Nonsequential Equivalence Class Without Special Gadgets Is Canonical Chain¶

Claim/Theorem¶

Let \((A,\vec X,B)\) be a 3-sequence in a 3-connected matroid with

\[ |X|\ge 3, \]

and assume \(X\) contains no clocks, no p-flans, and no p-coflans in the sense of Hall--Oxley--Semple.

Let

\[ T_1,T_2,\dots,T_n \]

be the maximal segments and maximal cosegments contained in \(X\).

Then:

there is a unique ordering of these sets such that

\[ (A,T_1,T_2,\dots,T_n,B) \]

is a 3-sequence;
every (A,B) 3-sequence in the same nonsequential equivalence class is obtained from this canonical one by:
- arbitrarily reordering the elements inside each \(T_i\), and
- at an interface between a maximal segment and a maximal cosegment, possibly swapping a single guts element with a single coguts element.
Equivalently, after removing the special gadget obstructions, a nonsequential exact-3-separation equivalence class collapses to a canonical ordered chain of maximal segments and maximal cosegments with only local reorder freedom.

The theorem is conditional on excluding clocks, p-flans, and p-coflans.
It applies only to the nonsequential exact-3-separation regime, i.e. intrinsic cut rank 2.
For Conjecture 3, the remaining work is therefore to show that the relevant Quantum Tanner parity-check matroids either exclude those special gadgets, or that those gadgets themselves are incompatible with expansion, LTC irreducibility, or Cayley symmetry.