The Forcing Spectrum: Toward a Base-Canonical,∗ Machine-Checkable Measure of Logical Strength over a Forced Arithmetic Base

Reverse mathematics calibrates the logical strength of a theorem against a stipulated base such as RCA₀, so its verdicts are relative: change the base and the answer can move. We propose a measure that is not relative. The discrete tower ℕ_δ → ℤ_δ → ℚ_δ is forced as the initial object of an explicit distinction signature, hence unique up to unique isomorphism, so the named posit cost of deriving a theorem T becomes a base-canonical, machine-checkable invariant whose upper half is an internal Lean term, the forcing spectrum σ(T), where "base-canonical" means free of any base-representative parameter while remaining relative to the chosen signature, the posit alphabet, and the ambient metatheory. We present the framework: the forced base, the choice-free discrete tower including a fully intrinsic total order on ℚ_δ with the Archimedean and density properties, the proved fact that the continuum is not forced, and the demarcation that forced base plus classical completeness entails an omniscience principle of LPO class. We define σ as a certificate discipline on a finite semilattice of named posits, argue it cannot be a total computable recognizer, and state honestly which half of a certificate is internal Lean and which is external metatheory. The continuous-integration-gated audit engine promised in the June version of this paper is now built and running: fourteen manifest rungs are checked against measured Lean reality with zero mismatches (seven FORCED, one NAMED, six BRIDGE) over a census of 4,067 tracked declarations, 2,316 of them choice-free and none containing sorry. The engine also delivers the first cross-domain measurement: two milestones of the Recognition Science physics spine (the eight-tick minimal period and dimension forcing D = 3) measure at σ = ⊥, choice-free. The first complete spectrum certificate SpectrumCert(T₀) remains the named next deliverable. The one status table printed here is machine-emitted; no machine output is invented.