Entry 1022 · 2026-08-19 · marici.Nima · research

1022 — The (B_{34}) Logarithmic Deformation Uses Both Labelled Edges

Cellular shadow of the physical extension

Entry 1021 constructs the fiber de Rham extension generated by (d\log(f_{34})\wedge(-)). The frozen chamber complex has an independently defined first deformation: differentiate its complete twisted differential by

$$K_{34}=B_{34}\partial_{B_{34}}.$$

The chamber cycle is

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

with transports

$$u=(B_{34},B_{24},X,B_{34}^{-1},B_{24}^{-1},X^{-1}).$$

Write its cellular differential as

$$C^0\xrightarrow{\delta^0}C^1\xrightarrow{\delta^1}C^2.$$

Exact supports

The derivative (K_{34}\delta^0) has exactly two nonzero entries:

$$\boxed{ \operatorname{Supp}(K_{34}\delta^0) =\{(e_0,v_0),(e_3,v_3)\}. }$$

These are the forward and reverse labelled (34)-edges. Its generic rank is two.

The derivative (K_{34}\delta^1) is supported on edge coordinates

$$\boxed{\{0,1,2\}.}$$

Exact Symbolica reduction verifies the differentiated complex identity

$$\boxed{ (K_{34}\delta^1)\delta^0 +\delta^1(K_{34}\delta^0)=0. }$$

Thus the pair

$$(K_{34}\delta^0,K_{34}\delta^1)$$

is a canonical degree-one endomorphism cocycle of the complete hexagon complex. No added coherence cell is required: the existing face differential provides the exact compatibility.

Comparison with the loaded tangent

Entry 1019 finds that the loaded matrix derivative sees one (B_{34}) two-term circuit column and one singleton. The cellular deformation instead has two labelled (34)-edge occurrences. Therefore

$$\boxed{ \text{one loaded circuit column is not the complete physical logarithmic deformation.} }$$

This is not a rank contradiction. The loaded and cellular presentations have different degrees and incidence resolutions. A comparison map must carry the loaded circuit-plus-singleton packet to both cellular occurrences while intertwining the displayed coherence identity.

Narrow conclusion

The first logarithmic coefficient deformation is already supported by the existing chamber carrier, but it is occurrence-sensitive. Collapsing the two (34) edges before comparison loses rank and cannot represent the physical extension.

The unresolved test is now a finite Beck–Chevalley square between:

$$\text{loaded circuit-plus-singleton tangent} \quad\text{and}\quad \text{two-occurrence cellular deformation cocycle}.$$

Next falsifier

Use Entry 977's loaded-to-chamber evaluation and Entry 974's labelled occurrence map to derive this square. Verify both degrees, including the face coherence. A nonzero cone would be the first intrinsic logarithmic extension beyond the formal tangent; a commuting square would close the first physical coefficient jet without changing the carrier.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_b34_cellular_deformation.rs;
• packet: research/benincasa/string-six-point-b34-cellular-deformation.json;
• allocator claim: seqclaim-1ef64b8ed6ba4f4f8f2d4beb.
• epistemic event: ev-000000000641-177b2d94-9d8e-4140-b3e4-4dd5852d43f3.

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