IndisputableMonolith.Cost.Jlog
The Jlog module supplies the logarithmic realization of the J-cost function used throughout Recognition Science. It defines Jlog(x) = cosh(log x) - 1 and proves its non-negativity, zero set, and strict monotonicity on the positive reals. Ledger models import these facts to track parity shifts under single debit or credit postings. The content consists of a closed-form definition followed by direct verifications from standard hyperbolic and logarithmic identities.
claimThe module introduces the function $J(x) = (x + x^{-1})/2 - 1 = e^{J(x)} - 1$ realized as $J(x) = Jlog(x) := cosh(log x) - 1$ for $x > 0$, together with the statements $Jlog(x) = 0$ if and only if $x = 1$, $Jlog(x) > 0$ for $x > 0$ with $x ≠ 1$, and $Jlog$ strictly increasing on $[1, ∞)$.
background
Recognition Science obtains the J-cost from the unique solution to the forcing functional equation at step T5 of the unified chain. This module works in logarithmic coordinates so that the composition law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) becomes elementary. The imported Mathlib trigonometric library supplies the identity cosh(log x) = (x + 1/x)/2 that converts the algebraic form of J into the explicit Jlog expression used by downstream ledger constructions.
proof idea
This is a definition module. Jlog is introduced by the closed-form equation Jlog x = cosh (log x) - 1. Each subsequent lemma is a one-line wrapper that applies the corresponding Mathlib identity for cosh, log, or the order properties of the reals; no custom tactics or inductive arguments appear.
why it matters in Recognition Science
Jlog supplies the concrete cost function required by the eight-tick octave and the D = 3 spatial-dimension step. It is imported by LedgerPostingAdjacency to guarantee that each single-account posting changes the debit-credit difference by exactly ±1 while preserving non-negative total cost. The module therefore bridges the abstract T5 uniqueness result to explicit ledger parity tracking.
scope and limits
- Does not derive the Recognition Composition Law.
- Does not address the phi-ladder or mass formulas.
- Does not prove uniqueness of the J solution.
- Does not connect to the alpha inverse interval.