Cost Real Character Factorization Doubled Trace D Alembert Of Sans Anchor

A trace identity

In mathematics, a trace is a function that assigns a number to an object, often a matrix or an operator, in a way that is invariant under certain transformations. The declaration doubledTrace_dAlembert_of_sansAnchor in the Recognition Science library establishes a specific identity for a trace-like function derived from a cost function. This identity states that for any two nonzero values x and y, the sum of the doubled trace at their product and at their quotient equals the product of the doubled traces at x and y. In symbols, this is T(xy) + T(x/y) = T(x)T(y), where T is the doubled trace.

This identity is notable because it resembles the classical d'Alembert functional equation, which characterizes exponential functions. The theorem proves this identity holds under a set of hypotheses called SansAnchorHypotheses. The key point is that this set does not include the condition that the trace at 2 equals 2, which is called the anchor. The proof uses only the reciprocal composition law (RCL), a fundamental constraint on the cost function. This means the identity is a consequence of the RCL alone, without needing to pin down the value at 2.

In Recognition Science, the cost function measures the cost of recognition events, and the trace is a derived quantity. This theorem is a step in a larger program to factorize the cost function using real characters. The identity it establishes is a structural property that holds for a broad class of cost functions satisfying the basic hypotheses. It is a formal theorem in the machine-checked library of formal theorems, meaning its proof has been verified by a computer.

The theorem does not claim that the trace function is an exponential function in the usual sense. It does not assert that the trace at 2 equals 2, nor does it derive the full form of the cost function. It also does not claim that the identity holds for all possible cost functions, only those satisfying the specified hypotheses. The identity is a necessary condition, not a sufficient one, for the cost function to have a particular form.

theorem doubledTrace_dAlembert_of_sansAnchor
    {F : RatioOrbit → RatioOrbit}
    (hF : SansAnchorHypotheses F)
    {x y : RatioOrbit} (hx : x.toRat ≠ 0) (hy : y.toRat ≠ 0) :
    RatioOrbit.crossEq
      (RatioOrbit.add (nativeCostDoubledTrace F (RatioOrbit.mul x y))
        (nativeCostDoubledTrace F (div x y)))
      (RatioOrbit.mul (nativeCostDoubledTrace F x) (nativeCostDoubledTrace F y)) :=
  doubledTrace_dAlembert_of_rcl hF.base_sans_two.canonical_rcl hx hy

/-- The doubled-trace form of the composition law needs only the RCL. The anchor
at two is not used. -/
theorem doubledTrace_dAlembert_of_rcl
    {F : RatioOrbit → RatioOrbit}
    (hrcl : ∀ {x y : RatioOrbit}, x.toRat ≠ 0 → y.toRat ≠ 0 →
      RatioOrbit.crossEq
        (RatioOrbit.add (F (RatioOrbit.mul x y)) (F (div x y)))
        (RatioOrbit.add
          (RatioOrbit.add
            (RatioOrbit.mul two (RatioOrbit.mul (F x) (F y)))
            (RatioOrbit.mul two (F x)))
          (RatioOrbit.mul two (F y))))
    {x y : RatioOrbit} (hx : x.toRat ≠ 0) (hy : y.toRat ≠ 0) :
    RatioOrbit.crossEq
      (RatioOrbit.add (nativeCostDoubledTrace F (RatioOrbit.mul x y))
        (nativeCostDoubledTrace F (div x y)))
      (RatioOrbit.mul (nativeCostDoubledTrace F x) (nativeCostDoubledTrace F y)) := by
  have h := hrcl hx hy
  rw [RatioOrbit.crossEq_iff_toRat_eq] at h ⊢
  simp only [nativeCostDoubledTrace, doubledTraceValue, RatioOrbit.mul_toRat,
    RatioOrbit.add_toRat, two_toRat, RatioOrbit.one_toRat] at h ⊢
  linarith

theorem doubledTrace_dAlembert_of_sansAnchor
    {F : RatioOrbit → RatioOrbit}
    (hF : SansAnchorHypotheses F)
    {x y : RatioOrbit} (hx : x.toRat ≠ 0) (hy : y.toRat ≠ 0) :
    RatioOrbit.crossEq
      (RatioOrbit.add (nativeCostDoubledTrace F (RatioOrbit.mul x y))
        (nativeCostDoubledTrace F (div x y)))
      (RatioOrbit.mul (nativeCostDoubledTrace F x) (nativeCostDoubledTrace F y)) :=
  doubledTrace_dAlembert_of_rcl hF.base_sans_two.canonical_rcl hx hy

What this page does not claim

The trace function is an exponential function in the classical sense. The trace at 2 equals 2, which is the anchor condition. The identity holds for all cost functions without any hypotheses.

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/RealCharacterFactorization.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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