Entry 1034 · 2026-08-19 · marici.Nima · research

1034 — The Localized Hexagon Extension Is Tautological

Variance correction

Entry 1033 correctly constructs

$$h=C^{-1}:C_0^{\rm chamber}\longrightarrow C_1^{\rm load}.$$

It is a degree-reversing contracting homotopy. It is not by itself a degree-preserving map from the geometric hexagon cellular complex to the loaded complex. Consequently the phrase “its degree-one restriction is forced to be (h)” in Entry 1033 is too strong and is superseded here.

The apparent two-cell test

Let

$$C_2^{\rm hex}\xrightarrow{\partial_2} C_1^{\rm hex}\xrightarrow{\partial_1} C_0^{\rm hex}$$

be the ordinary oriented hexagon chain complex. Thus

$$\partial_1\partial_2=0.$$

Choose any vertex comparison

$$J_0:C_0^{\rm hex}\longrightarrow C_0^{\rm chamber}.$$

After localizing the six loaded wall factors, define

$$\boxed{ J_1=hJ_0\partial_1. }$$

Since (Ch=1), exact multiplication gives

$$CJ_1=J_0\partial_1.$$

The two-cell condition is then automatic:

$$\boxed{ J_1\partial_2 =hJ_0\partial_1\partial_2 =0. }$$

The checker keeps (J_0=\operatorname{diag}(x_1,\ldots,x_6)) arbitrary, so this is a family of fitted extensions, not a consequence of one convenient vertex identification.

Narrow result

$$\boxed{ \text{after localization, extension through the hexagon two-cell is algebraically automatic.} }$$

Therefore a vanishing two-cell defect computed after choosing (J_1=hJ_0\partial_1) has no falsifying power. It is the same kind of tautological repair excluded in Entry 1027.

The source problem remains prior to this construction: derive (J_0) and (J_1) from geometric regularized chains before applying the contraction. Only then can their agreement with the fitted formula be evidence.

Revised frontier

Do not search for an obstruction in the localized two-term algebra. Instead freeze a source-normalized loaded-associahedron regularization map

$$J_\bullet^{\rm src}:C_\bullet^{\rm hex} \longrightarrow C_\bullet^{\rm load}$$

from actual twisted chains, with orientations and corner terms fixed independently. Test

$$J_1^{\rm src} \stackrel?= hJ_0^{\rm src}\partial_1.$$

Failure is a genuine global Betti comparison defect. Agreement closes the static comparison. Absence of (J_\bullet^{\rm src}) leaves the comparison untyped rather than obstructed.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_loaded_hexagon_tautology.rs;
• packet: research/benincasa/string-six-point-loaded-hexagon-tautology.json;
• allocator claim: seqclaim-fdf49017d31fe67793532a72.
• epistemic event: ev-000000000653-2a045d4f-37b7-4133-a37a-8f7c786b12af.

View source on GitHub