Masses Mass Genesis Null Vs Closed No Null Propagating Mode For Finite Cyclic Rh

The finite cyclic split

The theorem no_nullPropagatingMode_for_finite_cyclic_Rhat states a structural fact about a specific discrete model. In the framework's ledger, a discrete record of recognition events, a recognition event is a tick that updates a pattern. The model in question uses a finite cyclic surface, meaning the pattern space repeats after eight ticks. The theorem proves that on this surface, no LightPattern can satisfy the NullPropagatingMode predicate, which describes a zero-cost propagating mode. Every finite pattern has an eight-tick closure witness, so the null-propagating condition is empty here.

The classical counterpart is the split between light and matter. In the framework, a stable closed pattern with positive rest mass is the matter side. The theorem closed_nontrivial_restMass_pos proves that a localized, nontrivial neutral load yields a stable pattern with positive rest mass. The null side, by contrast, is carried by OpenNullLightMode, a zero-cost propagating mode on an injective infinite ray. The theorem open_null_light_exists proves such a mode exists, and open_null_light_restMass_zero proves its rest mass is zero.

What the theorem does not claim is broader. It does not say null modes are impossible in general; it says they are impossible on the finite cyclic surface specifically. It does not claim the null predicate is vacuous everywhere; the open carrier provides a non-vacuous null side. It does not derive the speed of light or any dynamical propagation law. It only records that in this finite model, the null-propagating predicate is empty, and the physical null role is represented separately by the open carrier.

/-- In the concrete finite cyclic Rhat surface, no `LightPattern` can satisfy
the current null-propagating predicate, because every finite pattern has the
eight-tick closure witness. -/
theorem no_nullPropagatingMode_for_finite_cyclic_Rhat
    (ψ : LightPattern Λ) :
    ¬ NullPropagatingMode ψ := by
  intro hnull
  exact hnull.2.1 (closedRHatOrbit_period_eight ψ)

theorem closed_nontrivial_restMass_pos
    (ψ : LightPattern Λ)
    (hloc : LocalizedSupport ψ)
    (hnontrivial : NontrivialNeutralLoad ψ) :
    0 < restMass ψ :=
  restMass_pos_of_stable ψ
    (closed_nontrivial_is_stable ψ hloc hnontrivial)

theorem open_null_light_exists :
    ∃ γ : OpenNullLightMode, openNullRestMass γ = 0 :=
  openNullLightMode_nonempty

theorem open_null_light_restMass_zero
    (γ : OpenNullLightMode) :
    openNullRestMass γ = 0 :=
  openNullRestMass_zero γ

What this page does not claim

The theorem does not claim null modes are impossible in all models, only on the finite cyclic surface. It does not claim the null predicate is non-vacuous; the open carrier is a separate construction. It does not derive the speed of light or any dynamical propagation law.

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/NullVsClosed.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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