Foundation Pair Kernel Constructed Source Covector No Constructed Source Scale E

The unit-scale impossibility

The declaration no_constructedSourceScale_eq_one_of_nonempty_depth proves, inside the Recognition Science framework's machine-checked library of formal theorems, that no faithful readout can have a constructed source scale equal to one at any nonempty depth. In plain language: an instrument that reports the scale of an event cannot return a gain of exactly 1 when there is any actual depth to the event. The proof runs from the definition of a faithful readout, an instrument field that must be positive and invert the transfer of the attachment it reads, to the theorem that the constructed scale at any depth beyond zero is strictly greater than one.

The statement is a theorem, not a definition or a hypothesis. It is proved in the module PairKernelConstructedSourceCovector.lean by combining the attenuation property of a carrier with the theorem that the scale is forced to a value greater than one. The proof is value-free in a specific sense: it never cites the fact that 1 is not equal to φ⁵. It does not need the numerical value of the golden ratio to rule out a unit scale. The impossibility is structural, coming from the requirement that a faithful instrument must amplify because the carrier attenuates.

What the declaration does not claim is as important as what it proves. It does not claim that the constructed scale is the only possible scale, nor that the law of gravity is derived from it. The docstring for the module is explicit: identifying the gravity source with this constructed covector remains a physical selection, a hypothesis labeled H4 in the reopen design. The module only builds the object and records forced values of its instrument field. It does not claim that a unit scale is impossible for all instruments, only for faithful ones, and only at nonempty depth.

The consequence for the framework is that a whole class of candidate scales, those with unit gain, is ruled out before any target constant is named. This is a step in a larger campaign to rebuild the Planck law derivation, but it is not the derivation itself. The theorem narrows the space of possible instruments, and the next stages of the campaign carry the normalization closure and the reproof of the Planck bridge.

/-- 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

/-- 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

The declaration does not claim the law of gravity is derived from the constructed source covector. The declaration does not claim a unit scale is impossible for all instruments, only for faithful ones at nonempty depth. The declaration does not claim the constructed scale is the only possible scale for an event.

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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