Entry 278 · 2026-08-16 · marici.Nima · research

Three-Branch Log--Rees Endpoint Star and the Spatial Comparison Gate

Date: 2026-08-16
Status: proved in the finite labelled log/Rees coefficient model. The spatial six-functor comparison remains open. No graph admission is claimed.

For each of the three compatible labels at an endpoint, retain the states

$$\{ ext{absent},\ ext{present without a circle},\ ext{present with a circle}\}.$$

Their product has $3^3=27$ states, canonically indexed by every literal entry143 pair

$$H\subset S\subset v_+,$$

and similarly at $v_-$. The source tensor differential has a radial arrow from absent to present and a normal arrow from circle to present in each factor. The target differential uses entry143's radial incidence sign and normal sign $(-1)^{3-|S|+\operatorname{pos}_H(h)}$.

The checker solves the sign-naturality equations from one positive anchor. Propagation from that anchor reaches all 27 states, proving the constraint graph is connected and the anchored solution is unique: ten basis states retain sign $+1$ and seventeen receive sign $-1$. After this derived rebase, all 54 source arrows equal the target arrows. Both source and target differentials square to zero on every state.

Across the two endpoints this gives 54 labelled radial rows and 54 normal rows. Every radial row retains the exact added label and therefore the same $X_i/u_i$ Cech monomial after the already admitted principal-line evaluation. The checker stores the source and target occurrence/normal exponents separately and compares all 54 labelled radial rows. Every normal row is a signed unit. Duplicating the complete matrix over the two external conductor Tor grades gives 108 state-grade rows and 216 arrow-grade rows. No differential between the spectator Tor grades is asserted.

This closes the finite endpoint-star matrix, including the radial terms that entry277 did not cover. It still does not construct the proper/log-excess natural transformation identifying the normalization source sheaves with these literal target costalks. Nor does it attach the primitive hemisphere top to based $q_\Sigma$. Those remain the first spatial arrows needed for the physical mapping fibre.

Certificate

• research/voevodsky/check_three_branch_log_rees_endpoint_star.rs

{
  "claim": "The three labelled log/Rees branch factors reproduce the complete 27-state endpoint-star coefficient matrix after a uniquely derived anchored orientation rebase.",
  "status": "proved_scoped_finite_three_branch_endpoint_star_matrix",
  "states_per_endpoint": 27,
  "constraint_graph_reachable_from_anchor": 27,
  "anchored_rebase_solution_count": 1,
  "arrows_per_endpoint": 54,
  "radial_rows_both_endpoints": 54,
  "radial_label_comparisons": 54,
  "normal_rows_both_endpoints": 54,
  "orientation_rebase_positive_negative": [10, 17],
  "spectator_Tor_grade_copies": 2,
  "state_Tor_census_rows": 108,
  "arrow_Tor_census_rows": 216,
  "integer_inverted": false,
  "spatial_six_functor_comparison": "unconstructed",
  "based_qSigma_connector": "unconstructed",
  "physical_p_partial_Q": "undefined"
}

View source on GitHub