Foundation Operator Core Coupled Recognition Cores Coupled Core Index

The coupled core index

The coupled core index is a label that counts the states of a two-core recognition system. In the Recognition Science framework, a recognition core is a discrete record of events, and coupling two such cores produces a system with four distinct states. The index is the number that names each of those four states, running from zero to three. The machine-checked library of formal theorems defines this index as an abbreviation for the underlying coupled-core construction, so the four-state labeling is a definitional choice, not a derived result.

The four states form a ququart, the four-level analogue of a qubit. The framework's library establishes that the ququart carries a pair of operators, X and Z, which satisfy the Weyl commutation relation XZ = ωZX, where ω is a fourth root of unity. This is the standard finite-dimensional form of the Heisenberg commutation relation, adapted to four levels. The library also proves that the four states are orthonormal, meaning each state is distinct and has unit norm, and that the monomials built from the X and Z operators form a complete basis for the space of operators on the ququart.

In Recognition Science, this four-state system is the smallest coupled core that supports the full operator algebra. The framework proves that the index set has cardinality four, and that the tensor product of two ququarts yields a sixteen-dimensional space. These facts are formal theorems in the library, checked by a machine. The practical consequence is that the framework has a concrete, finite object on which to build larger coupled systems, and the ququart serves as the atomic unit for that construction.

What the declaration does not claim is equally important. The coupled core index does not establish that any physical system realizes a ququart, nor does it assign energy levels or dynamical laws to the four states. It is purely a combinatorial and algebraic object: a labeling scheme with a proven operator relation. The framework does not claim that the ququart is the unique four-state system, nor that the Weyl relation is the only possible commutation law. Those would be further hypotheses, not consequences of this declaration.

abbrev CoupledCoreIndex := IndisputableMonolith.Foundation.CoupledRecognitionCores.CoupledCoreIndex

abbrev QuquartState := IndisputableMonolith.Foundation.CoupledRecognitionCores.QuquartState

abbrev tensorWeylMonomial_self_inner {N : ℕ} :=
  IndisputableMonolith.Foundation.CoupledRecognitionCores.tensorWeylMonomial_self_inner (N := N)

abbrev coupledCoreIndex_card := IndisputableMonolith.Foundation.CoupledRecognitionCores.coupledCoreIndex_card

What this page does not claim

The declaration does not claim that any physical system realizes a ququart. It does not assign energy levels or dynamical laws to the four states. It does not claim the ququart is the unique four-state system or that the Weyl relation is the only possible commutation law.

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/OperatorCore/CoupledRecognitionCores.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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