Verification Yardstick Assignment Principle

The yardstick assignment

The verification yardstick assignment principle is a rule in Recognition Science that connects each particle sector to a specific combinatorial feature of a cube. The cube has 8 vertices, 12 edges, and 6 faces. The principle assigns each sector a pair of numbers, B_pow and r0, which together set the sector's mass scale. The assignment is not arbitrary: it is built from the cube's structure, and the arithmetic identities that make the assignment consistent are proved.

The B_pow values are the exponents in a power-of-two mass factor. Leptons get -22, up quarks get -1, down quarks get 23, and the electroweak sector gets 1. These values partition into two pairs with equal magnitude: the absolute values of the lepton and electroweak exponents sum to 23, which is exactly the down quark exponent. The up quark and electroweak exponents are opposites, -1 and +1. These relations are proved as theorems, not assumed.

The r0 values are offsets that also come from the cube. They are 62 for leptons, 35 for up quarks, -5 for down quarks, and 55 for electroweak. Each r0 is a linear combination of W = 17, the number of wallpaper groups, and small integer corrections. The multipliers {4, 2, -1, 3} sum to 8, the cube's vertex count. The additive corrections {-6, 1, 12, 4} sum to 11, the number of passive edges. The four r0 values sum to 147, which equals 8 times 17 plus 11.

In Recognition Science, the framework models each particle sector as coupling to a distinct level of the cube's combinatorial hierarchy. The B_pow formula for each sector encodes how much its recognition boundary borrows from the cube's edge network. Leptons borrow heavily from passive edges, giving a large negative exponent and strong suppression. Down quarks amplify through total edge doubling, giving a large positive exponent. The r0 values modulate these exponents with wallpaper-group offsets.

The contribution here is to verify that the assignment satisfies four named constraints: sign duality between up and electroweak, complementarity between lepton and electroweak summing to down, vertex exhaustion by the multipliers, and passive-edge exhaustion by the corrections. These are proved in a machine-checked library, so the arithmetic is certified. The principle does not derive the values from first principles; it proves the structural relations that hold among the assigned values.

/-- Lepton and EW form a complementary pair: |B_pow(L)| + |B_pow(EW)| = B_pow(DQ). -/
theorem lepton_ew_complement_down :
    |B_pow .Lepton| + |B_pow .Electroweak| = B_pow .DownQuark := by
  simp only [B_pow_Lepton_eq, B_pow_Electroweak_eq, B_pow_DownQuark_eq]
  norm_num

/-- Structural interpretation: each sector's B_pow reflects its edge-coupling depth.
    - Leptons: 2 × passive edges (deep edge coupling, large suppression)
    - Up quarks: active edge (minimal coupling, sign = borrowing)
    - Down quarks: 2 × total edges − 1 (complementary amplification)
    - Electroweak: active edge (minimal coupling, sign = lending)
    B_pow(Up) = −B_pow(EW) reflects the sign duality of the active edge. -/
theorem up_ew_sign_duality :
    B_pow .UpQuark = -B_pow .Electroweak := by
  simp only [B_pow_UpQuark_eq, B_pow_Electroweak_eq]

/-- The W-multipliers in the r₀ formulas sum to V = 8.
    r₀ = m × W + c, where m ∈ {4, 2, −1, 3} and c ∈ {−6, 1, 12, 4}.
    Sum of m: 4 + 2 + (−1) + 3 = 8 = V. -/
theorem W_multipliers_sum_to_V : (4 : ℤ) + 2 + (-1) + 3 = 8 := by norm_num

/-- The additive corrections sum to E_passive = 11.
    c ∈ {−6, 1, 12, 4}: sum = −6 + 1 + 12 + 4 = 11 = E_passive. -/
theorem additive_corrections_sum_to_Ep : (-6 : ℤ) + 1 + 12 + 4 = 11 := by norm_num

/-- Structural sum identity used by O1 filters: `Σ r₀ = V*W + E_passive`. -/
theorem r0_sum_eq_V_mul_W_add_Epassive :
    r0 .Lepton + r0 .UpQuark + r0 .DownQuark + r0 .Electroweak =
      (cube_vertices D : ℤ) * (W : ℤ) + (E_passive : ℤ) := by
  calc
    r0 .Lepton + r0 .UpQuark + r0 .DownQuark + r0 .Electroweak = 147 := r0_sum
    _ = (cube_vertices D : ℤ) * (W : ℤ) + (E_passive : ℤ) := by native_decide

What this page does not claim

The module does not derive the B_pow and r0 values from first principles; it proves structural relations among assigned values. The assignment principle does not predict particle masses directly; it sets exponents and offsets in a mass-scale formula. The framework's claim that sectors couple to cube hierarchy levels is a model, not a theorem.

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/Verification/YardstickAssignmentPrinciple.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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