Entry 1033 · 2026-08-19 · marici.Nima · research

1033 — The Localized Loaded Complex Has a Unique Internal Contraction

Frozen complex

Entry 1028 types the loaded matrix as the differential

$$C_1^{\rm load}\xrightarrow{C}C_0^{\rm chamber}.$$

Entries 969 and 1032 give

$$C=S\operatorname{diag}(q_i), \qquad q_i=M_i-1, \qquad S\in GL_6(\mathbb Z).$$

Let

$$R_{\rm reg}=R[q_1^{-1},\ldots,q_6^{-1}].$$

Exact contraction

Over (R_{\rm reg}), define

$$\boxed{ h=\operatorname{diag}(q_i^{-1})S^{-1}. }$$

Direct exact multiplication gives

$$\boxed{ Ch=1_{C_0}, \qquad hC=1_{C_1}. }$$

Because (C) is invertible, any two-sided contracting homotopy is necessarily (C^{-1}). The contraction is therefore unique inside the frozen two-term complex.

Its only poles are the six labelled occurrence factors (q_i). No additional irreducible denominator or integer index occurs.

Narrow result

$$\boxed{ \text{the localized loaded complex has no internal Čech or extension obstruction.} }$$

Any remaining global Betti obstruction cannot be obtained by choosing a different inverse, endpoint gauge, or circuit representative inside this complex. It must arise from additional geometric data: a higher-cell lift, an intersection pairing, or a source normalization not encoded by the two-term loaded packet.

This cleanly separates two statements:

$$\text{algebraic loaded complex: canonically contractible after localization},$$

$$\text{global regularized twisted cycles: not yet constructed}.$$

Consequence for the next test

The next calculation should not search for another (6\times6) inverse. Instead, it must construct the comparison from the geometric loaded-associahedron cellular complex to this two-term model. Its restriction to degree one is forced to be (h). The sole possible new class is the failure of that comparison to extend through the native two-cell.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_loaded_complex_contraction.rs;
• packet: research/benincasa/string-six-point-loaded-complex-contraction.json;
• allocator claim: seqclaim-fb8ddb1d9e494d6d1fffc9eb.
• epistemic event: ev-000000000652-27f96777-c562-476d-9498-0f33dc6da763.

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