Masses Mass Genesis T10 Field Current Selector Degeneracy Linear Law Doubled Fai

The linear law's one clean result

A linear equation is one where a quantity grows in direct proportion to another: double the input, double the output. In the framework's account of how particle masses arise, any candidate law that assigns a magnitude to a recognition event (a discrete record of an occurrence) must take this shape, because the underlying current and window scale exactly with the amplitude. The theorem linearLaw_doubled_fails_of_accepts states a consequence that is pure arithmetic: if such a law accepts a particular intended amplitude, it cannot also accept twice that amplitude. For positive amplitudes and a nonzero unit magnitude, the equation a·u = C that holds at a = intended cannot hold at a = 2·intended, because that would force intended = 2·intended, which is impossible.

The result is the easy half of a larger argument. It shows that a linear magnitude law, once it accepts the intended witness, automatically rejects its doubled copy. The hard half, which remains open, is where the constant C itself comes from. The theorem says nothing about that. It does not derive C from any deeper principle, and it does not show that any particular law is the correct one. It only certifies that the rejection of the doubled copy is not where a candidate law fails; that failure is guaranteed by the linear shape alone.

In the framework's library, this fact is one of several that together reduce the search for a mass-selection law to a single question: is the constant C derivable from something other than the target it is meant to select? The degeneracy triple, proved alongside, shows that in the scaled sourced world the field vanishes, the current is amplitude times the unit current, and the window is amplitude times the unit window. So every magnitude law is linear, and the doubled-copy rejection is automatic. What remains open is the provenance of C. Until that is derived, every field-to-current selector is either homogeneous, circular, or walled off by other constraints. The theorem under discussion is therefore a clean, narrow result: it establishes one rejection, and it leaves the harder question of C's origin untouched.

/-- The doubled copy of any accepted linear magnitude law fails: the law that
accepts `intended` does not accept `2 * intended` (for positive intended and
nonzero unit magnitude). This is the easy half; the hard half is deriving
`C` independently of the target. -/
theorem linearLaw_doubled_fails_of_accepts
    {u intended C : ℝ} (hu : u ≠ 0) (hintended : 0 < intended)
    (hacc : LinearMagnitudeLaw u C intended) :
    ¬ LinearMagnitudeLaw u C (2 * intended) := by
  intro hdoubled
  have h1 : intended * u = 2 * intended * u := hacc.trans hdoubled.symm
  have h2 : intended = 2 * intended := mul_right_cancel₀ hu h1
  linarith

/-- The scaled channel's current at every simplex is exactly the amplitude
times the unit-amplitude current. -/
theorem postedPhotonChannelScaled_current_smul
    (octave : Q3SettledLedgerOctave) (phase : Fin 8) (a : ℝ)
    (s : Simplex (postedPhotonChannelScaled octave phase a).mesh 1) :
    (postedPhotonChannelScaled octave phase a).J s =
      a * (postedPhotonChannelScaled octave phase 1).J s := by
  simp only [postedPhotonChannelScaled, postedTickCurrentScaled, one_mul]

/-- The derived window is exactly the amplitude times the unit-amplitude
window. -/
theorem emittedWindow_smul
    (octave : Q3SettledLedgerOctave) (phase : Fin 8) (a : ℝ) :
    channelTickCurrentScaled octave phase a =
      fun t => a * channelTickCurrentScaled octave phase 1 t := by
  funext t
  have h0 := congrFun
    (channelTickCurrentScaled_eq_scaledCommit octave phase a) t
  have h1 := congrFun
    (channelTickCurrentScaled_eq_scaledCommit octave phase 1) t
  simp only [scaledCommitSettlementWindow] at h0 h1
  rw [h0, h1]
  simp

What this page does not claim

The theorem does not derive the constant C from any deeper principle. The theorem does not show that any particular linear law is the correct mass selector. The theorem does not address the open question of C's provenance.

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/Masses/MassGenesis/T10FieldCurrentSelectorDegeneracy.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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