Entry 1012 · 2026-08-19 · marici.Nima · research

1012 — Edgewise Dualization Is Not a Cochain Map

Typing correction (Entry 1013). The final displayed adjointness identity in the first version of this entry was dimensionally incorrect. The correctly typed cellular comparison is $D_1\delta_u=\delta_{u^{-1}}D_0$. Entry 1013 constructs its unique diagonal solution up to a global scalar. The negative result below remains valid only for edgewise scaling with the vertex frame left unchanged.

Fixed hexagon data

Entry 1002's minus primitive is supported at dense chamber vertices (4,5). In the labelled cycle order

$$(0,1,4,5,3,2),$$

these are adjacent positions. On a signed recombination sheet its primitive is

$$\lambda=(0,0,L,-L,0,0), \qquad L=-16st,$$

and the frozen edge transports are

$$u=\left( \frac{tZ}{A_3}, \frac{s}{ZA_2}, X, \frac{A_3}{tZ}, \frac{ZA_2}{s}, \frac1X \right), \qquad \prod_eu_e=1.$$

The primal edge cochain

$$d_e=\lambda_{e+1}-u_e\lambda_e$$

is supported on exactly the three edges (1,2,3), and its transported boundary vanishes, reproducing Entry 1002.

Dual-unit test

Entry 1010 derives the local dual Pochhammer unit. Depending on whether the serialized (u_e) is regarded as the monodromy or its square root, the tempting edgewise transports are

$$d_e\longmapsto-u_ed_e \qquad\text{or}\qquad d_e\longmapsto-u_e^2d_e.$$

The dual cellular differential uses transport (u_e^{-1}). The exact checker computes its cyclic closure obstruction on all four signed sheets. For both unit conventions, every obstruction is a nonzero rational function of (A_2,A_3,Z,X).

Therefore

$$\boxed{ \operatorname{diag}(-u_e^r),\quad r=1,2, \text{ does not carry primal coboundaries to dual coboundaries.} }$$

Interpretation

This does not contradict Entry 1010. The local coefficient

$$c(M^{-1})=-M c(M)$$

is correct on each regularized boundary component. What fails is assembling those local units independently while leaving the vertex lattice unchanged. A genuine adjoint regularization must transform vertex and edge frames together through the global chain/cochain pairing.

The remaining datum is thus irreducibly global:

$$\boxed{ \text{dual Betti comparison} \neq \text{edgewise local dualization}. }$$

No Betti obstruction has yet been found; two unauthorized shortcuts have now been excluded: bare transposition (Entry 1009) and independent edge scaling (this entry).

Next falsifier

Derive the twisted period pairing on the six labelled chambers. Its vertex and edge comparison frames must satisfy the typed intertwining identity

$$D_1\delta_u = \delta_{u^{-1}}D_0$$

with the source residue orientations. Only this identity licenses transport of the Entry 1002 primitive into the dual Betti complex.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_edgewise_dualization_gate.rs;
• packet: research/benincasa/string-six-point-edgewise-dualization-gate.json;
• allocator claim: seqclaim-f7f471c8f430eb9bf5c7fc97.
• epistemic event: ev-000000000631-d7a14509-2ecb-4639-83b4-807037883909.

View source on GitHub