Foundation Pair Kernel Constructed Source Covector Constructed Occurrence Source

The injectivity theorem

A covector is a mathematical object that assigns a number to each vector. In the Recognition Science framework, a constructed occurrence source covector is a specific covector built from two ingredients: a scale, which is a positive real number, and a pair of oriented poles, which are two distinct points with a designated direction between them. The framework defines this covector so that its value on any vector is the scale multiplied by the difference between the vector's values at the two poles. This construction is a definitional choice, a model, not a derived physical law.

The theorem constructedOccurrenceSourceCovector_scale_injective establishes that this construction is injective in its scale argument. In plain language: if two constructed covectors, built with the same poles but possibly different scales, are equal as functions, then their scales must be equal. The scale is recoverable from the covector. This is a formal theorem, proved in the framework's machine-checked library of formal theorems, and it holds because the two poles are distinct by construction, so the covector's action on a vector that distinguishes them pins down the scale uniquely.

The theorem's content is purely structural. It says nothing about what the scale should be, what physical process the covector represents, or whether this constructed object is the correct description of gravity or any other physical source. Those questions are settled elsewhere, if at all. The theorem's role is to guarantee that the construction is well-behaved: the scale is not lost information, and two different scales cannot produce the same covector.

In Recognition Science, this injectivity is a step in a larger campaign to rebuild the framework's account of a physical source without treating its scale as a free real parameter. The scale is instead an instrument field, a property of a faithful readout, and its values are forced by theorems. The injectivity theorem ensures that this instrument field is a genuine degree of freedom, one that can be recovered from the covector it produces. It is a consistency result about the construction, not a physical prediction.

/-- Scale injectivity of the underlying shape constructor, specialized to
oriented occurrence poles (distinct by construction). -/
theorem constructedOccurrenceSourceCovector_scale_injective
    {N : ℕ} [NeZero N] {d : ℕ}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N)
    {n₁ n₂ : ℕ} (k : Fin d) (side : Side)
    (h :
      constructedOccurrenceSourceCovector readout event n₁ k side =
        constructedOccurrenceSourceCovector readout event n₂ k side) :
    constructedSourceScale readout event n₁ =
      constructedSourceScale readout event n₂ :=
  pairKernelPhysicalSourceCovector_scale_injective
    (orientedPolesOfOwner k side).distinct h

/-- **Constructed occurrence source covector.**  The pair-kernel physical
source covector at the instrument gain and the oriented poles of the owner
occurrence `(k, side)`.  Scale slot is filled by `constructedSourceScale`,
never by a free real parameter. -/
def constructedOccurrenceSourceCovector
    {N : ℕ} [NeZero N] {d : ℕ}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (n : ℕ)
    (k : Fin d) (side : Side) :
    PhysicalSourceCarrier (2 * d) :=
  pairKernelPhysicalSourceCovector
    (constructedSourceScale readout event n)
    (orientedPolesOfOwner k side).source
    (orientedPolesOfOwner k side).sink

What this page does not claim

The theorem does not claim that this constructed covector is the correct description of any physical source. The theorem does not claim that the scale is forced to a specific value such as phi to the n. The theorem does not claim that the law 'source scale = phi to the fifth' has been derived.

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: