Foundation Pair Kernel Constructed Source Covector Constructed Source Scale Ne O

The forced gain

In the Recognition Science framework, a ledger (a discrete record of events) is read by an instrument. The declaration constructedSourceScale_ne_one_of_nonempty_depth is a theorem about that instrument's gain, which is the factor by which it multiplies a signal. The theorem states that on any nonempty depth of the ledger, meaning any level that actually contains an event, this gain cannot equal one. The proof is direct: the carrier attenuates, so a faithful instrument must amplify, and therefore its gain is strictly greater than one.

This result is a consequence of how the framework defines a faithful readout. Such a readout must preserve positivity and invert the transfer against the attachment's own block energy. The theorem does not depend on any particular value of the golden ratio or other constants; it is value-free. It rules out the existence of a faithful instrument with unit gain on a nonempty depth, which the framework calls the candidate-A instrument. The framework's machine-checked library of formal theorems proves this impossibility.

What the theorem does not claim is just as important. It does not say that the gain has any specific value, only that it is not one. It does not derive the law that the source scale equals phi to the fifth power; that is a separate, later stage of the framework's development. The theorem is a structural impossibility result, not a numerical prediction. It establishes a boundary condition on what a faithful instrument can be, without saying what it must be.

/-- Constructed scale cannot equal one on a nonempty depth. -/
theorem constructedSourceScale_ne_one_of_nonempty_depth
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (m : ℕ) :
    constructedSourceScale readout event (m + 1) ≠ 1 :=
  ne_of_gt (constructedSourceScale_gt_one_of_nonempty_depth
    readout event m)

/-- 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

/-- Constructed scale cannot equal one on a nonempty depth. -/
theorem constructedSourceScale_ne_one_of_nonempty_depth
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (m : ℕ) :
    constructedSourceScale readout event (m + 1) ≠ 1 :=
  ne_of_gt (constructedSourceScale_gt_one_of_nonempty_depth
    readout event m)

What this page does not claim

The theorem does not derive the numerical value of the gain. The theorem does not prove that the source scale equals phi to the fifth power. The theorem does not say that a unit-gain instrument is physically impossible outside the framework's definition of a faithful readout.

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)

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Derived articles

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