Masses Mass Genesis T10 Run Cone Terminal Wall Cone Pure Sqrt Two Div Four

A number inside the cone

The number √2/4 is the ratio of a square's diagonal to four times its side. In the Recognition Science framework, it appears as a quantity derived from the structure of a run record, a discrete ledger of posting events. The declaration conePure_sqrtTwoDivFour proves that this number belongs to the pure cone, the set of real numbers the framework can generate from run data alone, without invoking any cost term.

The cone is built from a small set of generators: the rational numbers, √2, and the operations of addition, subtraction, multiplication, division, and natural powers. The declaration shows that √2/4 is in this set, since it is the product of √2 and the inverse of 4. This is a theorem in the machine-checked library of formal theorems, meaning the proof is verified by a computer.

The significance lies in what the cone excludes. The framework's target mass value, intendedGapOneFactorAmplitude, is not in the pure cone. A separate theorem proves this: no element of the pure cone equals that target. The target requires the golden ratio φ, which is not a generator of the pure cone. This separation is the terminal wall, a boundary between what the run record can produce and what it cannot.

In Recognition Science, this wall is a step toward showing that the target mass is not a free choice but is forced by the framework's own structure. The conePure_sqrtTwoDivFour declaration is a small but necessary part of that proof: it confirms that √2/4 is available to the framework, while the target sits outside the pure cone's reach.

theorem conePure_sqrtTwoDivFour : RunCone false (Real.sqrt 2 / 4) := by
  have h4 : ((4 : ℚ) : ℝ) = 4 := by norm_num
  rw [div_eq_mul_inv, ← h4]
  exact RunCone.mul (RunCone.sqrtTwo false)
    (RunCone.inv (RunCone.rat false 4) (by norm_num))

/-- The registered run-record derivation cone. Generators: rational
settlement data at any marking, `√2` from the holonomy transport at any
marking, and the per-post J-cost `cosh 1 − 1` at the `true` marking only.
Operations: field operations and natural powers, the only operations the
run-real theorems use. -/
inductive RunCone : Bool → ℝ → Prop
  | rat (j : Bool) (q : ℚ) : RunCone j (q : ℝ)
  | sqrtTwo (j : Bool) : RunCone j (Real.sqrt 2)
  | jcost : RunCone true (Real.cosh 1 - 1)
  | add {j : Bool} {x y : ℝ} : RunCone j x → RunCone j y → RunCone j (x + y)
  | neg {j : Bool} {x : ℝ} : RunCone j x → RunCone j (-x)
  | mul {j : Bool} {x y : ℝ} : RunCone j x → RunCone j y → RunCone j (x * y)
  | inv {j : Bool} {x : ℝ} : RunCone j x → x ≠ 0 → RunCone j x⁻¹
  | pow {j : Bool} {x : ℝ} : RunCone j x → (n : ℕ) → RunCone j (x ^ n)

/-- No element of the pure run-record cone equals the factor amplitude. -/
theorem runConePure_ne_intended {x : ℝ} (h : RunCone false x) :
    x ≠ intendedGapOneFactorAmplitude :=
  fun hx => intended_not_in_QsqrtTwo (hx ▸ runConePure_inQsqrtTwo h)

What this page does not claim

The declaration does not show that √2/4 is the target mass. The declaration does not prove that the target mass is absent from the full cone that includes the J-cost term. The declaration does not establish that the golden ratio is a generator of the pure cone.

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/T10RunConeTerminalWall.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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