Foundation Pair Kernel Finite Heisenberg Weyl Event Operator Finite Weyl Event A

The nonnegative action

A cost function that could go negative would let a system lower its total cost without limit, so proving a cost stays at or above zero is a basic sanity check. The declaration finiteWeylEventAction27_nonneg establishes exactly that for a specific model: the recognition cost on a 27-site grid is never negative. The model places three recognition coordinates, each with three phases, giving 3^3 = 27 total sites. On each site, the cost combines a shift term, the squared difference between neighboring amplitude values, and a clock term, a nonnegative displacement factor times the squared amplitude. The theorem proves the sum of these nonnegative pieces is itself nonnegative for any choice of amplitude values and any chosen center site.

The proof is a direct application of basic facts: squares are nonnegative, and the clock displacement is defined as a squared complex norm, which is also nonnegative. The machine-checked library of formal theorems records this as a theorem, not an assumption. The same library also proves the action is zero when all amplitudes are zero, and that the shift and clock operators on each axis satisfy an exact Weyl relation, a standard commutation pattern in quantum mechanics. These results are packaged together in a certificate structure that collects the carrier size, the Weyl relation, the nonnegativity, and the derivative-based definitions of response and operator.

In Recognition Science, this action is a model, a definitional choice, not a derived consequence. The framework's own documentation states this plainly: the primary action is a MODEL, while the carrier size, the Weyl relation, and the nonnegativity are THEOREMs. The response and operator are defined as derivatives of this primary action, not as separate physical inputs. No kinetic, potential, or source coefficients are accepted into the action; it is coefficient-free by construction.

What the theorem does not claim is as important as what it proves. It does not claim that this action is the unique or forced cost for recognition events. Existing Recognition theory does not yet force the equal shift/clock weight, this specific action as the molecular action, its clock center as a realized posting endpoint, or a dimensioned continuum radius. The theorem only certifies that this particular model's cost is nonnegative, a necessary but not sufficient condition for a physical theory. It is a building block, not a final answer.

theorem finiteWeylEventAction27_nonneg
    (center : WeylEventSite27)
    (ψ : WeylEventAmplitude27) :
    0 ≤ finiteWeylEventAction27 center ψ := by
  unfold finiteWeylEventAction27
  apply Finset.sum_nonneg
  intro axis _
  apply Finset.sum_nonneg
  intro site _
  exact add_nonneg (sq_nonneg _)
    (mul_nonneg (Complex.normSq_nonneg _) (sq_nonneg _))

theorem weylEventSite27_card :
    Nat.card WeylEventSite27 = 27 := by
  change Nat.card (Fin 3 → ZMod 3) = 27
  rw [Nat.card_fun, Nat.card_zmod, Nat.card_fin]
  norm_num

/-- Exact finite Weyl relation on every production-torus axis. -/
theorem axis_weyl_relation27
    (ψ : WeylEventSite27 → ℂ)
    (axis : Fin 3)
    (site : WeylEventSite27) :
    clockAxis27 (shiftAxis27 ψ axis) axis site =
      omega3 * shiftAxis27 (clockAxis27 ψ axis) axis site := by
  have hval :
      ∀ j : ZMod 3,
        j.val = (j - 1).val + 1 ∨
          (j.val = 0 ∧ (j - 1).val = 2) := by
    decide
  have e :
      omega3 ^ (site axis).val =
        omega3 ^ ((site axis - 1).val + 1) := by
    rcases hval (site axis) with h | ⟨h0, h2⟩
    · rw [h]
    · rw [h0, h2, pow_zero]
      exact omega3_pow_three.symm
  simp only [clockAxis27, shiftAxis27, shiftSite27]
  simp only [Function.update_self]
  rw [e]
  simp [pow_succ, mul_comm, mul_assoc, mul_left_comm]

/-- Coefficient-free self-dual Weyl action. -/
def finiteWeylEventAction27
    (center : WeylEventSite27)
    (ψ : WeylEventAmplitude27) : ℝ :=
  Finset.univ.sum fun axis : Fin 3 =>
    Finset.univ.sum fun site : WeylEventSite27 =>
      (ψ (shiftSite27 axis site) - ψ site) ^ 2 +
        clockDisplacement27 center axis site * (ψ site) ^ 2

What this page does not claim

The action is not claimed to be the unique or forced cost for recognition events. The equal shift/clock weight is not derived from Recognition Theory. The clock center is not claimed to be a physically realized endpoint.

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/Foundation/PairKernelFiniteHeisenbergWeylEventOperator.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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