IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Factorization.SubstrateDichotomy
Formalizes a substrate dichotomy for integer factorization in the primitive recognition calculus: a coherent substrate can deliver a certified factor readout (Door B), while a definite ledger magnitude alone cannot extract factors. Packages both sides as a dichotomy certificate for the master factorization certificate. Structural separation of what the substrate may supply; no claim that the RS substrate realizes a factoring speedup.
claimEither a coherent substrate supplies a certified factor readout for some base modulo $N$ (Door B oracle), or a definite ledger magnitude cannot extract a nontrivial factor from magnitude data alone. The two sides are packaged as a substrate dichotomy certificate. Factoring speedup (uniform delivery below classical cost) is not asserted.
background
In the factorization branch of the primitive recognition calculus, one asks what a recognition substrate can certify about factors of an integer $N$. Upstream, physical period readout supplies the readout interface that a substrate may or may not implement. The local setting is Branch B: the antecedent that the substrate supplies a certified factor readout for some base modulo $N$, identified with the Door B oracle.
That antecedent is not proved for the RS substrate. A factoring speedup is the stronger claim that the antecedent can be delivered uniformly in $N$ at cost below classical factoring; that remains an open performance problem. The module therefore separates capability claims from performance claims.
Sibling content records the positive side (coherent substrate delivers a factor certificate), the negative side (definite ledger magnitude cannot extract a factor), and a bundled dichotomy certificate that MasterCertificate can import without committing to either branch as a theorem about RS physics.
proof idea
Definition-and-certificate module, not a single deep proof. One side states that coherence of the substrate yields a factorization delivery (Door B shaped). The other side states that a definite ledger magnitude cannot extract a nontrivial factor from magnitude data alone. Those two statements are wrapped into a substrate dichotomy certificate object. No reduction of classical factoring hardness is attempted; the structure is packaging and separation of antecedents for downstream master certification.
why it matters in Recognition Science
Feeds the factorization MasterCertificate module, which imports this dichotomy so the master certificate can cite a clean Branch B interface without smuggling an unproved RS speedup. In the Recognition framework this sits under Foundation: it keeps the forcing and readout story honest about what is structural (oracle shape, magnitude non-extraction) versus what is open (uniform sub-classical delivery on the RS substrate). Without the dichotomy, master certification would either overclaim Door B for RS or leave the factoring branch unscoped.
scope and limits
- Does not prove the RS substrate realizes Door B or any factor readout.
- Does not claim a factoring speedup below classical cost, uniformly in N.
- Does not reduce integer factoring hardness or give a classical algorithm.
- Does not identify which physical RS layer would implement coherence.
- Does not discharge MasterCertificate; only supplies the dichotomy import.