Cost Geometric Root Cosh Sub One Eq Two Sinh Sq Half

The cost identity

The hyperbolic cosine and sine functions, familiar from geometry and physics, satisfy a standard identity: cosh(t) - 1 = 2·sinh(t/2)². This is not a new discovery; it is a routine consequence of the addition formulas for these functions, and it holds for every real number t. The Recognition Science library contains a machine-checked proof of this identity, named cosh_sub_one_eq_two_sinh_sq_half, as part of its formal development of the cost function.

In this framework, the cost of a recognition event, written J(x), measures the price of distinguishing a value x from a baseline of 1. The library proves that this cost equals cosh(log x) - 1. Substituting t = log x into the hyperbolic identity gives J(x) = 2·sinh((log x)/2)². This is the chordal form: the cost is exactly twice the square of the hyperbolic sine of half the logarithmic distance from 1. The same quantity appears as (x-1)²/(2x), which is the squared distance between points on the upper half-plane model of hyperbolic geometry, measured along the imaginary geodesic.

The identity is not merely decorative. It feeds directly into the library's proof that the cost function is superadditive: for two distinctions a and b of the same sign, the cost of chaining them, J(e^(a+b)), is strictly greater than the sum J(e^a) + J(e^b). The excess is the product term plus the product of the hyperbolic sines, and the identity shows why this excess is always positive for nonzero same-sign steps. Chaining two distinctions always costs more than the sum of its parts, a fact that drives the framework's later results about the forced discreteness of the ledger.

What the identity does not do is establish the golden ratio, the eight-tick cycle, or three-dimensional space. Those results depend on additional assumptions, such as the closure of a geometric scale sequence, which the library states but does not derive from the cost function alone. The identity is a lemma, not a forcing theorem. It is a precise algebraic tool that the framework uses, but it does not by itself imply any physical or structural conclusion.

/-- In the additive coordinate `x = e^t`, `J = cosh t - 1` exactly.
This is the banked theorem restated through `log`. -/
theorem jcost_eq_cosh_log_sub_one {x : ℝ} (hx : 0 < x) :
    Jcost x = Real.cosh (Real.log x) - 1 := by
  have h := FunctionalEquation.Jcost_G_eq_cosh_sub_one (Real.log x)
  simpa [FunctionalEquation.G, Real.exp_log hx] using h

/-- The chordal form: `J(x) = (x-1)^2/(2x)`. On the upper half-plane this is
`cosh d(i, ix) - 1` with `d` the hyperbolic distance, via
`cosh d(z,w) - 1 = |z-w|^2/(2 Im z Im w)`. The geometric reading is the
docstring; the arithmetic content is this identity. -/
theorem jcost_eq_chordal {x : ℝ} (hx : 0 < x) :
    Jcost x = (x - 1) ^ 2 / (2 * x) := by
  unfold Jcost
  field_simp [hx.ne']
  ring

/-- **Strict superadditivity.** For same-sign nonzero `a, b`, chaining two
distinctions costs strictly more than the sum of the parts: the excess
`J(e^a)J(e^b) + sinh a sinh b` is positive. -/
theorem jcost_superadd_strict_same_sign {a b : ℝ} (h : 0 < a * b) :
    Jcost (Real.exp a) + Jcost (Real.exp b) < Jcost (Real.exp (a + b)) := by
  rw [jcost_chain_excess_identity]
  have hJa : 0 < Jcost (Real.exp a) := jcost_exp_pos (fun ha => by
    rw [ha] at h
    simp at h)
  have hJb : 0 < Jcost (Real.exp b) := jcost_exp_pos (fun hb => by
    rw [hb] at h
    simp at h)
  have hsinh : 0 < Real.sinh a * Real.sinh b := by
    rcases mul_pos_iff.mp h with ⟨ha, hb⟩ | ⟨ha, hb⟩
    · exact mul_pos (sinh_pos_of_pos ha) (sinh_pos_of_pos hb)
    · have h1 : Real.sinh a = - Real.sinh (-a) := by
        rw [Real.sinh_neg a]
        ring
      have h2 : Real.sinh b = - Real.sinh (-b) := by
        rw [Real.sinh_neg b]
        ring
      rw [h1, h2, neg_mul_neg]
      exact mul_pos (sinh_pos_of_pos (neg_pos.mpr ha)) (sinh_pos_of_pos (neg_pos.mpr hb))
  have h3 : 0 < Jcost (Real.exp a) * Jcost (Real.exp b) := mul_pos hJa hJb
  linarith [h3, hsinh]

What this page does not claim

The identity does not prove the golden ratio, the eight-tick cycle, or three spatial dimensions. The identity does not derive the closure condition on geometric scale sequences. The identity does not imply that the cost function is the only possible measure of recognition.

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/Cost/GeometricRoot.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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