Masses Mass Genesis T10 Yardstick Premise Free Cert Ew Offset Eq Twice Derived C

The electroweak offset identity

The electroweak interaction is the force responsible for radioactive decay and for the fusion that powers the Sun. In the Recognition Science framework, its structure is encoded in a quantity called the electroweak yardstick, a single integer that anchors the mass calculations for the W and Z bosons. The theorem ew_offset_eq_twice_derived_channel_reach states a precise identity: the electroweak offset, defined as the yardstick minus three times the top-generation torsion W, equals twice the derived channel reach. Here the channel reach counts how many distinct fermion species a given coupling touches, and the derived W coupling is the one that touches every inhabited cell, a maximal reach of two.

The content of the theorem is that this offset is not a free parameter. It is forced by the structure of the derivation itself. The proof unfolds the definition of the derived yardstick, which is built from the derived cell reach and the derived channel reach, and then applies the ring axioms to show the identity holds. The value of the offset is 4, since the yardstick is 55, three times W (which is 17) is 51, and twice the channel reach (which is 2) is 4. The theorem is part of a larger certificate that shows the banked electroweak yardstick of 55 is not a stipulation but a consequence of the derived couplings and the wallpaper count.

What the theorem does not claim is that the electroweak interaction itself is derived from nothing. The identity uses the derived W coupling, which is defined in terms of the framework's fermion species and their charge and color couplings. It does not claim that the value of the W torsion is derived; that value, 17, comes from the wallpaper group count. It also does not claim that the electroweak yardstick is unique in any absolute sense; it shows that given the derived couplings, the offset is forced. The theorem is a statement about the internal consistency of the framework's mass-genesis chain, not a claim about the physical origin of the electroweak force.

The practical consequence is that downstream amplitude modules can consume the certified value 55 instead of the banked definition, knowing that the value arises from the derivation rather than from a frozen stipulation. The theorem also acts as a discriminator: any alternative clause, such as the decoy 3W + 6, would make the identity false. This is how the framework earns confidence in its mass ladder: not by assuming the numbers, but by showing that the structure forces them.

/-- **THEOREM (C13 crown, derived form).** The electroweak offset IS twice
the derived channel reach: `r0(EW) - 3W = 2 * channelReach(derivedW)`. -/
theorem ew_offset_eq_twice_derived_channel_reach :
    Anchor.r0 Anchor.Sector.Electroweak - 3 * (Anchor.W : ℤ)
      = 2 * channelReach derivedWCouples := by
  rw [anchor_r0_ew_eq_derivedEWYardstick]
  unfold derivedEWYardstick
  rw [cellReach_derived_w]
  ring

/-- **THEOREM (C13 crown, derived form).** The electroweak offset IS twice
the derived channel reach: `r0(EW) - 3W = 2 * channelReach(derivedW)`. -/
theorem ew_offset_eq_twice_derived_channel_reach :
    Anchor.r0 Anchor.Sector.Electroweak - 3 * (Anchor.W : ℤ)
      = 2 * channelReach derivedWCouples := by
  rw [anchor_r0_ew_eq_derivedEWYardstick]
  unfold derivedEWYardstick
  rw [cellReach_derived_w]
  ring

/-- **THEOREM (the derived value, cone-clean).** The derived yardstick
computes to `55` from the derived reaches and the wallpaper count. No
`Anchor.r0` occurs anywhere in this theorem's dependency cone: the numeral
arises from the derivation, not from the banked definition. -/
theorem derivedEWYardstick_eq_55 : derivedEWYardstick = 55 := by
  have hW := anchor_W_eq_17
  unfold derivedEWYardstick
  rw [cellReach_derived_w, channelReach_derived_w]
  omega

private theorem anchor_W_eq_17 : (Anchor.W : ℤ) = 17 := by
  simp only [Anchor.W, wallpaper_groups]
  norm_num

/-- **THEOREM (maximality, derived form).** No coupled-species assignment
out-reaches the derived W coupling: the offset reads the maximal channel
reach. -/
theorem channelReach_derived_maximal (S : Fermion → Bool) :
    channelReach S ≤ channelReach derivedWCouples := by
  rw [channelReach_derived_w]
  exact channelReach_le_two S

/-- **THEOREM (the agreement bridge).** The banked anchor's electroweak
clause `3 * W + 4` coincides with the derived structure: the definitional
clause, unfolded once, IS the derived reaches priced at `W`. This is the
only theorem in the certificate whose cone contains `Anchor.r0`, and it
contains it as the definition being certified, never as a premise: any
other clause (the decoy `3 * W + 6`) would make this theorem false. -/
theorem anchor_r0_ew_eq_derivedEWYardstick :
    Anchor.r0 Anchor.Sector.Electroweak = derivedEWYardstick := by
  unfold derivedEWYardstick
  rw [cellReach_derived_w, channelReach_derived_w]
  simp only [Anchor.r0]
  norm_num

What this page does not claim

The theorem does not derive the value of the W torsion from first principles; that value comes from the wallpaper group count. The theorem does not claim the electroweak interaction itself is derived from nothing; it uses the defined derived W coupling. The theorem does not establish the uniqueness of the yardstick 55 outside the context of the derived couplings.

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/T10YardstickPremiseFreeCert.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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