A global-gauge-resistant Heisenberg separation

Exact statement

On the exact four-branch support $\{0a^m\#0,0a^m\$0,1a^m\#1,1a^m\$1:m\ge0\}$ , there is a fixed trim unambiguous $H_3(\mathbb Z)$ -valued target with $A_f^1(N)=2N$ for $N\ge2$ . After minimization over every single global gauge $L_{r,c}f(w)=r^{|w|}f(w)c$ , where $(r,c)$ may depend on the horizon but acts on all four branches, optimized raw height is $\Theta(\sqrt N)$ and optimized canonical height is $\Theta(N)$ .

StatusProved and independently audited

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Two marker-separated components have incompatible abelianized drifts. One global length gauge acts identically on both and cannot remove their linear difference.

Definitions

The four branches have values $X^m,X^mZ^m,Y^m,Y^mZ^m$ , with $Z=[X,Y]$ central and $|Z^m|=\Theta(\sqrt m)$ .

Proof or evidence

Equal-length branch delays remain conjugates of Z^m, giving the raw lower bound. Comparing the X- and Y-components in abelianization leaves m(-1,1) under every global gauge, forcing one canonical exit value to have linear norm.

Incompatible-drift separation →

Verification notes

The archived audit checks 400 random gauges and 6,421 exact word norms. The new independent review adds 13,122 global-delay checks, 6,561 abelianized-drift and canonical-lower-bound checks, 404 two-mode-collapse controls, and the exact four-state boundary at N=2.

Limitations

The separation is only against one global length-and-constant gauge chosen per horizon.
The explicit first-marker modes (X^{-1},X^2) and (Y^{-1},Y^2) cancel both component drifts, and an unrestricted letter-emitting transducer trivializes the proposed coordinate; neither family is covered by the theorem.

Open work

Determine the correct restricted family of presentation-aware corrections before proposing any broader gauge hierarchy.

Notes

This is not unrestricted gauge invariance. A two-mode length gauge collapses the example, and a general letter-emitting transducer baseline collapses the branch-delay argument. No state-indexed gauge hierarchy is inferred.