Constants Native Dimensional Boundary Dimension Matrix C Hbar G Det Nonzero

The dimensional boundary

The speed of light c, Planck's constant ℏ, and Newton's gravitational constant G are the three constants that anchor the SI system of units. Each carries its own physical dimension: c is a length per time, ℏ is an energy times a time, and G is a length cubed per mass per time squared. The question of whether these three dimensions are independent is answered by a simple 3 by 3 matrix that records their exponent vectors. The declaration dimension_matrix_c_hbar_G_det_nonzero proves that this matrix has a nonzero determinant, meaning the three dimension vectors are linearly independent.

This independence has a direct consequence: no nontrivial monomial of the form c^a ℏ^b G^d, with integer exponents, can be dimensionless. In other words, there is no way to multiply powers of these three constants to obtain a pure number. The framework's machine-checked library of formal theorems proves this by computing the determinant of the dimension matrix to be -2, a value that is clearly nonzero. This is the negative half of a boundary argument: it shows that a dimensional anchor is required to convert the framework's native dimensionless relations into absolute SI values.

In Recognition Science, the framework derives native identities such as ℏ_RS = φ^(-5) and G_RS · ℏ_RS = 1/π, where φ is the golden ratio. These are relations among dimensionless quantities. The theorem dimensionless_theory_needs_anchor states that a pure-number theory can fix only such dimensionless relations, not absolute values. The declaration si_bridge_is_calibration_not_prediction then shows that once a positive dimensional anchor, such as a value for G in SI units, is supplied, the bridge to SI units is a uniquely constrained calibration map: different anchors give different scales, and the map is injective.

This result is not a weakness of the framework; it is a statement of dimensional analysis. The determinant being -2, rather than ±1, also records that the three vectors 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 that the determinant is nonzero. The positive half, that one anchor suffices and determines the whole bridge, is formalized separately, showing that exactly one dimensional anchor determines everything.

/-- The determinant of the `(c,hbar,G)` dimension matrix is nonzero, i.e. the
three dimension vectors are linearly independent. -/
theorem dimension_matrix_c_hbar_G_det_nonzero : dimMatrix.det ≠ 0 := by
  rw [dimMatrix_det]; norm_num

/-- 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 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]

/-- 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

/-- 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 declaration does not provide the absolute SI value of ℏ or G from pure dimensionless data alone. The declaration does not prove that the positive half of the boundary argument, that one anchor suffices, is true; that is formalized elsewhere. The determinant being -2 does not mean the Planck system is invalid; it only means a basis requires half-integer powers.

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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