partitionFunctionFormula
The declaration defines the partition function in Recognition Science as the sum of exponentials over ledger J-costs normalized by J_0. Researchers modeling statistical mechanics within the phi-forcing chain or deriving e from summations would cite this normalization step. The definition is a direct string assignment that captures the exponential decay form without further computation.
claimThe partition function is $Z = ∑_i exp(-J_i / J_0)$, the normalization factor for probabilities over ledger configurations.
background
The Mathematics.Euler module targets derivation of Euler's number e from phi-related summations. It lists e as the base of the natural logarithm, the limit (1 + 1/n)^n, and the series sum 1/n!, while noting its role in the derivative fixed point. In Recognition Science, e emerges from J-cost exponential decay and 8-tick probability normalization, with connections explored through phi continued fractions even though no simple algebraic link between e and phi is known.
proof idea
This is a direct definition that assigns the string literal describing Z as the sum of exp(-J_i/J_0) terms for ledger normalization.
why it matters in Recognition Science
The definition supplies the RS-native form of the partition function that appears in the J-cost decay and 8-tick normalization steps of the forcing chain. It supports the module's goal of extracting e from phi-summations and precedes sibling definitions that treat e as a limit, series, or fixed point. No downstream uses are recorded, so it functions as an interface statement for later algebraic work on the exponential.
scope and limits
- Does not compute numerical values or bounds for Z.
- Does not derive e or phi from this expression.
- Does not reference the Recognition Composition Law or spatial dimension D=3.
- Does not connect to mass formulas or the alpha band.
formal statement (Lean)
154def partitionFunctionFormula : String :=
proof body
Definition body.
155 "Z = Σ exp(-J_i/J₀) = normalization for probabilities"
156
157/-! ## Provable Bounds on e and φ -/
158
159/-- e = exp(1) is positive. -/