Masses Mass Genesis Open Propagation Open Null Not Finite Closed

Open null modes

In the Recognition Science framework, a recognition event is a discrete record of a distinction the universe keeps. A ledger is that discrete record of events. The framework models physical structure from this starting point, and its machine-checked library of formal theorems derives consequences from it. One consequence concerns light: the framework needs a carrier for zero rest mass, and it finds it in an open null light mode.

An open null light mode is a zero-cost propagating recognition mode carried by an injective infinite ray. In plain terms: a mode that costs nothing to recognize, moving along a path that never repeats a step and never closes on itself. The ray is a function from natural numbers to natural numbers, and injective means each step lands on a new value. This is the recognition-level carrier for zero rest mass. Polarization and photon momentum are separate physics layers, not part of this carrier.

The declaration openNull_not_finite_closed establishes that such a ray cannot close with any positive period. If the ray had a period p, then after p steps it would return to the same value, violating injectivity. The theorem proves this for every open null light mode: for any positive p and any starting point n, the ray at n+p differs from the ray at n. The proof is a direct application of injectivity, so the claim is a THEOREM in the framework's library, axiom-clean.

This matters because finite LightPatterns close after the concrete eight-tick cyclic Rhat evolution. That is right for closed matter patterns, but it made the null-propagating mode predicate empty on the finite surface. The open null mode adds the missing carrier, and the non-periodicity proof ensures it cannot be mistaken for a finite localized closed matter pattern. The framework also provides a canonical example: the identity ray, which is injective and zero-cost, proving the carrier is nonempty.

The declaration does not claim that photons exist, that polarization is derived, or that momentum is derived. Those are separate physics layers. It does not claim that the infinite ray is physically realized; it is a mathematical carrier. It does not claim that zero rest mass is the same as zero cost in any other framework. The claim is narrow: within Recognition Science, a zero-cost mode on an injective infinite ray exists, has zero rest mass, and is not a finite closed pattern.

/-- An open null light mode is a zero-cost propagating recognition mode carried
by an injective infinite ray. This is the recognition-level carrier needed for
zero rest mass; polarization and photon momentum are separate physics layers. -/
structure OpenNullLightMode where
  mode : Physics.NullRecognitionMode.PropagatingMode
  zero_cost : Physics.NullRecognitionMode.totalModeCost mode = 0
  ray : ℕ → ℕ
  ray_injective : Function.Injective ray

theorem openNull_not_finite_closed (γ : OpenNullLightMode) :
    OpenNullNotFiniteClosed γ := by
  intro n p hp
  exact openNullRay_no_positive_period γ n p hp

theorem openNullLightMode_nonempty :
    ∃ γ : OpenNullLightMode, openNullRestMass γ = 0 :=
  ⟨canonicalOpenNullMode, openNullRestMass_zero canonicalOpenNullMode⟩

What this page does not claim

Photons exist as physical objects in the framework. Polarization or photon momentum are derived from the open null mode. The infinite ray is physically realized rather than a mathematical carrier.

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/OpenPropagation.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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