phi
The declaration supplies the explicit real value of the golden ratio as (1 + sqrt(5))/2. Researchers auditing the Recognition Science ledger uniqueness arguments cite this value when invoking the fixed-point solution for the J-cost function in discrete conservative systems. The definition is a direct arithmetic expression with no lemmas or tactics applied beyond the quadratic root formula.
claim$phi := (1 + sqrt(5))/2$
background
The Meta.LedgerUniqueness module addresses Gap 9 by resolving why any discrete conservative system must use specifically the golden ratio phi, the 3D cube Q3, and the 8-tick cycle. It states that phi is the unique positive fixed point of the equation x squared equals x plus 1 arising from the cost-function constraint J(x) equals J(1/x) with minimal curvature. The upstream point definition from Numerics.Interval.Basic constructs a singleton interval containing a single rational, supporting numeric interval arithmetic for constants in the module.
proof idea
This is a direct definition that sets phi to the closed-form expression (1 + sqrt(5))/2. The accompanying comment records that the value satisfies the fixed-point equation phi squared equals phi plus 1. No lemmas are applied and no tactics are used.
why it matters in Recognition Science
This definition anchors the uniqueness argument by supplying the concrete value for phi that solves the J-cost fixed point. It feeds the sibling results phi_satisfies_fixed_point and phi_unique_fixed_point. In the Recognition Science framework it corresponds to T6 where phi is forced as the self-similar fixed point from the J-uniqueness property in T5. The module's main theorem asserts that any discrete conservative system is isomorphic to the RS ledger using this phi together with Q3 and the 8-tick cycle.
scope and limits
- Does not establish uniqueness of the fixed point.
- Does not derive phi from the T0-T8 forcing chain.
- Does not provide interval bounds or numeric approximations.
- Does not address fixed points outside the positive reals.
formal statement (Lean)
54noncomputable def phi : ℝ := (1 + Real.sqrt 5) / 2
proof body
Definition body.
55
56/-- φ satisfies the fixed-point equation φ² = φ + 1. -/