Corridor Suspension Cocycle and the Graded Reflection-Lift Gate
Record
Date: 2026-08-15
Status: proved integral classification of the suspension-valued reflection required by entry320. The class is the nonzero element of (H^1(C_2;\mathbb Z_{\rm sign})\cong\mathbb Z/2). This does not construct the wall-supported geometric correspondence or the literal entry143 map. No graph admission is claimed.
The graded reflection equation
Reflection exchanges the ordinary and shifted corridor boundaries. Let (ain\mathbb Z) be the degree carried by the forward exchange. Because reflection acts on the shift lattice by sign, the cocycle equation is
Thus every integer is a cocycle. Changing the grading trivialization on one boundary changes (a) by
Consequently
The required exchange (P\leftrightarrow P[1]) has (a=1), hence is the nonzero class. The reverse exchange has degree (-1), so their composite has degree zero and reflection still squares to the identity. But the odd class cannot be removed by any integral change of boundary grading.
Geometric consequence
An ordinary reflection pullback has degree cocycle zero. It therefore cannot realize the entry320 exchange even after replacing a dualizing line by an ungraded two-term wall complex. The wall correspondence must additionally carry a graded lift of the reflection action groupoid whose forward and reverse arrows have degrees (+1) and (-1).
This is not the physical endpoint parity (p_{\partial,Q}). It is a local grading obstruction attached to the corridor reflection. Identifying the two would be circular: the physical class is defined only after the pointed endpoint/(Q) mapping fiber exists.
Minimal next datum
The smallest viable geometry now has two inseparable parts:
- a wall-supported two-term relative-dualizing/excess triangle restricting to (P) and (P[1]);
- a graded reflection-groupoid lift carrying the odd suspension cocycle and compatible with endpoint Beck--Chevalley maps.
Only after both are mapped to literal entry143 normal/Cech rows can the endpoint connector and generic-(Q) equations be assembled.
Executable evidence
Checker:
research/voevodsky/check_dp6_corridor_suspension_cocycle.rs
The checker verifies the cocycle equation, the Smith factor [2], the
nontrivial odd class, and cancellation of forward/reverse degrees.
SHA-256:
6fdc921b506f910088c7aec32fcb7b483901cd91b6baa6ecd080a8496ecd0f37
Fresh rustfmt --check, warnings-denied optimized compilation, runtime
assertions, and JSON output passed. Native PowerShell was used only because
the user-site structured-command surface cannot access this repository or
invoke rustc.
Outcome contract
{
"claim": "The suspension-valued corridor reflection is the nonzero integral class in H1(C2;Z_sign)=Z/2; it cannot be realized by an ordinary degree-zero geometric reflection or removed by an integral regrading.",
"status": "proved_scoped_corridor_suspension_cocycle",
"scope": "integral grading local system on the two-boundary reflection action groupoid",
"smith": [2],
"required_cocycle": 1,
"required_class_mod_2": 1,
"ordinary_reflection_class": 0,
"not_identified_with": "physical p_partial_Q",
"minimal_additional_datum": "a graded reflection-action-groupoid lift with odd suspension cocycle, coupled to the wall-supported two-term excess triangle",
"unconstructed": [
"wall-supported excess triangle",
"literal entry143 realization",
"endpoint connector cells",
"endpoint/Q mapping fiber",
"p_partial_Q and Bockstein",
"D8 and Jordan coherence"
],
"checker": "research/voevodsky/check_dp6_corridor_suspension_cocycle.rs",
"checker_sha256": "6fdc921b506f910088c7aec32fcb7b483901cd91b6baa6ecd080a8496ecd0f37"
}