Constants Dimensions Dimensioned Quantity

A quantity with its dimensions attached

A dimensioned quantity is a real number carrying a dimensional signature, which records the exponents of length, time, and mass. The speed of light, for instance, has signature [L¹T⁻¹M⁰]: one power of length, minus one power of time, no mass. This pairing is what lets an equation like E = mc² be checked for consistency before any numbers are plugged in. The framework's declaration DimensionedQuantity defines exactly this object: a value of type ℝ and a dimension made of three integers.

The classical idea is centuries old. Physicists have used dimensional analysis since Fourier's 1822 work on heat, and the Buckingham π theorem of 1914 formalized how to find dimensionless groups. The framework's contribution is to make the bookkeeping explicit and machine-checked. Its library defines the base dimensions, assigns signatures to c, ħ, and G, and provides operations for multiplying, dividing, and taking square roots of dimensioned quantities. These operations add or subtract the exponents, so multiplying a length by a time gives a quantity with signature [L¹T¹M⁰].

In Recognition Science, the fundamental units are the tick τ₀, the recognition length ℓ₀ = c·τ₀, and the golden ratio φ. The framework models physical constants as dimensioned quantities built from these primitives. The declaration itself is a definitional choice, not a theorem: it establishes the data structure, not any physical law. What it does is give the framework a way to state that ħ has dimensions [L²T⁻¹M¹] and G has [L³T⁻²M⁻¹], and to verify that Planck units have the right signatures.

The payoff is that dimensional consistency becomes a formal property. The library checks that the Planck length formula √(ħG/c³) has dimension [L¹], that the Planck time has [T¹], and that the Planck mass has [M¹]. These checks are documented, not proved as theorems about the world. They show the framework's internal arithmetic is coherent, which is a necessary condition for any further claim, but not evidence that the framework's constants match measured physics.

/-- A dimensioned physical quantity with its value and dimensional signature. -/
structure DimensionedQuantity where
  value : ℝ
  dim : Dimension

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

/-- 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

The declaration does not prove that any physical constant has a particular numerical value. The declaration does not establish that the framework's fundamental units exist in nature. The declaration does not derive the dimensions of c, ħ, or G; it assigns them by definition.

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)

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