mul_val
plain-language theorem explainer
The underlying value of a product of two unit residues equals the ordinary product of those values. Anyone building multiplicative characters or the unit group of residues mod N cites this projection lemma. The equality is definitional: it unfolds from the construction of multiplication on the UnitResidue carrier.
Claim. Let $N$ be a distinction natural and let $u,v$ be unit residues modulo $N$ (residues coprime to $N$, packaged with their unit witness). Writing $\mathrm{val}$ for the underlying residue, one has $\mathrm{val}(u\cdot v)=\mathrm{val}(u)\,\mathrm{val}(v)$.
background
In the Primitive Recognition Calculus factorization layer, residues live in DistinctionNat. A unit residue modulo $N$ is a residue whose value is coprime to $N$, packaged as a structure with a value field and a unit-witness field. The module doc frames these as the carrier for finite multiplicative character theory.
Multiplication on unit residues is defined componentwise on the value field (with a separate closure proof that the product remains a unit). The present lemma isolates the value projection of that product. It sits beside the identity unit residue and the corresponding identity-value lemma, forming the basic monoid interface before certificates and character theory are built.
proof idea
One-line definitional proof by rfl. Once multiplication on unit residues is defined so that the value of the product is the product of the values, the stated equality is judgmentally true and needs no further rewriting or lemmas.
why it matters
This is bookkeeping infrastructure for the unit-residue monoid used in finite multiplicative character theory inside the factorization strand of Primitive Recognition Calculus. It makes the value projection a monoid homomorphism on the nose, so later group or character constructions can reason about ordinary multiplication of distinction naturals while staying inside the unit-residue type.
No downstream dependents are recorded yet; siblings include the multiplication operation, the identity, the identity-value lemma, and the unit-group certificate. In the broader Recognition stack this supports the arithmetic side of the foundation (residue orbits and factorization), not a direct T0–T8 forcing step.
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