IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Factorization.PhysicalPeriodReadout
Defines certified period and factor readouts for a base $a$ modulo $N$, packaging the data that a recognition lower-bound argument has extracted a genuine multiplicative order or nontrivial factor. Factorization and substrate-dichotomy developments cite it as the interface between period-finding and factor certificates. The module is mostly structure and conversion lemmas, not a deep existence proof.
claimA certified period readout for base $a$ modulo $N$ is a package asserting that a measured period is the multiplicative order (or a controlled multiple) of $a$ in $(\mathbb{Z}/N\mathbb{Z})^\times$. A certified factor readout is the corresponding package that converts such a period into a nontrivial factor of $N$, together with a certificate type witnessing the physical-period readout.
background
In the Primitive Recognition Calculus factorization stack, one does not treat classical period-finding as a black box. A recognition lower bound supplies quantitative control on how much structure must be present before a period (or factor) can be read out. This module sits immediately above that lower-bound layer and names the readout objects themselves.
The central notions are a certified period readout for base $a$ mod $N$, and a certified factor readout obtained from it. The conversion lemma links a factor readout to a nontrivial factorization of $N$. A separate certificate type records that the readout is the physical one used downstream, rather than an abstract number-theoretic witness alone.
Notation is ordinary modular arithmetic: base $a$, modulus $N$, and the multiplicative order in the units group when it exists. The module imports only Mathlib and the recognition lower-bound development.
proof idea
This is primarily a definition and interface module. It introduces structures for certified period and factor readouts, a conversion from factor readout to nontrivial factorization, and a physical-period readout certificate with its constructor or witness lemma. Argumentative content is thin: conversion lemmas unpack the certificate fields and discharge the standard gcd/order conditions that turn a period into a factor. No deep analytic or forcing argument lives here; those sit in the imported lower-bound module and in downstream dichotomy or coordinate-transform files.
why it matters in Recognition Science
The module is the handoff point between recognition-bounded period extraction and concrete factorization claims. Downstream, PrimeCoordinateTransform and SubstrateDichotomy import it, so any theorem that routes factors through prime coordinates or that splits substrate cases on whether a physical period readout exists depends on these types.
In the broader Recognition Science foundation, factorization is not an isolated number-theory side quest: period structure on the discrete recognition side is part of how the calculus interfaces with arithmetic hardness and with substrate-level dichotomies. Packaging the readout as a certificate keeps later proofs from re-opening the lower-bound bookkeeping.
It does not itself force $\varphi$, the eight-tick octave, or $D=3$; those landmarks live in the unified forcing chain. Its role is local and infrastructural inside the factorization branch of Primitive Recognition Calculus.
scope and limits
- Does not prove that a period readout always exists for arbitrary $a,N$.
- Does not implement or analyze a quantum or classical period-finding algorithm.
- Does not derive prime factorization complexity bounds beyond the certificate interface.
- Does not connect directly to T5–T8 forcing, $\phi$-ladder masses, or $\alpha$ bounds.
- Does not replace the recognition lower-bound module; it only consumes that interface.