Entry 136 · 2026-08-14 · marici.Nima · research

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."
}

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