Constants Dimensions

Dimensional bookkeeping

Dimensional analysis is the habit of writing down not just a number but the kind of quantity it is: a length, a time, a mass, or some combination. The speed of light is not merely 299,792,458; it is a length divided by a time. The reduced Planck constant is an energy multiplied by a time. The gravitational constant carries length cubed over mass times time squared. These signatures are the units' skeleton, and they let a physicist check an equation before trusting it: if both sides do not carry the same skeleton, the equation is wrong.

In Recognition Science, this bookkeeping is made explicit and machine-checked. The framework's library of formal theorems defines a dimension as a triple of integers, one exponent each for length, time, and mass. A length is (1, 0, 0), a time is (0, 1, 0), a mass is (0, 0, 1). Multiplying two quantities adds their exponents; dividing subtracts them. The speed of light gets (1, -1, 0), the reduced Planck constant gets (2, -1, 1), and the gravitational constant gets (3, -2, -1). These are definitions, not discoveries: the framework chooses how to encode standard physics, and the choice is recorded in the library.

The framework's real work is showing that its own fundamental units are dimensionally consistent. The framework builds from a fundamental tick, a unit of time, and a recognition length defined as the speed of light times that tick. From these it derives the reduced Planck constant and the gravitational constant. The library checks that the Planck length, Planck time, and Planck mass formulas, built from hbar, G, and c, come out with the right dimensions: a length, a time, a mass. It also checks that the framework's tick tau_0, defined through hbar, G, pi, and c, is genuinely a time. These are documentation checks, written as comments in the source, not as proved theorems; they confirm that the framework's constants are not just numbers but quantities with the right physical kind.

What this establishes in plain language is discipline. A reader of the framework can see, in one place, what dimensions every constant carries and how those dimensions survive the formulas that define Planck units. The framework does not derive the values of hbar or G; it records their dimensional signatures and checks the algebra around them. It is a foundation stone for the framework's larger claims, the part that keeps the rest of the building from floating free of physical meaning.

/-- Dimensional signature: [Length, Time, Mass] exponents.
    Used to track physical dimensions through calculations. -/
structure Dimension where
  L : ℤ  -- Length exponent
  T : ℤ  -- Time exponent
  M : ℤ  -- Mass exponent
  deriving DecidableEq

/-- Speed of light dimension: [L¹T⁻¹M⁰] -/
def dim_c : Dimension := ⟨1, -1, 0⟩

/-- Reduced Planck constant dimension: [L²T⁻¹M¹] -/
def dim_hbar : Dimension := ⟨2, -1, 1⟩

/-- Gravitational constant dimension: [L³T⁻²M⁻¹] -/
def dim_G : Dimension := ⟨3, -2, -1⟩

/-- Summary of dimensional analysis module. -/
def dimensions_status : String :=
  "✓ Dimension structure [L, T, M] defined\n" ++
  "✓ Physical constant dimensions (c, ℏ, G) specified\n" ++
  "✓ Planck unit dimensions documented\n" ++
  "✓ τ₀ dimension documented as [T]\n" ++
  "✓ DimensionedQuantity algebra defined"

What this page does not claim

This framework derives the numerical values of hbar, G, or c; it only records their dimensional signatures. The dimensional consistency checks are documentation, not proved theorems in the library. The framework does not establish the physical existence of the fundamental tick or recognition length.

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/Dimensions.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

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