J_eq_zero_imp_one
The result shows that the recognition cost J vanishes only at the identity ratio x=1 when x is positive. Researchers deriving mass rungs or ledger factors from the phi-ladder cite it to exclude trivial transitions from event counts. The proof unfolds the definition of J and applies linear and nonlinear arithmetic to obtain a contradiction unless x equals 1.
claimIf $x > 0$ and $J(x) = 0$, then $x = 1$, where the recognition cost is given by $J(x) = (x + x^{-1})/2 - 1$.
background
The functional J is the recognition cost J(x) = (1/2)(x + x^{-1}) - 1 defined on positive reals. In the module, every integer baseline traces to the single input D = 3 via cube combinatorics on Q_3. Item B-20 upgrades non-triviality to derived status: any transition with x = 1 has zero cost and leaves no ledger record, so identity transitions are excluded from the physical event count. Upstream results supply the cost interpretation of recognition events and the identity event at the J minimum.
proof idea
The tactic proof unfolds the definition of J to obtain the equation x + x^{-1} = 2. Linear arithmetic isolates the inverse as x^{-1} = 2 - x. Multiplication by x and substitution of the known product x * x^{-1} = 1 then yields a quadratic relation that nonlinear arithmetic shows holds only when x = 1.
why it matters in Recognition Science
This declaration upgrades the non-triviality assumption B-20 to derived status from T1 plus T5 J-uniqueness. It guarantees that only non-unit ratios contribute to ledger records, supporting the mass formula yardstick * phi^(rung - 8 + gap(Z)) and the exclusion of zero-cost events. The result sits inside the baseline rung derivations that feed octave offset and neutrino baseline calculations from D = 3.
scope and limits
- Does not address the case x ≤ 0.
- Does not compute the explicit value of J at any point other than the zero set.
- Does not extend uniqueness to complex numbers or other algebraic structures.
- Does not derive further properties of J such as convexity or minimum location.
formal statement (Lean)
72theorem J_eq_zero_imp_one (x : ℝ) (hx : 0 < x) (hJ : J x = 0) : x = 1 := by
proof body
Tactic-mode proof.
73 unfold J at hJ
74 have hsum : x + x⁻¹ = 2 := by linarith
75 have hxx : x * x⁻¹ = 1 := mul_inv_cancel₀ (ne_of_gt hx)
76 have hinv : x⁻¹ = 2 - x := by linarith
77 have : x * x⁻¹ = x * (2 - x) := by rw [hinv]
78 rw [hxx] at this
79 nlinarith [this]
80
81/-- **B-20 DERIVED**: Non-triviality is a consequence of T1 + T5.
82 Any transition with x = 1 has zero cost and leaves no ledger record.
83 Therefore identity transitions are excluded from the physical event count. -/