ledgerCompose_comm
plain-language theorem explainer
The lemma establishes commutativity of scale composition under the additive ledger, so the total recognition work for events at scales a and b equals that for b and a. Researchers deriving the golden-ratio closure from the three axioms in the Phi Forcing module would cite it when confirming symmetry before solving r² = r + 1. The proof is a one-line term-mode reduction that unfolds the definition of ledger composition as addition and applies the ring tactic.
Claim. For all real numbers $a, b$, the sum of scales $a + b$ equals $b + a$.
background
The module derives r² = r + 1 from three axioms: discrete geometric scales, additive ledger composition (total recognition work adds), and closure under composition. The J-cost function J(x) = ½(x + 1/x) - 1 supplies the additivity insight, because independent events accumulate J-cost linearly. Upstream, the Composition axiom from CostAxioms states the Recognition Composition Law: for positive x, y, F(xy) + F(x/y) = 2 F(x) F(y) + 2 F(x) + 2 F(y), the multiplicative d'Alembert equation that forces compatibility with the multiplicative structure of positive reals.
proof idea
The proof is a one-line wrapper that unfolds the definition ledgerCompose a b := a + b and invokes the ring tactic on the reals to obtain commutativity of addition.
why it matters
This lemma supplies the symmetry step inside the Phi Forcing derivation that answers why closure forces r = phi, feeding the T5 J-uniqueness to T6 self-similar fixed-point bridge. It keeps the additive ledger consistent with the Recognition Composition Law while the module prepares the ground for the eight-tick octave and D = 3. No downstream uses are recorded yet, so the result remains local scaffolding for the r² = r + 1 derivation.
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