pion_triplet_mod
plain-language theorem explainer
The modular identity 8 mod 5 equals 3 encodes the alignment between the eight-tick octave and the five-step structure used for pion rung assignment on the phi-ladder. Physicists deriving hadron masses within Recognition Science would cite this when fixing the triplet placement relative to the forcing chain. The proof is a direct native_decide evaluation of the arithmetic with no additional lemmas.
Claim. $8 mod 5 = 3$ holds as the modular signature connecting the eight-tick octave to the phi-ladder rung for the pion triplet.
background
The PionMasses module derives the masses of the charged and neutral pions from Recognition Science by placing them on the phi-ladder as quark-antiquark bound states. The eight-tick octave (period 2^3) is the T7 step in the UnifiedForcingChain, while the phi-ladder assigns masses via yardstick times phi to the power of (rung minus 8 plus gap(Z)). The supplied doc-comment states that 3 relates to the 8-tick via the relation 8 mod 5 = 3.
proof idea
The proof is a one-line wrapper that applies the native_decide tactic to evaluate the arithmetic equality directly.
why it matters
This identity anchors the pion triplet rung calculation to the eight-tick octave landmark (T7) of the forcing chain and supports the mass predictions listed in the module doc-comment. It supplies the modular step required for consistent phi-ladder placement of the pion triplet without invoking the Recognition Composition Law or explicit mass formulas. No downstream theorems are listed, so the result closes a local arithmetic interface for the P-013 derivation.
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