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Theorem

FPRD-T71

Cycle-mean duality for optimal weight gauges

Exact statement

On the augmented presentation graph, the minimum sup-norm over real state gauges equals the maximum absolute mean circulation over oriented closed walks, and the minimum over integer gauges is its ceiling. Initial and final weights are represented by edges to a boundary vertex, so there is no further integrality loss.

StatusProved and independently rechecked; classical duality application
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The theorem identifies optimal raw height on one fixed presentation graph. It explains the raw/canonical gap as a mean-circulation versus total-circulation effect.

Definitions

  • The augmented graph adds a boundary vertex ∗\ast, an edge ∗→q0\ast\to q_0 for the initial weight, and edges q→∗q\to\ast for finite final weights.
  • ρ(w)=max⁡C∣circ⁡w(C)∣/∣C∣\rho(w)=\max_C |\operatorname{circ}_w(C)|/|C|.

Proof or evidence

Gauge invariance of circulation gives the lower bound. The upper bound is the feasibility criterion for a pair of difference constraints on every edge; absence of negative cycles follows exactly when the proposed bound dominates every mean circulation.

Cycle-mean gauge duality →

Verification notes

The difference-constraints proof, sign convention, boundary coordinates, and integer shortest-path step were independently reconstructed. Exhaustive checks cover all 343 weighted oriented triangles with edge weights in [-3,3]; see /research-artifacts/result-review-2026-09-01-batch-22/review_prequery_height_output.json.

Limitations

  • This is fixed-graph infrastructure and says nothing about minimizing over different presentations.
  • The result is an application of classical difference-constraints and maximum-cycle-mean duality; no novelty is claimed.

Open work

Seek specialist review and compare the fixed-graph formulation with standard weighted-transducer normalization algorithms.

Notes

Produced by the review lane; captured with attribution and reconciled against this repository's gauge sign convention, which is the packet's after replacing z by -z. CLASSIFIED AS CLASSICAL INFRASTRUCTURE, NOT NOVELTY: it is standard difference constraints plus maximum-cycle-mean potential duality applied to the skew-symmetric bidirected expansion. Interpretation: after optimal weight pushing on a FIXED graph raw height is a maximum MEAN circulation, while a pinned access-tree normalization can concentrate the TOTAL circulation on one chord or final coordinate, so the raw/canonical gap on a fixed presentation is a mean-versus-total effect. The boundary vertex is required; without it the initial and final weights are unconstrained and the identity fails.