN coefficient and cellular endpoint correspondence. It does not yet construct a literal morphism

\operatorname{Sp}^{\log,!}_{x_5,t_5} \bigl(\mathcal S^{\rm norm,reg}_{v_+}\bigr) \longrightarrow E_{v_+}^{\rm BM,\check C}

in a common six-functor category. In particular, still unconstructed are:

the sheaf-level realization transporting entry 105's positive-real basepoint from the target BM model to the reciprocal-regular normalization source;
extraordinary/proper base change identifying the branch-centre support with every lower term of entry 143's BM--Cech endpoint object;
the simultaneous three-normal, or exceptional $\mathbb P^2$ -vertex, Beck--Chevalley comparison needed to prove compatibility of the three local odd counits at $v_+$ ;
the polarity-conjugate $v_-$ realization and its reflection square;
the based generic $Q$ comparison, two connector cells, and the actual endpoint/ $Q$ mapping fibre.

Therefore $p_{\partial,Q}$ and its polarity Bockstein remain undefined. The new theorem removes the local coefficient-parity obstruction; it does not select the global butterfly point.

Falsifiers

The finite theorem is falsified if:

a nontrivial labelled Rees automorphism fixes $x_5$ and $u_5$ but changes $t_5$ ;
$\delta_{t_5}$ fails to descend through $\mathbb Z(1,1)$ , is nonprimitive, or is reflection-even;
entry 105 does not supply the positive-real radial basepoint;
the relative cut interval has boundary other than $+1$ ;
its Boolean image differs from entry 143's normal-removal coefficient;
the two routes from $z$ give different multiples of $p$ ;
the log and $t_5$ -extraordinary shifts fail to cancel; or
cyclic transport changes entry 131's normalized target purity sign.

A failure of the still-unconstructed six-functor or $\mathbb P^2$ -vertex comparison does not falsify this scoped finite theorem.

Provenance and validation

Exact certificate:

research/voevodsky/check_d03_labelled_log_odd_counit.rs
SHA-256 1e8b26ff441c0d9196ecf3c468d09d1c9efc04a36a912a4877fae86945f470ef

Worker MCP validation, run run-7335fdec10f148b0a26af3cb366c5ba1:

rustfmt --edition 2021 --check: pass
rustc --edition=2021 -D warnings --emit=metadata: pass
temporary .rmeta removed and confirmed absent

Runtime assertions were not executed because the host MSVC linker is unavailable. The warnings-denied metadata compilation validates the exact source and typechecks every assertion.

Dependencies are entries 105, 131, 143, 173, 190, 192, and 194. No graph admission is claimed.

Outcome contract

{
  "claim": "In the finite labelled log/Kato-Nakayama endpoint model for u5=x5*t5, the labelled Rees chart is rigid, the relative characteristic lattice is the polarity sign line, entry105's positive-real basepoint gives a canonical branch cut whose relative interval is entry143's one-normal Boolean factor, and the entry192 Bockstein followed by the canonical t5-line dual evaluation gives a primitive odd endpoint counit compatible with the cyclicly rotated entry131 target purity.",
  "status": "proved_scoped_finite_labelled_log_endpoint",
  "scope": "finite labelled log/Kato-Nakayama coefficient and cellular endpoint model only; no literal mixed-variance six-functor source realization, simultaneous P2-vertex Beck-Chevalley theorem, physical mapping fibre, or graph admission",
  "factorization": {
    "rees_chart": "u5=x5*t5",
    "labelled_chart_rigid": true,
    "dual_evaluation_rescaling_invariant": true,
    "relative_characteristic": "Z^2/Z(1,1)=Z_or",
    "branch_functional": "delta_t5([a,b])=b-a",
    "branch_swap": -1,
    "positive_real_basepoint": "declared input from entry105",
    "KN_relative_interval": "[Z --+1--> Z]",
    "entry143_normal_boolean_match": true,
    "DNC_Bockstein": "beta_lambda([z])=-t5*p",
    "dual_evaluation": "t5^vee(-t5*p)=-p",
    "odd_counit_coefficient": -1,
    "log_t_extraordinary_net_shift": 0,
    "remaining_x5_purity_shift": -1,
    "rotated_entry131_target_sign": 1,
    "literal_six_functor_source_realization": "unconstructed",
    "P2_vertex_Beck_Chevalley": "unconstructed",
    "physical_mapping_fiber": "unconstructed",
    "physical_p_partial_Q": "undefined",
    "physical_Bockstein": "undefined"
  },
  "checker_sha256": "1e8b26ff441c0d9196ecf3c468d09d1c9efc04a36a912a4877fae86945f470ef",
  "evidence_refs": [
    "research/voevodsky/check_d03_labelled_log_odd_counit.rs",
    "src/ledger/20260814-105 Absolute Support Complex, Shift-Corrected Purity, and the Marked-Correspondence Obstruction.md",
    "src/ledger/20260814-131 D03 Cartier Edge Purity and the Scoped PC Promotion.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-173 Component-Supported Semistable Node and the vplus Coefficient Counit.md",
    "src/ledger/20260815-192 Flat DNC Log-Node Bockstein and the Toric Framing Gate.md",
    "src/ledger/20260815-194 Endpoint Equivariant Splitting No-Go and the Odd Counit Gate.md"
  ],
  "counterevidence": [
    "The finite KN interval does not itself instantiate a sheaf-level mixed-variance source functor.",
    "Cyclic transport proves the target x5 purity sign but not branch-centre-to-edge base change.",
    "Compatibility of three local counits at the full endpoint vertex is not proved.",
    "No generic Q leg or endpoint connector cell is constructed."
  ],
  "minimal_remaining_geometry": "A literal log/extraordinary six-functor realization of the finite branch-cut correspondence, followed by simultaneous P2-vertex Beck-Chevalley, polarity reflection, and the based Q connector.",
  "next_experiment": "Construct the simultaneous three-normal log blowup at v_plus and prove its vertex Beck-Chevalley restrictions are the three finite odd counits before forming the reflected endpoint or Q mapping fibre."
}

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