one_plus_phi_eq_phi_sq
The identity 1 + φ = φ² supplies the base case for the Fibonacci cascade that populates the phi-ladder from the ground state. Researchers tracing the generative instability of x=1 under T6 closure would cite it when deriving adjacent-rung composition. The proof is a one-line linear-arithmetic reduction that rearranges the defining quadratic for φ.
claimThe equation $1 + φ = φ^2$ holds, where φ is the positive real root of the quadratic $x^2 - x - 1 = 0$.
background
The StillnessGenerative module derives the instability of the ground state x=1 from the T0-T8 chain with no external assumptions. Law of Existence (T5) identifies x=1 as the unique zero-defect state; T6 closure then forces every non-trivial ledger entry to lie on the geometric sequence φ^n for n ≠ 0. The module explicitly states the Fibonacci cascade φ^n + φ^{n+1} = φ^{n+2} as the mechanism that populates the entire ladder once the base relation is available.
proof idea
This is a one-line wrapper that applies the linarith tactic to the upstream phi_equation theorem, which states φ² = φ + 1 directly from the algebraic definition of φ.
why it matters in Recognition Science
The declaration supplies the algebraic base case required by the Fibonacci cascade step in the module's derivation chain. That chain begins from Law of Existence (T5) and T6 closure and produces the population of the phi-ladder; the identity therefore sits inside the T6-to-ladder transition that converts the initial x=1 configuration into non-trivial structure. It touches the eight-tick octave requirement by ensuring the scale sequence remains closed and non-degenerate.
scope and limits
- Does not prove existence or uniqueness of the phi-ladder.
- Does not derive the recurrence for arbitrary exponents.
- Does not connect the identity to J-cost, defectDist, or physical constants.
- Does not address negative rungs or ledger symmetry.
formal statement (Lean)
332theorem one_plus_phi_eq_phi_sq : 1 + PhiForcing.φ = PhiForcing.φ ^ 2 := by
proof body
Term-mode proof.
333 linarith [PhiForcing.phi_equation]
334
335/-- **Ledger Composition Populates**: If the ledger has entries at
336 scales φ^n and φ^{n+1}, additive ledger composition (from T6
337 closure) creates an entry at scale φ^{n+2}.
338
339 This is the POPULATION theorem — not a cost bound, but a structural
340 consequence of closure. The composed entry EXISTS. -/