Encyclopedia Masses Masses Mass Genesis T10 Multi Quantum Posted Emission

ARTICLE 6 claims 5 theorems 1 model

Masses Mass Genesis T10 Multi Quantum Posted Emission

A machine-checked proof shows how a free amplitude parameter lets the recognition ledger emit matter, with one honest residual choice left open.

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The scaled emission mechanism

In Recognition Science, the ledger is a discrete record of recognition events, and the cost is the forced price of each recognition. The T10 module addresses a specific gap: after the Boolean unit emission, the next mechanism must allow a charged species to be emitted with zero cost. The answer is scaled occupation posting, where a free positive amplitude parameter `a` scales the settlement window. This is not a bare assertion; it is a construction with a machine-checked proof of existence.

The core structure is `EmittedForcedMatterInterfaceScaled`, which packages a readout with an amplitude that matches a topology factor. The theorems prove that this interface has a settled anchor, a zero load recognition cost, and a rest mass equal to the predicted mass. The amplitude squared equals the predicted mass divided by sixteen, a relation that ties the free parameter directly to the mass law. The module also proves that the emission window cannot be the gap-two mode, because that mode lacks adjacent-pair support, a shape constraint that is independent of amplitude.

In Recognition Science, the framework models this as a scaled emission channel. The key equation is `a² = predictedMass / 16`, which shows how the amplitude determines the mass. The existence theorem `emittedForcedMatterInterfaceScaled_nonempty` proves that for any settled octave, such an interface exists. The certificate `t10MultiQuantumPostedEmissionCert` is a theorem, not an axiom, and the entire construction has no `sorry` and no new Lean axioms.

The honest packaging is the residual `PostedAmplitudeMatchesTopologyFactor`. Choosing the amplitude equal to the topology factor amplitude yields sigma-zero, but this choice is not forced by the Boolean ledger. It is named as a residual wall, meaning the framework proves the mechanism exists but does not claim the specific amplitude is uniquely determined. This is the difference between a theorem and a forced result.

The practical consequence is that the framework now has a proven path from a free amplitude to a mass, with a clear boundary between what is derived and what remains a named choice. This lets the next stage of the forcing chain proceed without pretending the amplitude is fixed by the ledger alone.

MODEL EmittedForcedMatterInterfaceScaled · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean
EmittedForcedMatterInterfaceScaled · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:589 
/-- **Honest scaled interface.** Scaled origin emission discharges settled-
anchor shape at free amplitude `a`. Named residual:
`a = primitivePositiveStationaryFactorAmplitude` (equivalently pattern
sigma-zero on that settled anchor). -/
structure EmittedForcedMatterInterfaceScaled where
  readout : EmittedScaledSettledReadoutCarrier
  amplitude_matches_factor :
    readout.amplitude =
      primitivePositiveStationaryFactorAmplitude readout.pattern
THEOREM settled_anchor · pattern_sigma_zero · restMass_eq_predictedMass · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean
settled_anchor · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:401 
theorem settled_anchor (c : EmittedScaledSettledReadoutCarrier) :
    SettledAnchorWindow c.amplitude c.pattern :=
  reads_originEmittedScaled_settledAnchor
    c.octave c.amplitude c.amplitude_pos c.reads_emitted
pattern_sigma_zero · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:605 
theorem pattern_sigma_zero (iface : EmittedForcedMatterInterfaceScaled) :
    loadRecognitionCost iface.readout.pattern = 0 := by
  obtain ⟨E⟩ := iface.readout.q3_carrier
  exact (settledAnchor_loadRecognitionCost_zero_iff_amplitude
    E iface.settled_anchor (le_of_lt iface.readout.amplitude_pos)).2
    iface.amplitude_matches_factor
restMass_eq_predictedMass · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:618 
theorem restMass_eq_predictedMass
    (iface : EmittedForcedMatterInterfaceScaled) :
    restMass iface.readout.pattern =
      predictedMass iface.readout.pattern := by
  obtain ⟨E⟩ := iface.readout.q3_carrier
  exact loadNormalizedToTopology_forces_massLaw E iface.loadNormalized
THEOREM amplitude_sq_eq_predictedMass_div_sixteen · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean
amplitude_sq_eq_predictedMass_div_sixteen · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:625 
theorem amplitude_sq_eq_predictedMass_div_sixteen
    (iface : EmittedForcedMatterInterfaceScaled) :
    iface.readout.amplitude ^ 2 =
      predictedMass iface.readout.pattern / 16 := by
  have hnorm := settledAnchor_neutralizedNormSq iface.settled_anchor
  have hsite :
      normSq8 (neutralize (iface.readout.pattern.window 0)) =
        predictedMass iface.readout.pattern / 8 :=
    loadNormalizedToTopology_site0_load_eq_predictedMass_div_eight
      iface.loadNormalized
  rw [hnorm] at hsite
  linarith
THEOREM window_ne_gapTwo · gapTwo_not_adjacentPairSupport · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean
window_ne_gapTwo · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:234 
theorem window_ne_gapTwo (e : EmittedPhotonStateScaled) :
    e.window ≠ gapTwoTwoPhaseMode := by
  rw [window_eq_scaledCommit]
  exact scaledCommitSettlement_ne_gapTwo e.amplitude e.phase
gapTwo_not_adjacentPairSupport · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:60 
/-- Gap-two occupies ticks `{0,2}`, which is never an adjacent tick pair. -/
theorem gapTwo_not_adjacentPairSupport (phase : Fin 8) :
    ¬ RawPhotonAdjacentPairSupport phase gapTwoTwoPhaseMode := by
  intro hsup
  -- Adjacent support forces every off-pair tick to vanish; gap-two is
  -- nonzero at both 0 and 2, so both must lie in `{phase, phase+1}`.
  have h0mem : (0 : Fin 8) = phase ∨ (0 : Fin 8) = phase + 1 := by
    by_contra hnin
    push_neg at hnin
    have := hsup 0 hnin.1 hnin.2
    simp [gapTwoTwoPhaseMode] at this
  have h2mem : (2 : Fin 8) = phase ∨ (2 : Fin 8) = phase + 1 := by
    by_contra hnin
    push_neg at hnin
    have := hsup 2 hnin.1 hnin.2
    simp [gapTwoTwoPhaseMode] at this
  -- `{0,2}` cannot be an adjacent pair: their difference is 2, not 1.
  match h0mem, h2mem with
  | Or.inl h0p, Or.inl h2p =>
      exact absurd (h0p.trans h2p.symm) (by decide : ¬(0 : Fin 8) = 2)
  | Or.inl h0p, Or.inr h2c =>
      have hphase : phase = 0 := h0p.symm
      have : (2 : Fin 8) = (1 : Fin 8) := by
        simpa [hphase] using h2c
      exact absurd this (by decide)
  | Or.inr h0c, Or.inl h2p =>
      have hphase : phase = 2 := h2p.symm
      have : (0 : Fin 8) = (3 : Fin 8) := by
        simpa [hphase] using h0c
      exact absurd this (by decide)
  | Or.inr h0c, Or.inr h2c =>
      exact absurd (h0c.trans h2c.symm) (by decide : ¬(0 : Fin 8) = 2)
THEOREM emittedForcedMatterInterfaceScaled_nonempty · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean
emittedForcedMatterInterfaceScaled_nonempty · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:665 
theorem emittedForcedMatterInterfaceScaled_nonempty
    (octave : Q3SettledLedgerOctave) :
    Nonempty EmittedForcedMatterInterfaceScaled :=
  ⟨EmittedForcedMatterInterfaceScaled.ofGapOneFactorAmplitude octave⟩
THEOREM t10MultiQuantumPostedEmissionCert · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean
t10MultiQuantumPostedEmissionCert · IndisputableMonolith/Masses/MassGenesis/T10MultiQuantumPostedEmission.lean:713 
theorem t10MultiQuantumPostedEmissionCert :
    T10MultiQuantumPostedEmissionCert where
  scaled_tick_eq_scaled_commit := channelTickCurrentScaled_eq_scaledCommit
  scaled_ne_gap_two := channelTickCurrentScaled_ne_gapTwo
  gap_two_not_adjacent := gapTwo_not_adjacentPairSupport
  emitted_scaled_ne_gap_two := EmittedPhotonStateScaled.window_ne_gapTwo
  amplitude_equation_reexport := fun E hwin ha =>
    settledAnchor_loadRecognitionCost_zero_iff_amplitude E hwin ha
  factor_choice_sigma_zero :=
    scalePattern_gapOne_at_factorAmplitude_sigmaZero
  residual_wall := postedAmplitudeMatchesTopologyFactor_wall
  scaled_interface_inhabited := emittedForcedMatterInterfaceScaled_nonempty
  scaled_interface_mass_law := fun iface => iface.restMass_eq_predictedMass
  honest_world := trivial

What this page does not claim

The amplitude `a` is uniquely determined by the ledger; the match to the topology factor is a named residual, not a forced result. The module derives a specific numerical value for any particle mass; it proves the structural relation between amplitude and mass. The scaled emission mechanism is the only possible way to emit charged matter; it is one proven construction.

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/T10MultiQuantumPostedEmission.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.

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