H_ThresholdFromPhi
The declaration encodes the hypothesis that a unit consciousness threshold emerges from φ-quantization via an eight-tick mechanism. Researchers tracing constant emergence in Recognition Science would cite it when connecting J-cost minimization to discrete thresholds at the octave scale. The definition proceeds by asserting existence of a map from tick count and golden ratio to the value 1, without supplying the map explicitly.
claimThere exists a function $m : ℕ → ℝ → ℝ$ such that $m(8, φ) = 1$, where $φ$ is the golden ratio and the argument 8 indexes the eight-tick octave.
background
The PhiEmergence module develops the golden ratio as the fixed point of J-cost minimization under the Recognition Composition Law. Upstream, the reciprocal automorphism from CostAlgebra supplies the inverse operation that closes the J-automorphism group, while LedgerForcing.reciprocal and DAlembert.LedgerFactorization.of calibrate the ratio structure on positive reals. NucleosynthesisTiers.of and SpectralEmergence.of place physical quantities on discrete φ-tiers, and the eight-tick octave (period 2^3) supplies the discrete scale at which thresholds are expected to appear.
proof idea
This is a direct definition of a Prop that asserts existence of a mechanism function. The body simply states the existential claim with the concrete substitution mechanism 8 φ = 1; no lemmas are applied and no tactics are invoked beyond the declaration itself.
why it matters in Recognition Science
The definition supplies the missing link that H_StableIffPhiLadder requires to equate stable positions with the φ-ladder. It addresses the open derivation of the consciousness threshold C = 1 from the eight-tick structure (T7) and φ self-similarity (T6), potentially closing the gap between J-cost integration over one octave and the emergence of unit thresholds in the forcing chain.
scope and limits
- Does not exhibit an explicit mechanism function.
- Does not prove that the threshold equals unity.
- Does not derive the claim from J-cost integration.
- Does not connect the threshold to specific observables.
formal statement (Lean)
141def H_ThresholdFromPhi : Prop :=
proof body
Definition body.
142 ∃ (mechanism : ℕ → ℝ → ℝ),
143 mechanism 8 φ = 1 -- Some function of 8 ticks and φ gives threshold 1
144
145/-- 1/φ = φ - 1 (golden ratio reciprocal property) -/