Foundation Pair Kernel Affine Weyl Event Action Finite Weyl Shift Occupation Cos

The nonnegative shift cost

In the Recognition Science framework, an event ledger records discrete recognition events, and each event carries a cost. The cost is not chosen freely; it is forced by five plain conditions. One component of that cost is the shift occupation cost, which measures how much a proposed amplitude pattern deviates from a reference shift. The declaration finiteWeylShiftOccupationCost27_nonneg is a theorem in the framework's machine-checked library of formal theorems. It states that for any amplitude pattern, the shift occupation cost is greater than or equal to zero. In plain language: the cost of recording a shift can be zero, but it can never be negative.

The theorem is proved by expanding the definition of the cost and applying the fact that a square of a real number is nonnegative. The cost is a sum over axes and sites of squared terms, so each term is nonnegative, and the sum of nonnegative terms is nonnegative. This is a simple but load-bearing property: it guarantees that the total action, which combines shift and clock costs, is bounded below, so minimizing it is a well-posed problem. The companion theorem finiteWeylClockOccupationCost27_nonneg proves the same for the clock cost, and affineWeylEventAction27_nonneg extends it to the full action with a positive dilation factor.

What the theorem does not claim is just as important. It does not say that the shift cost is positive; it allows zero cost for some patterns. It does not say that the cost is physically meaningful as a measured energy; the framework explicitly leaves OPEN the connection to physical Fourier exchange, dimensioned length, and SI units. The cost is a dimensionless mathematical quantity defined on a finite set of sites. It also does not claim that the shift cost alone determines the dynamics; the full action combines it with the clock cost, and the balance between them is set by a dilation coordinate that is itself a dynamical variable.

The practical consequence is that the framework can proceed with variational calculations: the action has a lower bound, so stationarity conditions are meaningful. The theorem is a small but necessary step in building a reversible carrier for events, without yet attaching it to physical reality.

theorem finiteWeylShiftOccupationCost27_nonneg
    (ψ : WeylEventAmplitude27) :
    0 ≤ finiteWeylShiftOccupationCost27 ψ := by
  unfold finiteWeylShiftOccupationCost27
  apply Finset.sum_nonneg
  intro axis _
  apply Finset.sum_nonneg
  intro site _
  exact sq_nonneg _

What this page does not claim

The shift cost is positive, only nonnegative. The shift cost is a measured physical energy. The shift cost alone determines event dynamics.

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/PairKernelAffineWeylEventAction.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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