# Quantum algebra, photonics, and mechanics

[All sections](../README.md)

## Modules

| Module | Verified result |
|---|---|
| [SpinPhotonicWaveguide](SpinPhotonicWaveguide.en.md) | Nonnegative four-coordinate energy, preservation under specified mixing, and phase and geometric-profile formulas. |
| [NonHermitianPhotonicEP](NonHermitianPhotonicEP.en.md) | The sign of κ²−γ², its vanishing at κ=γ, and real splitting 2√Δ. No operator or complex spectrum is constructed. |
| [QuantumTransmonEngine](QuantumTransmonEngine.en.md) | A lower bound for a prescribed two-level gap and exact deviations from ng=1/2. |
| [QuantumHeatEngine](QuantumHeatEngine.en.md) | Prescribed scalar heat and work in an Otto cycle, efficiency in (0,1), and a conditional comparison with Carnot efficiency. |
| [QuantumStabilizerCodes](QuantumStabilizerCodes.en.md) | Phase-free Boolean Pauli representation, symmetry of its product, and four distinct syndromes for a three-qubit bit-flip code. |
| [QuantumCliffordTableau](QuantumCliffordTableau.en.md) | H, S, and CNOT preserve a Boolean symplectic form; basis actions and involutions in the phase-free representation. |
| [QuantumToricCode](QuantumToricCode.en.md) | Parity of a prescribed overlap, energy formulas, and addition of a defect pair by definition. No lattice is constructed. |
| [NonAbelianHolonomy](NonAbelianHolonomy.en.md) | Real skew-symmetric 3×3 matrices: closure under commutators, so(3) relations, and Jacobi. Parallel transport along loops is not defined. |
| [SpintronicTransport](SpintronicTransport.en.md) | A prescribed antisymmetric response matrix, zero scalar work, and an inverse matrix for a nonzero coefficient. |
| [SymplecticHamiltonianDynamics](SymplecticHamiltonianDynamics.en.md) | The prescribed symplectic Euler step has determinant one and preserves a modified quadratic energy, nonnegative when h²k < 4m. |
| [QuantumMajoranaChain](QuantumMajoranaChain.en.md) | Explicit complex 2×2 matrices: Majorana algebra, CAR, a number projection, and involutive parity. No spatial chain is constructed. |

| [QuantumPhaseFlipCode](../07_quantum_physics_and_optics/QuantumPhaseFlipCode.en.md) | HZH = X, Z² = I, and an invertible four-label syndrome table; no state-space recovery theorem. |

## Scope

The results concern the explicit matrices, transformations, and scalar energy models. Their correspondence to a physical device or a full many-body system is not established by compilation.

[Contributing](../CONTRIBUTING.en.md) · [Validation](../VERIFICATION.en.md)

- [QuantumTeleportationProtocol](../07_quantum_physics_and_optics/QuantumTeleportationProtocol.en.md): exact correction of four prescribed complex amplitude pairs.

- [QuantumDeutschJozsa](../07_quantum_physics_and_optics/QuantumDeutschJozsa.en.md): exact one-bit truth-table classification by prescribed amplitudes.

- [QuantumGroverSearch](../07_quantum_physics_and_optics/QuantumGroverSearch.en.md): exact one-step search among four basis targets and unit squared overlap.

- [QuantumSuperdenseCoding](../07_quantum_physics_and_optics/QuantumSuperdenseCoding.en.md)

- [QuantumNoCloningTheorem](../07_quantum_physics_and_optics/QuantumNoCloningTheorem.en.md)

- [QuantumBB84Protocol](../07_quantum_physics_and_optics/QuantumBB84Protocol.en.md)

- [QuantumBernsteinVazirani](../07_quantum_physics_and_optics/QuantumBernsteinVazirani.en.md)

- [QuantumBitFlipCode](../07_quantum_physics_and_optics/QuantumBitFlipCode.en.md)

- [QuantumBellCHSH](../07_quantum_physics_and_optics/QuantumBellCHSH.en.md)

- [SolarisMithraCore](../07_quantum_physics_and_optics/SolarisMithraCore.en.md)

- [OpticalLossAttenuation](../07_quantum_physics_and_optics/OpticalLossAttenuation.en.md)

- [ThermoOpticPhaseDrift](../07_quantum_physics_and_optics/ThermoOpticPhaseDrift.en.md)

- [SolarisPhysicalModels](../07_quantum_physics_and_optics/SolarisPhysicalModels.en.md)

- [ChipLayoutGeometry](../07_quantum_physics_and_optics/ChipLayoutGeometry.md): universal translation invariance and exact rational placement bounds; concrete data are certified separately.

- [QuantumOptomechanicalCoupling](../03_quantum_physics_and_optics/QuantumOptomechanicalCoupling.md): scalar energy/force identities, affine dispersion, prescribed damping and relaxation; no quantum cooling theorem.

- [ChipPlacementCertificate](../07_quantum_physics_and_optics/ChipPlacementCertificate.md): exact 256-node bounds, positive dimensions, unique IDs, and pairwise disjoint closed rectangles; external provenance check.

- [MithraicPhaseCollapse](../03_quantum_physics_and_optics/MithraicPhaseCollapse.md): scalar visibility bounds, fixed-maximum contrast decay, and a threshold specification; no quantum decoherence or automatic dump-port routing.

- [QuantumStandardQuantumLimit](../03_quantum_physics_and_optics/QuantumStandardQuantumLimit.md): exact scalar noise minimum, unique optimizer, and an explicitly calibrated SQL-shaped bound.

- [QuantumBeamSplitterTransform](../03_quantum_physics_and_optics/QuantumBeamSplitterTransform.md): Real orthogonal mode transformation, normalized finite two-photon lift and ideal balanced HOM suppression; no full Fock-space or detector model.

[QuantumPhaseEstimation](../03_quantum_physics_and_optics/QuantumPhaseEstimation.md) — exact complex two-bit QPE, controlled powers, Fourier cancellation and deterministic weights.

[QuantumGroverMultipleTargets](../03_quantum_physics_and_optics/QuantumGroverMultipleTargets.md) — exact subset dynamics, invariant plane and marked-state weights for all iteration counts at N=4.

[QuantumDeutschJozsaGeneral](../03_quantum_physics_and_optics/QuantumDeutschJozsaGeneral.md) — general real Hadamard/XOR circuit, promise separation and one-bit compatibility.

- [QuantumGroverArbitraryPhase](../03_quantum_physics_and_optics/QuantumGroverArbitraryPhase.md) — complex unitarity, arbitrary-phase invariant plane and exact one-step probability.

- [QuantumSimonsAlgorithm](../03_quantum_physics_and_optics/QuantumSimonsAlgorithm.md) — complex XOR circuit, exact Simon sampling and rank-conditional recovery.

- [QuantumPhaseEstimationGeneral](../03_quantum_physics_and_optics/QuantumPhaseEstimationGeneral.md) — general controlled-power QPE, unitary Fourier transform, exact dyadic recovery and nearest-sample probability at least 4/π².

- [QuantumShorOrderFindingCore](../03_quantum_physics_and_optics/QuantumShorOrderFindingCore.md) — padded modular multiplication, orthonormal orbit spectrum, accessible |1⟩ input and the derived QPE probability mixture.

- [QuantumShorContinuedFractions](../03_quantum_physics_and_optics/QuantumShorContinuedFractions.md) — executable continued fractions, strict Legendre recovery and reduced-order candidate boundaries.

- [QuantumShorOrderRecoveryGCD](../03_quantum_physics_and_optics/QuantumShorOrderRecoveryGCD.md) — conditional LCM/GCD order recovery and candidate-provenance boundaries.

- [QuantumDeutschJozsaNBitBalanced](../03_quantum_physics_and_optics/QuantumDeutschJozsaNBitBalanced.md) — Exact balanced-oracle cardinality and circuit interference.

- [PhotonicsBeamSplitterPhaseShift](../03_quantum_physics_and_optics/PhotonicsBeamSplitterPhaseShift.md) — Complex unitary beam splitter, MZI intensities and visibility.

- [PhotonicsHongOuMandelInterference](../03_quantum_physics_and_optics/PhotonicsHongOuMandelInterference.md) — Normalized two-photon HOM interference, phase bridge and distinguishable baseline.

- [PhotonicsCPhaseGateUnitary](../03_quantum_physics_and_optics/PhotonicsCPhaseGateUnitary.md) — Six-mode unitary scattering, postselected CZ with success probability 1/9 and graph-state entanglement.

- [PhotonicsBellStateAnalyzer](../03_quantum_physics_and_optics/PhotonicsBellStateAnalyzer.md) — Optical Bell amplitudes, derived detector POVM and prior-dependent discrimination.

- [PhotonicsQuantumTeleportation](../03_quantum_physics_and_optics/PhotonicsQuantumTeleportation.md) — Optical teleportation instrument, exact state recovery and no-signalling.

- [QuantumBB84ProtocolQBER](../03_quantum_physics_and_optics/QuantumBB84ProtocolQBER.md) — Born-derived BB84 QBER, normalized intercept-resend channel and finite-sample detection.

- [QuantumE91ProtocolBellCHSH](../03_quantum_physics_and_optics/QuantumE91ProtocolBellCHSH.md) — Born-derived E91 correlations, CHSH bounds and key routing.

- [QuantumDepolarizingChannel](../03_quantum_physics_and_optics/QuantumDepolarizingChannel.md) — Qubit depolarization, complete positivity and noisy E91 correlations.

- [QuantumPOVMNaimarkDilation](../03_quantum_physics_and_optics/QuantumPOVMNaimarkDilation.md) — Finite POVMs, unitary Naimark dilation and Lueders reduction.

- [QuantumNoCloningUnitary](../03_quantum_physics_and_optics/QuantumNoCloningUnitary.md) — Exact unitary no-cloning, phase-invariant overlap constraints and CNOT action.

- [Mixed-state no-broadcasting (BCFJS)](../03_quantum/QuantumNoBroadcastingMixed.md): complete CPTP equivalence, singular-state SLD/SOS necessity, and pure-state regressions.

- [Informationally complete POVMs and exact tomography](../03_quantum/QuantumPOVMInformationallyComplete.md)

- [POVM information entropy](../03_quantum_information/QuantumPOVMInformationEntropy.md)

- [Classical POVM postprocessing and KL contraction](../03_quantum_information/QuantumChannelClassicalDataProcessing.md)

- [Holevo–Helstrom state discrimination](../03_quantum_information/QuantumHelstromStateDiscrimination.md)

- [Trace distance and measurement contraction](../03_quantum_information/QuantumChannelTraceDistanceContract.md)

| [QuantumCPTPChannelContract](../03_quantum_information/QuantumCPTPChannelContract.md) | Kraus channels, dual effects and CPTP trace-distance contraction; unitary equality, reset and strict dephasing loss. |

- [Stinespring isometric dilation](../03_quantum_information/QuantumStinespringDilationContract.md) — rectangular Kraus families, partial trace, dual maps and redundant environments.

- [Complementary channel and environment state](../03_quantum_information/QuantumComplementaryChannelContract.md) — Kraus construction, both reductions and explicit environment isometries.

- [Knill-Laflamme environment criterion](../03_quantum_information/QuantumKnillLaflammeEnvironmentCriterion.md) — Trace-pairing equivalence, environment density and index convention.

- [Orthogonal error recovery](../03_quantum_information/QuantumOrthogonalErrorRecoveryContract.md) — Explicit CPTP recovery, zero weights and completion on the unused output space.

- [Recovery with certified unitary diagonalization](../03_quantum_information/QuantumUnitaryKrausDiagonalizationRecoveryContract.md) — Complex Kraus mixing, derived spectral weights and exact recovery with a supplied diagonalization certificate.

- [General Knill-Laflamme recovery](../03_quantum_information/QuantumKnillLaflammeGeneralRecoveryTheorem.md) — Spectral construction and exact CPTP recovery without an external diagonalization certificate.

- [Knill-Laflamme recovery necessity](../03_quantum_information/QuantumKnillLaflammeRecoveryNecessity.md) — Necessity, exact recovery equivalence and impossibility of full-qubit dephasing recovery.

- [Reference-preserving entanglement recovery](../03_quantum_information/QuantumReferencePreservingRecoveryContract.md) — Arbitrary references, environment decoupling and a normalized maximally entangled test.

- [Subsystem erasure correctability](../03_quantum_information/QuantumErasureCorrectabilityContract.md) — Known erased factor, recovery/privacy equivalence and explicit auxiliary-pair example.

- [Qutrit two-of-three secret sharing](../03_quantum_information/QuantumQutritTwoOfThreeSecretSharingContract.md) — Explicit isometric code, singleton privacy, recovery from every pair and reference preservation.

- [Explicit qutrit decoder](../03_quantum_information/QuantumQutritSecretSharingExplicitDecoder.md) — Basis permutation, full-state factorization and three explicit recovery Kraus operators.

- [Qutrit decoder circuit refinement](../03_quantum_information/QuantumQutritDecoderCircuitRefinement.md): executable three-gate interpreter, exact matrix/channel refinement, and incorrect-circuit witnesses.

- [HCTK scalar partition](../03_quantum_physics_and_optics/HCTKScalarPartition.md) — formal scalar functions, transfer conservation and bounded regularization; no physical topology claim.

- [PrometheusPiModelContract](PrometheusPiModelContract.md).
- [PrometheusNoiseAudit](PrometheusNoiseAudit.md).
- [PrometheusUCNAudit](PrometheusUCNAudit.md).
- [PB11EnergyBalance](PB11EnergyBalance.md).

- [PrometheusChannelAndInjectorAudit](PrometheusChannelAndInjectorAudit.md).
- [PrometheusFiniteBlockAudit](PrometheusFiniteBlockAudit.md).
- [PrometheusDecoderQueue](PrometheusDecoderQueue.md).
- [PrometheusPipelinedQueue](PrometheusPipelinedQueue.md).
- [PrometheusMetrologyJitter](PrometheusMetrologyJitter.md).
- [PrometheusNQKDAudit](PrometheusNQKDAudit.md).
- [PrometheusTAKLAudit](PrometheusTAKLAudit.md).

- [AlgebraQuaternionSpinor](../05_mathematics_and_geometry/AlgebraQuaternionSpinor.md) — explicitly scoped contract; see its assumptions and limitations.

- [PhotonicsMachZehnderChiralContract](PhotonicsMachZehnderChiralContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [SpintronicSkyrmionSynapseContract](../04_spintronics_and_hardware/SpintronicSkyrmionSynapseContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [QuantumDiamondNVInterfaceContract](../03_quantum_and_hardware/QuantumDiamondNVInterfaceContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [BiohybridOrganoidSynapseContract](../02_biological_and_wetware/BiohybridOrganoidSynapseContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [FungalMyceliumMemristorContract](../02_biological_and_wetware/FungalMyceliumMemristorContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [OptospintronicTransducerContract](../04_spintronics_and_hardware/OptospintronicTransducerContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [QuantumIVCAccumulatorContract](../01_cryptography_and_protocols/QuantumIVCAccumulatorContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [QuantumStateAvailabilityContract](../03_quantum_and_hardware/QuantumStateAvailabilityContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [QuantumEntanglementSwappingPurificationContract](../03_quantum_and_hardware/QuantumEntanglementSwappingPurificationContract.md) — explicitly scoped contract; see its assumptions and limitations.

- [QuantumSteaneCodeErrorCorrectionContract](../03_quantum_and_hardware/QuantumSteaneCodeErrorCorrectionContract.md) — explicitly scoped contract; see its assumptions and limitations.

## Additional conditional Prometheus audits

| Module | Scope |
|---|---|
| [PrometheusVariableQueue](PrometheusVariableQueue.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusLifetimeInformation](PrometheusLifetimeInformation.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusCascadedFECBound](PrometheusCascadedFECBound.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusMultiLaneQueue](PrometheusMultiLaneQueue.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusLifetimeIdentifiability](PrometheusLifetimeIdentifiability.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusJitterWindow](PrometheusJitterWindow.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusRSCycleModel](PrometheusRSCycleModel.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusNQKDFiniteKey](PrometheusNQKDFiniteKey.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |
| [PrometheusThresholdTomography](PrometheusThresholdTomography.md) | Explicit conditional mathematics; see the card for assumptions and unproved bridges. |