Cost Symplectic Action Conserves Sigma Iff Defect Zero

The conservation condition

In linear algebra, a 2 by 2 matrix acts on pairs of numbers (v₀, v₁) by the usual rule of matrix multiplication. One geometric question is whether such a map preserves the area of parallelograms: if you take two vectors, form the parallelogram they span, apply the matrix to both, and compute the new area, do you get the original area back? The answer is governed by a single number attached to the matrix, its determinant. A matrix preserves area exactly when its determinant equals 1. This is a classical fact of linear algebra, known since the nineteenth century.

The Recognition Science declaration conservesSigma_iff_defect_zero restates this classical fact in the framework's own vocabulary. The framework models a ledger, a discrete record of events, as a two-dimensional phase space whose states are pairs (debit, credit). A recognition event, a single update to the ledger, is a linear map of that space. The framework defines the sigma defect of a map as its determinant minus 1. The declaration proves, as a theorem in the machine-checked library of formal theorems, that a map conserves sigma, meaning it creates no net imbalance, if and only if its sigma defect is zero. In ordinary language: the ledger's conservation law is exactly the condition that the map preserves area.

This equivalence is the first step in a longer chain. The framework proves that area preservation is equivalent to the determinant being 1, and that on the group of such maps a trace identity holds: for any two area-preserving matrices A and B, the trace of A times B plus the trace of A times B inverse equals the trace of A times the trace of B. Specializing this identity to diagonal matrices of the form diag(x, x⁻¹) reproduces the Recognition Composition Law, the functional equation that forces the cost function J(x) = ½(x + x⁻¹) − 1. The declaration itself, however, is only the equivalence between conservation and zero defect; the later steps are separate theorems.

What the declaration does not claim is any physical interpretation. It does not say that real ledgers obey this law, nor that the framework's model of recognition events is the correct description of any actual physical process. It is a purely mathematical statement about 2 by 2 matrices. The bridge from this algebra to physics, the identification of the cost function with a physical cost, is a separate, documented step in the framework, not part of this theorem. The declaration also does not generalize to higher dimensions; it is specific to two-dimensional matrices, where the symplectic group and the special linear group coincide.

theorem conservesSigma_iff_defect_zero (M : Matrix (Fin 2) (Fin 2) ℝ) :
    ConservesSigma M ↔ sigmaAreaDefect M = 0 := by
  unfold ConservesSigma sigmaAreaDefect
  constructor <;> intro h <;> linarith

/-- **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 SL(2,ℝ) trace identity of the area-preserving ledger group.**  When the
"reverse event" `B⁻¹` exists (σ = 0, i.e. `det B = 1`), the adjugate is the
inverse and the trace identity becomes
`tr(A·B) + tr(A·B⁻¹) = tr A · tr B`.  This is the Fricke/SL(2) identity; below it
specializes to the Recognition Composition Law. -/
theorem trace_identity_of_conservesSigma (A B : Matrix (Fin 2) (Fin 2) ℝ)
    (hB : ConservesSigma B) :
    (A * B).trace + (A * B⁻¹).trace = A.trace * B.trace := by
  have hdet : B.det = 1 := hB
  have hadj : B⁻¹ = B.adjugate := by
    rw [Matrix.inv_def, hdet]; simp
  rw [hadj]
  exact trace_mul_add_trace_mul_adjugate A B

What this page does not claim

The declaration does not claim that any real ledger or physical process obeys the conservation law. The declaration does not generalize to matrices of dimension other than two. The declaration does not itself establish the full cost function J(x); that requires additional theorems.

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

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