ledgerCompose_assoc
The lemma establishes associativity of the ledger composition operation on real numbers. Researchers deriving the golden ratio from closure axioms in the Recognition Science scale sequence cite it to confirm that multi-event groupings remain unambiguous. The proof is a one-line algebraic reduction that unfolds the definition to ordinary addition and applies the ring tactic.
claimFor all real numbers $a$, $b$, $c$, $(a + b) + c = a + (b + c)$, where $+$ denotes the ledger composition of scales.
background
In the PhiForcingDerived module the ledger tracks total recognition work through additive composition of scales. The operation ledgerCompose is defined directly as the sum of two scale values, consistent with the additive property of the J-cost function J(x) = ½(x + 1/x) - 1. This choice follows from the module's three axioms: discrete geometric scales, additive ledger composition, and closure under composition.
proof idea
The proof unfolds the definition of ledgerCompose to expose ordinary real addition, then invokes the ring tactic to verify the associativity identity for addition.
why it matters in Recognition Science
The lemma supplies the associativity needed for the closure condition in the derivation of r² = r + 1. It supports the sibling theorem closure_forces_golden_equation that extracts the golden ratio from the geometric scale sequence. In the framework it aligns with the Recognition Composition Law (Axiom 2) by guaranteeing that additive ledger operations are unambiguous, thereby bridging the T5 J-uniqueness step to the forcing of phi.
scope and limits
- Does not prove commutativity of ledgerCompose.
- Does not derive the ratio r from closure.
- Does not establish existence of a closed scale sequence.
- Does not connect ledgerCompose to the J-cost function explicitly.
formal statement (Lean)
92lemma ledgerCompose_assoc (a b c : ℝ) :
93 ledgerCompose (ledgerCompose a b) c = ledgerCompose a (ledgerCompose b c) := by
proof body
Term-mode proof.
94 unfold ledgerCompose; ring
95
96/-! ## Axiom 3: Closure Under Composition
97
98The composed scale must be in the scale sequence.
99For a geometric sequence, this means 1 + r must equal some rⁿ.
100-/
101
102/-- A scale sequence is closed if composing the first two scales
103 gives the third scale (minimal closure condition) -/
104def GeometricScaleSequence.isClosed (S : GeometricScaleSequence) : Prop :=
105 ledgerCompose (S.scale 0) (S.scale 1) = S.scale 2
106
107/-! ## The Main Derivation -/
108
109/-- **THEOREM**: Closure forces the golden ratio equation.
110
111If a geometric scale sequence is closed under additive composition,
112then the ratio r must satisfy r² = r + 1. -/