Foundation Pair Kernel Constructed Source Covector Constructed Source Scale At P

The instrument's scale

In the Recognition Science framework, a ledger (a discrete record of events) is read by an instrument called a faithful channel readout. The theorem `constructedSourceScaleAtPhysicalCarrier_eq_gain` states that the scale assigned to a physical source at its own carrier dimension is exactly the instrument's gain at that depth. It is a definitional equality: the constructed scale is defined to be that gain, so the proof is immediate by unfolding definitions.

The significance is that the scale is not a free parameter. The framework models the scale as a structure field of the instrument, and the theorem establishes that this field is the same object as the readout's gain. This is the first stage of a carrier rebuild, where the source scale is built from the instrument's properties rather than inserted by hand.

The theorem itself does not claim that the instrument's gain has any particular numerical value. It does not assert that the gain equals φ, the golden ratio, or any other constant. Those values are derived in separate theorems, such as `constructedSourceScale_eq_phi_pow`, which states the scale at depth n equals φ^n, and `constructedSourceScale_completeCarrier_eq_nativeActionQuantumInv`, which applies on a coherence-complete carrier.

In Recognition Science, this result is a step toward showing that physical scales are forced by the structure of recognition, not chosen. The theorem's role is to tie the abstract scale to the concrete instrument reading, making the later forced values meaningful as properties of the readout itself.

theorem constructedSourceScaleAtPhysicalCarrier_eq_gain
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (physical : PostingPhysicalChannelCarrier3 N)
    (event : RealizedPostingEvent3 N) :
    constructedSourceScaleAtPhysicalCarrier readout physical event =
      readout.gain event
        (physicalPostingCarrierDimension3 physical event) :=
  rfl

/-- **Constructed source scale.**  The faithful readout's gain at carrier
depth `n`.  Defining fields are target-blind: the scale is an instrument
structure field (`gain`), not a free real and not a named target constant.
Values `φⁿ`, `nativeActionQuantumInv`, and `uniqueCotangentCoordinate`
arrive only as theorems below. -/
def constructedSourceScale
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (n : ℕ) : ℝ :=
  readout.gain event n

/-- At depth `n`, the constructed scale is forced to `φⁿ`. -/
theorem constructedSourceScale_eq_phi_pow
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (n : ℕ) :
    constructedSourceScale readout event n =
      Constants.phi ^ (n : ℤ) :=
  gain_forced readout event n

What this page does not claim

The theorem does not claim the instrument's gain has any specific numerical value. The theorem does not claim the gravity source is identified with this constructed covector; that remains a physical selection. The theorem does not claim the Planck law is derived; that is a target for later stages.

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/PairKernelConstructedSourceCovector.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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