Canonical AW-Cap Roof and the Endpoint-Connector Gap

Record

Date: 2026-08-14

Status: proved at carrier level. The canonical integral (D_3)-equivariant AW/cap roof exists and is independent of the front/back convention. It does not select a direct full-cone lift; the endpoint-compatible butterfly connector remains open.

Entry 135 proved that the minimal strict projection is obstructed modulo three while the full augmented cone has an affine rank-nine lattice of integral lifts. The question was whether relative Alexander–Whitney geometry selects one of those nine parameters. The exact answer is no. It canonically selects the derived roof common to every lift.

Claim

Let

[ U=\operatorname{Cone}!\left( C_(v_+)\longrightarrow C_(B_{\rm short}) \right), \qquad T=\operatorname{Cone}!\left( P_{\rm road}\xrightarrow{\epsilon}\mathbb Z \right). ]

Define

[ C_{\rm tag}= \left[ \mathbb Z_{\rm or}\langle\omega\rangle \xrightarrow{N} P_{\rm tag} \right], \qquad N\omega=e_{14}+e_{03}+e_{25}. ]

There is a canonical integral (D_3)-equivariant roof

[ \boxed{ \mathcal R_{\rm AD}^{\rm car}: \quad U\xleftarrow[\sim]{,g_{\rm cap},} C_{\rm tag} \xrightarrow{,m,} T } ]

with

[ g_2(\omega)=-S,\qquad g_1(e_i)=c_i=\partial F_i,\qquad m_1=M_{\rm AD}=R-R^2,\qquad m_2=m_0=0, ]

where (S) is the oriented sum of the six short facets. Both legs induce the saturated complementary-boundary Alexander isomorphism on (H_1). This roof is the canonical AW/cap output. It is not a canonical strict map (U\to T).

Exact AW/cap census

The labelled barycentric boundary of (K_6) has an oriented fundamental cycle with 84 flag triangles. For each long facet (F_i):

  • the positive-normalized front cap is a closed 8-edge cycle equal to the subdivision of the oriented boundary (c_i);

  • the back cap is a second closed 8-edge dual loop;

  • a 16-triangle collar (H_i) satisfies

    [ \partial H_i=z_i^{\rm back}-z_i^{\rm front}. ]

Rotation cycles the three packets. Reflection sends each packet to minus the correspondingly reflected packet. Consequently

[ g_{\rm back}-g_{\rm front}=dH+Hd ]

strictly and (D_3)-equivariantly. Changing the AW convention changes only the representative of one roof; it cannot toggle the unresolved reflection parity.

The left leg is integrally saturated. The matrix ([d_{B,2}\mid c_{14}\mid c_{03}]) has an explicit (8\times8) minor of determinant (-1), and the third cycle has only the norm relation. Thus

[ H_1(C_{\rm tag})=\operatorname{coker}N \xrightarrow{\sim}H_1(U) ]

integrally. On the right, (M_{\rm AD}N=0), (\epsilon M_{\rm AD}=0), and the induced (A_2)-matrix has determinant (+1).

Non-selection theorem

For every solution (F:U\to T) of entry 135’s frozen full-cone system, the nine peripheral equations are exactly

[ \boxed{F,g_{\rm cap}=m.} ]

The coefficient rank remains 71 in 80 variables, so the solution space remains affine rank nine. Every full-cone lift factors the same canonical roof and AW/cap fixes none of the remaining affine parameters.

The next object is therefore not a preferred matrix in the rank-nine family. It is an endpoint-coherent pointing of the roof in the arrow/two-extension category.

Sharp blocker

To turn (\mathcal R_{\rm AD}^{\rm car}) into a pointed butterfly one needs a (D_3)-equivariant contraction, or equivalent connector 2-cells, for the acyclic complement of (g_{\rm cap}), compatible with the endpoint maps in (U) and (T). Relative AW/cap supplies the cap-direction map but not such an inverse or connector.

A strict endpoint-identity inverse would impose (F_0(v_+)=1), while the exact full-cone equations force (F_0(v_+)=3k). The endpoint identity must therefore be carried by butterfly/homotopy data rather than a strict degree-zero component.

This entry does not prove that every additional geometric SDR algorithm is impossible. It proves only that the canonical AW roof, including its front/back homotopy, does not choose one. Until an endpoint-compatible connector is constructed, no canonical direct representative or reflection parity is defined.

Formula objective

The immediate objective is

[ \boxed{ \widehat{\mathcal R}{\rm AD}^{\rm car} \in \operatorname{Lift}{\operatorname{Arr}^2_{D_3}} \left( \mathcal R_{\rm AD}^{\rm car}; \mathbb E_F,\mathbb E_\triangle \right). } ]

Equivalently, construct a pointed butterfly whose underlying roof is the proved (\mathcal R_{\rm AD}^{\rm car}), with endpoint identities and both connector coherences explicit. Only after this pointing exists should one compute its mod-two reflection class and attempt the loaded extraordinary lift toward (d_{\rm sp,sc}) and (G_{03}^{\rm Cousin}).

Evidence

Exact certificate:

  • research/voevodsky/check_k6_strict_ad_chain_map.rs
  • SHA-256 02b2a4691719501aee5d3535a209dbe131534c67048d05ada55d6ff062ed521c

Verification:

rustfmt --edition 2021 --check
rustc --edition 2021 -D warnings -O
executable exit 0
JSON output parses
git diff --check

Outcome contract

{
  "claim": "The labelled relative barycentric AW/cap construction canonically gives a saturated integral D3-equivariant roof U<-C_tag->T with right leg M_AD=R-R^2. Front and back representatives are D3-equivariantly homotopic. Every integral full-cone lift factors this roof, so the roof does not select a point in the affine rank-nine lift lattice.",
  "status": "proved",
  "assumptions": [
    "The K6 incidence signs, ambient orientation, and D3 actions are those reconstructed from the labelled face poset.",
    "The physical road order is F14, F03, F25 and M_AD=R-R^2.",
    "The entry-135 full-cone equations and their frozen peripheral values are retained integrally.",
    "No rational splitting or endpoint-unit normalization is imposed."
  ],
  "evidence_refs": [
    "research/voevodsky/check_k6_strict_ad_chain_map.rs",
    "src/ledger/20260814-135 Strict Alexander Projection No-Go and the Integral Butterfly Objective.md",
    "src/ledger/20260814-115 Boundary-Triad Tate Realization and the Multi-Rees Cartier Bicomplex.md"
  ],
  "factorization_test": {
    "barycentric_fundamental_cycle": "84 oriented flag triangles",
    "front_caps": "three closed 8-edge cycles equal to subdivided oriented long-facet boundaries",
    "back_caps": "three closed 8-edge B-side dual loops",
    "front_back_homotopy": "three 16-triangle collars with boundary back-front",
    "D3_covariance": "rotation-covariant and reflection-odd for front, back, collars, and top chains",
    "left_leg": "g_cap is an integral saturated quasi-isomorphism; explicit determinant -1 minor",
    "right_leg": "m1=M_AD; mN=0; epsilon*m=0; induced A2 determinant +1",
    "canonical_roof": "proved",
    "full_cone_factorization": "all nine frozen peripheral equations are exactly F*g_cap=m",
    "full_cone_dimension": "affine rank 9 remains unchanged",
    "canonical_direct_point": "not selected",
    "reflection_parity": "undefined"
  },
  "counterevidence": [
    "All nine full-cone affine directions factor the same roof, so AW/cap cannot distinguish them.",
    "Front and back caps are explicitly D3-chain-homotopic, so convention reversal does not select or toggle a point.",
    "A strict endpoint-identity inverse would require 3k=1.",
    "No exhaustive no-go is claimed for additional geometric SDR or connector constructions."
  ],
  "sharp_blocker": "Construct endpoint-compatible D3 connector 2-cells, or an equivalent pointed butterfly, over the canonical AW roof.",
  "next_experiment": "Build the endpoint-compatible lift of the canonical roof in the arrow/two-extension category, compute its reflection class, and only then load that same pointed object with occurrence, multi-Rees, reciprocal/BM, and PC/Cousin data."
}