Constants Native Dimensional Boundary Calibrated Tick Square Pos

The calibration boundary

The declaration `calibratedTickSquare_pos` is a small but load-bearing result inside the Recognition Science framework. It states that if you supply any positive number as the value of Newton's constant G, then the formula for the squared tick scale, the framework's native unit of time, produces a positive number. The formula is `calibratedTickSquare(G_input) = π · hbar_SI · G_input / c_SI^5`, where `hbar_SI` and `c_SI` are the framework's stored values for Planck's constant and the speed of light in SI units. The theorem is proved in the machine-checked library of formal theorems, with no unproved assumptions beyond the standard axioms of the underlying type theory.

The point of the theorem is not the positivity itself, which is obvious to any reader of the formula. The point is what the formula represents: a dimensional bridge. The framework can derive dimensionless relations among its native constants, such as `hbar_RS = φ^(-5)` and `G_RS · hbar_RS = 1/π`. But it cannot, from pure dimensionless data alone, output the absolute SI value of hbar or G. The reason is dimensional analysis: the dimensions of c, hbar, and G are independent, so no nontrivial monomial `c^a hbar^b G^d` is dimensionless. A dimensional bridge needs a dimensional anchor, and the theorem `calibratedTickSquare_pos` is part of the formal record that one anchor suffices and determines the whole bridge uniquely.

In Recognition Science, the framework models reality as a discrete record of recognition events, and the cost of recognition is forced by a proved uniqueness theorem. The tick is the framework's native unit of time, and the tick square is the scale that converts native time to seconds. The declaration `calibratedTickSquare_pos` establishes that this conversion is a calibration map: for every positive supplied anchor `G_input`, the bridge assigns a positive tick-square scale, and different anchors give different scales. The companion theorem `si_bridge_is_calibration_not_prediction` packages this as: the bridge calibrates, it does not predict.

What the declaration does not claim is just as important. It does not claim that the framework predicts the measured value of G from first principles. It does not claim that the SI value of hbar or G is derived. It does not claim that the dimensional anchor is unnecessary. The theorem is honest about the boundary: the framework can force dimensionless relations, but the absolute SI scale requires one measured input, and the bridge then determines everything else. This is not a weakness of the framework; it is dimensional analysis. The positive half, that one anchor suffices and determines the whole bridge uniquely, is formalized elsewhere in the library, and the capstone result shows that T0-T8 plus the single anchor τ₀ yields the electron mass in SI kilograms with the mass audit at 0 Hypothesis / 0 Open / 0 External.

/-- Any positive supplied `G_input` gives a positive calibrated tick square. -/
theorem calibratedTickSquare_pos {G_input : ℝ} (hG : 0 < G_input) :
    0 < calibratedTickSquare G_input := by
  unfold calibratedTickSquare
  exact div_pos (mul_pos (mul_pos Real.pi_pos hbar_SI_pos) hG) (pow_pos c_SI_pos 5)

/-- No nontrivial monomial in `c`, `hbar`, and `G` is dimensionless. -/
theorem no_nontrivial_dimensionless_monomial {a b d : ℤ}
    (h : cHbarGDimension a b d = (0, 0, 0)) :
    a = 0 ∧ b = 0 ∧ d = 0 := by
  unfold cHbarGDimension at h
  simp only [Prod.mk.injEq] at h
  rcases h with ⟨hL, hT, hM⟩
  omega

/-- The SI bridge is a calibration map: for every positive supplied dimensional
anchor `G_input`, the bridge assigns a positive tick-square scale, and different
anchors give different scales. -/
theorem si_bridge_is_calibration_not_prediction :
    (∀ G_input : ℝ, 0 < G_input → 0 < calibratedTickSquare G_input) ∧
      Function.Injective calibratedTickSquare :=
  ⟨fun _ hG => calibratedTickSquare_pos hG, calibratedTickSquare_injective⟩

What this page does not claim

The framework predicts the measured SI value of Newton's constant G from first principles. The declaration `calibratedTickSquare_pos` by itself proves that one anchor suffices; that is a separate result in the library. The dimensional anchor is unnecessary; the theorem explicitly requires a positive supplied `G_input`.

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/Constants/NativeDimensionalBoundary.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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