GaugeEquivalent
plain-language theorem explainer
Two eight-tick propagating modes are ratio-gauge equivalent when they share the same recognition ratio at every tick of the cycle. Downstream uniqueness theorems cite this relation to identify all zero-cost modes with the canonical null recognition mode. The definition is a pointwise equality of ratios over Fin 8; no cost algebra is involved.
Claim. Two propagating modes $M$ and $N$ (each an assignment of a carrier event to every tick of the eight-tick cycle) are gauge-equivalent when, for every tick $i\in\{0,\ldots,7\}$, the recognition ratio of $M$ at $i$ equals the recognition ratio of $N$ at $i$.
background
The module isolates the null recognition mode (NRM): the unique zero-cost propagating mode of the eight-tick recognition cycle, up to ratio-gauge equivalence. It deliberately withholds photon labels (spin, polarization, lightlike kinematics); those live in a later photon module.
A propagating mode is a map from the eight ticks (Fin 8) to carrier events. Each carrier event carries a positive recognition ratio; the identity event sits at ratio $1$, the J-cost minimum. The eight-tick period is the fundamental RS evolution octave (forcing chain T7).
Gauge equivalence here is purely ratio-level: modes that agree tick-by-tick on ratios are identified, independent of any other event data. Total mode cost is summed per-tick reciprocal recognition cost; zero total cost forces every ratio to $1$, which is how uniqueness later lands on the canonical NRM.
proof idea
Definitional, not a proof. The predicate is the universal quantification over the eight ticks that the two modes' carrier-event ratios coincide. No lemmas are applied; downstream theorems unfold this Prop and discharge the pointwise equalities (typically via a ratio-equals-one lemma when total cost vanishes).
why it matters
This is the equivalence relation that makes the NRM unique "up to gauge." The uniqueness theorem states that any propagating mode with total cost zero is gauge-equivalent to the canonical NRM (identity ratio at every tick). The certificate structure packages existence of a zero-cost mode, canonical zero cost, and this uniqueness clause.
In the framework, the eight-tick octave (T7) is the discrete stage on which modes propagate; zero J-cost at the identity ratio is the recognition-theoretic vacuum channel. Keeping the relation ratio-only separates kinematic gauge from later physical identification of the channel with the photon.
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