delta_factorization_character_theory_certificate
plain-language theorem explainer
The factorization character-theory lane is fully certified: every named sub-ledger (chart transition, residue orbits, unit group, period spectrum, finite multiplicative characters, recognition lower bound, period readout, prime-coordinate transform, goal closure, coordinate uniqueness, period factor/existence, even-period gap, and substrate dichotomy) holds simultaneously. Anyone citing the closed factorization ledger uses this single Prop. The proof is a pure structure assembly of the already-proved component certificates.
Claim. There exists a completed ledger asserting that the factorization character-theory package holds: chart transitions preserve product magnitude, residue orbits and unit groups are certified, the period spectrum and finite multiplicative characters exist, a recognition lower bound is in force, physical period readout and prime-coordinate transform interfaces are discharged, goal closure and coordinate uniqueness hold, period factors exist, even-period gaps force nontrivial factorization, and the substrate dichotomy is settled.
background
In the Primitive Recognition Calculus factorization lane, each major claim is packaged as a small certificate structure (a Prop bundling the named lemmas that close that subgoal). The master ledger DeltaFactorizationCharacterTheoryCertificate is the conjunction of those certificates: chart transition, residue orbit, unit group, period spectrum, finite multiplicative character, recognition lower bound, physical period readout, prime-coordinate transform, goal closure, coordinate uniqueness, period factor, period existence, even-period gap, and substrate dichotomy.
Upstream, chart transition records that factor-pair products display the same natural magnitude and exhibits an explicit ambiguous product; finite multiplicative characters supply a principal character that is multiplicative on units; goal closure names the residual, pins its provenance, and shows the prime-coordinate readout commitment is exact, closed, and would recover a prime divisor; coordinate uniqueness equates prime orbits with ordinary natural primes and factors data accordingly; even-period gap reduces existence questions to nontrivial factorization of a witness.
The local setting is the Foundation factorization module: a character-theoretic account of how recognition periods and prime coordinates factor, prior to any physics readout.
proof idea
Term-mode structure construction. Each field of the master ledger is filled by the corresponding named certificate theorem already proved in a sibling module: chart transition, residue orbit, unit group, period spectrum, finite multiplicative character, recognition lower bound, physical period readout, prime-coordinate transform, goal closure, coordinate uniqueness, period factor, period existence, even-period gap, and substrate dichotomy. No new algebra is performed; the proof is pure aggregation of those component certificates into one Prop.
why it matters
This is the master certificate for the factorization character-theory lane inside Primitive Recognition Calculus. It closes the ledger that packages chart transitions, multiplicative characters on residue/unit data, period spectrum and existence, even-period gap factorization, coordinate uniqueness, and goal closure into a single citeable Prop. Downstream consumers (none yet wired in the graph) can depend on one object rather than fifteen separate certificates.
In the broader Recognition Science forcing picture, factorization and period structure sit under the eight-tick octave and the discrete recognition substrate that later force $D=3$ and the $\varphi$-ladder mass formula. The certificate does not itself derive $J$-uniqueness or $\varphi$; it certifies the number-theoretic factorization layer those later steps rely on. With used_by_count = 0, this is presently a terminal ledger node rather than an intermediate lemma.
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