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Entry 1027 · 2026-08-19 · marici.Nima · research

1027 — The Minimal Loaded Support Cone Splits Tautologically

The full corner-complex lift is impossible

Let KcorK_{\rm cor} be Entry 962's rank-six kernel and let

C:Kcor⟶VhexC:K_{\rm cor}\longrightarrow V_{\rm hex}

be Entry 967's generically invertible occurrence comparison. Any lift from the full 18→1218\to12 corner complex into the twisted hexagon complex would send source cycles to target cycles. Entry 1025 gives

rank⁡(δC)=5.\operatorname{rank}(\delta C)=5.

Therefore such a chain map cannot restrict to CC on KcorK_{\rm cor}.

Minimal supported cone

The smallest algebraic target that forces the six graph classes to close is

Vhex⊕Kcor→DEhex,D=(δ,−δC).V_{\rm hex}\oplus K_{\rm cor} \xrightarrow{D} E_{\rm hex}, \qquad D=(\delta,-\delta C).

Indeed,

D(Ck,k)=0D(Ck,k)=0

for every k∈Kcork\in K_{\rm cor}.

This construction looks like the desired support-sensitive repair, but its extension content must be tested before interpretation.

Exact triangular splitting

Apply the invertible triangular change of frame

T(v,k)=(v−Ck,k).T(v,k)=(v-Ck,k).

Then

D(v,k)=δ(v−Ck),D(v,k)=\delta(v-Ck),

and hence

DT−1=(δ,0).\boxed{ DT^{-1}=(\delta,0). }

Thus the cone is canonically isomorphic, once CC is fixed, to the direct sum of the ordinary twisted hexagon complex and six zero-differential support generators.

Generically,

dim⁡H0=6+1=7,dim⁡H1=1,\dim H^0=6+1=7, \qquad \dim H^1=1,

with

H0≃Kcor⊕ker⁡δ.H^0\simeq K_{\rm cor}\oplus\ker\delta.

Narrow conclusion

the minimal cone built only from δC is tautologically split and supplies no comparison coherence.\boxed{ \text{the minimal cone built only from }\delta C\text{ is tautologically split and supplies no comparison coherence.} }

It packages the six occurrence classes but does not explain their coupling to chamber transport. Interpreting its zero-differential summand physically would be exactly the prohibited operation of adjoining support generators solely to kill an obstruction.

Revised frontier

The required lower cell must be independently source-derived and cannot be the formal graph cone of CC. The finite search is now:

  1. return to the two-step pivot-transition path of Entry 966;
  2. retain its intermediate pivot endpoint rather than only the composite coefficient MAMB−1M_AM_B-1;
  3. assemble the resulting bar/group-cohomology differential on all six occurrences;
  4. test whether its comparison to the hexagon produces δC\delta C by a native homotopy.

Only that pre-existing path homotopy can turn the rank-five boundary into derived coherence without a post hoc support summand.

Durable evidence

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