SansAnchorRealCharacterFactorizationInput
plain-language theorem explainer
Alias for the anchor-free real-character factorization target: every ledger map on ratio orbits that meets the sans-anchor hypotheses factors through a real-valued PRC ratio character with rational traces on positive small bases. Cost and PRC authors cite it when stating the input side of the factorization problem. The body is a one-line abbreviation of that target proposition.
Claim. The input proposition is identical to the corrected factorization target: for every map $F$ on ratio orbits satisfying the sans-anchor hypotheses, there exists a real-valued character $\chi$ on ratio orbits that is a PRC real ratio character and whose traces at positive rational bases are rational (so the rational-exponent step applies).
background
In the Cost module, recognition cost is tied to real characters on ratio orbits rather than to an external anchor. The sibling target proposition states the corrected factorization goal: every inhabitant of the anchor-free ledger factors through a real-valued character whose traces at the small bases are rational, so the exponent step can fire.
Sans-anchor hypotheses package the structural assumptions on $F$ (ledger compatibility without a fixed basepoint). A PRC real ratio character is the real-display form of the native cost character on ratio orbits. The module sits between real-trace roots, rational-exponent traces, and PRC native-cost uniqueness, and feeds the doubled-trace d'Alembert identities for the sans-anchor case.
Upstream display and quotient machinery (finite Hilbert display of amplitudes; object-level orbit quotients) supply the real and rational readings of the doubled trace that the character must match.
proof idea
One-line definitional abbreviation: the input name is definitionally equal to the target proposition. No tactics, no lemmas, no proof obligations beyond unfolding the alias.
why it matters
Gives a stable input-side name for the corrected factorization target inside RealCharacterFactorization, separating the statement that must be assumed or proved from downstream constructions such as cost-from-real-character and the sans-anchor doubled-trace d'Alembert route. That factorization is the bridge from anchor-free ledger maps to real characters with rational small-base traces, which is what lets the rational-exponent and native-cost uniqueness layers fire.
In the broader Recognition chain this is cost-side infrastructure for the unique $J$-cost (T5) realized on ratio orbits without fixing an external anchor, consistent with RCL and the native cost calculus. No used-by edges are recorded yet; the abbrev is organizational scaffolding for stating the input cleanly.
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