Constants Native Dimensional Boundary Calibrated Tick Square Injective

The calibration map

The declaration calibratedTickSquare_injective establishes a precise fact about the Recognition Science framework's bridge to ordinary physics units. The framework works with dimensionless, native constants: it can derive relations like the reduced Planck constant being phi to the minus fifth, but it cannot, from pure dimensionless data alone, output an absolute SI value for hbar or G. That requires a dimensional anchor, a single measured quantity in seconds, meters, or kilograms. The theorem in question concerns the tick square, the squared time scale that the SI bridge assigns.

The theorem states that the function which maps a supplied value of Newton's constant G to a calibrated tick square is injective. In plain language: different positive inputs for G always produce different tick-square scales. The formula is calibratedTickSquare(G_input) = pi * hbar_SI * G_input / c_SI^5. The proof is machine-checked, meaning it is verified by a computer from axioms, not merely argued informally. This injectivity is the mathematical content of the statement that the bridge is a calibration, not a prediction.

What the theorem does not claim is equally important. It does not say that the framework predicts the value of G, hbar, or c. It does not say that any particular measured value is correct. It establishes only a structural property: the map from a dimensional anchor to a tick scale is one-to-one. The framework's own documentation is explicit that the absolute SI value of hbar or G cannot be output from pure dimensionless data alone, because the dimensions of c, hbar, and G are independent. No nontrivial monomial c^a hbar^b G^d is dimensionless, a separate proved theorem.

The practical consequence is that the framework's bridge to SI units is uniquely constrained once a single anchor is supplied. The positive half, that one anchor suffices, is formalized elsewhere; this result proves the negative half, that an anchor is required. The injectivity theorem is the certificate that different anchors give different bridges, so the calibration is honest and reproducible. It is a statement about the structure of the bridge, not about the values it carries.

/-- Changing the supplied `G` changes the calibrated tick square.  Thus the SI
bridge depends on the dimensional anchor; it does not predict that anchor. -/
theorem calibratedTickSquare_injective :
    Function.Injective calibratedTickSquare := by
  intro G₁ G₂ h
  unfold calibratedTickSquare at h
  have hcoeff : Real.pi * hbar_SI / c_SI ^ 5 ≠ 0 := by
    exact div_ne_zero (mul_ne_zero Real.pi_ne_zero hbar_SI_pos.ne') (pow_ne_zero 5 c_SI_pos.ne')
  have hlin : (Real.pi * hbar_SI / c_SI ^ 5) * G₁ =
      (Real.pi * hbar_SI / c_SI ^ 5) * G₂ := by
    calc
      (Real.pi * hbar_SI / c_SI ^ 5) * G₁
          = Real.pi * hbar_SI * G₁ / c_SI ^ 5 := by ring
      _ = Real.pi * hbar_SI * G₂ / c_SI ^ 5 := h
      _ = (Real.pi * hbar_SI / c_SI ^ 5) * G₂ := by ring
  exact (mul_left_cancel₀ hcoeff hlin)

/-- A pure-number theory can fix only dimensionless/native relations among
`c`, `hbar`, and `G`; an absolute SI value needs a dimensional anchor. -/
theorem dimensionless_theory_needs_anchor {a b d : ℤ}
    (hDimensionless : cHbarGDimension a b d = (0, 0, 0)) :
    a = 0 ∧ b = 0 ∧ d = 0 :=
  no_nontrivial_dimensionless_monomial hDimensionless

/-- 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

What this page does not claim

The theorem does not predict any numerical value for G, hbar, or c from first principles. The theorem does not establish that the framework's bridge produces correct SI values, only that it is one-to-one. The theorem does not prove that a dimensional anchor is sufficient, only that it is necessary.

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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: