Constants Native Dimensional Boundary Dimensionless Theory Needs Anchor

The dimensional boundary

Dimensional analysis is the discipline of tracking what kind of quantity a number stands for: length, time, mass, or some combination. The speed of light c has dimensions of length over time. Planck's constant ℏ has dimensions of mass times length squared over time. Newton's constant G has dimensions of length cubed over mass over time squared. These three dimensions are independent: no product of powers of c, ℏ, and G can collapse into a pure number. The determinant of their dimension matrix is -2, a nonzero value that records this independence.

That fact has a consequence for any theory that works only with dimensionless ratios, which are pure numbers with no units attached. Such a theory can fix relations like ℏ_RS = φ⁻⁵ or G_RS · ℏ_RS = 1/π, where φ is the golden ratio. What it cannot do is produce the absolute SI value of ℏ or G, because those values carry dimensions. The theorem dimensionless_theory_needs_anchor states this precisely: if a monomial c^a ℏ^b G^d is dimensionless, then a = b = d = 0. Only the trivial case survives; no nontrivial combination is unit-free.

This is not a weakness of the Recognition Science framework; it is dimensional analysis itself. The framework's library proves the negative half: an anchor is required. The positive half, that one anchor suffices and determines the whole bridge uniquely, is formalized separately. Given the c, ℏ, G constraints, the tick conversion factor is uniquely a_T = √π · τ_Planck. From one scalar τ₀ in seconds, the full external calibration follows: meters per voxel from SI c, joules per coherence from SI ℏ. The electron mass in SI kilograms then emerges from the framework's T0–T8 chain plus that single anchor, with no remaining external inputs.

The library also proves the bridge is a calibration, not a prediction. For any positive supplied value of G, the calibrated tick square is positive, and different supplied values give different tick squares. The function is injective: the anchor is not determined by the theory, it is chosen by measurement. This is the honest boundary: native constants are first-principles objects up to the dimensional boundary, and SI conversion is a uniquely constrained calibration once an anchor is supplied.

/-- 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 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]

/-- 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

/-- 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 absolute SI value of ℏ or G without any measured input. The theorem shows that one anchor suffices; that positive half is formalized elsewhere. The framework's native constants are SI values; they are dimensionless relations until calibration.

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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