Foundation Coupled Recognition Cores Added Config Eq Added Config Iff Left

The cancellation law

The declaration addedConfig_eq_addedConfig_iff_left establishes a cancellation law for a specific arithmetic on codes made from four symbols, which the framework calls a ququart. The operation, named addedConfig, combines two codes position by position using modular addition modulo 4. The theorem states that if adding a fixed code s to two codes a and a' produces the same result, then a and a' must themselves be identical. In symbols, add4 a s = add4 a' s ↔ a = a'. This is the left-cancellation property: the added code s can be removed from both sides of an equality without changing its truth.

The proof is a finite case check. Since each position holds one of four values, the declaration's proof script examines all 4 × 4 × 4 = 64 possible combinations of the three codes and confirms the equivalence in every case. The result is a theorem in the framework's machine-checked library of formal theorems, not an assumption or a model choice. It is one of several small lemmas that support larger constructions in the same file, such as the fact that shifting a configuration and then adding it back recovers the original configuration.

In Recognition Science, this lemma matters because it guarantees that the operation of combining configurations is faithful: no information is lost when a fixed configuration is added. The framework models a coupled system of N recognition cores, each core carrying one of four states, and addedConfig is the pointwise combination rule. The cancellation law ensures that the map sending a configuration a to addedConfig a s is injective for any fixed s, so the combined configuration uniquely determines the original one. This is a structural guarantee that the framework's composition operation does not merge distinct states.

The declaration does not claim anything about the physical meaning of the four symbols, nor does it assert that this cancellation law holds for other arithmetic operations or for codes with a different number of symbols. It is a purely formal statement about the specific add4 operation defined in the file. The theorem also does not establish that the framework's model of coupled cores is physically realized; it only states a property of the formal objects the framework defines.

/-- For fixed `s`, the shift label `a` is recoverable from `add4 a s`. -/
theorem add4_eq_add4_iff_left (a a' s : Fin 4) :
    add4 a s = add4 a' s ↔ a = a' := by
  exact add4_eq_add4_iff_left_core a a' s

/-- For fixed `s`, the shift label `a` is recoverable from `add4 a s`. -/
theorem add4_eq_add4_iff_left (a a' s : Fin 4) :
    add4 a s = add4 a' s ↔ a = a' := by
  exact add4_eq_add4_iff_left_core a a' s

/-- For fixed `s`, the left addend is recoverable from `add4 a s`. -/
theorem add4_eq_add4_iff_left_core (a a' s : Fin 4) :
    add4 a s = add4 a' s ↔ a = a' := by
  fin_cases a <;> fin_cases a' <;> fin_cases s <;> decide

/-- For fixed `s`, the shift label `a` is recoverable from `add4 a s`. -/
theorem add4_eq_add4_iff_left (a a' s : Fin 4) :
    add4 a s = add4 a' s ↔ a = a' := by
  exact add4_eq_add4_iff_left_core a a' s

What this page does not claim

The declaration does not claim any physical meaning for the four symbols. The declaration does not claim the cancellation law holds for other arithmetic operations or for codes with a different number of symbols. The declaration does not establish that the framework's model of coupled cores is physically realized.

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/CoupledRecognitionCores.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: