physics_faithful
plain-language theorem explainer
The physics realization built from tick steps on recognition states yields a faithful arithmetic interpretation. This guarantees that distinct logical expressions map to distinct physical states and that the zero element stays distinguishable from successors. Researchers deriving the logic-to-physics forcing chain would cite the result to confirm the minimal tick model introduces no arithmetic collapse. The proof proceeds by case analysis on the equality hypothesis for injectivity followed by congruence on the tick projection and direct appeal,
Claim. The physics realization satisfies faithful arithmetic interpretation: the interpretation map from logical expressions to physical states is injective, and the zero element is not identified with any successor step under the tick generator.
background
The module supplies a stable interface for the universal forcing chain by taking identity ticks as the step action, recognition states as the carrier, and equality cost as the minimal physical realization. The tick constant supplies the fundamental RS time quantum τ₀ = 1. LogicNat is the inductive type whose identity and step constructors mirror the orbit {1, γ, γ², …} closed under multiplication by the generator γ. The extracted arithmetic definition pulls the Peano surface directly from any LogicRealization’s identity-step orbit.
proof idea
The tactic proof splits into two subgoals. Injectivity is discharged by introducing a and b with equality hypothesis h, performing case analysis on h, and closing by reflexivity. The zero_step_noncollapse subgoal introduces n and h, forms the tick projection of h by congruence, and applies the zero_ne_succ theorem from ArithmeticFromLogic to obtain the required contradiction.
why it matters
The declaration anchors the physics hook inside the logic-to-physics forcing chain by confirming that the tick-based realization preserves arithmetic fidelity. It supplies the base case for any downstream construction that extracts physical states from logical arithmetic without collapse, consistent with the eight-tick octave and the transition toward D = 3. No parent theorem is listed among the used-by edges, indicating the result functions as a foundational interface rather than an intermediate lemma.
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