unit_normalization_forced
The theorem records that the normalization hypothesis J(1) = 0 directly entails the same equality as its conclusion. Recognition Science workers closing the self-comparison axioms in the ledger reconstruction would cite it when converting R2 from axiom to theorem. The proof is a one-line term that returns the supplied hypothesis without additional steps.
claimLet $J : ℝ → ℝ$ satisfy $J(1) = 0$. Then $J(1) = 0$.
background
The Closed Observable Framework module absorbs R1, R2, R5 and R6 as structure fields rather than axioms, leaving only the Regularity Axiom for finite-description content. The J-cost function enters via self-comparison, with its normalization at unity fixed by the hypothesis. Upstream results supply the unit elements in LogicInt and LogicRat that anchor the integer and rational layers of the phi-ladder.
proof idea
The proof is a one-line term that directly applies the hypothesis h_unit.
why it matters in Recognition Science
This declaration converts the R2 self-comparison axiom into a theorem inside the Closed Observable Framework, feeding the ledger reconstruction path. It aligns with T5 J-uniqueness in the forcing chain before the phi fixed point and eight-tick octave are derived. The module doc notes that the remaining Regularity Axiom now encodes the finite-description obligations.
scope and limits
- Does not derive the explicit form of J from other axioms.
- Does not extend the normalization to functions outside ℝ.
- Does not discharge the Regularity Axiom that remains in the framework.
- Does not supply numerical values or empirical calibration.
formal statement (Lean)
54theorem unit_normalization_forced
55 (J : ℝ → ℝ)
56 (h_unit : J 1 = 0) :
57 J 1 = 0 := h_unit
proof body
Term-mode proof.
58
59/-- Legacy regularity bundle.
60
61This compatibility structure is kept for downstream users that still expect one
62record, but the public reconstruction path below now prefers the split
63finite-description obligations. -/