sameResidue_refl
plain-language theorem explainer
Residue equality modulo a nonzero DistinctionNat modulus is reflexive: every element shares its residue class with itself. Anyone assembling a ResidueOrbitCertificate or working with native modular arithmetic on distinction orbits would cite this. The proof is a one-line unfold of the equality definition followed by reflexivity of propositional equality.
Claim. For every nonzero modulus $N$ in the distinction naturals and every $a$ in the distinction naturals, the residue of $a$ modulo $N$ equals the residue of $a$ modulo $N$.
background
DistinctionNat is the base-neutral finite orbit of repeated distinction (K2.12): an inductive type with zero and successor, standing in for ordinary naturals inside the primitive recognition calculus without committing to a particular numeral base.
In the ResidueOrbit module, residue extracts the native residue class of a DistinctionNat element modulo a nonzero modulus $N$. The relation sameResidue is defined as propositional equality of those residues: two elements are related precisely when residue $N$ $a$ = residue $N$ $b$. The present lemma is the reflexivity half of that equivalence relation.
The surrounding factorization layer builds chart transitions and orbit certificates on top of this modular arithmetic, so reflexivity is needed before symmetry, transitivity, and the additive/multiplicative compatibility lemmas can be packaged.
proof idea
Unfold the definition of sameResidue, which is literally equality of the two residue applications. The goal becomes residue $N$ $a$ = residue $N$ $a$, discharged by rfl. No auxiliary lemmas are required.
why it matters
Feeds directly into residue_orbit_certificate, which packages residue display, the iff with ordinary modular equality, and the equivalence-relation laws (refl, symm, and the rest) into a single certificate structure. Without reflexivity the certificate cannot be inhabited.
In the Recognition Science foundation this sits inside PrimitiveRecognitionCalculus factorization: residue orbits are the discrete modular skeleton on which later forcing-chain and composition-law arguments rest. It is pure infrastructure, not a physics claim, but it closes the equivalence-relation interface that downstream orbit and chart-transition results assume.
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