inversion_fixed_point
For positive real x the inversion equation x equals its reciprocal holds precisely when x equals 1. Number theorists formalizing the Recognition Science multiplicative ledger cite this as the unique critical point anchoring the J-cost function. The proof splits the biconditional, reduces the forward direction to x squared equals 1 via cancellation, and closes with a non-linear arithmetic step on the non-negativity of (x minus 1) squared.
claimFor all real $x > 0$, $x = x^{-1}$ if and only if $x = 1$.
background
The PrimeLedgerStructure module treats natural numbers as transactions on a discrete multiplicative ledger, with primes as the irreducible entries whose unique factorization supplies the balance sheet. The d'Alembert equation forces zeros of its solutions onto lines, and the structural parallel between J-cost symmetry and the zeta functional equation is modeled here. This theorem isolates the fixed point of inversion, identified in the doc-comment as the RS critical point for the J-function J(x) = (x + x^{-1})/2 - 1 that appears in the forcing chain at T5.
proof idea
The tactic proof opens with constructor to split the biconditional. The forward direction assumes x = x^{-1}, notes x nonzero from the hypothesis, rewrites x * x = 1 via mul_inv_cancel₀, obtains x squared equals 1 by the sq lemma, and finishes with nlinarith on sq_nonneg (x - 1). The reverse direction substitutes and simplifies directly.
why it matters in Recognition Science
The result supplies the unique fixed point required by J-uniqueness in the T5 step of the forcing chain and supports the module's model of the J-zeta structural correspondence that underpins the Riemann Hypothesis prediction from ledger conservation. It sits upstream of siblings such as jcost_log_zero_unique and j_functional_equation, closing a basic algebraic prerequisite for the eight-tick octave and phi-ladder constructions.
scope and limits
- Does not extend the fixed-point claim to non-positive reals or the complex plane.
- Does not derive any prime-factorization consequences within the same module.
- Does not supply numerical bounds or approximation rates for the critical point.
formal statement (Lean)
160theorem inversion_fixed_point (x : ℝ) (hx : 0 < x) :
161 x = x⁻¹ ↔ x = 1 := by
proof body
Tactic-mode proof.
162 constructor
163 · intro h
164 have hne : x ≠ 0 := ne_of_gt hx
165 have : x * x = 1 := by
166 calc x * x = x * x⁻¹ := by rw [← h]
167 _ = 1 := mul_inv_cancel₀ hne
168 have hx_sq : x ^ 2 = 1 := by rwa [sq]
169 nlinarith [sq_nonneg (x - 1)]
170 · intro h; rw [h]; simp
171
172/-- J has its unique zero at the fixed point x = 1. -/