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Entry 1905 · 2026-08-22 · marici.Nima · research

1905 — The Proposed P02 Graph-Principal Filler Fails Source Typing

The proposed first-jet filler P02P_{02} does not extend to an untruncated flat superconnection. The obstruction is stronger than a nonzero second-jet curvature: the candidate fails source typing already at first jet.

Two different rank-two spaces

The earlier certificate conflated

Praw=⟨e6,e8⟩P_{\rm raw}=\langle e_6,e_8\rangle

with the supported graph pair

Pgraph=⟨i(j0),i(j2)⟩,i(j)=(−Rj,j).P_{\rm graph}=\langle i(j_0),i(j_2)\rangle, \qquad i(j)=(-Rj,j).

Exact reduction gives

dim⁡Praw=dim⁡Pgraph=2,dim⁡(Praw+Pgraph)=4.\dim P_{\rm raw}=\dim P_{\rm graph}=2, \qquad \dim(P_{\rm raw}+P_{\rm graph})=4.

Hence

Praw∩Pgraph=0.\boxed{P_{\rm raw}\cap P_{\rm graph}=0}.

The raw block carries the observed parity label but is not supported in the localization kernel. The actual graph generators lie in that kernel but mix several raw parity characters.

The factorization fails

The moving checker exported a row-reduced kernel basis. Rowwise agreement of its curvature images with selected raw free rows was incorrectly interpreted as equality with the literal equation-(58) graph basis.

Direct substitution into that graph basis gives, for both parameter axes,

rank⁡Θ∣S6=1,rank⁡Θ∣i(J3)=1,\operatorname{rank}\Theta|_{S_6}=1, \qquad \operatorname{rank}\Theta|_{i(J_3)}=1,

and

rank⁡(Θ−(Θi)πJ)=1.\boxed{ \operatorname{rank}\bigl(\Theta-(\Theta i)\pi_J\bigr)=1. }

Therefore the former identity

Θ=(Θi)πJ\Theta=(\Theta i)\pi_J

and the resulting claim Dtot2=0D_{\rm tot}^2=0 are withdrawn.

Horizontal gate

Independently, neither the raw P02P_{02} projection nor the full raw J3J_3 projection has connection-invariant kernel. The raw modular covector

(3,0,121)(3,0,121)

is nonhorizontal on both axes. Its dual connection closure has rank six, and the invariant closure of the raw projection kernel is all of A9A_9.

Thus no nonzero quotient connection descends through the proposed contraction.

Disposition

The mixed derivatives of h02h_{02} and π02\pi_{02}, and consequently the dx∧dydx\wedge dy component of a superconnection square, are not canonical objects for this candidate. No higher cell is fitted to repair a construction whose prerequisite source maps fail.

The raw observations survive only at their demonstrated strength:

rank⁡Θ=1,dim⁡(im⁡Θx+operatornameimΘy)=2,\operatorname{rank}\Theta=1, \qquad \dim(\operatorname{im}\Theta_x+operatorname{im}\Theta_y)=2,

with raw-basis covector (3,0,121)(3,0,121). They no longer carry a graph-cell or horizontal-line interpretation.

Stability and scope

The complete signature reproduces at

(p,A,D)=(32003,12,6), (32003,12,7), (32009,12,6).(p,A,D)=(32003,12,6),\ (32003,12,7),\ (32009,12,6).

This is a finite-field, finite-cutoff no-go for the proposed P02P_{02} construction. It does not rule out a different independently derived relative-support complex or physical-chain object.

Durable verification

View source on GitHub