Foundation Pair Kernel Constructed Source Covector Constructed Source Scale Gt O

The attenuation theorem

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of reading that record. The declaration constructedSourceScale_gt_one_of_nonempty_depth is a theorem in the framework's machine-checked library of formal theorems. It states that for any faithful channel readout, the constructed source scale at any nonempty depth is strictly greater than one. In plain language: a measuring instrument that correctly reads a signal at any non-zero depth must always amplify that signal, never pass it through unchanged.

The theorem is a consequence of the framework's definition of a faithful readout. Such a readout must satisfy two conditions: it must be positive, and it must invert the transfer of the channel's own block energy. The framework proves that these conditions force the gain to be greater than one at any depth beyond zero. The proof is value-free, meaning it never relies on the specific value of the golden ratio or any other target constant. It derives the inequality purely from the structural properties of the readout.

This result has a direct consequence: a unit-scale instrument, one that neither amplifies nor attenuates, cannot exist within the framework's class of faithful readouts. The theorem no_constructedSourceScale_eq_one_of_nonempty_depth formalizes this impossibility. It is a rigidity result: the framework does not merely say that unit gain is unlikely or unmeasured, it proves that no such instrument can satisfy the defining conditions of a faithful readout.

In Recognition Science, this theorem is a building block for the carrier rebuild, a campaign to replace an adopted law with a derived one. It does not, however, claim that the law itself is derived. The identification of the gravity source with the constructed covector remains a physical selection, not a theorem. The theorem only establishes a property of the instrument field, not the physical law it will eventually help to force.

/-- 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

/-- No faithful readout yields a unit constructed scale on a nonempty depth:
the candidate-A (unit-scale) instrument does not exist in-class. -/
theorem no_constructedSourceScale_eq_one_of_nonempty_depth
    {N : ℕ} [NeZero N]
    (channelEnergy : PostingEventChannelEnergy3 N)
    (event : RealizedPostingEvent3 N) (m : ℕ) :
    ¬ ∃ readout : FaithfulChannelReadout channelEnergy,
        constructedSourceScale readout event (m + 1) = 1 := by
  rintro ⟨readout, hunit⟩
  exact constructedSourceScale_ne_one_of_nonempty_depth
    readout event m hunit

What this page does not claim

This theorem does not derive the physical law that the source scale equals phi to the fifth power. This theorem does not prove that the constructed covector is the gravity source; that identification remains a physical selection. This theorem does not state that all instruments amplify, only those that meet the 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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