Which changes of frame preserve a source together with the records it can carry? For the native eight-state cell of Recognition Science, we classify the unitary transformations preserving its six-dimensional contact space and admitting covariance of the full populated-record source. The source forces commutation with the literal half-cycle. In the contact coordinates this is the conformal symplectic equation BᵀJB = wJ. The represented group is (Sp(2) × U(1))/ℤ₂, with an eleven-dimensional Lie algebra. We derive the source commutant, the faithful representation and its observable kernel, and show how one shared control transports both retained references while leaving the readers unchanged in operator closure. The mechanism is explicit: record mixing identifies three source components, and the half-cycle turns that identification into an alternating form on the contact carrier.
Which changes of frame preserve a source together with the records it can carry? For the native eight-state cell of Recognition Science, we classify the unitary transformations preserving its six-dimensional contact space and admitting covariance of the full populated-record source. The source forces commutation with the literal half-cycle. In the contact coordinates this is the conformal symplectic equation BᵀJB = wJ. The represented group is (Sp(2) × U(1))/ℤ₂, with an eleven-dimensional Lie algebra. We derive the source commutant, the faithful representation and its observable kernel, and show how one shared control transports both retained references while leaving the readers unchanged in operator closure. The mechanism is explicit: record mixing identifies three source components, and the half-cycle turns that identification into an alternating form on the contact carrier.
A symmetry specifies which interactions are compatible with a system; it does not specify which interactions its controls can produce. We resolve this distinction for a finite recognition apparatus with a main signal, a retained record and an independent clock. An explicit unitary change of coordinates turns the two signal ports into four equivalent copies of a four-dimensional symplectic representation. The invariant source Hamiltonians are therefore exactly the Hermitian 4×4 coefficient matrices acting between those copies. The original controls preserve one decoded record component, excluding even a simple exchange allowed by the symmetry. Adding forward waits under one fixed positive-strength recorded-exchange Hamiltonian removes this obstruction: every invariant source unitary, with identity on the clock, lies in the operator-norm closure of the resulting finite programs. The proof derives the required operations from the stated inventory: a clock echo removes unwanted clock motion, and a record-restoring subroutine supplies control of the main register. We also carry the original energy readout through the same programs, retaining its reference and positivity conditions.
A symmetry specifies which interactions are compatible with a system; it does not specify which interactions its controls can produce. We resolve this distinction for a finite recognition apparatus with a main signal, a retained record and an independent clock. An explicit unitary change of coordinates turns the two signal ports into four equivalent copies of a four-dimensional symplectic representation. The invariant source Hamiltonians are therefore exactly the Hermitian 4×4 coefficient matrices acting between those copies. The original controls preserve one decoded record component, excluding even a simple exchange allowed by the symmetry. Adding forward waits under one fixed positive-strength recorded-exchange Hamiltonian removes this obstruction: every invariant source unitary, with identity on the clock, lies in the operator-norm closure of the resulting finite programs. The proof derives the required operations from the stated inventory: a clock echo removes unwanted clock motion, and a record-restoring subroutine supplies control of the main register. We also carry the original energy readout through the same programs, retaining its reference and positivity conditions. The deposit includes the paper and its LaTeX source with six vector schematics. AI-use disclosure: OpenAI Codex assisted with mathematical formalization, manuscript drafting and editing, schematic code, and source and layout checks. The results are supported by explicit mathematical arguments and machine-checked proofs in the author's research project.
Recognition Science describes one ledger of recognition events, yet experience arrives from inside separate perspectives. We ask what in the ledger's own dynamics makes a perspective, for one explicit operation: the native recognition step, in which two eight-state clock channels exchange occupation, post a three-component current and move an integer loop flux. Boundaries. For the algebra of native events, every closed invariant subspace of histories is a join of atoms, the atoms are exact superselection sectors with no environment involved, and the atoms an actual history occupies are fixed at its beginning. Neither a symmetric rule nor an element of that algebra selects among them, although a larger read algebra crosses these walls. Participants. Inside a boundary the postings select. Exactly a four-dimensional subspace of the 64-dimensional cell never acquires neutral weight under any record: two persistent loop currents and an occupation oscillation. Three postings expose every other state, and the selected states are exactly those that pass the eight-tick window on the loop edge with certainty. On every network of contacts in its round order the kept states obey a parity law, the kept projection commutes with every round word, so composition creates no kept weight, and participation on a connected network is joint. Other timings act through one token mechanism, which decides births, fair and gated participation, random timing and what a participant's own record reads of its world. Records. A participant's own record decides its status; the conditional present purifies; its long-run motion is a classical chain that is reversible exactly on the participant classes and carries a stationary circulation elsewhere. Every record reads each site only within the same twelve directions, and composites of participants add only crossed products of their members' reads. What is supplied. The book locates every stated premise and identification: the step and its loop-edge routing, the window's account and tick, the genesis state, contact locality, orientation and posting order, the gate and the independence of attempts, the relay's extra terms, least-cost booking of postings, the capture apparatus, the photon dictionary, the present and the experiential reading of a sure subject. On the ledger's torus, locality, symmetry and the existence of a contact place the contacts on nearest-neighbour pairs, and the native posting is the product of eighth-turns of the relay's photon-free terms. It gives the case for the experiential reading and states every open problem exactly. Full manuscript, 402 pages. Preliminary manuscript shared for discussion. It has not been peer reviewed.
Born probabilities determine the distribution of individual records, but leave their temporal relations open. In the U(1) M-event model a transported circle phase carries path memory; fixed transports on the same fiber nevertheless commute. We extend the event selector to a nontrivial compact connected finite-dimensional Lie group within the physical–observation dual-axis structure. One action governs the selector, its conjugate charge and their coupled motion on the two-axis base. A Haar initial selector, independent of the other initial data and prescribed controls, preserves the Born probabilities at every observation. Non-Abelian transport changes the relations between records. For balanced hemisphere readout on the full SU(2) group, we derive the exact law Pmis = α/π, relating the mismatch probability to the transport’s conjugacy angle. In a U(1) × SU(2) realization, a central charge sustains the original M response, while a periodic potential preserves its orbit and phase integral and makes finite deviations exactly harmonic. A classical pointer action gives hemisphere readout and a two-period SU(2) return for the stated apparatus preparation. With the faithful fixed-axis coupling, four blocks of 25 periods produce α25 = 1.012373567 . . .. If readout retains the selector after either outcome, the third supercycle differs from the initial record with probability 96.6746%, and the sixth agrees with it with probability 93.3491%. Independent response and matrix measurements fix these values before the event sequence is observed. Scans of axis angle and block length test the relation between noncommuting transport and event memory.
How a finite real cycle acquires a complex structure and a rule for composition Quantum theory uses a common imaginary unit when individual systems become a composite. This framework constructs that compatibility from finite real source histories. On the complete contrast sector of a three-cycle, the forward and backward comparisons give J=3P−P2,J2=−I The complex structure appears after the real comparison is specified. At a qualified unmarked boundary, opposite reference shifts select QAB=2I−JA⊗JB The range is the balanced complex tensor product. Its many-source composition is associative. A finite controlled contact supplies a nonfactorizable complex-linear operation, and exact residual bounds quantify departure from the selected sector. A source model that produces the contact A reversible controller compares operation orders on 24 proper signed frames. Its ordered, arrow-insensitive axis detector produces the retained cyclic response. The contact algebra gives [[G,G],[G,G]][G,G]≅C3. The construction prints the controller, scratch cleanup, full contrast probes, admissible operations and refusal states. A triaxial real response tensor realizes the detector and provides an explicit map between material contrasts and event-count contrasts. A record must survive the test of future operations A present tensor can hide a material marker that a later operation reveals. The framework derives independent force-gradient calibration and a delayed odd-field release. With a nonzero release field, energy and vector torque reconstruct the entire marker change: DU=−Δp⋅E,Dτ=Δp×E.Δp=∣E∣2E×Dτ−DUE The test includes shared calibration errors, stray fields, finite release pulses, marker relaxation and a complete operation inventory. It distinguishes detection of a retained record from agreement with its frozen prediction and from discrimination between physical theories. From proved structure to existing experiments The paper adds a conditional probability-correspondence theorem for the balanced composite, examines independent-source quantum-network experiments, and reproduces all eight measured rows of a public torque dataset. These supply composition benchmarks and demonstrated apparatus capabilities. Their published significances are kept attached to the hypotheses they actually test. The calibrated material release law is also a conventional electrostatic prediction; it is not presented as a QTT-exclusive signal. One material coefficient, two readouts A new archival calculation transfers a published same-particle libration calibration to a separate precession measurement. In the qualified rotor regime, the cross-readout relation is Ξ=ωlib2EpωsΩElcosϕl=alibaprec. An unchanged dipole-to-inertia coefficient gives unity. The calculation yields a slope of 2.909 from the libration calibration, compared with the published precession slope of 2.950, a central difference of 1.4135%. This shared electrostatic consistency relation eliminates the separate dipole and inertia; unequal field gains, dynamical corrections and shared uncertainty remain explicit. Reversal and long-term dipole-stability observations identify complementary laboratory capabilities. The complete matched-tensor delayed-release test remains an experimental target. When do two descriptions differ in any future experiment? The finite-reference extension follows a controlled source contact through shared references, fresh references, inverse operations, intermediate measurements and feedback. Its complete order-three response is gn=∑k=02pkζkn,ζ=e2πi/3,g3=1,g2=g1. A finite observable-span algorithm tests two fully specified models across every permitted future word. It either certifies agreement or returns the preparation, operation sequence, terminal outcome and exact probability difference. In the populated compatible-instrument benchmark, the complete quantum representation gives the same records without an additional target fit. A source-selected physical coupling or preparation remains a distinct route to predictive value. The framework includes nine vector figures, the retained source proofs and countermodels, complete reference-history benchmarks, calibration and vector-reconstruction theorems, runnable exact and numerical checks, published-input receipts, a data assessment and an experimental packet specification. The introduction defines the QTT terms needed to follow the construction without prior familiarity with the corpus. The probability-interface theorem, finite-memory bounds and covariance-weighted witness specify which further laboratory comparisons can distinguish physical models. Source anchors QTT Main Book, especially pp. 56, 248-250, 352 and 827-829; Artian's Completed-Event Four-Capacity Theorem; Frame-cycle derivation map, completed-event lexicon, and Artian's Universe and laboratory readout. Stable citation: 10.5281/zenodo.22882084.
In quantum information theory and statistical physics, symmetries of multiple copies, or replicas, of a system play a pivotal role. For unitary ensembles, these symmetries are encoded in the replicated commutant: the algebra of operators commuting with the ensemble across k replicas. Determining the commutant is straightforward for the full unitary group, but remains a major obstacle for structured, computationally relevant circuit families. We solve this problem for matchgate circuits, which prepare fermionic Gaussian states on n qubits. Using a Majorana fermion representation, we show that operators coupling different system copies generate the orthogonal Lie algebra so(k), endowing the space of invariants with rich and tractable structure. This underlying symmetry decomposes the matchgate commutant into irreducible sectors, which we completely resolve via a Gelfand--Tsetlin construction. We provide an explicit orthonormal basis of the matchgate commutant for all k and n, together with a formula for its dimension that grows polynomially in n. Furthermore, we characterize the commutant of the Clifford--matchgate subgroup, showing that restricting to signed permutations of Majorana modes yields a commutant that qualitatively diverges from the matchgate case for k≥4 replicas. Ultimately, our orthonormal basis turns algebraic classification into a working toolbox. Using it, we derive closed-form expressions for matchgate twirling channels and a fermionic analogue of Weingarten calculus, the projector encoding all moments of the Gaussian state orbit, state and unitary frame potentials, the average nonstabilizerness of fermionic Gaussian states, a systematic hierarchy of non-Gaussianity measures, and a fermionic de Finetti theorem.