Foundation Pair Kernel Constructed Source Covector Constructed Source Scale Comp

The instrument's forced scale

A ledger is a discrete record of events, and in Recognition Science every event carries a scale: a number that multiplies the difference between two bookkeeping entries. The declaration constructedSourceScale_completeCarrier_eq_uniqueCotangentCoordinate pins that scale down. It says that when a physical channel is complete, meaning it can carry every possible posting event, the scale of a faithful readout equals a single fixed coordinate, uniqueCotangentCoordinate. The proof is a chain of equalities: the constructed scale equals the inverse native action quantum, and that inverse equals the unique cotangent coordinate.

The content is best understood as a statement about instruments. A faithful readout is one that inverts the channel's own energy transfer, so it reports what actually happened instead of a distorted version. The theorem says any two such readouts on a complete carrier must agree on the scale, and that scale is forced to be the unique cotangent coordinate. This is a rigidity result: the instrument cannot be tuned to a different value and still be faithful. The scale is not a free parameter chosen by the experimenter; it is a structure field of the instrument itself.

In Recognition Science, this matters because it replaces an earlier assumption. Previously the source scale was taken to be the fifth power of the golden ratio, phi, as a law. This theorem rebuilds that claim: the scale is now derived as the gain of a faithful readout, and the value phi to the fifth power, or equivalently the unique cotangent coordinate, arrives as a forced consequence instead of a postulate. The module is the first of four stages in a campaign to rederive the Planck law from this constructed source.

What the declaration does not claim is just as important. It does not say that the constructed source covector is the gravity source; that identification remains a physical selection, a hypothesis labeled H4 in the campaign design. The theorem only builds the object and records the forced values of its instrument field. It also does not claim that the Planck bridge is complete; that closure lives in later stages. The theorem is value-free in one sense: it never cites the specific fact that one is not equal to phi to the fifth power, so the impossibility of a unit scale on a nonempty depth is a separate result.

/-- Same complete-carrier specialization equals `uniqueCotangentCoordinate`. -/
theorem constructedSourceScale_completeCarrier_eq_uniqueCotangentCoordinate
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    constructedSourceScaleAtPhysicalCarrier readout physical event =
      uniqueCotangentCoordinate := by
  rw [constructedSourceScale_completeCarrier_eq_nativeActionQuantumInv
      readout hcomplete event,
    uniqueCotangentCoordinate_eq_nativeActionQuantumInv]

/-- Any two faithful readouts of the same attachment give the same
constructed scale at every depth. -/
theorem constructedSourceScale_unique
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout₁ readout₂ : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (n : ℕ) :
    constructedSourceScale readout₁ event n =
      constructedSourceScale readout₂ event n :=
  faithful_readout_unique readout₁ readout₂ event n

/-- **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

/-- Same complete-carrier specialization equals `uniqueCotangentCoordinate`. -/
theorem constructedSourceScale_completeCarrier_eq_uniqueCotangentCoordinate
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    constructedSourceScaleAtPhysicalCarrier readout physical event =
      uniqueCotangentCoordinate := by
  rw [constructedSourceScale_completeCarrier_eq_nativeActionQuantumInv
      readout hcomplete event,
    uniqueCotangentCoordinate_eq_nativeActionQuantumInv]

What this page does not claim

The constructed source covector is not claimed to be the gravity source; that identification is a physical selection. The Planck bridge is not claimed to be complete; that closure is a target of later stages. The theorem does not claim that a unit scale is impossible on a nonempty depth; that is a separate theorem.

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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