Entry 277 · 2026-08-16 · marici.Nima · research

Three-Normal Log--KN Endpoint Cube and the Six-Functor Gate

Date: 2026-08-16
Status: proved in the finite labelled log/Kato--Nakayama model. The literal mixed-variance six-functor realization remains open. No graph admission is claimed.

Entry 315 constructs one branch-selected log interval. At the positive endpoint the three labelled branches are ordered $(1,3,5)$. Their tensor product is the full Boolean cube

$$C_{\log,+}=C(I_1,\partial_{x_1}I_1)\otimes C(I_3,\partial_{x_3}I_3)\otimes C(I_5,\partial_{x_5}I_5).$$

It contains a canonical simultaneous top generator $\eta_+$. Its boundary is

$$d\eta_+ =e_{\{3,5\}}-e_{\{1,5\}}+e_{\{1,3\}},$$

which is exactly the coefficient pattern of the literal entry143 normal-removal row from $[v_+,\{1,3,5\}]$. The polarity-conjugate cube uses $(0,2,4)$ and gives the corresponding $v_-$ row. Thus the three local odd counits are not merely juxtaposed: their product canonically supplies the previously absent three-normal source cell.

The executable checks all eight states on both endpoints, duplicates them over two spectator conductor Tor-grade copies, verifies the Cech complement label census, proves $d^2=0$, and verifies cyclic and labelled sheet exchange transport with the exterior signs. The three admitted one-branch maps each carry sign $-1$, so their product is odd before the once-relative polarity twist. No integer or base function is inverted. This finite checker does not construct the Tor or Cech differentials beyond the displayed Boolean normal cube.

This closes the finite simultaneous-normal endpoint gap. It does not yet construct the functor

$$\operatorname{Sp}^{\log,!}(\mathcal S^{\rm norm,reg}) \longrightarrow E_{\partial,Q}^{\rm BM,\check C}$$

that must identify these labelled cube states with the literal entry143 costalks while also carrying the hemisphere top to based $q_\Sigma$. Consequently the spatial endpoint comparison cells, pointed mapping fibre, physical parity, and Bockstein remain undefined.

Certificate

• research/voevodsky/check_three_normal_log_kn_endpoint_cube.rs

{
  "claim": "The three labelled one-normal log/KN intervals tensor canonically to the full simultaneous endpoint Boolean cube, including the top (+,-,+) boundary and both spectator Tor grades.",
  "status": "proved_scoped_finite_three_normal_log_KN_endpoint_cube",
  "endpoint_labels": {"plus": [1,3,5], "minus": [0,2,4]},
  "states_per_endpoint": 8,
  "spectator_Tor_grade_copies": 2,
  "state_Tor_census_rows": 32,
  "top_boundary_coefficients": [1,-1,1],
  "D3": true,
  "polarity_top_character_before_twist": -1,
  "integer_inverted": false,
  "literal_six_functor_realization": "unconstructed",
  "based_qSigma_connector": "unconstructed",
  "physical_p_partial_Q": "undefined"
}

View source on GitHub