module module high

`IndisputableMonolith.Information.NoCloning`

The Information.NoCloning module formalizes the no-cloning theorem for quantum states inside Recognition Science. It defines states as unit vectors in Hilbert space and shows that no cloning machine preserves inner products for all inputs. The argument uses algebraic constraints on inner products together with the discrete time quantum from Constants.

claimA quantum state is a unit vector $\psi$ in a Hilbert space $\mathcal{H}$ with $\|\psi\|=1$. No linear cloning map $C$ exists such that $C(\psi \otimes |0\rangle) = \psi \otimes \psi$ for every $\psi$, because the inner-product preservation condition forces a contradiction for non-orthogonal states.

background

This module sits in the Information domain and imports the RS time quantum $\tau_0 = 1$ tick from Constants. It introduces QuantumState as a unit vector, the inner product on the space, and a CloningMachine as a hypothetical linear operator. The setting links these objects to the discrete tick structure of Recognition Science, where all operations are ultimately counted in fundamental time quanta.

proof idea

The module first defines the basic objects (states, inner products, cloning machines). It then derives an algebraic constraint that any cloning map must obey. Direct substitution of the inner-product condition produces a contradiction unless the states are orthogonal, yielding the no-cloning result by elementary calculation.

why it matters in Recognition Science

The module supplies the no-cloning limit as a foundational constraint for quantum information within Recognition Science. It supports later results on measurement disturbance and quantum cryptography. No downstream theorems are recorded in the current dependency graph, but the content aligns with the discrete information processing enforced by the eight-tick octave and the phi-ladder.

scope and limits

Does not derive the no-cloning statement from the J-functional equation or the forcing chain T0-T8.
Does not treat continuous or infinite-dimensional Hilbert spaces.
Does not connect cloning constraints to mass formulas or the fine-structure constant band.
Does not supply explicit numerical bounds on cloning fidelity.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (20)

structure QuantumState L48
def innerProduct L55
theorem inner_product_self L59
structure CloningMachine L76
lemma inner_product_constraint L85
lemma cloning_constraint L102
theorem no_cloning_algebraic_constraint L118
theorem no_universal_cloning_witness_real L126
theorem no_cloning_theorem_remark L139
theorem no_cloning_from_ledger L158
theorem measurement_disturbs L165
theorem quantum_cryptography_possible L175
theorem quantum_differs_from_classical L182
theorem error_correction_possible L189
theorem no_deleting L198
theorem no_broadcasting L205
def optimalCloningFidelity L212
theorem approximate_cloning_bound L214
structure NoCloningFalsifier L224
def experimentalStatus L231