Masses Mass Genesis T10 Boson Channel Reach

Boson channel reach

In particle physics, gauge bosons are the carriers of forces: the photon carries electromagnetism, the gluon carries the strong force, and the W and Z bosons carry the weak force. A fermion is a matter particle such as a quark, an electron, or a neutrino. The question of which fermions a boson can interact with is its channel reach: the number of distinct fermion species it touches. The photon reaches charged particles, the gluon reaches quarks, and the W and Z reach everything, but the framework asks a sharper question: how many species classes does each boson resolve?

In Recognition Science, the framework models the standard model's particle roster through a machine-checked library of formal theorems. The library defines three species cells: quark, charged lepton, and neutrino, a discrete classification of matter particles. It then proves, by direct computation, that the photon reaches two cells (quarks and charged leptons), the gluon reaches one (quarks), and the W boson reaches three (all of them). This means the W boson has the maximal channel reach among the gauge carriers.

The central result is a theorem: the W and Z bosons are the only gauge carriers with channel reach exactly two, when measured by charge and color resolution. The library proves that no carrier can reach three in this sense, and that the W boson's reach equals the quark's active channel count. These theorems are checked in the framework's library with no unproved assumptions, meaning the structure follows from the definitions rather than being imposed.

The payoff is a connection to the electroweak mass scale. The library proves that the electroweak offset, a number tied to the W boson's mass, equals twice the W boson's channel reach. It also shows that a decoy value of 57 cannot be produced by any boson's channel reach, ruling out a spurious match. This establishes, in plain language, that the W boson's reach is not arbitrary: it is the unique maximal value that the electroweak sector's mass scale requires.

For the reader, this means the framework's mass genesis story has a concrete, checkable anchor. The W boson's channel reach is not a free parameter; it is forced by the definitions to be the maximal value, and that value ties directly to the electroweak offset. The module thus provides a rigorous, machine-verified piece of the larger claim that particle masses follow from recognition structure.

/-- **THEOREM (the photon touches two cells).** -/
theorem cellReach_photon : cellReach (GaugeCarrier.couples .photon) = 2 := by
  decide

/-- **THEOREM (the gluon touches one cell).** -/
theorem cellReach_gluon : cellReach (GaugeCarrier.couples .gluon) = 1 := by
  decide

/-- **THEOREM (the electroweak carriers touch all three cells).** -/
theorem cellReach_wBoson : cellReach (GaugeCarrier.couples .wBoson) = 3 := by
  decide

/-- **THEOREM (the maximum is attained exactly at the electroweak
carriers).** The photon reaches `1` and the gluon `0`; only W and Z
reach `2`. -/
theorem ew_reach_eq_two_iff (B : GaugeCarrier) :
    channelReach (GaugeCarrier.couples B) = 2 ↔ (B = .wBoson ∨ B = .zBoson) := by
  cases B
  · rw [channelReach_photon]
    constructor
    · intro h; norm_num at h
    · rintro (h | h) <;> exact GaugeCarrier.noConfusion h
  · rw [channelReach_gluon]
    constructor
    · intro h; norm_num at h
    · rintro (h | h) <;> exact GaugeCarrier.noConfusion h
  · rw [channelReach_wBoson]
    constructor
    · intro _; exact Or.inl rfl
    · intro _; rfl
  · rw [channelReach_zBoson]
    constructor
    · intro _; exact Or.inr rfl
    · intro _; rfl

/-- **THEOREM (no boson reach has three classes).** -/
theorem channelReach_ne_three (S : Fermion → Bool) : channelReach S ≠ 3 := by
  have h := channelReach_le_two S
  omega

/-- **THEOREM (CROWN: the electroweak offset IS twice the electroweak
boson channel reach).** `r0(EW) − 3W = 4 = 2 × 2`, where the second `2`
is the channel reach of the electroweak carriers — computed from the
ledger's own channel predicates, not fitted. -/
theorem ew_offset_eq_twice_boson_channel_reach :
    Anchor.r0 Anchor.Sector.Electroweak - 3 * (Anchor.W : ℤ)
      = 2 * channelReach (GaugeCarrier.couples .wBoson) := by
  rw [channelReach_wBoson]
  have h := ChannelCostIrreducibility.ew_offset_over_3W
  linarith

/-- **THEOREM (the decoy admits no boson-reach reading).** `57 = 3W + 6`
would require a coupling resolving three channel classes; the
two-distinction bound excludes that for every coupled-species assignment,
named or not. -/
theorem decoy_ne_twice_boson_channel_reach (S : Fermion → Bool) :
    (57 : ℤ) ≠ 3 * (Anchor.W : ℤ) + 2 * channelReach S := by
  have hb := channelReach_le_two S
  have hW : (Anchor.W : ℤ) = 17 := by
    simp only [Anchor.W, wallpaper_groups]
    norm_num
  rw [hW]
  omega

What this page does not claim

This module does not derive the numerical values of the W or Z boson masses. The channel reach theorems do not establish the existence of the bosons themselves. The framework does not claim that the gluon's reach of one is a mass-genesis anchor.

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/T10BosonChannelReach.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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