Constants Native Dimensional Boundary Si Bridge Is Calibration Not Prediction

The dimensional boundary

In physics, the speed of light c, the reduced Planck constant ℏ, and Newton's gravitational constant G carry independent dimensions: length per time, mass times length squared per time, and length cubed per mass per time squared. No product of powers of these three, such as c²ℏ/G, can be dimensionless, because the three dimension vectors are linearly independent. The Recognition Science framework's machine-checked library of formal theorems proves this fact directly: the determinant of the dimension matrix is −2, which is nonzero, and any monomial whose exponents sum to zero in all three dimensions must have all exponents zero.

This single algebraic fact draws a sharp line around what a first-principles theory can say. A theory that works only with pure numbers can fix dimensionless relations among the constants, such as ℏ_RS = φ⁻⁵ or G_RS · ℏ_RS = 1/π, where φ is the golden ratio. It cannot, from dimensionless data alone, output the absolute SI value of ℏ or G. The framework's declaration si_bridge_is_calibration_not_prediction (a machine-checked theorem, not a conjecture) states this boundary precisely: for every positive supplied value of G as a dimensional anchor, the bridge assigns a positive tick-square scale, and different anchors give different scales. The bridge is a calibration map, not a prediction of the anchor itself.

This is not a weakness of the framework; it is dimensional analysis. The dimensions of c, ℏ, and G are independent, so no nontrivial monomial is dimensionless. The framework's library proves the negative half, that an anchor is required, and points to a positive half formalized elsewhere: one scalar anchor, such as the tick duration τ₀ in seconds, suffices to determine the entire bridge uniquely. From that single anchor, the framework derives the electron mass in SI kilograms with a mass audit at zero hypotheses, zero open items, and zero external inputs. The calibration theorem is the honest boundary marker: exactly one dimensional anchor, and it determines everything.

What the declaration does not claim is as important as what it proves. It does not claim that the framework can predict the measured value of G, ℏ, or c from pure theory. It does not claim that the bridge is a prediction at all; it is a uniquely constrained conversion once an anchor is supplied. It also does not claim that the determinant being −2 rather than ±1 is physically significant; the only fact the boundary argument needs is that the determinant is nonzero. The declaration is a statement about what a pure-number theory can and cannot do, and it is a theorem in the framework's library, not an empirical check.

/-- 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⟩

/-- The determinant of the `(c, hbar, G)` dimension matrix is `-2` (a real
`Matrix.det`, not a free-floating numeral). The value `-2` (rather than `±1`)
also records that `(c, hbar, G)` span an index-`2` sublattice of the integer
dimension lattice, so the Planck system is a basis only up to half-integer
powers; but the only fact the boundary argument needs is `det ≠ 0`. -/
theorem dimMatrix_det : dimMatrix.det = -2 := by
  simp [dimMatrix, Matrix.det_fin_three, Matrix.of_apply,
    Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,
    Matrix.cons_val_fin_one]

What this page does not claim

The framework can predict the absolute SI value of G, ℏ, or c from pure dimensionless theory. The bridge is a prediction of the dimensional anchor itself; it is a uniquely constrained calibration once an anchor is supplied. The determinant −2 has physical meaning beyond the fact that it is nonzero.

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.

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