Foundation Pair Kernel Constructed Source Covector

The constructed source covector

A covector is a linear map that takes a vector and returns a number. In Recognition Science, the constructed source covector is a specific covector that assigns a scale to a physical source, such as a particle or an event. Its defining feature is that this scale is not chosen freely. It is derived from the gain of a faithful readout, a hypothetical instrument that records events without distortion. The gain is a structure field of the instrument, meaning it is a property the instrument has, not a number inserted by hand.

The construction proceeds in two stages. First, it defines the constructed source scale as the gain of a faithful readout at a given depth, a measure of how many layers of processing the signal has passed through. Second, it feeds this scale into an existing shape constructor, the pair kernel physical source covector, at the oriented poles of an owner occurrence. The result is a covector whose scale slot is always filled by the instrument field, never by a free real parameter.

The main results are forced values and rigidity. At depth n, the scale must equal φⁿ, where φ is the golden ratio. On a complete carrier, it equals the inverse of the native action quantum. These values are not assumptions; they are consequences of the definitions and the forcing chain. Also, any two faithful readouts of the same attachment give the same scale and covector. There is no ambiguity in the construction.

Another key result is that a unit scale is impossible on a nonempty depth. Since the carrier attenuates, a faithful instrument must amplify, so its gain is always greater than one. This rules out a whole class of candidate instruments. Finally, evaluating the constructed covector on a unit potential-drop probe gives the occurrence boundary record times the constructed scale, matching the sign convention of the bridged covector.

In Recognition Science, this construction is stage 1 of a four-stage campaign to rebuild the law that sets the source scale. It does not claim to derive the law itself. Identifying the gravity source with this constructed covector remains a physical selection, not a theorem. The construction only builds the object and records the forced values of its instrument field.

/-- 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

/-- 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

/-- Hence any two faithful readouts give the same constructed covector. -/
theorem constructedOccurrenceSourceCovector_unique
    {N : ℕ} [NeZero N] {d : ℕ}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout₁ readout₂ : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (n : ℕ)
    (k : Fin d) (side : Side) :
    constructedOccurrenceSourceCovector readout₁ event n k side =
      constructedOccurrenceSourceCovector readout₂ event n k side := by
  simp only [constructedOccurrenceSourceCovector]
  rw [constructedSourceScale_unique readout₁ readout₂ event n]

/-- Constructed scale on a nonempty depth is strictly greater than one:
the carrier attenuates, so a faithful instrument must amplify.  Value-free
(never cites `1 ≠ φ⁵`). -/
theorem constructedSourceScale_gt_one_of_nonempty_depth
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (m : ℕ) :
    1 < constructedSourceScale readout event (m + 1) :=
  gain_gt_one_of_attenuating readout event m

/-- On a coherence-complete physical carrier, the constructed scale at the
event's own carrier dimension equals `nativeActionQuantumInv`. -/
theorem constructedSourceScale_completeCarrier_eq_nativeActionQuantumInv
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    constructedSourceScaleAtPhysicalCarrier readout physical event =
      nativeActionQuantumInv := by
  simpa [constructedSourceScaleAtPhysicalCarrier, constructedSourceScale] using
    completeCarrier_gain_eq_nativeActionQuantumInv
      readout hcomplete event

What this page does not claim

This construction does not claim the source scale law is derived; it only builds the object. This construction does not claim the gravity source identification is a theorem; it is a physical selection. This construction does not claim the constructed covector is the only possible source covector.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: