Masses Mass Genesis T10 Boson Channel Reach Ew Offset Eq Twice Boson Channel Rea

The electroweak offset

The electroweak sector of the standard model of particle physics is the part that governs the weak nuclear force and electromagnetism, carried by the W and Z bosons and the photon. In the Recognition Science framework, this sector is assigned a number called the electroweak offset, written Anchor.r0 Anchor.Sector.Electroweak. The framework's ledger, a discrete record of recognition events, assigns this offset as part of its mass-generation scheme. The theorem ew_offset_eq_twice_boson_channel_reach states that this offset, minus three times the wallpaper-group constant W (which equals 17), is exactly twice the channel reach of the W boson. Channel reach counts how many of two basic fermion properties, electric charge and color charge, a carrier can distinguish among the fermions it couples to.

The W boson couples to all fermions, so it resolves both charge and color, giving it a channel reach of 2. The theorem therefore computes the electroweak offset as 3 times 17 plus 2 times 2, which is 55. The machine-checked library of formal theorems proves this equality by unfolding the definition of channel reach and using a prior result about the offset. The proof is a finite calculation, checked by the kernel, with no hidden assumptions beyond the standard three axioms of the ambient type theory. The same library also proves that the W boson's channel reach is maximal among gauge carriers, and that the photon reaches 1 and the gluon reaches 0.

What the theorem does not claim is as important as what it proves. It does not say that the electroweak offset is derived from first principles; the offset itself is an input, not an output. It does not claim that the W boson's channel reach explains the observed masses of particles. It does not even say that the channel reach is a physical quantity in the conventional sense; it is a combinatorial count defined inside the framework. The theorem is a structural identity, a consistency check that the ledger's numbers line up with the framework's own definitions. It is not a prediction about the standard model or a measurement of any particle property.

The value of this theorem is that it shows the framework's internal arithmetic is coherent. The electroweak offset is not an arbitrary number floating free; it is tied, by a proved equality, to a simple count derived from the W boson's couplings. A reader can verify the calculation by hand: 3 times 17 is 51, plus 2 times 2 is 4, giving 55. The theorem guarantees that this equality holds in the framework's formal system, and that is all it guarantees. It is a small but solid brick in the larger mass-generation edifice, one that a skeptic can check and a believer can build on.

/-- **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 electroweak reach is `2`).** -/
theorem channelReach_wBoson : channelReach (GaugeCarrier.couples .wBoson) = 2 := by
  unfold channelReach
  rw [if_pos wBoson_resolvesCharge, if_pos wBoson_resolvesColor]
  norm_num

/-- **THEOREM (the electroweak reach is maximal among the carriers).** -/
theorem ew_reach_maximal (B : GaugeCarrier) :
    channelReach (GaugeCarrier.couples B) ≤ channelReach (GaugeCarrier.couples .wBoson) := by
  rw [channelReach_wBoson]
  exact channelReach_le_two _

/-- **THEOREM (the photon reach is `1`).** -/
theorem channelReach_photon : channelReach (GaugeCarrier.couples .photon) = 1 := by
  unfold channelReach
  rw [if_neg photon_not_resolvesCharge, if_pos photon_resolvesColor]
  norm_num

/-- **THEOREM (the gluon reach is `0`).** Its coupled set is a single
channel cell: it resolves no channel distinction. -/
theorem channelReach_gluon : channelReach (GaugeCarrier.couples .gluon) = 0 := by
  unfold channelReach
  rw [if_neg gluon_not_resolvesCharge, if_neg gluon_not_resolvesColor]
  norm_num

What this page does not claim

The electroweak offset is not derived from first principles but is an input to the framework. The theorem does not connect the channel reach to any measured particle mass. Channel reach is a combinatorial count defined inside the framework, not a conventional physical observable.

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.

Derived articles

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