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Behavioral Memory under Symmetry in One-Way Quantum Automata

Zeyu Chen

arXiv:2609.01451v1cs.FLquant-ph

TL;DR

Symmetry reduces observable behavior to invariant operator algebras, but their dimensions do not directly determine classical memory. The paper separates reachable-observable geometry, dynamical capacity, and operational state cost, obtaining an exact commutative-versus-noncommutative one-state dichotomy and symmetry-dependent polynomial, exponential, and Catalan memory laws.

  • Problem

    Invariant operator algebras under compact symmetry do not by themselves identify which coordinates are dynamic, observable, or already classical memory.

    Source: §1 · p. 1

  • Method

    The theory filters reachable-observable geometry, symmetry-constrained structural capacity, and strict-cutpoint operational realization using Hankel rank and commutant structure.

    Source: §1 · p. 2, §2 · p. 2, §3 · p. 7

  • Results

    Fully mobile commutative invariant algebras cost their dimension, while a noncommutative multiplicity block raises unrestricted strict-cutpoint cost by exactly one state; symmetry can also yield polynomial, exponential, or Catalan scales.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework provides a symmetry-aware memory calculus whose exact unrestricted state cost is determined by commutant dimension and commutativity in the fully mobile case.

    Source: §1 · p. 2

    Geometrically local or hardware-native gate restrictions can prevent the saturated profile law from being attainable.

    Source: §4.2 · p. 13

Abstract

from arXiv · show

Under compact symmetry, observable behavior reduces to an invariant operator algebra, but its dimension is not yet classical memory: some coordinates are dynamically frozen, some invisible to threshold tests, and some already classical. We develop an operator-algebraic theory that separates these effects through three filters. For one automaton, behavior is the Hilbert--Schmidt pairing between prefix-reachable states and suffix-observable effects, whose rank equals the real Hankel rank without controllability or observability assumptions. Maximizing this invariant over a symmetry-constrained dynamical class gives a structural capacity controlled by the symmetry commutant: its center stores isotypic populations frozen by reversible dynamics, its traceless multiplicity blocks carry movable noncommutative coordinates, dissipation removes the unary spectral loss inside those blocks, and covariant mobility releases relative populations subject to component conservation. Operational realization then determines which surviving coordinates force probabilistic states. For a fixed nontrivial invariant readout, full mobility gives an exact dichotomy in worst-case state cost: a commutative invariant algebra costs exactly its dimension, whereas a noncommutative multiplicity block raises the unrestricted cost by exactly one state. Thus noncommutativity has a one-state worst-case classical price. The known four-letter quadratic-plus-one law at trivial symmetry is the fully mobile endpoint of this principle. Schur--Weyl duality further shows that different preserved symmetries on the same tensor-power Hilbert space can change the worst memory scale from polynomial to exponential, while fixed-weight modules give an exact Catalan law at half filling, with structural capacity equal to the Catalan count minus its central-sector correction.

1 Introduction · 2 Instance geometry: the exact reachable–observable invariant

The paper separates symmetry-aware behavioral memory into exact instance geometry, dynamically activatable coordinates, and operational threshold cost. For the instance-geometric layer, Hankel rank is an exact reachable–observable invariant, while stochastic realization adds at most a codimension-one state overhead.

1 Introduction

  • Under compact symmetry, observable behavior lives in an invariant operator algebra, but frozen, unreadable, or already-classical coordinates need not impose equivalent classical memory.The scope is strict-cutpoint recognition by real PFAs, preserving the threshold language rather than word-by-word acceptance probabilities.Source: §1 · pp. 1–2
  • The broader theory shows that commutative fully mobile invariant algebras are already classical, whereas a noncommutative multiplicity block adds exactly one unrestricted-alphabet state.At trivial symmetry, the familiar four-letter N^2 + 1 law is the fully mobile whole-matrix endpoint.Source: §1 · pp. 1–2
  • Three filters distinguish invariant-algebra dimension from classical memory: reachability and observability, dynamical mobility, and threshold realization.Symmetry fixes the invariant algebra, dynamics selects movable coordinates, and threshold geometry determines which visible directions require probabilistic states.Source: §1 · p. 2

2.1 Model and sector reduction

  • Sector dephasing removes cross-sector coherences from acceptance, while mixed reduced multiplicity states remain necessary even though extremal unitary capacities can use pure physical inputs.The resulting behavioral-memory space is distinct from operational strict-cutpoint PFA memory.Source: §2.1 · p. 3

2.2 The reachable–observable pairing

  • The exact instance invariant is the rank of the Hilbert–Schmidt pairing between prefix-reachable states and suffix-observable effects, requiring no controllability or observability assumptions.Quotienting states by directions annihilated by every suffix effect separates failure to reach an operator direction from failure to observe it.Source: §2 · p. 2, §2.2 · p. 4

2.3 Representation-theoretic resolution

  • Representation-theoretic decomposition resolves the exact rank into irreducible operator channels, with each channel weighted by its irreducible dimension and the rank of its effective contracted coefficient space.The effective rank need not equal the number of nonzero multiplicity components.Source: §2.3 · pp. 4–5

2.4 From Hankel rank to probabilistic states

  • A k-dimensional real linear representation at cutpoint zero embeds into an alphabet-preserving (k + 1)-state PFA, while a general cutpoint costs at most k + 2 states.A normalized left or right constant mode centers the cutpoint within the original representation and restores the k + 1 bound.Source: §2.4 · pp. 5–6
  • The stochastic construction preserves the strict-cutpoint language, not numerical acceptance probabilities, so β(A) is an exact linear invariant but only an upper bridge to operational PFA memory.The resulting PFA compares acceptance against a shifted cutpoint to reproduce the language exactly.Source: §2.4 · p. 6

3 Structural capacity: the center–commutator split

Structural capacity is determined by which invariant operator coordinates symmetric dynamics can write and which readout can observe. Haar reduction exposes multiplicity spaces, while the center–commutator split identifies frozen central populations and movable traceless modes.

3.1 Haar reduction to multiplicity spaces

  • Compact symmetry reduces behavior to multiplicity spaces: irreducible dimensions assemble the physical Hilbert space, while multiplicities carry word-dependent acceptance behavior.The reduction is standard representation theory, but here it connects the Hermitian noncommutative operator space to Hankel rank and probabilistic state complexity.Source: §3.1 · p. 7

3.2 The noncommutative commutant law

  • The commutant law separates read-only isotypic populations in the center from all word-dependent directions in traceless multiplicity blocks.Under the stated nonzero-component, nontrivial-readout, and independent-control conditions, the Hilbert–Schmidt pairing reaches the central direction plus the traceless-block dimension; the upper bound is attainable in every case.Source: §3.2 · p. 9

3.3 The universal sector cap

  • The universal sector cap counts only jointly active sectors, and equality requires both reachable state-orbit and observable effect-orbit spaces to fill their ambient behavior-relevant spaces.A nonempty active profile can attain the cap under independent sector control, while an empty active set has instance cap δ ∈ {0, 1}.Source: §3.3 · p. 10
  • Locked controls can still saturate the structural cap when their multiplicity channels span the relevant operator modes, but probabilistic lower bounds depend on the accepting measurement.The cap measures the largest blockwise behavior space compatible with fixed sector traces, whereas the measurement induces a potentially smaller operational capacity.Source: §3.3 · p. 11

4 Operational realization: finite and dynamic shattering

This section connects movable operator coordinates to probabilistic state complexity through finite sign witnesses, dynamic Markov-limit effects, and alphabet-capacity transitions. It establishes how sector-preserving and noncommuting dynamics realize noncommutative capacity, while deriving tight probabilistic-state laws governed by commutant structure.

4.1 Differential sign witnesses

  • Finite Jacobian shattering converts m visible tangent coordinates into an affine complete-sign witness, forcing every equivalent real PFA to use at least m + 1 states.The dynamic lift can instead force d + 2 states because centering prefix distributions at a recurrent Markov limit creates one extra state beyond finite sign-rank; a one-sided positive anchor is essential.Source: §4.1 · p. 12

4.2 Two generators for independent sector control

  • Two sector-preserving unitaries generate a dense subgroup of every compact semisimple symmetry-commutant profile, attaining full noncommutative capacity with a binary alphabet.This is an abstract global-control result; locality or hardware-native restrictions may prevent saturation, although the instance-rank and tangent bounds remain applicable.Source: §4.2 · p. 13

4.3 A binary prepare–test construction

  • For every active rank profile, a binary prepare–test construction yields a strict-cutpoint language requiring at least κ + 2 PFA states, with uniform margins and dynamic carrier recurrence preserving the obstruction.For rank-one readouts, curvature supplies an independent finite witness: the sign matrix itself expands, producing the same κ + 2 lower bound without a Markov limit or infinite carrier orbit.Source: §4.3 · pp. 14–16

4.4 Exact alphabet-capacity phase diagram

  • The unary case has an exact alphabet-capacity phase diagram: its capacity is analyzed block by block through the Hankel ranks of exponential sequences generated by the reduced transition.Distinct nonzero exponentials contribute their number of terms to complex and real Hankel rank, enabling the exact unary calculation.Source: §4.4 · pp. 16–17
  • A unary automaton attains the upper bound rank 1 + ν_K, and no automaton can exceed it.Generic nonzero eigenphase differences can be chosen globally distinct, with nonzero coefficients ensuring the Hankel rank reaches the bound.Source: §4.4 · p. 17
  • The threshold from one to two symbols is an exact noncommutative alphabet transition, because a second noncommuting transition releases the traceless diagonal directions invisible to unary dynamics.Central phases do not reduce this measure-once capacity, whereas dissipation can make relative central populations observable coordinates.Source: §4.4 · p. 17

4.5 Tight binary profile laws

  • For every active binary profile, the minimum equivalent real-PFA state count is characterized by tight fixed-profile bounds.The lower bound follows from the finite sign-rank certificate or theorem 4.7, while the upper bound uses the centered representation and theorem 2.5.Source: §4.5 · p. 17
  • Balanced readouts expose a constant fraction of B(D), while the codimension-one stochastic embedding adds only one probabilistic state.The resulting finite-alphabet cost remains at the scale of sectorwise operator capacity, and the pair (B, κ) tracks changes under relaxed symmetry constraints.Source: §4.5 · p. 18
  • When ν_K > 0, the worst strict-cutpoint probabilistic state cost is controlled up to universal constants by the noncommutative commutant dimension; when ν_K = 0, one state suffices and is necessary.The upper bound comes from a constant-normalized behavior representation of dimension at most 1 + ν_K, while zero noncommutative dimension makes every acceptance function constant.Source: §4.5 · p. 18

5 Dissipative release: channels and central mobility

This section examines how dissipation and mobility determine which invariant coordinates become behavioral memory and establishes exact laws for both channel rank and probabilistic state cost. It distinguishes dynamical notions of covariance, separates Hankel capacity from state cost, and identifies when noncentral movable directions attain the unrestricted upper endpoint.

5.2 Mobility components and the channel commutant law

  • Mobility partitions interpolate between reversible read-only centers and full mobility, with each merged conserved-component group releasing one relative population coordinate.For a fixed nontrivial readout, only active components contribute their trace-zero directions, while all component baselines contribute one shared constant mode; one input symbol suffices for full capacity.Source: §5 · p. 18, §5.2 · pp. 20, 22

5.1 Model and exact channel rank

  • Covariant general channels have behavioral rank equal to the Hilbert–Schmidt pairing rank between reachable states and observable suffix effects.The invariant-boundary condition is essential: without an invariant initial or final boundary, covariance alone need not confine behavior to the commutant.Source: §5.1 · p. 19

5.3 Kraus-wise charge-zero symmetry

  • Kraus-wise charge-zero channels conserve every isotypic population and attain the reversible bound, whereas general covariant channels can release relative populations without Kraus operators lying in the commutant.A qubit example separates the notions: charge-zero capacity is one, while a covariant channel with charged off-diagonal Kraus operators reaches rank two; if both boundaries break symmetry, rank can exceed the commutant bound.Source: §5.3 · pp. 23–24

5.4 Strict-cutpoint probabilistic state cost

  • The unary Hankel-capacity construction does not by itself establish probabilistic state lower bounds, so a separate finite sign witness is required for strict-cutpoint language cost.The probabilistic-state quantity is defined as the supremum of minimum real-PFA sizes over arbitrary finite alphabets and symmetry-compatible automata.Source: §5.4 · p. 25
  • Every nonzero readout-visible component contributes an exact binary sign-rank obstruction, while stochastic positivity adds at most one state.If the readout or active set is absent, both costs collapse to one state.Source: §5.4 · p. 25
  • The binary construction realizes a sign matrix of sign-rank d + 1 = MΠ,P, forcing every equivalent real PFA to use at least MΠ,P states.At the cutpoint, suffixes encode the prefix bits, while a final suffix is negative for every prefix.Source: §5.4 · p. 26
  • With an active component having νC∗ > 0, unrestricted-alphabet state cost reaches the upper endpoint.The construction adapts dynamic shattering to the fixed-weight symmetry-reduced slice while preserving component constraints through covariant channels.Source: §5.4 · p. 26
  • A witness needs at most 2s + 3 letters, reduced to 2s + 2 when the selected readout projector is noncentral in its component center.Here s = |AΠ,P|, and the reduction exploits a noncentral readout direction.Source: §5.4 · p. 26
  • The dynamic witness combines orthant coding with an irrational commutant rotation in a two-dimensional multiplicity corner, allowing the prescribed functional to detect the required coordinates.When the readout is central, an invertible contracted decoder map first makes the functional nonzero on both rotated directions.Source: §5.4 · p. 27

C and TCRT

The section establishes exact state-cost laws for component-conserving, charge-zero, and fully mobile symmetry-reduced automata. Noncommutative multiplicity blocks add exactly one state in the fixed-readout unrestricted setting, while commutative reductions retain their dimension cost.

Component-conserving state law

  • The binary component-conserving law gives matching state-cost bounds from the invariant dimension M_Π and active nontrivial components.When M_Π = 1, both costs equal one; otherwise the construction attains the lower bound while universal upper bounds match it.Source: C and TCRT · p. 27

Charge-zero state law

  • Charge-zero dynamics preserve the same quadratic-plus-one obstruction whenever a nontrivial multiplicity block is available, but otherwise every invariant-readout behavior is constant.For P ≠ 0 and Λ_P ≠ ∅, the fixed-readout cost is M_K0,P + 1; if P = 0 or Λ_P = ∅, it is one.Source: C and TCRT · p. 28

Full-mobility noncommutativity dichotomy

  • Under full mobility and a fixed nontrivial invariant readout, commutative reductions cost D_K states, whereas a noncommutative multiplicity block costs exactly D_K + 1.The witness uses at most five letters, or four when the projector is noncentral; with no multiplicity block, the reduced automaton is a D_K-state PFA.Source: C and TCRT · pp. 28–29

Fixed-rank general-channel law

  • For trivial symmetry, every nontrivial fixed rank 1 ≤ r < N retains the exact quadratic-plus-one unrestricted state law, with four input letters sufficient.The rank-one case recovers the quadratic state law of [21], while the extension uses only a nonzero trace-zero component of the readout.Source: C and TCRT · p. 29

Finite-test complexity

  • The block-cyclic binary witnesses use prefixes and suffixes of length at most M_Π,P − 1, with strict-cutpoint margin at least ε M_Π,P − 1/4.The component, charge-zero, and full-mobility corollaries inherit this linear word-length bound, although margins may shrink with dimension.Source: C and TCRT · p. 29

6 Representation-theoretic consequence: Schur–Weyl inversion

Schur–Weyl duality shows that preserving different symmetries on the same tensor-power Hilbert space produces polynomial versus exponential worst-case behavioral memory. The inversion is therefore caused by the preserved symmetry, not ambient quantum dimension, and becomes logarithmic versus linear classical indexing cost.

6.1 Permutation-equivariant automata

  • Permutation-equivariant automata have polynomial state capacity and probabilistic simulation cost, with the commutant identified through the symmetric tensor power of End(C^d).For fixed d, the center contributes only polynomially, lower order than the binomial-coefficient capacity.Source: §6.1 · pp. 29–30

6.2 Collective-unitary-equivariant automata

  • Collective-unitary-equivariant automata have capacity and simulation cost governed by the commutant dimension Ad(n), whose fixed-d asymptotics are exponential.The center is polynomial in n and negligible relative to the asymptotic commutant expression, for fixed d ≥2.Source: §6.2 · p. 30

6.3 The inversion on a common Hilbert space

  • On the same physical Hilbert space, the two generally non-nested automaton classes exhibit a polynomial–exponential inversion because they preserve different members of the Schur–Weyl dual pair.Each scale is maximized over its own symmetry-constrained class, so the contrast isolates preserved symmetry rather than ambient dimension.Source: §6 · p. 29, §6.1 · p. 29, §6.3 · pp. 30–31
  • 3 log2 n + O(1) bits versus 2n −3 2 log2 n + O(1) bits index worst-case simulator states for permutation versus collective-unitary symmetry.Thus the state-count inversion becomes a logarithmic-versus-linear separation in classical memory bits.Source: §6.3 · p. 31

7 Readout geometry and symmetry release

The section identifies readout capacity with the geometry of the accepting-projector orbit, distinguishing total behavior directions from those directly visible to threshold tests. It then characterizes symmetry release through branching and block mixing, showing that only newly exposed accepting–rejecting coherences directly enlarge readout-visible capacity.

7.1 The accepting-orbit capacity

  • The accepting-orbit capacity equals the dimension of the projector’s conjugation orbit, while a central readout yields only constant reversible behavior.Under full sector control, independently movable readout sectors contribute additively; sectors with accepting rank zero or full rank are behaviorally constant.Source: §7.1 · pp. 31–32

7.2 Intrinsic Fisher geometry of the readout orbit

  • The same orbit dimension is the intrinsic Fisher rank of tangent projector variations, but circuit-coordinate Fisher rank matches it only when the circuit differential covers the orbit tangent space.In a Hilbert–Schmidt orthonormal tangent frame, every nonzero Fisher eigenvalue is 2/RP.Source: §7.2 · p. 32

7.3 Visible and dark behavior directions

  • Readout-visible directions are accepting–rejecting coherences, whereas same-outcome coherences and the traceless block-population direction are dark curvature directions.The total capacity controls maximal Hankel rank, while κ measures directions directly convertible into threshold tests; an additional finite sign witness is needed for operational state lower bounds.Source: §7.3 · p. 33

7.4 Symmetry release along subgroup chains

  • Symmetry release increases noncommutative capacity according to representation branching, with the net gain determined by matrix-direction growth relative to changes in the commutant center.Block merging adds cross-block Hermitian coherences and a relative population direction; only the coherences crossing the accepting–rejecting boundary are directly visible.Source: §7.4 · pp. 34–35
  • The visible-release fraction is a geometric property of the chosen readout, not by itself a fixed-language or task-performance gain.A particular automaton’s gain depends on its reachable–observable pairing, and threshold advantage additionally requires a finite sign witness.Source: §7.4 · p. 35

8 Fixed-weight memory and the critical transition

Fixed-weight permutation modules convert symmetry-controlled memory into an exact representation-theoretic count. Half filling yields a Catalan law, while endpoint-dominated and critical regimes describe the transition to it.

8.1 Multiplicity-free sector decomposition

  • Multiplicity-free Schur–Weyl sectors make the squared-dimension sum the maximal behavior capacity, up to subtracting the nontrivial-sector count, and determine the binary simulation-cost scale.The construction realizes the abstract generators in the unitary group of the image algebra, without claiming physical permutation commutation or shallow local implementation.Source: §8.1 · pp. 35–36

8.2 Away from half filling

  • For every fixed filling density α < 1/2, endpoint terms remain geometrically dominant, preserving the full quadratic simulation-cost scale in the Hilbert-space dimension.This endpoint mechanism fails at half filling, where a growing window of irreducible sectors contributes on the same scale.Source: §8.2 · p. 36

8.3 The Catalan law at half filling

  • At half filling, the squared-sector-dimension sum is exactly Catalan, while structural capacity equals the Catalan count minus its central-sector correction.The half-filled point is smaller than the naive quadratic scale by a factor of order √n.Source: §8.3 · p. 37

8.4 The critical window

  • When the distance from half filling is of order √n, the sector sum converges to a nontrivial interpolation function between the macroscopic and half-filled laws.The critical scaling matches the onset of the macroscopic law, and balanced nontrivial accepting ranks attain the corresponding lower-order simulation cost.Source: §8.3 · p. 37, §8.4 · pp. 37–38
  • All three regimes arise from one representation-theoretic sum: endpoint domination away from half filling, Catalan accumulation at half filling, and a Gaussian critical window between them.A finite S4 calculation provides a consistency check through capacity, Fisher rank, and prepare–test margin, not evidence for the asymptotic law.Source: §8.4 · p. 38

9 Intermediate halting and nonhalting behavioral memory

This section develops symmetry-compatible models of intermediate halting and nonhalting behavior, distinguishing absorbing dynamics from general covariant channels. It characterizes how mobility components and nonhalting operator dimensions govern behavioral capacity and exact worst-case real-PFA state requirements.

9 Intermediate halting and nonhalting behavioral memory

  • Absorbing channels preserve every word’s acceptance probability, lie in the commutant, and conserve the total weight of each isotypic component.Intermediate measurements are replaced by trace-preserving absorbing dynamics whose accepting trace equals cumulative measure-many acceptance.Source: §9 · pp. 38–39
  • The absorbed model retains the exact reachable–observable Hankel invariant, while nonhalting survivor weights become dynamical coordinates without releasing central commutant coordinates.Fixing the halting profile therefore closes the structural capacity through the accumulated acceptance coordinate and surviving nonhalting blocks.Source: §9 · pp. 39–40
  • One input symbol attains the exact profile capacity, and the corresponding strict-cutpoint language has probabilistic state cost at most one more than that capacity.The active nonhalting blocks contribute their full Hermitian operator spaces, including trace directions; an empty active set yields capacity zero and exact state cost one.Source: §9 · pp. 39–41
  • A dynamic phase upgrade is not automatic: a scalar active profile has exact probabilistic state cost two, despite the general upper bound allowing three.The compiled weak-leak construction can retain a nonzero affine offset, preventing the required recurrent cutpoint crossing; a higher-dimensional nonhalting corner and new test frame are needed for any phase-assisted upgrade.Source: §9 · pp. 41–42

9.1 Covariant general channels between halting measurements

  • Covariant general channels conserve mobility components rather than individual isotypic labels, and generic component-wise contractions can realize the full nonhalting operator rank.Absorption remains completely positive, trace preserving, and covariant while preserving the prescribed mobility components.Source: §9.1 · p. 42
  • The covariant measure-many channel model has an exact component-profile capacity attained with one input symbol, with inactive components contributing no acceptance behavior.The Hankel matrix factors through accumulated acceptance and the active nonhalting invariant blocks; when the active set is empty, component conservation makes acceptance identically zero.Source: §9.1 · p. 43
  • The behavioral rank of the constructed family is 1 + P_C because the constant mode and all component modes are nonzero and spectrally disjoint.The Vandermonde factorization proves equality.Source: §9.1 · p. 43
  • The mobility partition is essential: labels with pλ = 0 can still contribute when channels move weight within their mobility component, yielding 1 + Dnon for one fully mobile component.For singleton components, the result reduces to the active-label law.Source: §9.1 · p. 44
  • Theorem 9.8 gives the exact worst-case real-PFA state cost for every nonempty active component profile.The lower bound is realized with a finite alphabet, while the upper bound follows from the preceding capacity results.Source: §9.1 · p. 44
  • If the active set is empty, the exact worst-case state cost is one.This is realized by a one-state threshold language.Source: §9.1 · p. 44
  • General covariant channels close both measure-many profile capacity and its strict-cutpoint state scale: witness alphabet size follows addressable labels, whereas state cost follows full nonhalting operator dimension.The construction uses covariant, trace-preserving, component-conservative channels and compiled preparation and test words.Source: §9.1 · pp. 43–44

10 Bridges and limits of operational realization

The section separates instance geometry, structural capacity, and operational state cost, then identifies the bridges and limits connecting them. It shows that readout-orbit geometry and alphabet constraints can restrict operational lower bounds even when structural capacity is larger.

Three operational layers

  • The three filters measure distinct quantities: instance geometry is an equality, structural capacity is a dynamical-class maximum, and operational state cost concerns cutpoint signs.A stochastic embedding provides the general upper bridge between these layers.Source: §10 · p. 44

Bridges and endpoints

  • Task-visible orbit geometry, rather than total structural capacity alone, controls the available lower obstruction, and the bridge inequalities do not identify their endpoint quantities.A recurrent noncommutative phase reaches the measure-once upper endpoint, while the scalar measure-many construction reaches the lower endpoint.Source: §10 · pp. 44–45

Reversible fixed readouts

  • Reversible fixed-readout witnesses face an operational ceiling: affine-complete shattering is limited by the accepting-projector orbit dimension, so full Hankel capacity need not be certified by one sign witness.The ceiling applies to every affine-complete witness drawn from the readout orbit, not merely one construction.Source: §10 · p. 45
  • For one rank-one block, κ = 2(m −1) versus full capacity m2, so affine-complete witnesses certify only O(m log m) states; the quadratic fixed-readout gap remains open for balanced ranks.A recurrent affine lift adds one state without changing this asymptotic ceiling.Source: §10 · p. 45

Alphabet boundary

  • The measure-many lower construction requires two nonleaking controls and one weak-leak letter per active label, so it proves a finite-alphabet result rather than a binary theorem.Within the terminal-reject weak-leak architecture, powers of one test letter cannot realize the complete sign frame for dimension at least two, leaving binary measure-many automata as a distinct semigroup problem.Source: §10 · p. 45

Resource scope

  • The resource counted is PFA states, whereas word length, gate depth, and hardware locality are separate; finite witnesses have positive margins and finite word lengths, while universal claims are analytic.The fixed-weight group-algebra realization is representation-sector based, and finite S4 calculations audit concrete formulas and certificates.Source: §10 · p. 45

11 Conclusion

Symmetry does not simply reduce quantum-automaton memory: behavioral memory emerges by filtering invariant operator coordinates through reachability, observability, dynamical mobility, and threshold realization. The resulting framework separates instance memory, class capacity, and operational state cost, yielding an exact commutative/noncommutative dichotomy under full mobility and identifying scale-dependent and operationally unresolved cases.

Behavioral memory

  • The three-stage theory classifies invariant coordinates by reachable–observable pairing, center–commutator and mobility structure, and threshold realization, rather than by ambient Hilbert-space dimension.These stages determine one-instance memory, dynamical-class capacity, and which surviving coordinates require probabilistic states.Source: §11 · p. 45

Full mobility

  • For a fixed nontrivial invariant readout, full mobility costs exactly the algebra dimension for commutative invariant algebras and exactly one additional state when a noncommutative multiplicity block is present.The quadratic-plus-one law at trivial symmetry is the fully mobile endpoint; reversible dynamics, component conservation, and intermediate halting explain why broader restrictions matter.Source: §11 · p. 45

Representation-theoretic scale

  • Schur–Weyl commutants on the same tensor-power Hilbert space yield polynomial versus exponential worst-case memory laws, while half filling gives a Catalan squared-sector-dimension sum and Catalan-minus-center structural capacity.Readout-orbit Fisher geometry and subgroup branching quantify task-sensitive visibility and symmetry release.Source: §11 · p. 45

Open cases

  • Open cases concern operational realization: rank-one reversible readouts need stronger lower-bound invariants, while binary channel endpoints and phase-assisted measure-many constructions require different finite semigroup realizations.These are realization problems, not uncertainty about the operator spaces made available by symmetry.Source: §11 · p. 45

A Binary Lie generation

The construction separates adjacent-edge frequencies using distinct two-adic valuations, enabling polynomial isolation of each edge’s Lie generators. These generators yield su(Dα) in every nontrivial block, with no central u(1) contribution.

A Binary Lie generation

  • Distinct two-adic valuations make all adjacent-edge frequency differences |∆e| distinct, enabling frequency separation.For b > a and d > c, equal differences force a = c and b = d.Source: A Binary Lie generation · p. 46
  • Lagrange polynomials isolate each edge eigenvalue, so the resulting iterated commutators belong to the generated Lie algebra.The interpolation polynomials are one at a selected edge eigenvalue and zero at all others; ∆e ≠ 0 then yields Ae.Source: A Binary Lie generation · p. 46
  • Each adjacent edge provides two off-diagonal quadratures and a traceless diagonal difference, while commutators propagate these generators along each path.For example, [Ej,j+1, Ej+1,j+2] = Ej,j+2 generates longer-range matrix units.Source: A Binary Lie generation · p. 46
  • The generated Lie algebra is the direct sum of su(Dα) over blocks with Dα ≥ 2; one-dimensional blocks contribute su(1) = {0}.Trace subtraction in H0 and the zero diagonal of H1 exclude central u(1) directions.Source: A Binary Lie generation · p. 46

B Fisher geometry of the readout orbit · C Critical-window asymptotics

The readout orbit has a positive-definite Hilbert–Schmidt geometry of dimension κG(P). Critical-window asymptotics are established by combining local central-limit estimates, endpoint-safe expansions, Gaussian bounds, and successive limiting arguments.

B Fisher geometry of the readout orbit

C Critical-window asymptotics

D Measure-many contraction and test constructions · E Operational boundary arguments

The constructions realize exact threshold-sign tests by combining spectrally rich measure-many contractions, compiled rank-one leak frames, and controlled approximation. They support both commuting-unitary and component-conservative covariant realizations with explicit finite alphabets and preserved operational constraints.

D Measure-many contraction and test constructions

  • Diagonalizable defect-one contractions provide nonzero, pairwise-distinct product spectra and arise as compressions of one higher-dimensional unitary.Their rank-one defect construction also yields a singular-value dilation with V as the compression to the first n coordinates.Source: D Measure-many contraction and test… · pp. 47–48
  • A compiled weak-leak frame realizes exact finite prefix–suffix sign matrices using at most |L| + 2 symbols, despite allowing approximation only internally.The construction uses two nonleaking controls and one weak leak per label, with invariant accepting exits and component-conservative covariant variants under prescribed seed weights.Source: D Measure-many contraction and test… · pp. 48, 50
  • Each n-dimensional multiplicity block uses 2n^2 − n rank-one projections whose real span is Herm(C^n), forming the operator frame needed for arbitrary tests.Pairwise differences recover real and imaginary off-diagonal matrix units, while diagonal projections recover the diagonal units.Source: D Measure-many contraction and test… · p. 49
  • The compiled tests maintain uniform control error while preserving a fixed number of control approximations determined by the frame size.After rational dualization and integer scaling, the relevant construction parameters and frame-dependent quantities are O(R) once the initial rational data are fixed.Source: D Measure-many contraction and test… · p. 49
  • Leak sequences prepare the required survivor weights and directions, with exact interval filling for blocks of dimension at least two and O(δ) target-weight error for one-dimensional blocks.Choosing R, δ, and the dense-control error in sequence makes the signal dominate construction errors and yields the desired sign matrix with a positive anchor.Source: D Measure-many contraction and test… · pp. 49–50
  • The covariant implementation replaces each unitary leak with a completely positive, trace-preserving, K-covariant, Π-conservative channel while leaving the preparation, test, and error analysis unchanged.The channel uses an explicit survivor and a measure–prepare branch into a normalized invariant accepting state in the same mobility component.Source: D Measure-many contraction and test… · p. 50

E.1 Affine-complete shattering ceiling … G Finite S4 certificate and numerical validation

The merged sections establish polynomial shattering ceilings and architecture-specific limits, then delimit common-character capacity and validate the structural formulas through a finite S4 witness. The example confirms dimensions, Fisher rank, margins, and state-cost certificates while leaving universal scaling laws analytic.

E.1 Affine-complete shattering ceiling

  • Grassmannian charts convert threshold tests into polynomial sign conditions, enabling Warren’s theorem to bound realizable patterns across the invariant orbit.Each active block contributes real graph coordinates, while cleared denominators yield bounded-degree real polynomials; coordinate-subspace charts cover the orbit.Source: §E.1 · p. 50

E.2 Why the compiled weak-leak frame is not directly binary

  • Power tests in the compiled weak-leak architecture cannot realize the complete binary frame because each row’s sign changes at most once, producing at most D + 1 patterns.This obstruction is architecture-specific; survivor effects at the right endmarker or other semigroup constructions are not ruled out.Source: §E.2 · p. 51

F Common-character Kraus boundary

  • Common-character Kraus operators permute occurring irreducible labels and conserve populations on each resulting orbit, yielding the universal component-based capacity bound.The conclusion follows from intertwining, trace preservation, invertibility of character tensoring, and orbit summation.Source: F Common-character Kraus boundary · p. 51
  • The component bound need not be tight: for Z2 × Z2 with four one-dimensional labels, behaviors depend only on word-length parity and have Hankel rank at most two.Although the component bound equals 3 in this example, deterministic paired-label transport prevents automatic saturation.Source: F Common-character Kraus boundary · p. 51

G Finite S4 certificate and numerical validation

  • The Hamming-weight-two S4 module supplies a concrete certificate with sector dimensions 1, 3, and 2, using the one-dimensional sector as an always-accepting spectator.The construction activates rank-one readouts in the two nontrivial sectors and instantiates the paper’s finite structure-to-witness chain.Source: G Finite S4 certificate and numerical… · p. 51
  • The active Lie algebra has dimension 11, its projector stabilizer dimension is 5, and the resulting orbit dimension is 6, matching κ.The same example’s intrinsic SLD Fisher metric has rank six with six equal nonzero eigenvalues.Source: G Finite S4 certificate and numerical… · pp. 51–52
  • The explicit certificate gives a seven-state sub-obstruction, while the curved rank-one witness yields an eight-state lower bound; the conservative first-order margin is γ = 1/288.The finite construction uses strict prefix–suffix signs and a spectator-weighted cutpoint, with active derivatives retaining weight p = 1/3.Source: G Finite S4 certificate and numerical… · p. 52
  • Finite calculations check dimension, margin, branching, Catalan, and certificate formulas, but they diagnose implementations rather than prove universal capacity, state-cost, or asymptotic efficiency results.The paper explicitly distinguishes these concrete checks from its analytic scaling laws and universal theorems.Source: G Finite S4 certificate and numerical… · p. 52
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