StrengthTag
plain-language theorem explainer
Five-level strength taxonomy for Primitive Recognition Calculus claims, from distinction-and-repetition alone up through classical logic. Auditors and foundation papers cite it to label how much structure a PRC result actually uses. It is a plain inductive enumeration with decidable equality; no proof content.
Claim. A strength tag is one of five labels on a PRC claim: (1) forced by distinction and finite repetition alone; (2) uses completed orbits or completed stable trace families; (3) uses selection of witnesses from stable families; (4) uses controlled subtrace-class or power-class formation; (5) uses excluded middle or full classical reasoning as an extension.
background
Primitive Recognition Calculus (PRC) is the foundation layer that builds arithmetic and recognition structure from bare distinction and finite repetition, before the full Recognition Science forcing chain (T0–T8). Claims in this layer are not all equally strong: some follow from the minimal apparatus, others quietly import orbit completion, choice of witnesses, power-class formation, or classical logic.
The module packages a small audit vocabulary so each claim can carry an explicit strength label. Sibling material includes an audit record pairing a claim name with its tag, and inequality lemmas separating the constructors (e.g. distinction-only is not trace-closure).
Upstream arithmetic-from-logic order on LogicNat and related foundation imports supply the ambient language of finite repetition; they are not used inside the inductive itself.
proof idea
No proof. The declaration is an inductive type with five nullary constructors and derived DecidableEq and Repr. Each constructor is documented by a one-line strength gloss; equality of tags is decidable by construction.
why it matters
Recognition Science aims to force physics from a minimal functional equation and the T0–T8 chain (J-uniqueness, phi, eight-tick octave, D = 3). That program is only credible if foundation lemmas state how much logic they consume. This tag is the K1 audit hook: every PRC claim can be marked delta-only versus trace-closure, choice, power comprehension, or classical extension.
No downstream theorems are wired yet in the graph, but the sibling audit record and constructor-distinctness lemmas show the intended use: classify claims, refuse silent classical upgrades, and keep the forcing chain honest about its hypotheses. It does not itself prove any physical constant or mass formula; it polices the logical budget of the foundation.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.