Cost Symplectic Action Jcost Forced By Symplectic Action

The symplectic forcing

The declaration jcost_forced_by_symplectic_action is a theorem in the framework's machine-checked library of formal theorems. It states that any cost function F satisfying five plain conditions, reciprocity, normalization, a composition law, calibration, and continuity, must equal J(x) = ½(x + x⁻¹) − 1. The composition law is the equation F(x·y) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y). This is not a new primitive; the theorem derives the composition law from a physical principle, the conservation of the ledger's area form.

The physical principle is this: a double-entry ledger is a two-dimensional phase space, a state being a pair (debit, credit). A recognition event is a linear map of this space. The conservation law σ = 0, meaning no net imbalance is created, is geometrically area preservation, det M = 1. The theorem conservesSigma_iff_preservesArea proves this equivalence. On this area-preserving group, the trace identity tr(A·B) + tr(A·B⁻¹) = tr(A)·tr(B) holds, and specializing to the split torus diag(x, x⁻¹) reproduces the composition law as a theorem, not a primitive. The cost of an event is the calibrated trace functional traceCost M = ½tr M − 1, which vanishes on the identity, the balanced ledger.

The theorem jcost_exp_eq_cosh_sub_one shows J(eᵗ) = cosh t − 1, so the cost is the hyperbolic cosine of the log-eigenvalue t, the Hamiltonian action of the event, uniquely minimized at t = 0. This closes the loop: σ = 0 implies symplectic, which implies the composition law, which with the other conditions forces F = J. The framework proves that the cost of recognition is the action of the unique area-preserving dynamics, derived from a physical principle that is itself ledger-forced.

/-- **The recognition cost is forced to be `J` by the symplectic action.**  Any
reciprocal, normalized, calibrated, continuous cost whose composition law is the
symplectic trace identity (`SatisfiesCompositionLaw`, here supplied by the
area-preserving ledger group) equals `J`.  This composes the σ = 0 ⇒ symplectic
⇒ RCL derivation of this module with the cost-shape uniqueness theorem
`law_of_logic_forces_jcost`, closing the documented bridge. -/
theorem jcost_forced_by_symplectic_action (F : ℝ → ℝ)
    [FunctionalEquation.AczelSmoothnessPackage]
    (hRecip : FunctionalEquation.IsReciprocalCost F)
    (hNorm : FunctionalEquation.IsNormalized F)
    (hComp : FunctionalEquation.SatisfiesCompositionLaw F)
    (hCalib : FunctionalEquation.IsCalibrated F)
    (hCont : ContinuousOn F (Set.Ioi 0)) :
    ∀ x : ℝ, 0 < x → F x = Cost.Jcost x :=
  FunctionalEquation.law_of_logic_forces_jcost F hRecip hNorm hComp hCalib hCont

/-- **σ = 0 is exactly symplectic (area-preserving).**  A ledger event conserves
σ iff it preserves the ledger area form, iff `det = 1`. -/
theorem conservesSigma_iff_preservesArea (M : Matrix (Fin 2) (Fin 2) ℝ) :
    ConservesSigma M ↔
      ∀ v w : Fin 2 → ℝ, areaForm (M.mulVec v) (M.mulVec w) = areaForm v w := by
  unfold ConservesSigma
  constructor
  · intro hdet v w
    rw [areaForm_mulVec, hdet, one_mul]
  · intro h
    have h01 := h ![1, 0] ![0, 1]
    rw [areaForm_mulVec] at h01
    have hbase : areaForm (![1, 0] : Fin 2 → ℝ) ![0, 1] = 1 := by
      simp [areaForm]
    rw [hbase, mul_one] at h01
    exact h01

/-- **The symplectic action cost is `J`.**  On the split torus the calibrated
trace functional equals the canonical recognition cost. -/
@[simp] theorem traceCost_diagSL (x : ℝ) : traceCost (diagSL x) = Cost.Jcost x := by
  unfold traceCost Cost.Jcost
  rw [diagSL_trace]

/-- The action cost is `cosh` of the generator's log-eigenvalue: with `x = eᵗ`,
`J(eᵗ) = cosh t − 1`.  `t` is the Hamiltonian action of the event; the cost is
minimized at the balanced ledger `t = 0`. -/
theorem jcost_exp_eq_cosh_sub_one (t : ℝ) :
    Cost.Jcost (Real.exp t) = Real.cosh t - 1 := by
  simp only [Cost.Jcost, Real.cosh_eq, Real.exp_neg]

What this page does not claim

This theorem does not claim that the composition law alone forces J; it requires the other four conditions. It does not claim that the ledger is a physical system in the sense of classical mechanics; it is a mathematical model. It does not claim that the symplectic action is the only possible physical interpretation of J.

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/SymplecticAction.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: