phi_ladder_div_closed
The theorem shows that the set of all integer powers of the golden ratio is closed under division by φ itself. Researchers constructing discrete spectra from J-cost minimization in Recognition Science would cite this algebraic closure when building the ladder of stable states. The proof is a direct term-mode reduction that extracts the witnessing integer exponent, decrements it by one, and rewrites the quotient via the power-subtraction identity.
claimIf $x$ belongs to the set of all integer powers of the golden ratio, then $x / φ$ also belongs to that set.
background
The module derives the golden ratio from J-cost minimization. PhiLadder is the set of all φ^n for n ∈ ℤ, representing stable positions under the Recognition Composition Law. Upstream results include the cost function induced by multiplicative recognizers, the J-cost of recognition events, and energy conservation under time-translation symmetry, all of which presuppose the ladder as the discrete spectrum of allowed states.
proof idea
The term proof obtains the integer n such that x = φ^n, then supplies n-1 as the witness for the quotient. It rewrites via zpow_sub_one₀ (using phi_pos) followed by div_eq_mul_inv to obtain φ^{n-1}.
why it matters in Recognition Science
This closure property confirms the self-similar structure of the φ-ladder required by T6, where φ is the fixed point of the forcing chain. It supplies the algebraic foundation for mass formulas of the form yardstick · φ^(rung-8+gap(Z)) and supports the eight-tick octave (T7) and D=3 emergence (T8). The result has no recorded downstream uses yet, indicating it functions as an early scaffolding lemma for later unification theorems such as the Yang-Mills mass gap.
scope and limits
- Does not prove closure under addition or subtraction.
- Does not address convergence of infinite sums over the ladder.
- Does not connect the ladder to specific constants such as α or G.
- Does not establish density or completeness in the reals.
formal statement (Lean)
104theorem phi_ladder_div_closed (x : ℝ) (hx : x ∈ PhiLadder) :
105 x / φ ∈ PhiLadder := by
proof body
Term-mode proof.
106 obtain ⟨n, rfl⟩ := hx
107 use n - 1
108 rw [zpow_sub_one₀ (ne_of_gt phi_pos)]
109 rw [div_eq_mul_inv]
110
111/-! ## J-cost at φ-Ladder Positions -/
112
113/-- J-cost formula applied to φ -/