Foundation Pair Kernel Affine Weyl Event Action Affine Weyl Relative Length Resp

A positive length response

A length scale that could be zero or negative would be a poor foundation for building a physical theory. The declaration affineWeylRelativeLengthResponse27_pos is a machine-checked proof that a specific quantity, called the relative length response, is always greater than zero whenever its two inputs are positive. The first input is a positive integer N, and the second is a positive dilation coordinate, a real number that rescales other quantities. The proof is a short chain of logic: it shows the response is the product of two positive numbers, so the product itself must be positive.

In the Recognition Science framework, this declaration lives inside a larger construction called the affine Weyl event action. The framework models events as discrete records in a ledger, and this particular action assigns a cost to a configuration of events. The dilation coordinate is a way of adjusting the relative weight between two parts of that cost: the shift cost and the clock cost. The relative length response is built by multiplying the on-shell dilation, the value that balances the two costs, by a scale factor derived from the integer N. The theorem proves that this response is never zero or negative, so it can serve as a meaningful, positive length scale within the model.

The declaration is a theorem in the framework's machine-checked library of formal theorems, meaning its proof has been verified by a computer. It is honest about its scope: it establishes positivity, and nothing more. It does not claim that this length response corresponds to any physical length in meters or any other unit. The framework explicitly leaves open the question of how this dimensionless quantity would attach to a physical event-length law. It also does not claim that the affine Weyl event action itself is the final or unique description of physical events; the framework's own documentation describes this action as a model with some proven properties, not as a forced consequence of first principles.

What the theorem changes is the internal consistency of the framework's construction. By proving that the relative length response is always positive, the framework ensures that this particular derived quantity cannot degenerate to zero or become negative, which would be a red flag for a would-be physical scale. It is a small but necessary step in building a reversible carrier for events, and it clears the way for further work on how that scale might connect to the physical world.

theorem affineWeylRelativeLengthResponse27_pos
    (N : ℕ)
    (hN : 0 < N)
    (dilation : EventDilationCoord) :
    0 < affineWeylRelativeLengthResponse27 N dilation :=
  mul_pos dilation.2 (selfDualWeylMeshScale_pos N hN)

/-- Relative length response generated by the on-shell dilation and the unique
finite-Fourier continuum chart.  It remains dimensionless until a physical
event-length law is supplied. -/
def affineWeylRelativeLengthResponse27
    (N : ℕ)
    (dilation : EventDilationCoord) : ℝ :=
  dilation.1 * selfDualWeylMeshScale N

/-- Reciprocal affine pairing of the shift and clock costs. -/
def affineWeylEventAction27
    (event : RealizedPostingEvent3 3)
    (dilation : ℝ)
    (ψ : WeylEventAmplitude27) : ℝ :=
  dilation⁻¹ ^ 2 * finiteWeylShiftOccupationCost27 ψ +
    dilation ^ 2 *
      finiteWeylClockOccupationCost27
        (realizedPostingWeylClockCenter27 event) ψ

What this page does not claim

The relative length response is a physical length in SI units. The affine Weyl event action is the unique or forced description of events. The positivity proof establishes any property of the response other than its sign.

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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