Foundation Topological Conservation Charge Count Equals Face Pairs

Charge from linking

A conserved quantity is something that stays the same as a system changes. The Recognition Science framework models these as charges, discrete values attached to a configuration that cannot change under the allowed dynamics. The declaration charge_count_equals_face_pairs proves, within the framework, that the number of independent charges in three dimensions equals the number of face pairs on a cube, which is three.

The framework's library of machine-checked theorems defines a charge as a function from a configuration to an integer, with a proof that the value is unchanged along any valid trajectory. This is a topological conservation law: charge is a linking number, an integer that counts how many times one structure winds around another. Linking numbers cannot change under continuous deformation, so they are exactly conserved. The framework contrasts this with Noether's theorem, which derives conservation from continuous symmetries and allows real-valued, non-quantized charges.

The theorem charge_count_equals_face_pairs states that the number of independent charges at dimension three equals the number of face pairs, which is three. This is a definitional identity: the framework defines independent_charge_count 3 to be 3, and face_pairs 3 also evaluates to 3. The theorem is a formal statement of this equality, proved by reflexivity. It is part of a larger result, topological_conservation_certificate, which bundles together the quantization of charges, their conservation along trajectories, the count of three at dimension three, zero at other dimensions, and the bijection between the three charges and the three axes of a cube.

The framework identifies the three charges with electric, baryon, and lepton number. The declaration SMCharge is an inductive type with exactly three constructors: electric, baryon, and lepton. The theorem sm_charge_count proves this type has cardinality three. The theorem charge_to_axis_bijective proves that the map from these three charges to the three axes of a cube is both injective and surjective, meaning there is a perfect one-to-one correspondence.

The theorem does not claim that these three charges are the only conserved quantities in the actual universe, nor does it derive their specific numerical values. It establishes a structural fact within the framework: in three dimensions, the framework's model of linking supports exactly three independent integer-valued charges, and these correspond to the three axes of a cube. The framework's claim is that this topological structure is the origin of conservation, not that it predicts the measured values of electric, baryon, or lepton number.

/-- Charge count = face pairs = colors = generations. -/
theorem charge_count_equals_face_pairs :
    independent_charge_count 3 = face_pairs 3 := rfl

/-- The three conserved charges of the Standard Model. -/
inductive SMCharge where
  | electric : SMCharge
  | baryon : SMCharge
  | lepton : SMCharge
  deriving DecidableEq, Fintype

theorem sm_charge_count : Fintype.card SMCharge = 3 := by decide

theorem charge_to_axis_bijective : Function.Bijective charge_to_axis :=
  ⟨charge_to_axis_injective, charge_to_axis_surjective⟩

What this page does not claim

This theorem does not derive the numerical values of electric, baryon, or lepton charge. This theorem does not prove that the only conserved quantities in the universe are these three charges. This theorem does not claim that Noether's theorem is false, only that it does not require integer values.

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/Foundation/TopologicalConservation.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: