Papers Gcic Discrete Gauge

The discrete gauge

The discrete gauge is a rule that treats two recognition states as equivalent when their log-ratios differ by an integer multiple of ln φ. In Recognition Science, the log-ratio r tracks the position of a system along a recognition cycle. The rule r ~ r + n·ln φ for any integer n says that shifting r by such a step does not change the physical situation. This module, DiscreteGauge, proves that this rule is a consequence of two earlier results in the forcing chain, not an extra assumption.

The two earlier results are T6 and T7. T6 states that each tick of the recognition cycle changes the log-ratio by an integer multiple of ln φ. T7 states that the sum of eight consecutive changes is zero, a property called 8-tick neutrality. The module defines a valid trajectory as a sequence of eight steps, each an integer multiple of ln φ, whose sum is zero. It then proves the key theorem: for any integer n, there exists a valid trajectory that starts at r and ends at r + n·ln φ. The simplest such trajectory moves forward by n, then backward by n, and stays still for the remaining six steps.

Because such a trajectory exists for every integer n, any two log-ratios differing by n·ln φ are dynamically connected by valid recognition steps. The identification r ~ r + n·ln φ is therefore not imposed by hand; it is forced by the dynamics. The module also defines the compact phase Θ = r / (ln φ) mod 1, which lies in [0, 1). This phase is invariant under the gauge rule: shifting r by n·ln φ leaves Θ unchanged. The compact phase gives a well-defined coordinate on the equivalence classes of the gauge.

The result closes a gap in the GCIC paper, where the discrete identification had been treated as an explicit model input. The module upgrades that input to a theorem. The structure mirrors Bloch's theorem in solid-state physics: the φ-lattice from T6 acts like a periodic potential, and the 8-tick periodicity from T7 acts like the crystal period, together producing a compact phase space without imposing periodicity by hand.

THEOREM discrete_gauge_forced · IndisputableMonolith/Papers/GCIC/DiscreteGauge.lean

THEOREM discrete_gauge_forced · IndisputableMonolith/Papers/GCIC/DiscreteGauge.lean

THEOREM eight_tick_compactification · IndisputableMonolith/Papers/GCIC/DiscreteGauge.lean

THEOREM compactPhase_gauge_invariant · IndisputableMonolith/Papers/GCIC/DiscreteGauge.lean

What this page does not claim

This answer does not claim the discrete gauge is a conventional gauge symmetry in the sense of field theory. This answer does not claim the module derives the value of φ itself. This answer does not claim the module establishes any empirical predictions.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Papers/GCIC/DiscreteGauge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: