Foundation Phi Forcing

Phi forcing

Phi forcing is a theorem in Recognition Science: if a ledger (a structure that keeps a record of recognition events with J-cost) is discrete and can reference itself at different scales, then the scale ratio must be the golden ratio, phi = (1 + sqrt(5))/2. The module IndisputableMonolith/Foundation/PhiForcing.the kernel-checked library proves this in the kernel-checked library 4. The proof chains from the forced J-cost function to the golden equation, and the result is a milestone in the unified forcing chain that also yields the eight-tick recognition cycle, 2^3, and three spatial dimensions.

The core argument is simple. A self-similar ledger has a scale ratio r greater than zero and not equal to one. If the cost structure is the same at every scale, then the scale transformation must be cost-free in the ledger sense. The only positive solution to the resulting golden constraint r^2 = r + 1 is phi. The theorem phi_forced states: for any discrete ledger L and real r, if L is self-similar with ratio r, then r = phi. Uniqueness is established by golden_constraint_unique and phi_unique_self_similar.

The module also defines two positive constants from phi: the coherence quantum E_coh = phi^(-5) and the minimum cost bit J_bit = log phi. These are definitions, not derived physical constants, but they are positive and used later in the forcing chain. The theorem phi_forcing_principle packages the golden equation, uniqueness, and positivity of both constants into one statement.

/-- **PHI FORCING THEOREM**: In a self-similar discrete ledger, the scale ratio is φ.

If:
1. L is a discrete ledger (from DiscretenessForcing + LedgerForcing)
2. L is self-similar with scale ratio r
3. r satisfies the compositional constraint r² = r + 1

Then: r = φ = (1 + √5)/2 -/
theorem phi_forced (L : DiscreteLedger) (r : ℝ) (hr : is_self_similar L r) : r = φ := by
  rcases hr with ⟨S, rfl⟩
  exact golden_constraint_unique S.ratio_pos (self_similar_forces_golden_constraint S)

/-- The golden constraint characterizes φ among positive reals. -/
theorem golden_constraint_unique {r : ℝ} (hr_pos : 0 < r) (hr_eq : satisfies_golden_constraint r) :
    r = φ := by
  -- r² = r + 1 has solutions (1 ± √5)/2
  -- Only (1 + √5)/2 is positive
  simp only [satisfies_golden_constraint] at hr_eq
  have h : r^2 - r - 1 = 0 := by linarith
  -- Use quadratic formula and positivity
  -- The solutions are (1 ± √5)/2, and only (1 + √5)/2 > 0
  have h5 : Real.sqrt 5 > 2 := by
    have h4 : (4 : ℝ) < 5 := by norm_num
    have hsqrt4 : Real.sqrt 4 = 2 := by
      rw [show (4 : ℝ) = 2^2 by norm_num, Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2)]
    calc Real.sqrt 5 > Real.sqrt 4 := Real.sqrt_lt_sqrt (by norm_num) h4
      _ = 2 := hsqrt4
  -- The positive root is (1 + √5)/2
  have hsq5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 5)
  -- Verify φ satisfies the equation
  have hphi_satisfies : φ^2 = φ + 1 := phi_equation
  -- Both r and φ satisfy x² = x + 1, and both are positive
  -- The polynomial x² - x - 1 has exactly two roots
  -- Since r > 0 and φ > 0, and the other root is negative, we have r = φ
  nlinarith [sq_nonneg (r - φ), sq_nonneg (r + φ - 1), phi_pos, hsq5]

/-- The coherence quantum: E_coh = φ⁻⁵. -/
noncomputable def E_coh : ℝ := φ^(-5 : ℤ)

/-- The minimum non-trivial cost: J_bit = ln(φ). -/
noncomputable def J_bit : ℝ := Real.log φ

What this page does not claim

Not claiming that phi is derived from physical constants or measurements. Not claiming that the constants E_coh and J_bit are measured physical quantities. Not claiming that the self-similarity condition is physically realized; it is a structural premise.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PhiForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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