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IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Factorization.RecognitionLowerBound

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Magnitude-only observables on a factorized recognition pair see only the product orbit, never the separate left or right factors. The module packages that invisibility as a lower-bound certificate: no postprocessing of the product magnitude recovers either factor. Anyone working the factorization layer of primitive recognition calculus would cite it before reading physical period out of product data. The argument is definitional plus short non-existence lemmas against left/right extractors.

claimA magnitude-only observable on a recognition factorization depends only on the product orbit position. In particular, there is no magnitude-only postprocessing of the product that recovers the left factor or the right factor. The module records this as a recognition lower-bound certificate.

background

Primitive recognition calculus treats observables on multiplicative structure (characters, orbits, factorizations). Upstream, the finite multiplicative character layer supplies the orbit language used here: product orbits versus separate left and right factors.

A magnitude-only observable is one that is insensitive to phase or signed factor data and responds only to absolute scale along the product. The product-magnitude observable is the canonical example; left-factor and right-factor observables are shown not to be magnitude-only.

The local setting is factorization of recognition data: if the only accessible readout is a function of the product magnitude, individual factors are information-theoretically hidden. That is the lower bound the module names and certifies.

proof idea

The module introduces the magnitude-only predicate, the product-magnitude observable, and a product-magnitude postprocess class. Short lemmas show the product observable (and its postprocesses) are magnitude-only, while bare left- and right-factor observables are not. Non-extraction theorems then state that no product-magnitude postprocess recovers either factor. Those facts are bundled into a recognition lower-bound certificate object and a witness term that the certificate holds.

why it matters in Recognition Science

Physical period readout imports this module: before a period can be read from product-scale data, one must know that product magnitude cannot smuggle out separate factors. The lower-bound certificate is the formal guardrail for that step in the factorization stack of primitive recognition calculus. It sits under the Foundation domain and supports the claim that recognition of structured product data is strictly coarser than full factor recovery, which is the right granularity for period and octave-style readouts downstream.

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