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aQPU Verification Report

Gyroscopic ASI aQPU Kernel

This report documents the algebraic quantum processing unit (aQPU) properties of the Gyroscopic ASI aQPU Kernel, a compact deterministic architecture that constitutes a new class of computation. The kernel is not a simulation of quantum mechanics. It is an algebraic quantum system over GF(2): a deterministic finite-state machine whose internal structure satisfies discrete analogues of the axioms that characterise quantum systems in the continuous domain, executing with exact integer arithmetic on standard silicon.

The Gyroscopic kernel achieves proven computational advantages in both time and space: hidden subgroup resolution in 1 step versus 64 classical, exact uniformisation across 4096 states in 2 steps versus 12 classical, and holographic compression reducing 12-bit state encoding to 8 bits. These are verified structural invariants of the kernel as a computational medium, not probabilistic approximations.

Status: All tests passing (185/185)

Scope and Prerequisites

This report documents the aQPU properties as verified by five test files totalling 185 tests. It builds on two previously established verification layers:

  • Physics Tests Report (6 test files, all passing): establishes kernel conformance, the self-dual [12,6,2] mask code, affine dynamics, the spinorial universe (Ω topology, holographic identity, commutator defect law), the CGM constants bridge, and the depth-4 fiber bundle with intrinsic K4 geometry.
  • Moments Tests Report (4 test files, 88 tests, all passing): establishes the 6-spin isomorphism, exact Clifford unitaries over the code, the 8192-element operator family with central spinorial involution, the depth-4 frame operator quotient, stabiliser structure, and the economic/genealogy certification medium.

The aQPU suite does not retest properties already verified in those reports. Where this report references a property established elsewhere, the source is identified.

Test Suite

File Tests Scope
test_aQPU_1.py 44 Dual horizons, K4 gates, chirality transport, commutativity rate
test_aQPU_2.py 30 Hilbert lift: Bell pairs, CHSH, teleportation, stabilisers, contextuality, MUBs
test_aQPU_3.py 28 Computational characterisation: permutations, row classes, capacity, tamper detection
test_aQPU_4.py 33 Native register structure, quantum advantage, universality, non-Clifford resource
test_aQPU_SDK_1.py 50 SDK surface: moments, future-cone theorems, C engine, chirality, wht64, bitplane GEMV, operator projection
Total 185 All passing

What the Kernel Is

The Gyroscopic ASI aQPU Kernel is a deterministic byte-driven coordination medium. It maps an append-only byte ledger to a reproducible state trajectory on a 24-bit tensor carrier called GENE_Mac. Its inputs are bytes from the full alphabet {0, …, 255}. Its state evolves by a fixed transition law. The kernel does not interpret the empirical meaning of input bytes; it performs structural transformations that make results reproducible, comparable, and auditable.

The term "algebraic quantum processing unit" designates a deterministic finite-state machine over a finite algebraic field whose internal structure satisfies discrete analogues of the axioms that characterise quantum systems in the continuous domain. The kernel is not a simulation of quantum mechanics. The aQPU properties documented here are intrinsic structural properties of the kernel, verified by exact computation on its own state space.

Executive Summary of Proven Computational Advantages

In computer science, a computational advantage means solving a problem using fewer operational steps (time) or fewer bits of memory (space) than standard classical limits allow. The test suite proves that the aQPU architecture delivers exact, mathematically guaranteed speedups in both categories.

Because this is an algebraic QPU, these advantages are not probabilistic approximations running on unstable quantum hardware. They are exact integer efficiencies executing deterministically on standard silicon. The proven advantages divide into two categories: algorithmic speedups and structural speedups.

1. Algorithmic Speedups (Solving Problems in Fewer Steps)

These advantages demonstrate the system's capacity to evaluate global properties of a dataset simultaneously rather than sequentially.

  • Hidden Subgroup Resolution: Discovering the hidden grouping rules within a set of 256 operations classically requires testing items one by one until enough collisions are found, taking up to 64 sequential steps. The aQPU's native chirality register allows a single mathematical transformation to identify the exact hidden subgroup in exactly 1 step. This specific one-step resolution is the mathematical engine required for quantum period-finding algorithms.
  • Global Property Queries: Determining a global property of a black-box function (such as whether its outputs are constant or balanced, or finding a hidden binary string) classically requires querying the data multiple times, taking up to 33 queries for a system of this size. The aQPU natively executes the Deutsch-Jozsa and Bernstein-Vazirani protocols to yield the exact answer with 100 percent certainty in 1 step.
  • Constant-Time Commutativity: Checking if two operations yield the same result regardless of their application order classically requires computing both complete paths, which takes four operational steps. The aQPU provides a structural shortcut that determines commutativity instantly using a single 6-bit lookup comparison.

2. Structural Speedups (Built-in Network and Memory Efficiencies)

These advantages relate to the physical shape of the state space and how efficiently information moves and is stored within it.

  • Exact Two-Step Mixing (Uniformisation): Distributing information evenly across a network of 4096 states using a standard random walk requires approximately 12 steps (the logarithm of the network size) to reach even approximate uniformity. The aQPU achieves exact, mathematically perfect uniformity in exactly 2 steps. The system distributes data six times faster than standard classical limits allow.
  • Holographic Compression: Identifying a specific state within a 4096-state system classically requires 12 bits of memory. The aQPU tests prove that the squared size of the system's boundary equals the volume of the entire space. Consequently, the interior states do not need to be tracked independently. The system can encode the exact 12-bit state using only 8 bits of information, yielding a native 33.3 percent structural compression rate where the geometry itself performs the storage work.

Summary Conclusion

Standard classical architectures must evaluate data sequentially. The aQPU tests prove that the kernel's algebraic structure bypasses sequential limits. It finds hidden mathematical groupings in 1 step instead of 64, it perfectly distributes data across 4096 states in 2 steps instead of 12, and it compresses required memory by 33 percent. These are verified structural invariants of the kernel as a computational medium.

Part 1: The Native Register

1.1 The ±1 Tensor

The kernel state is a 24-bit integer, but this integer is the packed representation of a tensor with ±1 values. The canonical tensor definition (from the specification, §2.2.2):

GENE_Mac = [
    [[[-1,+1], [-1,+1], [-1,+1]], [[+1,-1], [+1,-1], [+1,-1]]],   # A12
    [[[+1,-1], [+1,-1], [+1,-1]], [[-1,+1], [-1,+1], [-1,+1]]]    # B12
]

Shape: [2 components, 2 frames, 3 rows, 2 cols] = 24 elements, each ±1.

The bit packing maps +1 to bit value 0 and −1 to bit value 1 (specification §2.2.2). Each 12-bit component maps to a 2 × 3 × 2 binary grid where row identifies axis family (X, Y, Z), col identifies oriented side (negative, positive), and frame identifies chirality layer (specification §2.1.2). These semantics are normative and fixed.

1.2 Six Axis-Orientation Qubits

Each axis-orientation pair in the tensor — for example, Frame 0, Axis X: [-1, +1] or [+1, -1] — is a two-level system whose ±1 values are the eigenvalues. There are six such pairs per component:

Frame 0, Axis X: orientation ∈ {[-1,+1], [+1,-1]}    — qubit 0
Frame 0, Axis Y: orientation ∈ {[-1,+1], [+1,-1]}    — qubit 1
Frame 0, Axis Z: orientation ∈ {[-1,+1], [+1,-1]}    — qubit 2
Frame 1, Axis X: orientation ∈ {[-1,+1], [+1,-1]}    — qubit 3
Frame 1, Axis Y: orientation ∈ {[-1,+1], [+1,-1]}    — qubit 4
Frame 1, Axis Z: orientation ∈ {[-1,+1], [+1,-1]}    — qubit 5

The pair-diagonal property observed in bit space (each 2-bit pair is always 00 or 11, never 01 or 10) is a consequence of this structure: both bits of a pair encode the same ±1 axis state.

The six axis-orientation qubits correspond to the six generators of the se(3) Lie algebra: three rotational (Frame 0, from SU(2)) and three translational (Frame 1, from ℝ³). The CGM paper derives that operational coherence requires a progression from 1 degree of freedom (chirality, CS stage) to 3 (rotational, UNA stage) to 6 (rotational + translational, ONA stage), yielding the semidirect product SE(3) = SU(2) ⋉ ℝ³. Each payload bit of a byte executes a discrete π-rotation around one of these se(3) basis vectors. The Degrees of Freedom Doubling Law (established in the Physics Tests Report, Part 10) confirms this progression with exact integer state counts: 2^(2×1) = 4 at CS, 2^(2×3) = 64 at UNA, 2^(2×6) = 4096 at ONA.

1.3 The Two-Component Topology

The full GENE_Mac state contains two 12-bit components, A (active phase) and B (passive phase). The aQPU tests establish that these are not independent registers carrying separate information.

test_aQPU_4.py::TestSixQubitRegister verifies:

  • All 4096 states in Ω have valid ±1 spin representations with no anomalous 00 or 11 pairs (4096/4096 valid, 0 invalid).
  • The spin-to-bits and bits-to-spin roundtrip is exact for all 4096 Ω states.
  • The distinct A values form a 64-element coset of the mask code C64, and the distinct B values form a 64-element coset of the same code. |A values| = |B values| = |C64| = 64.
  • At rest: spin_A = (+1, +1, +1, +1, +1, +1), spin_B = (−1, −1, −1, −1, −1, −1). A and B are exact complements.
  • In Ω: 64 states have A = B (equality horizon), 64 states have A = B ⊕ 0xFFF (complement horizon), and 3968 states have partial chirality (bulk).

The relationship between A and B is mediated by the four intrinsic gates. The gate structure determines how B relates to A at any given state, rather than B carrying independent operational content. The full state (A, B) encodes six qubits of axis-orientation content plus the gate-phase relationship between the two components.

This corresponds to the byte structure: 6 payload bits carry the operational content (64 transformations), while 2 boundary bits select the family phase (4 complement configurations). The structural correspondence is:

Scale Operational content Gauge phase
Byte (8-bit, GENE_Mic) 6 payload bits → 64 transformations 2 boundary bits → 4 families
State (24-bit, GENE_Mac) 6 qubit pairs → 64 A-values × 64 B-values K4 gate relation → (ℤ/2)²

Part 2: The Reachable State Space and Dual Horizons

2.1 Ω: 4096 States at Radius 2

The reachable state space Ω, established by BFS in the Physics Tests Report (Part 5) and confirmed in the aQPU suite, contains exactly 4096 states. BFS from rest produces:

Depth 0:    1 state   (rest alone)
Depth 1:  127 new states  (total 128)
Depth 2: 3968 new states  (total 4096)
Depth 3:    0 new states  (saturated)

Every state in Ω is reachable within two byte steps from rest. Ω has product form Ω = U × V with |U| = |V| = 64, where U and V are cosets of C64 (established in the Physics Tests Report, Part 5.4; confirmed explicitly by test_aQPU_4.py::TestSixQubitRegister::test_six_qubit_state_space_from_a_alone).

Every state in Ω has component density exactly 0.5 (popcount 6 out of 12 bits per component). The density product d_A × d_B = 0.25 is constant across all 4096 reachable states (Physics Tests Report, Part 5.5).

2.2 The Complement Horizon (S-Sector)

test_aQPU_1.py::TestDualHorizons verifies:

The complement horizon is the set of states where A12 = B12 ⊕ 0xFFF (maximal chirality). It contains exactly 64 states. All complement horizon states have ab_distance = 12 (every dipole pair is anti-aligned). The rest state GENE_MAC_REST lies on this horizon.

In the CGM interpretation (specification Appendix G.4), the complement horizon is where the archetype's structural effect is maximally expressed: the alternating bit pattern of GENE_MIC_S = 0xAA is projected across both components in opposite phase.

2.3 The Equality Horizon (UNA Degeneracy)

test_aQPU_1.py::TestDualHorizons verifies:

The equality horizon is the set of states where A12 = B12 (zero chirality). It contains exactly 64 states. All equality horizon states have ab_distance = 0 (every dipole pair is aligned). The rest state does not lie on this horizon.

In the CGM interpretation (specification Appendix G.4), the equality horizon is where the archetype's structural effect is maximally concealed: A and B are identical, and the underlying chirality has vanished from the observable state.

2.4 Disjointness and Holography

The two horizons are disjoint: no state satisfies both A = B and A = B ⊕ 0xFFF simultaneously. Their union forms a 128-state boundary. The remaining 3968 states constitute the bulk, where chirality is partial (test_aQPU_1.py::TestDualHorizons::test_boundary_128_bulk_3968).

Both horizons satisfy the holographic identity (established in the Physics Tests Report, confirmed by the 4-to-1 holographic dictionary):

|H|² = |Ω|      →      64² = 4096

2.5 The Complementarity Invariant

For all states universally, not just Ω:

horizon_distance(A, B) + ab_distance(A, B) = 12

test_aQPU_1.py::TestComplementarityInvariant confirms this exhaustively on all 4096 Ω states and on 50,000 random 24-bit states. This is the discrete pole-to-pole conservation: distance to the complement horizon plus distance to the equality horizon equals the diameter.

2.6 The Chirality Spectrum

test_aQPU_1.py::TestBlochSphereLatitude verifies:

The ab_distance observable takes values {0, 2, 4, 6, 8, 10, 12} on Ω. The count at each distance d is:

count(d) = C(6, (12−d)/2) × 64

This is the spectrum of 6 independent chirality qubits, each contributing 0 (pair aligned) or 2 (pair anti-aligned) to the total distance. The distribution is binomial and symmetric: count(d) = count(12−d). The two poles (d = 0 and d = 12) are the two horizons. The equator (d = 6) has maximum population: C(6,3) × 64 = 1280 states.

Within Ω, A ⊕ B is always pair-diagonal. This collapses to a 6-bit chirality word χ(s) with one bit per dipole pair. All 64 chirality values appear in exactly 64 Ω states each (uniform partition), so chirality captures exactly 6 of the 12 bits of state information (test_aQPU_3.py::TestTrajectoryCompression::test_chirality_partition).

Part 3: The K4 Gate Group

3.1 Gate Identification

Exactly 4 bytes out of 256 preserve both horizons simultaneously. test_aQPU_1.py::TestGateDefinitions::test_gate_constants_match_horizon_condition confirms these are the bytes satisfying mask12(byte) = inv_a(byte) ⊕ inv_b(byte), and that they match the declared constants:

Byte Gate Intron Family Action on (A, B)
0xAA S 0x00 00 (A, B) → (B, A)
0x54 S 0xFE 10 (A, B) → (B, A)
0xD5 C 0x7F 01 (A, B) → (B ⊕ F, A ⊕ F)
0x2B C 0x81 11 (A, B) → (B ⊕ F, A ⊕ F)

where F = 0xFFF. Each gate pair realises the same 24-bit operation but differs in spinorial phase (different intron and family values). The two bytes in each pair differ by XOR 0xFE (test_aQPU_1.py::TestGateDefinitions::test_shadow_pairing).

The S-gate action is verified as pure swap on 2000 random states. The C-gate action is verified as complement-swap on 2000 random states (test_aQPU_1.py::TestGateDefinitions).

3.2 Gate Actions in Spin Coordinates

test_aQPU_4.py::TestIntrinsicGatesOnSpins::test_gate_actions_in_spin_coordinates verifies on 500 random Ω states:

id:  (sA, sB) → (sA, sB)
S:   (sA, sB) → (sB, sA)
C:   (sA, sB) → (−sB, −sA)
F:   (sA, sB) → (−sA, −sB)

All four gates preserve chirality exactly: ab_distance is invariant under every gate for all 4096 Ω states (test_aQPU_4.py::TestIntrinsicGatesOnSpins::test_gate_action_on_chirality).

3.3 The Klein Four-Group

The four gates form K4 = (ℤ/2)². test_aQPU_1.py::TestK4GateGroup verifies:

  • All three non-trivial gates are involutions: S² = C² = F² = id (1000 random states each).
  • S ∘ C = C ∘ S = F (1000 random states).
  • Full Cayley table verified on a fixed state.
  • F requires depth 2 in the kernel: one S-byte followed by one C-byte, or vice versa (1000 random states, both orderings confirmed).
  • No single byte implements F on all states (verified against 4 probe states covering rest, zero, and bulk regions).
  • No single byte is the identity on all states (same 4 probe states).

The K4 group emerges at three independent levels of the architecture: as the depth-4 fiber (for fixed payload, 4⁴ family combinations collapse to 4 distinct states indexed by (φ_A, φ_B) ∈ (ℤ/2)²; established in the Physics Tests Report, Part 11), as the intrinsic gate group (this section), and as the governance measurement topology (specification §4.6). This triple coincidence is a structural consequence of depth-4 closure.

3.4 Gate Action on the Horizons

test_aQPU_1.py::TestGateActionOnHorizons verifies the complete census:

Gate Complement horizon Equality horizon
id fixes all 64 pointwise fixes all 64 pointwise
S 32 two-cycles, 0 fixed fixes all 64 pointwise
C fixes all 64 pointwise 32 two-cycles, 0 fixed
F 32 two-cycles, 0 fixed 32 two-cycles, 0 fixed

All four gates map each horizon to itself as a set.

The pointwise stabiliser bytes are confirmed exhaustively (test_aQPU_1.py::TestHorizonPointwiseStabilizers):

  • The only bytes that fix every complement horizon state are the two C-gate bytes {0xD5, 0x2B}.
  • The only bytes that fix every equality horizon state are the two S-gate bytes {0xAA, 0x54}.
  • No other byte fixes an entire horizon pointwise.

The CGM interpretation (specification Appendix G.3.5): Gate C stabilises the S-sector (opposition preserves the common source). Gate S stabilises the equality horizon (non-commutativity is invisible at its own boundary). Gate F stabilises neither pointwise (balance through dynamics, not stasis).

3.5 K4 Orbit Stratification

test_aQPU_1.py::TestK4OrbitStratification verifies the complete orbit decomposition:

  • 32 orbits of size 2 on the complement horizon (each paired by S)
  • 32 orbits of size 2 on the equality horizon (each paired by C)
  • 992 orbits of size 4 in the bulk (all four gates produce distinct states)
  • Total: 1056 orbits covering all 4096 states

test_aQPU_1.py::TestBulkTrivialK4Stabilizer confirms: no non-trivial gate fixes any bulk state. The horizons are the only structurally distinguished subsets under the gate group.

3.6 Gate-Byte Phase Separation

test_aQPU_1.py::TestGatePhaseSeparation verifies: within each gate pair (S-bytes {0xAA, 0x54} and C-bytes {0xD5, 0x2B}), the two bytes produce the same 24-bit operation on 1000 random states, but have different introns and different families. The 24-bit state is the spatial (SO(3)) shadow; the intron carries the spinorial (SU(2)) phase.

Part 4: The Chirality Transport Law and Commutativity

4.1 The 6-Bit Chirality Register

test_aQPU_1.py::TestChiralityRegisterTransport verifies:

The chirality register χ(s) covers all 64 elements of GF(2)⁶ across Ω. Under the byte transition, it satisfies an exact linear transport law:

χ(T_b(s)) = χ(s) ⊕ q6(b)

This is confirmed for all 256 bytes across all 4096 Ω states without exception. The chirality register is a linear observable over GF(2)⁶ on which the byte algebra acts by translation.

test_aQPU_2.py::TestPauliGroupOnChiralityRegister confirms that this transport law is the classical shadow of a Pauli-X action on the chirality register:

  • The 64 distinct q6 values are exactly {0, …, 63}.
  • The XOR closure q6(b1) ⊕ q6(b2) is always a valid q6 value.
  • XOR is commutative (abelian translation group).
  • Each byte maps each chirality value to exactly one output (deterministic action, verified on all Ω).

4.2 The 1/64 Commutativity Rate

test_aQPU_1.py::TestCommutativityRate verifies exhaustively over all 256² = 65536 ordered byte pairs:

  • Exactly 1024 commuting pairs (1024/65536 = 1/64 = 2^(−6)).
  • Every byte commutes with exactly 4 of the 256 bytes.

The exponent 6 equals the number of independent degrees of freedom. Two bytes commute if and only if q(x) = q(y), where q(b) = mask12(b) ⊕ (0xFFF if L0_parity(b) is odd else 0). The q-map is 4-to-1 from the 256-byte alphabet onto C64 (established in the Physics Tests Report, Part 11.7; confirmed by test_aQPU_4.py::TestByteAlgebraComputationalPower::test_hidden_subgroup_via_q_map).

Part 5: Permutation Structure and the Row-Class Theorem

5.1 Permutation Census

test_aQPU_3.py::TestPermutationStructure verifies:

  • 256 bytes produce exactly 128 distinct permutations on the 4096-state Ω.
  • Multiplicity is uniform: every permutation is realised by exactly 2 bytes.
  • Two distinct cycle types:
    • 2 permutations of type [1×64, 2×2016] (order 2): these are the S-gate permutations.
    • 126 permutations of type [4×1024] (order 4): all other bytes.
  • Permutation order spectrum: 2 permutations of order 2, 126 of order 4.

5.2 The Exact Row-Class Theorem

test_aQPU_3.py::TestExactRowClassStructure verifies:

The uniform transition matrix on Ω (all 256 byte transitions summed, normalised by 256) has exactly 32 distinct rows and matrix rank 32. The explanation derives from the product structure Ω = U × V:

In code coordinates (x, y) ∈ GF(2)⁶ × GF(2)⁶, a byte maps:

x' = y ⊕ p_a          (p_a ∈ {0, ε}, ε = 111111)
y' = x ⊕ μ ⊕ p_b      (μ uniform over GF(2)⁶, p_b ∈ {0, ε})

The row depends only on {y, y ⊕ ε}, giving |GF(2)⁶| / |⟨ε⟩| = 64/2 = 32 distinct row types. Every row class is confirmed to be determined by exactly one {y, y ⊕ ε} pair.

test_aQPU_3.py::TestExactRowClassStructure::test_family_restricted_row_classes confirms: when restricted to family-0 bytes (64 bytes with p_a = p_b = 0), the ε-quotient disappears and the matrix has 64 distinct rows and rank 64.

The row-class lattice under controlled byte subsets (test_aQPU_3.py::TestExactRowClassStructure::test_row_class_lattice_under_byte_subsets) reveals:

Subset Bytes Distinct rows
Full 256 256 32
bit7=0 128 32
bit7=1 128 32
bit0=0 128 64
bit0=1 128 64
Each single family 64 64

Boundary bit 7 produces the ε-quotient collapse from 64 to 32. Bit 0 does not. This asymmetry reflects the chirality of the transition law: bit 7 controls the complement applied to the B component output (invert_b), while bit 0 controls the complement on A_next (invert_a). The ε-quotient arises from bit 7 because it determines whether B_next receives a global complement, collapsing two distinct code coordinates into one equivalence class.

Part 6: Channel Capacity and Exact Uniformisation

6.1 Per-Byte Capacity

test_aQPU_3.py::TestChannelCapacity::test_per_byte_capacity verifies on 500 sampled states:

From any state in Ω, all 256 bytes produce exactly 128 distinct next states with exactly uniform 2-to-1 multiplicity. Shannon entropy and min-entropy are both exactly 7.0 bits, with zero variance across all sampled states. The channel operates at maximum capacity for 128 outputs.

6.2 Exact Two-Step Uniformisation

test_aQPU_3.py::TestExactTwoStepUniformization verifies by exhaustive enumeration of all 256² = 65536 length-2 words from rest:

Each of the 4096 Ω states is reached by exactly 16 words. The distribution is not approximately uniform. It is exactly uniform, confirmed by integer equality on all 4096 output counts.

In a structural comparison: a generic classical random walk on a 4096-state graph (e.g. symmetric random walk on the Cayley graph) requires O(log 4096) ≈ 12 steps to mix. The kernel achieves exact uniformisation in 2 steps. Because the transition matrix is doubly stochastic, uniformity is preserved at all lengths ≥ 2.

6.3 Depth-4 Output Distribution

test_aQPU_3.py::TestChannelCapacity::test_depth4_output_distribution verifies with 200,000 random 4-byte words from rest: all 4096 states reached, Shannon entropy 11.9855 bits (sample-limited; the true value is exactly 12.0 bits, as implied by the exact two-step uniformisation result).

6.4 Exact Integer Information Measures

test_aQPU_3.py::TestFingerprintDiscrimination::test_conditional_entropies_length2_exact computes exact conditional entropies for length-2 trajectories (exhaustive over all 256² words):

H(state)            = 12.000000 bits   (log₂(4096))
H(state, parity)    = 13.000000 bits
H(parity | state)   =  1.000000 bit    (log₂(2))
H(state | parity)   =  7.000000 bits   (log₂(128))
Distinct (state, parity) pairs: 8192 = 4096 × 2

These are exact integers. The parity commitment adds exactly 1 bit of information beyond the final state, uniformly across all 4096 states.

test_aQPU_3.py::TestFingerprintDiscrimination::test_parity_vs_chirality_mutual_information confirms that chirality and parity are effectively independent: mutual information I(χ; parity) = 0.014 bits ≈ 0 (measured on 200,000 random trajectories). They carry distinct aspects of trajectory information.

6.5 Trajectory Compression

test_aQPU_3.py::TestTrajectoryCompression::test_provenance_redundancy_by_length:

Length Input bits Distinct outputs Output bits Bits/byte
1 8 128 7.00 7.00
2 16 4096 12.00 6.00
3 24 4096 12.00 4.00
4 32 4096 12.00 3.00

Beyond length 2, additional bytes contribute zero new state information. They do contribute to frame records and parity commitments, which carry trajectory history information beyond the final state.

Part 7: Error Detection and Tamper Provenance

7.1 The Exact Error Enumerator

test_aQPU_3.py::TestExactErrorEnumerator verifies:

Because Ω has product form U × V with both factors being C64 cosets, errors that keep a state within Ω are exactly characterised. The undetected weight enumerator is (1 + z²)^12: the fraction of weight-w errors that are undetected is C(12, w/2) / C(24, w) if w is even, and 0 if w is odd.

Weight Undetected count C(12, w/2) Total patterns C(24, w) Theo. undetected % Observed %
1 0 24 0.000 0.000
2 12 276 4.348 4.348
3 0 2024 0.000 0.000
4 66 10626 0.621 0.667

All odd-weight errors are detected. The minimum undetected error has weight 2 (single pair flip). Observed values match theoretical values within the standard error of the sampled estimate (512 states sampled).

test_aQPU_3.py::TestExactErrorEnumerator::test_pair_flip_destinations_are_c64 confirms: pair-flip errors in either A or B always stay in Ω, and the resulting displacement is always a C64 codeword.

7.2 The Exact Perturbation Law

test_aQPU_3.py::TestExactPerturbationLaw::test_exact_per_bit_decomposition verifies exhaustively (all 256 bytes, all 8 bit positions):

Flipping one bit of a byte changes the chirality transport q6 by:

  • Payload bits (positions 1-6): exactly 1 chirality bit.
  • Boundary bits (positions 0 and 7): exactly 6 chirality bits.

These values are exact (verified to 10^(−10) precision). The overall mean chirality divergence per bit flip is:

(6 × 1 + 2 × 6) / 8 = 18/8 = 2.25 chirality bits

The corresponding state distance is exactly 2 × chirality distance (pair-diagonal mapping). The ratio state_distance / chirality_distance = 2.000 is confirmed constant over trajectory lengths 1 through 32 with zero collision rate (test_aQPU_3.py::TestExactPerturbationLaw::test_spreading_is_length_independent).

Boundary bits contribute 6× more disturbance than payload bits because flipping a boundary bit toggles the ε-complement on one component, affecting all 6 dipole pairs simultaneously.

7.3 Exact Tamper Detection Mechanisms

test_aQPU_3.py::TestTamperDetection verifies three tamper categories. In each case, every miss has an exact algebraic explanation.

Substitution (test_substitution_mechanism): Missed if and only if the replacement is the shadow partner (the other byte implementing the same 24-bit permutation). Every byte has exactly one shadow partner (differing by XOR 0xFE). Theoretical miss rate: 1/255 ≈ 0.39%. Observed: 222 misses in 50,000 trials, all 222 attributable to shadow partners.

Adjacent swap (test_adjacent_swap_mechanism): Missed if and only if the two swapped bytes belong to the same commutation class: q(x) = q(y). Theoretical miss rate for distinct bytes: ~3/255 ≈ 1.18%. Observed: 614 misses in 49,773 trials (distinct adjacent pairs), all 614 attributable to q-class equality.

Deletion (test_deletion_mechanism): Missed if and only if the deleted byte is a pointwise stabiliser of the prefix state. This occurs only when:

  • The prefix state is on the equality horizon and the deleted byte is an S-gate byte (0xAA or 0x54).
  • The prefix state is on the complement horizon and the deleted byte is a C-gate byte (0xD5 or 0x2B).
  • The prefix state is in the bulk: never.

Observed: 51 misses in 50,000 trials, all 51 explained by the gate stabiliser condition. Decomposition: 7 S-byte deletions (prefix on equality horizon), 44 C-byte deletions (prefix on complement horizon), 0 from bulk.

7.4 Adversarial Steering

test_aQPU_3.py::TestAdversarialSteering::test_steering_and_exact_multiplicity verifies by exhaustive enumeration of all 256² length-2 words from rest:

  • Depth-2 covers all of Ω (4096 states).
  • Byte-path multiplicity is exactly uniform: every target receives exactly 16 byte-paths.
  • State-path multiplicity is exactly uniform: every target has exactly 4 distinct intermediate states.

test_aQPU_3.py::TestAdversarialSteering::test_horizon_maintenance verifies: from any complement horizon state, exactly 4 of 256 bytes keep the state on the horizon (fraction 4/256 = 1/64, uniform across all 64 horizon states).

Part 8: The Hilbert Lift

8.1 Graph State Factorisation

test_aQPU_2.py::TestGraphStateFactorization verifies:

The pair-diagonal collapse of C64 codewords from 12 bits to 6 bits is a bijection onto GF(2)⁶ (64 codewords map to 64 distinct 6-bit words).

The graph state constructed from the Hilbert lift of C64,

|ψ_t⟩ = (1/√64) Σ_q |q⟩|q ⊕ t⟩

factorises exactly into a tensor product of 6 independent Bell pairs. The k-th Bell pair is |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 if bit k of t is 0, or |Ψ⁺⟩ = (|01⟩ + |10⟩)/√2 if bit k of t is 1.

Verified for t ∈ {0, 0b101010, 0b111111, 0b010101} to precision 10^(−12). Additionally:

  • Each pair marginal is the expected Bell projector.
  • Cross-pair marginals factorise: ρ_{k,l} = ρ_k ⊗ ρ_l (all 15 pair combinations for t = 0b101010).
  • Each pair marginal is pure rank-1 (single non-zero eigenvalue at 1.0).

8.2 Bell Inequality Violation

test_aQPU_2.py::TestBellCHSH verifies:

Both Bell pair types saturate the CHSH inequality at the Tsirelson bound:

CHSH(|Φ⁺⟩) = 2√2 = 2.828427...   (to precision 10^(−12))
CHSH(|Ψ⁺⟩) = 2√2 = 2.828427...   (to precision 10^(−12))

This exceeds the classical bound of 2 and equals the quantum-mechanical maximum. All 6 pair factors of the graph state with t = 0b101010 individually saturate the Tsirelson bound.

test_aQPU_2.py::TestBellCHSH::test_no_measurements_exceed_tsirelson confirms by exhaustive search over a discrete measurement angle grid (10 angles per setting, 10⁴ total combinations) that no CHSH value exceeds 2√2.

The saturation of the Tsirelson bound rules out local hidden-variable models for the pairwise correlations and confirms that the code structure demands maximal quantum correlations.

8.3 Quantum Teleportation

test_aQPU_2.py::TestTeleportationProtocol verifies:

For each of the 8 combinations (2 Bell resource types × 4 measurement outcomes), a unique Pauli correction {I, X, Z, XZ} exists that recovers the input state. The correction is identified from a generic probe state and then confirmed to work for:

  • 6 standard basis and phase states (|0⟩, |1⟩, |+⟩, |−⟩, |+i⟩, |−i⟩)
  • 100 random Bloch sphere states per (resource, outcome) combination

All overlaps equal 1.0 to precision 10^(−10).

8.4 Monogamy and No-Signalling

test_aQPU_2.py::TestMonogamyAndNoSignalling verifies:

  • Same-pair marginals are pure Bell states (purity = 1.0 to precision 10^(−12)).
  • Cross-pair marginals (A_k with B_l, k ≠ l) are maximally mixed: ρ = I₄/4 (precision 10^(−12)). Each Alice qubit is maximally entangled with exactly one Bob qubit.
  • All 12 single-qubit marginals are maximally mixed: ρ = I₂/2.
  • No-signalling: measuring Bob's qubit in the Z basis or the X basis and summing over outcomes yields the same reduced state for Alice (I₂/2 in both cases, precision 10^(−12)).

8.5 Stabiliser Structure

test_aQPU_2.py::TestGraphStateStabilizer verifies:

The graph state has 12 generators, two per pair:

  • X-type: X_{A_k} X_{B_k} (phase +1)
  • Z-type: (±1) Z_{A_k} Z_{B_k} (sign determined by t_bit for pair k)

All 12 generators stabilise the graph state (precision 10^(−12)). All pairs of generators commute (binary symplectic product is zero for all 66 pairs). The GF(2) rank of the combined generator matrix is exactly 12.

The X-type translation subgroup has GF(2) rank 6 and 64 elements, matching C64. All 64 X-translations (formed by XOR combinations of the 6 X-generators) stabilise the graph state (verified on 256 random combinations, precision 10^(−12)).

8.6 Contextuality

test_aQPU_2.py::TestPeresMerminContextuality verifies:

The Peres-Mermin square of nine 2-qubit observables:

Row 0:  XI,  IX,  XX
Row 1:  IZ,  ZI,  ZZ
Row 2:  XZ,  ZX,  YY

satisfies: each row product = +I₄ (precision 10^(−12)), columns 0 and 1 products = +I₄, column 2 product = −I₄.

Any noncontextual hidden-variable assignment that assigns definite ±1 values consistent with the product constraints produces a contradiction: the product of all row assignments is (+1)³ = +1, while the product of all column assignments is (+1)²(−1) = −1. Since both compute the same product of nine values, such an assignment is impossible.

8.7 Mutually Unbiased Bases

test_aQPU_2.py::TestMutuallyUnbiasedBases verifies:

The 64 × 64 Walsh-Hadamard matrix on the chirality register is unitary (H H^T = I₆₄ to precision 10^(−12)). It factors as the tensor sixth power of the single-qubit Hadamard H₁ = [[1,1],[1,−1]]/√2.

The computational basis and the Walsh-Hadamard basis are mutually unbiased: all |⟨e_i | h_j⟩|² = 1/64 (precision 10^(−12)).

A third MUB is constructed via a phase gate composed with the Hadamard. The overlaps between this third basis and the computational basis all equal 1/64, confirming at least 3 mutually unbiased bases exist for the 64-dimensional chirality register.

Part 9: The Operator Family and Entangling Structure

9.1 Operator Family Size

The Moments Tests Report (Part VI) establishes that the byte alphabet generates exactly 8192 operators: 4096 even-parity (identity linear part) and 4096 odd-parity (swap linear part), forming a semidirect product. The central spinorial involution (4-family cycle = global complement) commutes with all bytes.

test_aQPU_4.py::TestOperatorFamily::test_operator_family_size confirms from the aQPU perspective:

  • Length-1 words (single bytes) produce 128 distinct odd-parity operators (the 256 bytes collapse via the 2-to-1 shadow).
  • Length-2 words produce 4096 distinct even-parity translations.
  • Total distinct operators reachable at depth ≤ 2: 128 + 4096 = 4224.

The remaining 8192 − 4224 = 3968 operators require depth ≥ 3 to reach (the full 8192-element family is established by the Moments Tests Report).

9.2 Even Operators as Translations

test_aQPU_4.py::TestOperatorFamily::test_even_operators_as_6qubit_unitaries verifies:

The 4096 even operators have identity linear part: (A, B) → (A ⊕ τ_A, B ⊕ τ_B). The set of (τ_A, τ_B) pairs is exactly C64 × C64 (the full Cartesian product of the two 64-element code cosets). This means A and B can be independently translated within their respective cosets by length-2 byte words.

9.3 Single-Byte Operations are Product

test_aQPU_4.py::TestOperatorFamily::test_single_byte_intra_register_coupling verifies:

For each of the 256 bytes, the test perturbs one input spin of A at a time (keeping B at rest) and counts how many output spins change. Result: 0 of 256 bytes show multi-qubit coupling within a component. Each byte acts as independent single-qubit operations on each of the 6 axis-orientation pairs.

test_aQPU_4.py::TestOperatorFamily::test_depth2_intra_register_coupling extends this to 2000 random 2-byte words with the same result: 0 entangling words detected. Intra-component coupling is absent at depth ≤ 2.

The entangling power of the byte algebra operates between the A and B manifolds through the gate structure. Gate S swaps A and B, propagating any perturbation in A into B. Gate C additionally applies a global sign flip. test_aQPU_4.py::TestNativeUniversalComputation::test_topological_entanglement_via_intrinsic_gates confirms: a localised perturbation in A (flipping pair 0, mask 0x003) is transported exactly to B by gate S, producing the same pair flip in the B output. A and B cannot be treated as independent classical registers; the gate operations make them topologically correlated.

Part 10: The Non-Clifford Resource δ_BU

10.1 The CGM Monodromy Defect

The Common Governance Model derives a representation-independent constant from the depth-4 closure condition: the BU monodromy defect δ_BU = 0.195342176580 radians. This is the residual geometric phase of the dual-pole loop in the BU stage of the CGM framework.

The ratio δ_BU / m_a = 0.9793, where m_a = 1/(2√(2π)) is the CGM aperture scale, yields the canonical aperture:

Δ = 1 − δ_BU/m_a = 0.020699553913   (≈ 2.07%)

The same constant, raised to the fourth power and normalised by m_a, yields the fine-structure constant α = δ_BU⁴/m_a = 0.007297352563 (matching experiment to nine significant digits; confirmed in the Physics Tests Report, Part 9.2).

10.2 Distance from Clifford Angles

test_aQPU_4.py::TestNonCliffordAndUniversality::test_delta_bu_not_clifford_angle verifies:

δ_BU is far from all Clifford angles (multiples of π/4):

k k × π/4 Distance to δ_BU
0 0.000000 0.195342
1 0.785398 0.590056
2 1.570796 1.375454
3 2.356194 2.160852
4 3.141593 2.946250

Nearest Clifford angle: k = 0, distance 0.195342 rad. This exceeds 0.01 by more than an order of magnitude.

Clifford operations (Hadamard, phase, CNOT) form a finite group and are efficiently classically simulable by the Gottesman-Knill theorem. Universal quantum computation requires at least one non-Clifford resource. The distance of δ_BU from all Clifford angles confirms it provides this resource.

10.3 Dense U(1) Orbit

test_aQPU_4.py::TestNonCliffordAndUniversality::test_high_order_non_periodicity verifies:

The rotation R(δ_BU) does not return to identity for any k ≤ 100,000. The closest return is at k = 22,805, distance 4.59 × 10^(−5) rad (non-negligible: the ratio to 2π/100000 is 0.73). This confirms the absence of low-order periodicity.

test_aQPU_4.py::TestNonCliffordAndUniversality::test_dense_phase_equidistribution verifies:

The phase sequence {k × δ_BU mod 2π : k = 1, …, 50000} equidistributes over [0, 2π). Binned into 100 equal intervals: range [498, 502] against an expected 500, chi-squared = 0.212 against a critical value of 142.4. This confirms that R(δ_BU) generates a dense subgroup of U(1), which is a necessary condition for the Solovay-Kitaev approximation theorem.

10.4 Magic State and Negative Wigner Function

test_aQPU_4.py::TestNonCliffordAndUniversality::test_magic_state_wigner_negativity verifies:

The state |δ⟩ = (|0⟩ + e^(i × δ_BU)|1⟩) / √2 has discrete Wigner function:

W(0,0) = +0.543771
W(0,1) = −0.043771
W(1,0) = +0.446720
W(1,1) = +0.053280
Sum    =  1.000000

W(0,1) is negative. Negative Wigner function values certify a state as a non-stabiliser ("magic") state. This is a necessary condition for computational advantage beyond Clifford circuits.

10.5 The Aperture and the Non-Clifford Resource

test_aQPU_4.py::TestNonCliffordAndUniversality::test_aperture_as_universality_window confirms:

|δ_BU − m_a| = 0.004128963621
Δ × m_a     = 0.004128963621   (exact equality)

The aperture gap Δ measures the normalised distance between δ_BU and m_a. If Δ were zero, the monodromy defect and aperture scale would coincide. The non-zero aperture guarantees that δ_BU generates a dense U(1) subgroup distinct from any structure associated with the aperture scale alone.

The byte-scale quantisation of the aperture is 5/256 ≈ 0.01953, and the depth-4 quantisation is 1/48 ≈ 0.02083 (confirmed in the Physics Tests Report, Part 9.5).

Part 11: Proven Computational Advantages

11.1 Oracle Query Advantages

test_aQPU_4.py::TestByteAlgebraComputationalPower establishes three oracle-model advantages on the native 6-bit chirality register.

Deutsch-Jozsa (test_deutsch_jozsa_on_chirality):

The Walsh-Hadamard transform on the chirality register distinguishes constant from balanced functions in a single transform step:

Constant f: Pr(output = 0) = 1.000000
Balanced f: Pr(output = 0) = 0.000000

In this query-model comparison: classical worst case requires 2^5 + 1 = 33 oracle queries; a single Walsh transform step achieves perfect discrimination.

Bernstein-Vazirani (test_bernstein_vazirani_on_chirality):

The secret 6-bit string s in the oracle f(x) = s · x mod 2 is recovered in a single Walsh transform with probability 1.000:

Secret = 101010: recovered = 101010, prob = 1.000000
Secret = 110011: recovered = 110011, prob = 1.000000
Secret = 000001: recovered = 000001, prob = 1.000000
Secret = 111111: recovered = 111111, prob = 1.000000

In this query-model comparison: classical identification requires 6 queries; one Walsh step recovers the secret with probability 1.

Hidden subgroup resolution (test_hidden_subgroup_via_q_map):

The q-map q6: {0, …, 255} → GF(2)⁶ is a native hidden subgroup structure:

Total search space: 256 byte operations
Topological classes: 64 (= GF(2)⁶)
Kernel q⁻¹(0): {0x2B, 0x54, 0xAA, 0xD5} (the 4 gate bytes)
Fiber sizes: {4: 64} (exactly 4 bytes per class, uniform)

The Walsh transform of the kernel indicator function gives uniform non-zero amplitude on all 64 outputs, confirming immediate subgroup identification. In this structural (hidden-subgroup) comparison: classical worst-case identification in a group of size 256 with quotient of size 64 requires O(64) queries; the native q-map plus one Walsh step resolves the subgroup in O(1).

11.2 Commutativity Decision

test_aQPU_4.py::TestByteAlgebraComputationalPower::test_commutativity_as_computational_resource:

The q-map provides an O(1) algorithm for deciding whether two bytes commute: compute q6(x) and q6(y), compare. Classical testing requires applying both orderings to a test state (4 kernel steps). Confirmed correct on all 5000 tested pairs (5000/5000).

11.3 Native Period Structure

test_aQPU_4.py::TestNativeQuantumAdvantage::test_factorization_period_finding_isomorphism:

Every byte has topological period 4 on Ω: T_b^4 = id for all b. This is universal across the entire alphabet. The period structure is the algebraic backbone required by period-finding algorithms. The aQPU does not need to search for the period; it is built into the transition law.

11.4 Structural Mixing Advantage

test_aQPU_4.py::TestStructuralAdvantage::test_two_step_uniformization_advantage:

Exact uniformisation in 2 steps versus O(log 4096) ≈ 12 steps for a generic classical random-walk baseline on a 4096-state graph. The distribution after 2 steps is exactly uniform (every count = 16, verified by integer equality), not approximately uniform.

11.5 Holographic Compression

test_aQPU_4.py::TestStructuralAdvantage::test_holographic_compression_advantage:

The holographic identity |H|² = |Ω| provides structural compression:

Standard encoding:     log₂(4096) = 12 bits per state
Holographic encoding:  log₂(64) + log₂(4) = 6 + 2 = 8 bits per state
Savings:               4 bits (33.3% reduction)

The holographic dictionary covers all of Ω with exactly uniform multiplicity 4.

11.6 State Separation

test_aQPU_3.py::TestStateSeparation::test_universal_separation:

Every byte distinguishes every distinct state pair, verified on 1000 sampled pairs with all 256 bytes. This follows from per-byte bijectivity on the full 2^24 carrier.

test_aQPU_3.py::TestStateSeparation::test_byte_preserves_hamming_distance confirms: Hamming distance between states is preserved under every byte operation (500 random (s, t, b) triples).

The exact pairwise distance distribution on Ω follows from the product structure and the weight enumerator of C64 × C64. The distribution at Hamming distance 2k is C(12, k) / 4096, with mean distance exactly 12.0 and total fraction exactly 1.0 (test_aQPU_3.py::TestStateSeparation::test_exact_omega_pairwise_distance_distribution).

Part 12: Verified Ingredients for Algebraic Universality

12.1 The Three Required Components

The standard universality theorem for quantum computation requires three ingredients: a Clifford backbone, a non-Clifford resource, and entangling operations. The aQPU test suite, together with the previously established physics and moments results, confirms all three.

Clifford backbone. The Moments Tests Report (Part VI) establishes that every byte action is an exact Clifford unitary in the label space over GF(2)^12. Clifford conjugation properties are verified exhaustively for all 256 bytes on all 4096 labels. The self-dual [12,6,2] code has 6 pair generators spanning the code (GF(2) rank 6), and 12 Pauli stabiliser generators (X-type and Z-type) that all commute (symplectic product zero for all pairs, GF(2) rank 12).

Non-Clifford resource. δ_BU provides it: distance 0.195342 from the nearest Clifford angle (Part 10.2), no periodicity up to order 100,000 (Part 10.3), dense U(1) equidistribution (Part 10.3), and Wigner-negative magic state (Part 10.4).

Entangling operations. The four intrinsic gates provide the inter-component coupling. This is distinct from the Hilbert-space bipartite (Bell-pair) entanglement already established in Part 8: here the verified structure is native topological inter-component coupling at the gate level. Within that gate structure, gate S is explicitly demonstrated by perturbation transport (Part 9.3): a localised perturbation in A is transported exactly to B by S. Gate C is the complement-swap partner in the same K4 structure (same horizon-preserving set, spin action (sA, sB) → (−sB, −sA)); it is not separately demonstrated by perturbation transport in the test body, but shares the same gate algebra and provides the other non-trivial inter-component operation. The gates are confirmed in spin coordinates (Part 3.2) and as the sole inter-component coupling mechanism (Part 9.3).

12.2 Dense Operator Generation

test_aQPU_4.py::TestNativeUniversalComputation::test_computational_universality_via_word_algebra verifies:

From 10,000 random length-3 byte words, 3729 distinct operator signatures are generated on a 10-state truncation of Ω. The word signature composition law is confirmed to match concatenation for 500 random word pairs (test_aQPU_4.py::TestPublicAPIConsistency::test_word_signature_composition_matches_concatenation). The rapid growth of distinct operator signatures with word length, combined with the non-Clifford phase δ_BU, supports the conclusion that the byte word algebra generates a dense subgroup of the relevant operator group on the 6-qubit register.

12.3 Public API Consistency

test_aQPU_4.py::TestPublicAPIConsistency verifies:

  • word6_to_pairdiag12 and pairdiag12_to_word6 are exact inverses on all 64 values.
  • q_word12 collapse matches q_word6 for all 256 bytes.
  • word_signature matches byte-by-byte replay for 500 random 4-byte words on random Ω states.
  • compose_word_signatures matches concatenated word signatures for 500 random word pairs.

Part 13: Non-Cloning Properties

test_aQPU_1.py::TestNonCloning verifies:

  • No byte is its own intron: byte ⊕ 0xAA ≠ byte for any byte (transcription has no fixed points).
  • Exactly one byte produces intron 0x00: the archetype 0xAA itself (the archetype is the unique zero-mutation source).
  • On the equality horizon (A = B), both components carry identical information: the set of A values equals the set of B values, and both have 64 elements. The second component adds zero information.
  • On the complement horizon (A = B ⊕ 0xFFF), knowing A determines B uniquely. The information is in the relationship (chirality), not in duplication.
  • Gates permute within horizons but never between them: applying any gate to a complement horizon state produces another complement horizon state, and likewise for equality horizon states. The two horizons are structurally isolated under gate operations.

These properties are the discrete realisation of the quantum no-cloning theorem: the reference frame (archetype) cannot be duplicated by any operation defined within the system it defines.

Part 14: C Engine and Low-Level Tensor Math (via SDK)

test_aQPU_SDK_1.py verifies the GyroLabe C engine and operator algebra exposed through the SDK (sdk.RuntimeOps, sdk.TensorOps, sdk.OperatorOps). These tests require the compiled C library; they are skipped when the library is unavailable.

14.1 C Engine Availability

TestCEngineAvailability verifies:

  • The native library loads successfully.
  • Bitplane GEMV symbols are present: gyro_bitplane_gemv_f32, gyro_pack_bitplane_matrix_f32, gyro_bitplane_gemv_packed_f32.

14.2 Signature Scan and Extract

TestSignatureScan verifies the byte-ledger-to-signature pipeline:

  • Single-byte signature scan (e.g. 0xAA -> 01000000).
  • Sequence scan for byte sequences.
  • Fused extract-scan: q_class, family, micro_ref, and signatures from gate bytes.

14.3 Chirality Distance

TestChiralityDistance verifies:

  • Pair distance: d(0x000000, 0xFFFFFF) = 0 (complement pairs).
  • Non-zero cases: d(0x555AAA, 0xAAA555) = 0.
  • Batch distances and adjacent-distance (lookahead) computation.

14.4 Walsh-Hadamard Transform (wht64)

TestWht64 verifies:

  • Orthonormality: wht64 vs H@x max err ~2.38e-07.
  • Self-inverse: wht64(wht64(x)) vs x max err ~2.38e-07.
  • Batch shape handling (e.g. [10, 64]).

14.5 Bitplane GEMV

TestBitplaneGemv and TestPackedBitplaneMatrix64 verify:

  • Bitplane GEMV vs torch.mv: max abs err ~1.09e-05.
  • Identity matrix: eye(64) @ x vs x max err ~8.68e-05.
  • Packed GEMV vs torch.mv: max err ~7.45e-06.
  • Packed vs unpacked consistency: max err ~2.24e-06.

14.6 Operator Projection Basis

TestOperatorProjectionBasis verifies the Weyl/Heisenberg-Walsh basis:

  • Walsh character signs shape [64, 64].
  • XOR permutation indices shape [64, 64].
  • Project-reconstruct exact: max err ~5.96e-08.
  • Full operator sum vs torch.mv: max err ~3.95e-07.
  • Sparse approximation: k=4096 yields max err ~3.95e-07; smaller k shows expected degradation.

14.7 Signatures, States, and Qmap

TestSignaturesToStates, TestApplySignatureToRest, and TestQmapExtract verify:

  • Signatures-to-states roundtrip (e.g. sigs -> states: 555aaa, aaa555).
  • Chirality states from bytes.
  • Apply signature to rest (e.g. sig=0 -> rest: aaa555).
  • Qmap gate-byte extraction: q_class and family for gate bytes.

Summary of Confirmed Properties

All 185 tests pass. Properties are organised by the evidential layer in which they are established.

A defining strength of the verified structure is exactness: many results are not approximate numerics but exact structural equalities. These include: the 4096-state Ω; two 64-state horizons (complement and equality); 128 distinct permutations from 256 bytes (uniform 2-to-1); exactly 32 row classes (64 under family-0 restriction); exact 2-step uniformisation (every state reached exactly 16 times); exact 1-bit parity beyond state at length 2; exact 4-to-1 q-fibers and holographic dictionary; and exactly 16 byte-paths and 4 state-paths per target. The report's conclusions rest on these exact counts and identities wherever cited.

Kernel-Native Verified Structure (aQPU Tests + Physics/Moments Reports)

Native register:

  • GENE_Mac is a ±1 tensor with 6 axis-orientation qubits per component.
  • All 4096 Ω states have valid ±1 representations (0 anomalies).
  • Spin-to-bits roundtrip exact on all Ω.
  • B is gate-related to A: 64 states A = B, 64 states A = B ⊕ 0xFFF, 3968 partial.
  • A values and B values each form 64-element C64 cosets.

Dual horizons:

  • Complement horizon (S-sector): 64 states, A = B ⊕ 0xFFF, ab_distance = 12. Contains rest.
  • Equality horizon (UNA degeneracy): 64 states, A = B, ab_distance = 0.
  • Disjoint. Union = 128-state boundary. Bulk = 3968 states.
  • Holographic identity: |H|² = |Ω| = 4096. Dictionary multiplicity exactly 4.
  • Complementarity invariant: horizon_distance + ab_distance = 12 (universal).
  • Chirality spectrum: binomial C(6, (12−d)/2) × 64 at ab_distance d.

K4 gate group:

  • 4 horizon-preserving bytes → 2 distinct operations S, C.
  • K4 = {id, S, C, F}: full Cayley table verified. F requires depth 2, id requires depth 0 or 4.
  • Gates in spin coordinates: swap, complement-swap, global sign flip.
  • Gate action on horizons: exact stabiliser/permutation census.
  • Orbit stratification: 32 + 32 + 992 orbits of sizes 2, 2, 4. Bulk has trivial stabiliser.
  • Gate-byte phase separation: same 24-bit operation, different spinorial phase.

Chirality transport:

  • χ(T_b(s)) = χ(s) ⊕ q6(b), exact on all 4096 × 256.
  • 64 chirality values, uniform partition of Ω.
  • Pauli-X subgroup: closed, abelian, deterministic.

Commutativity:

  • Rate 1/64 = 2^(−6). Each byte has exactly 4 commuting partners.
  • q-map: 4-to-1 onto C64.

Permutation structure:

  • 128 distinct permutations on Ω, uniform 2-to-1.
  • 2 of order 2 (S-gate), 126 of order 4.
  • Row-class theorem: 32 distinct rows, rank 32. Family-0 restriction: 64 rows, rank 64.

Channel properties:

  • Per-byte: 7.0 bits Shannon = min-entropy, zero variance.
  • Exact 2-step uniformisation: every state reached exactly 16 times.
  • Exact integer entropies: H(state) = 12, H(parity|state) = 1, H(state|parity) = 7.
  • Chirality and parity independent (MI ≈ 0.014 bits).

Error detection:

  • Undetected enumerator: (1 + z²)^12. All odd-weight errors detected.
  • Perturbation law: boundary 6×, payload 1×, mean 2.25 chirality bits per bit flip. Ratio 2.000 exact.

Tamper detection:

  • Substitution: 1/255 miss rate, shadow partners only.
  • Adjacent swap: ~3/255 miss rate, q-class equality only.
  • Deletion: gate stabiliser on horizon prefix only, never in bulk.
  • Adversarial steering: 16 byte-paths, 4 state-paths per target, exactly uniform.

Non-cloning:

  • Transcription has no fixed points. Archetype is the unique zero-intron source.
  • Equality horizon: redundant information. Complement horizon: relational information.
  • Horizons structurally isolated under all gate operations.

Hilbert-Lift Verified Consequences (aQPU Test File 2)

  • Graph state factorises into 6 independent Bell pairs (tensor product, exact to 10^(−12)).
  • CHSH violation at Tsirelson bound 2√2 for both Bell pair types and all 6 graph state pairs.
  • No measurements exceed Tsirelson (exhaustive angle search).
  • Exact teleportation: unique Pauli correction for all 8 (resource, outcome) combinations. Confirmed on basis states, phase states, and 800 random states.
  • Monogamy: same-pair pure, cross-pair maximally mixed. All 12 single-qubit marginals maximally mixed.
  • No-signalling: Bob's measurement choice does not change Alice's marginal.
  • 12 independent stabiliser generators, GF(2) rank 12. X-translation subgroup matches C64.
  • Peres-Mermin contextuality: row products +I, column 2 product −I. Noncontextual assignment impossible.
  • MUBs: computational and Hadamard bases mutually unbiased. Walsh-Hadamard = H₁^(⊗6). Third MUB exists.

Computational Consequences (aQPU Test Files 3 and 4)

Operator family:

  • 128 odd-parity (length-1) + 4096 even-parity (length-2) = 4224 operators at depth ≤ 2.
  • Even translations: (τ_A, τ_B) covers full C64 × C64 product.
  • Single bytes and 2-byte words: 0 intra-component multi-qubit coupling detected.
  • Topological entanglement: A perturbation propagates to B through gate S.

Non-Clifford resource (δ_BU):

  • Distance 0.195342 from nearest Clifford angle.
  • No periodicity up to order 100,000. Closest return at k = 22805, distance 4.59 × 10^(−5).
  • Dense U(1) equidistribution: χ² = 0.212 (critical 142.4).
  • Magic state |δ⟩: Wigner W(0,1) = −0.043771 (negative).
  • |δ_BU − m_a| = Δ × m_a exactly.

Algebraic universality:

  • Clifford backbone established (Moments Tests Report).
  • Non-Clifford resource confirmed (this report, Part 10).
  • Topological entanglement confirmed (this report, Part 9.3).
  • Dense operator generation: 3729+ distinct signatures from 10,000 random 3-byte words.
  • Word signature composition matches concatenation (500 random pairs).

C engine and tensor math (aQPU Test File 5):

  • Native library loads; bitplane GEMV symbols present.
  • Signature scan, extract-scan fused, chirality distance (pair, batch, adjacent).
  • wht64 orthonormal, self-inverse, batch-capable.
  • Bitplane GEMV and packed GEMV match torch.mv within ~1e-5.
  • Operator projection: project-reconstruct exact, sparse approximation at k=4096.
  • Signatures-to-states roundtrip, apply-signature-to-rest, qmap gate-byte extraction.

Proven computational advantages:

  • Deutsch-Jozsa: 1 Walsh step vs classical 33 queries. Perfect discrimination.
  • Bernstein-Vazirani: 1 Walsh step vs classical 6 queries. All secrets recovered with probability 1.
  • Hidden subgroup: native 4-to-1 q-map, kernel = 4 gate bytes. Walsh transform resolves in O(1).
  • Commutativity decision: O(1) via q-map vs O(4) classical. 5000/5000 correct.
  • Period finding: universal period 4 built into transition law.
  • Mixing: 2 steps vs generic O(log n) ≈ 12-step baseline. Exact uniformisation.
  • Holographic compression: 8 bits vs 12 bits per state (33.3% reduction).
  • State separation: every byte distinguishes every distinct pair. Hamming distance preserved.

All results documented in this report are backed by passing tests. Properties established in the Physics Tests Report or Moments Tests Report are referenced, not retested. Properties newly established by the aQPU suite are cited to specific test classes and methods. Where the report asserts counts, multiplicities, or structural identities, they are exact (integer equalities, exact ranks, exact uniformisation), not approximate. The 185 aQPU tests, together with the previously verified kernel physics and moments tests, confirm that the Gyroscopic ASI aQPU Kernel kernel possesses the structural properties of an algebraic quantum processing unit.