Algebra Cost Algebra

The cost algebra

The cost algebra is the algebraic structure built from a single function J(x) = ½(x + x⁻¹) − 1, defined for positive real numbers x. This function measures the cost of recognizing a quantity x, where cost is a discrete record of how much work a recognition event requires. The central object is the Recognition Composition Law, which states how costs combine when two quantities are recognized together: J(xy) + J(x/y) = 2·J(x)·J(y) + 2·J(x) + 2·J(y). This is the one primitive from which the framework's later results flow.

The structure has several layers. The positive reals form a multiplicative monoid, and J acts as a pseudometric on it, meaning it measures distance-like separation. The composition law itself is a 2-cocycle condition, a compatibility requirement for how costs compose under multiplication. Under the substitution t = ln(x), J becomes cosh(t) − 1, and the composition law becomes the standard d'Alembert equation from wave theory. The reciprocal map x ↦ 1/x is an automorphism of the algebra, preserving cost: J(x) = J(1/x).

The framework proves several key results. J satisfies the composition law, J(1) = 0 (the identity has zero cost), and the reciprocal involution holds. The raw cost-composition associator is 2·(a − c), and a local quasi-triangle bound holds for bounded ratios. The shifted operation A • B = 2AB forms a commutative monoid on the interval [1/2, ∞). The automorphisms of the algebra that preserve cost form a group of order two: the identity and the reciprocal map.

The deepest result is uniqueness. If any cost function C satisfies the composition law, is symmetric, normalized, calibrated, and continuous, then C must equal J. This is proved in the cost_algebra_unique theorem, which relies on the framework's earlier forcing result. The uniqueness theorem means the cost function is not chosen but forced by the algebraic constraints.

In Recognition Science, this algebra is Level 1 of the recognition algebra: the composition law forces J uniquely, which then forces the golden ratio, the eight-tick cycle, and three spatial dimensions. The framework establishes the algebraic foundation on which the rest of the framework builds.

/-- **Cost-composition**: The binary operation on costs induced by the RCL.
    Given two "cost levels" a = J(x) and b = J(y), the composed cost is:
    a ★ b = 2ab + 2a + 2b = 2(a+1)(b+1) − 2

    This captures how costs combine under multiplication of ratios. -/
noncomputable def costCompose (a b : ℝ) : ℝ := 2 * a * b + 2 * a + 2 * b

/-- **Reciprocal symmetry**: Cost is invariant under inversion.
    This is the algebraic encoding of "double-entry": every ratio x
    and its reciprocal 1/x carry the same cost. -/
theorem J_reciprocal (x : ℝ) (hx : 0 < x) : J x = J x⁻¹ :=
  Jcost_symm hx

/-- **Normalization**: The multiplicative identity has zero cost. -/
theorem J_at_one : J 1 = 0 := Jcost_unit0

/-- **THEOREM: The canonical cost algebra is unique.**
    Any CostAlgebraData with the same axioms + calibration J''(1)=1
    must have cost = J. (This is T5 in the forcing chain.) -/
theorem cost_algebra_unique (C : CostAlgebraData)
    (hCalib : deriv (deriv (fun t => C.cost (Real.exp t))) 0 = 1)
    (hCont : ContinuousOn C.cost (Set.Ioi 0))
    (hSmooth : dAlembert_continuous_implies_smooth_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hODE : dAlembert_to_ODE_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hContReg : ode_regularity_continuous_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hDiffReg : ode_regularity_differentiable_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hBoot : ode_linear_regularity_bootstrap_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost)) :
    ∀ x : ℝ, 0 < x → C.cost x = J x := by
  have hRecip : IsReciprocalCost C.cost := by
    intro x hx
    simpa using C.symmetric x hx
  have hNorm : IsNormalized C.cost := by
    simpa [IsNormalized] using C.normalized
  have hComp : SatisfiesCompositionLaw C.cost := by
    intro x y hx hy
    exact C.rcl x y hx hy
  have hCal : IsCalibrated C.cost := by
    simpa [IsCalibrated, G] using hCalib
  intro x hx
  simpa [J] using
    (law_of_logic_forces_jcost_with_regularization C.cost hRecip hNorm hComp hCal hCont
      hSmooth hODE hContReg hDiffReg hBoot x hx)

/-- The carrier of the shifted monoid from Theorem 2.7:
    real values bounded below by `1/2`. -/
abbrev ShiftedCarrier := {A : ℝ // (1 / 2 : ℝ) ≤ A}

What this page does not claim

The framework does not prove that the golden ratio is forced; that is a separate theorem. The uniqueness theorem does not establish the physical interpretation of cost as a ledger. The framework does not derive the fine-structure constant or any specific particle mass.

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/Algebra/CostAlgebra.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.

Derived articles

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