phi_ladder
The phi-ladder maps each integer rung n to the real value phi raised to n, where phi is the golden-ratio fixed point forced at T6. Nucleosynthesis tiers and ontology predicates cite it to place all mass-luminosity values and recognition states on the discrete geometric skeleton. The declaration is a direct one-line abbreviation of the exponential map on the imported PhiForcing constant.
claimThe map sending each integer $n$ to the real number $phi^n$, where $phi$ is the unique positive self-similar fixed point of the recognition composition law.
background
Recognition Science derives the golden ratio phi from the T6 closure step as the unique ratio satisfying self-similarity under the J-cost functional J(x) = (x + 1/x)/2 - 1. The phi-ladder is the resulting discrete sequence of all non-trivial states generated from the zero-defect ground state x = 1, required by the eight-tick cycle (T7) and the Fibonacci recurrence phi^n + phi^{n+1} = phi^{n+2}. The StillnessGenerative module shows that the initial configuration x = 1 is unstable under recognition forcing and therefore populates exactly these values for n ≠ 0, with bounded creation cost J(phi) < 1. Upstream results in NucleosynthesisTiers and OntologyPredicates reuse the same name for tier-indexed and set-valued versions of this ladder.
proof idea
One-line definition that directly unfolds the exponential power using the imported golden-ratio constant from PhiForcing.
why it matters in Recognition Science
This supplies the concrete values realizing the phi-ladder skeleton demanded by T6 and the Fibonacci cascade in the StillnessGenerative derivation chain. It is invoked by all_ml_on_phi_ladder to prove every population tier lies on the ladder, by ml_nucleosynthesis to define the derived M/L ratio equal to phi, and by one_mem_phi_ladder to locate the ground state at rung zero. The construction closes the generative path from T0-T8 by furnishing explicit states that satisfy the period-8 requirement and the ledger symmetry J(x) = J(1/x).
scope and limits
- Does not prove uniqueness of the ladder among all possible geometric sequences.
- Does not derive the numerical value of phi inside this module.
- Does not extend the ladder to non-integer exponents or continuous limits.
- Does not compute explicit J-costs or defect distances for individual rungs.
Lean usage
theorem ml_nucleosynthesis_eq_phi : ml_nucleosynthesis = φ := by unfold ml_nucleosynthesis phi_ladder tier_difference; simp [nuclear_tier_local, luminosity_tier_local, zpow_one]formal statement (Lean)
44noncomputable def phi_ladder (n : ℤ) : ℝ := PhiForcing.φ ^ n
proof body
Definition body.
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