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Correction-oriented technical companion / reproducible computational audit
Signed and Integral Lifts of
Correspondence Matrices
Phase quotients, Hamming-isometry no-go results,
and exact circuit encodings
Brian Droncheff
27 September 2026· Submission revision
Abstract
This technical companion asks a narrow question: what survives if the four rotational
positions of a correspondence matrix (CM) are treated as phase labels, and what must
change before those labels can represent signed quantum amplitudes? In the strict Boolean
convolution algebraF2[𝐶4], rotation is a faithful𝐶4 action but equal contributions cancel
under XOR. An elementary classification, corroborated by exhaustive enumeration of
all 65,536 two-branch F2[𝐶4]-linear operators, shows that global preservation of the
coefficient Hamming weight leaves only the32 branch permutations with independent
cyclicrotations;thefullintegerpair-correlationconditiongivesthesameclass. Aspecified
weight-four shell has512 preservers, but none produces the target four-sample fringe in
the tested interferometer.
The successful phase construction changes scalars. Lifting toZ[𝐶4] and quotienting
by 1+𝑔2 gives Z[𝑖] andtheexplicitintertwiningrelation 𝑄𝑃 =𝐽𝑄,togetherwithanexact
carry/overlap formula measuring the failure of XOR to become signed addition. These
identities provide a typed route from predicate CMs to reversible or phase oracles and a
CM-visible realization of standard exact Clifford+𝑇 arithmetic. The Clifford+𝑇 coefficient
ring, Boolean path-sum semantics, and general matrix-logic ideas are established prior
art; they are used here as comparison targets, not claimed as new. The contribution is
therefore a correction-oriented translation and boundary analysis, plus reproducible finite
classifications, rather than a new quantum theory, a new exact-synthesis algorithm, or a
simulation speedup.
Scope and provenance.This is a technical companion to the correspondence-matrix program, not a
replacement for the foundational CM manuscript. All finite classifications and regression summaries
are generated by the accompanying code and were rerun for this revision. The paper separates proved
identities, exhaustive finite classifications, regression tests, and proposals. Literature and priority checks
were refreshed through 27 September 2026; no claim of exhaustive historical priority is made.

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Signed-integral correspondence-matrix phase calculus Technical companion
Contents
1 Scope, claims, and the question being answered 3
2 Original correspondence matrices and their types 4
2.1 Boolean semantics and the basis convention . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The four atomic CMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Three different products must not be conflated . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Single states, tensor states, and operators . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 An explicit bridge to logical gate matrices . . . . . . . . . . . . . . . . . . . . . . . . 6
3 What the four rotations actually represent 6
4 XOR interference, counting, and the limits of measurement 7
4.1 A well-defined interference-like experiment . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 What can be derived about Hamming measurement? . . . . . . . . . . . . . . . . . . . 7
4.3 Integer correlation retains orientation but is not XOR-bilinear . . . . . . . . . . . . . . 8
5 The finite operator audit and no-go results 8
5.1 The exact search universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2 Why the global answer is exactly 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.3 The shell is richer, but still does not yield the target fringe . . . . . . . . . . . . . . . . 10
5.4 Two explicit failures that must not be hidden . . . . . . . . . . . . . . . . . . . . . . . 10
6 The integral lift: the precise bridge from CMs to𝐽 11
6.1 A common source, not an embedding of characteristic two . . . . . . . . . . . . . . . 11
6.2 The intertwining equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.3 How a general CM amplitude becomes an operator . . . . . . . . . . . . . . . . . . . 12
6.4 What is lost, and the exact correction for XOR . . . . . . . . . . . . . . . . . . . . . . 12
7 The signed phase norm and visible CM gate operations 13
7.1 A quotient norm, not a new Boolean Born rule . . . . . . . . . . . . . . . . . . . . . . 13
7.2 An 𝑆 gate that really rotates a CM amplitude . . . . . . . . . . . . . . . . . . . . . . . 14
7.3 A CM-native splitter and recombiner . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4 Tensor products and the location of entanglement . . . . . . . . . . . . . . . . . . . . 14
8 A direct compiler from original CM predicates to quantum operators 15
8.1 All sixteen two-input sign oracles have a simple normal form . . . . . . . . . . . . . . 15
8.2 Why the typed distinction matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9 Exact gates, coefficient domains, and the𝐶8 extension 16
9.1 The common-denominator 𝐶4 format . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.2 Four integer coordinates for exact45◦ phases . . . . . . . . . . . . . . . . . . . . . . 16
9.3 Closure and equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
9.4 Measurement and projective caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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Signed-integral correspondence-matrix phase calculus Technical companion
10 A Boolean phase-path theorem and its computational meaning 18
11 Verification of the lifted calculations 19
11.1 Independent implementations and exactness . . . . . . . . . . . . . . . . . . . . . . . 19
11.2 What the algorithm examples do and do not demonstrate . . . . . . . . . . . . . . . . 20
12 Literature review: what is already known, and what is distinctive? 20
12.1 Search scope and evidence standard . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
12.2 Matrix logic and vector logic: direct predecessors . . . . . . . . . . . . . . . . . . . . 20
12.3 Finite-field and set-based models: close to the Boolean stage . . . . . . . . . . . . . . 21
12.4 Hamming isometries and the finite-ring context . . . . . . . . . . . . . . . . . . . . . 21
12.5 Boolean paths and exact amplitudes: strong prior overlap . . . . . . . . . . . . . . . . 22
12.6 Graphical and categorical calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
12.7 Exact synthesis and known coefficient rings . . . . . . . . . . . . . . . . . . . . . . . 22
12.8 Existing computational bridges and a recent close neighbor . . . . . . . . . . . . . . . 23
12.9 A conservative novelty assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13 A focused next step: certified local phase rewrites 24
13.1 Benchmark contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.2 Representation compression has a precise safety condition . . . . . . . . . . . . . . . 24
14 Conclusion 24
A Model-boundary and correction ledger 25
B Reproducibility, verification boundaries, and data encoding 26
B.1 Files and commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B.2 Notation and normalization checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
B.3 Additional verified bridge facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References 28
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Signed-integral correspondence-matrix phase calculus Technical companion
1 Scope, claims, and the question being answered
The motivating question is concrete: can a logical calculus with AND and XOR in the roles of
multiplication and addition support something analogous to multiplication by𝑖, and can rotations of
correspondence matrices provide that operation? An answer must distinguish a four-cycle of pictures
from a representation of a group, and a group representation from an amplitude algebra with a positive
norm. It must also distinguish a numerical simulation of quantum equations from a physical quantum
computer.
The starting manuscript,Correspondence Matrices; Algorithms for Propositional Logic, develops
CMs, their rotations and complements, XOR composition, logical measurement matrices (LMs), and
higher-dimensional constructions [1]. A public 2018 version is available and supplies the historical
CM definitions used here. Its measurement discussion explicitly does not incorporate probability
densities. Here we retain those definitions when discussing the original calculus and mark every change
of coefficients, multiplication, or measurement interpretation. We verify the identities needed below; this
is not a line-by-line certification of every statement in the original manuscript.
Four levels of evidence. A proof establishes an identity for its stated mathematical domain. An
exhaustive checkcovers a precisely specified finite universe. Aregression testchecks representative
circuits but is not a universal proof. Aresearch proposalidentifies an untested possibility. The
accompanying scripts implement the finite checks and regression tests independently of the prose;
informal exploratory assertions are not treated as evidence.
Contribution status. The paper deliberately separates CM-specific results from standard algebra and
quantum-circuit facts. The distinction is summarized here so that theorem numbering is not mistaken for
a novelty claim.
Item Status in this paper Priority position
Strict F2[𝐶4] Hamming-isometry
classification and shell audit
Proved for the stated CM coefficient weight
and independently enumerated.
A specialized classification;
related Hamming-isometry
extension theory is classical.
Integral lift, quotient by1+𝑔2,
and𝑄𝑃 =𝐽𝑄
Proved explicitly in CM coordinates. Standard
group-ring/quotient algebra
used as a CM translation,
not claimed as new abstract
algebra.
XOR carry/overlap correction and
typed predicate/phase compiler
laws
Proved exactly and exhaustively checked for
two-input CMs.
Elementary
Boolean/inclusion–
exclusion identities;
contribution is the typed
CM interface.
Clifford+𝑇 coefficient arithmetic
and Boolean path sums
Re-derived and regression tested. Established prior art;
included only to validate the
translation.
Physical quantum model or
asymptotic simulation advantage
Not claimed. Outside the evidence
supplied here.
Table 1: Claim discipline for the submission revision.
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Signed-integral correspondence-matrix phase calculus Technical companion
Question Answer established here Evidence / limitation
Does CM rotation realize
𝐶4?
Yes, as a faithful linear permutation representation.Explicit matrix and proof.
Does XOR reproduce
complex addition?
No: identical terms cancel, and−1= 1 in
characteristic two.
Algebraic obstruction.
Can a phase-sensitive
quotient be defined?
Yes, after an integral lift and an added opposite-phase
relation.
𝑄𝑃 =𝐽𝑄 and quotient
proof.
Do the lifted gates
reproduce quantum circuits?
Yes, within the standard exact coefficient domain. Intertwining proof and
recomputed tests.
Is the Born rule derived
from logic?
No. A support measure is conditionally unique;
quantum probabilities use an additional Hilbert-space
interpretation.
Assumptions stated
explicitly.
Is a new algorithmic or
physical advantage shown?
No. A concrete compiler/rewrite research program is
available.
Benchmarks remain
future work.
Table 2: A map of the main conclusions. Positive representation results do not erase the negative results
or their assumptions.
2 Original correspondence matrices and their types
2.1 Boolean semantics and the basis convention
We use⊕ for XOR. The foundational manuscript uses a rotated-biconditional glyph for XOR; some text
renderings display that glyph as𝑚. It is not a new operation. Products of Boolean scalar entries mean
AND. In the original convention,
|1⟩=
1
0

, |0⟩=
0
1

, |𝑥⟩=
 𝑥
¬𝑥

. (2.1)
Consequently, thefirst/left column of a CM is indexed by𝑦= 1, and the second/right column by𝑦= 0.
The analogous row order is𝑥= 1, 0. For
𝐴=
𝑎 𝑏
𝑐 𝑑

, (2.2)
the associated predicate is
𝑓𝐴(𝑥,𝑦)=𝑎𝑥𝑦⊕𝑏𝑥¬𝑦⊕𝑐¬𝑥𝑦⊕𝑑¬𝑥¬𝑦=⟨𝑥|𝐴|𝑦⟩. (2.3)
For a fixed valuation, exactly one of the four minterms can be true. Thus the contraction reads one entry
of𝐴. This is the essential content of the truth-table representation in [1, Sections 2.2–3].
Proposition 2.1(Pointwise CM composition). Let 𝐴,𝐵 encode predicates on the same two ordered
inputs, and letΦbe any binary Boolean operation. ApplyingΦ entrywise gives
𝑓𝐴Φ𝐵(𝑥,𝑦)= 𝑓𝐴(𝑥,𝑦)Φ 𝑓𝐵(𝑥,𝑦). (2.4)
In particular the representation is linear for XOR.
Proof. At each fixed input pair, both sides applyΦ to the same two selected entries. Since the four input
pairs exhaust the truth table, the predicates agree. □
This proof also makes the scope clear: one must align the inputs and their ordering before combining
truth tables. A generic pair of quantum state vectors is not a pair of one-hot truth assignments, so the
entry-selection argument does not automatically extend to them.
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Signed-integral correspondence-matrix phase calculus Technical companion
2.2 The four atomic CMs
For phase discussions it is useful to order the atomic features clockwise:
𝐸0=[∧] =
1 0
0 0

, 𝐸 1=[⇑] =
0 1
0 0

, (2.5)
𝐸2=[¬∨] =
0 0
0 1

, 𝐸 3=[⇓] =
0 0
1 0

. (2.6)
Then
𝐴=𝑎0𝐸0⊕𝑎1𝐸1⊕𝑎2𝐸2⊕𝑎3𝐸3, 𝐴 =
𝑎0 𝑎1
𝑎3 𝑎2

. (2.7)
The reordered coordinates are𝑣(𝐴)=(𝑎0,𝑎 1,𝑎 2,𝑎 3)𝑇 =(𝑎,𝑏,𝑑,𝑐 )𝑇. The existence of the four basis
features and the XOR expansion are source results [1, Figure 1 and Section 3.2.1]; the cyclic ordering is
a presentational choice.
2.3 Three different products must not be conflated
The same array can inhabit different algebras. Its shape alone does not specify its multiplication. A
Operation Definition / role Multiplicative unit
Entrywise AND𝐴∧𝐵 Combines truth predicates pointwise. T=
1 1
1 1

AND/XOR matrix product
𝐴𝐵
(𝐴𝐵)𝑖𝑗 = É
𝑘𝐴𝑖𝑘𝐵𝑘𝑗. Acts on two-component
Boolean vectors.
𝐼2=
1 0
0 1

Cyclic convolution𝐴★𝐵 𝐸 𝑗★𝐸 𝑘 =𝐸𝑗+𝑘 mod 4, extended distributively. Added
in this paper’s phase interpretation.
𝐸0=
1 0
0 0

Table 3: Three distinct algebras on related arrays. Neither the identities nor the products are interchange-
able.
Boolean truth-table CM is generally neither invertible nor unitary. The all-ones truth table, the ordinary
identity matrix, and the convolution identity are three different objects. In particular, an equation such as
“𝑖 AND𝑖” cannot be reinterpreted as convolution or gate composition without saying so.
2.4 Single states, tensor states, and operators
The tensor product, not XOR, combines two registers:
|10⟩=|1⟩⊗| 0⟩=(0, 1, 0, 0)𝑇 (2.8)
in the source’s ordering(11, 10, 01, 00). More generally
|𝑋⟩⊗| 𝑌⟩=(𝑋𝑌,𝑋¬𝑌,¬𝑋𝑌,¬𝑋¬𝑌)𝑇. (2.9)
The outer product|𝑋⟩⟨𝑌| is instead a2× 2operator-shaped array. The two contain related products but
have different types. A2× 2 CM cannot directly multiply a four-component ket without a specified
conversion. Later quantum computations use bitstring labels and the standard ascending storage order
(00, 01, 10, 11); this is only a permutation of coordinates, not a change of truth convention.
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Signed-integral correspondence-matrix phase calculus Technical companion
2.5 An explicit bridge to logical gate matrices
The same predicate has a truth-valued linear gate representation
𝑀𝑓𝐴=
 𝑎 𝑏 𝑐 𝑑
¬𝑎 ¬𝑏 ¬𝑐 ¬𝑑

, 𝑀 𝑓𝐴(|𝑥⟩⊗| 𝑦⟩) =|𝑓𝐴(𝑥,𝑦)⟩. (2.10)
The tensor input order is(11, 10, 01, 00). Each column is a one-hot truth vector. The identity follows by
selecting the column indexed by(𝑥,𝑦), and converting back to the CM simply reshapes the first row.
Thus there is a bijection between the sixteen predicate CMs and these sixteen2× 4 deterministic logical
gate matrices. This is a concrete translation into the dyadic vector-logic viewpoint [2]; it is not merely a
visual resemblance. The rectangular gate can erase its inputs and is generally not reversible. The oracle
construction in section 8 retains those inputs to obtain a unitary permutation embedding.
3 What the four rotations actually represent
Clockwise rotation is
R
𝑎 𝑏
𝑐 𝑑

=
𝑐 𝑎
𝑑 𝑏

, 𝑣 (R𝐴)=𝑃𝑣(𝐴), 𝑃 =
©­­­
«
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
ª®®®
¬
. (3.1)
ThusR𝐸𝑗 =𝐸𝑗+1 mod 4. The matrix has𝑃4=𝐼4 and𝑃2≠𝐼4 over bothF2 and Z.
Proposition 3.1(Faithful rotation representation). The map𝐶4→𝐺𝐿(4, F2) sending a generator to𝑃
is faithful. With cyclic convolution, the sixteen arrays form
A2= F2[𝐶4]  F2[𝑔]/(𝑔4− 1), 𝐸 𝑗↔𝑔𝑗, (3.2)
and𝑔★𝐴 =R𝐴.
Proof. The orbit of𝐸0 has length four, so no smaller positive power of𝑃 is the identity. The convolution
rule is precisely multiplication of the four group elements, extended overF2. □
The new product is circular convolution: if𝐴= Í𝑎𝑗𝑔𝑗 and𝐵= Í𝑏𝑗𝑔𝑗, then
(𝐴★𝐵)𝑘 =
3Ê
𝑗=0
(𝑎𝑗∧𝑏𝑘−𝑗 mod 4). (3.3)
Although multiplication by𝑔 is an invertible linear map, it is not a unital algebra automorphism of this
convolution algebra. The phrase “rotation symmetry” should not obscure that distinction.
Transpose induces the involution𝑔↦→𝑔−1. It is a natural phase-reversal operation for convolution,
not a claim that Boolean complement is complex conjugation. The Boolean ring has
𝑔4− 1=(𝑔+ 1)4, (3.4)
so it contains nilpotents. Its order-four units are not four distinct complex scalar eigenphases; in
characteristic two,−1= 1.
There is also a purely logical reading of the same rotation:
𝑓R𝐴(𝑥,𝑦)= 𝑓𝐴(¬𝑦,𝑥). (3.5)
Equivalently, a feature indexed by(𝑥,𝑦) moves to(𝑦,¬𝑥). That motion is a classical basis permutation,
expressibleasaswapfollowedbyaNOT.Itbecomesaphaseoperationonlyafteranadditionalamplitude
interpretation is specified. This provides a useful guard against reasoning from the geometry of the
drawing alone.
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Signed-integral correspondence-matrix phase calculus Technical companion
4 XOR interference, counting, and the limits of measurement
4.1 A well-defined interference-like experiment
For two CMs, define
𝐼𝑘(𝐴,𝐵)= 𝐴⊕R 𝑘𝐵, 𝑘 ∈ Z/4Z. (4.1)
This is a useful Boolean cancellation experiment. It is not yet a quantum superposition. Coincident
features cancel, since1⊕ 1= 0; two contributions in different positions survive separately. Surviving
distinct features should not be described as amplitude doubling.
Let wt(𝐴) count the ones using ordinary integers. Directly from𝑎⊕𝑏=𝑎+𝑏− 2𝑎𝑏 for bits,
wt(𝐼𝑘)= wt(𝐴)+ wt(𝐵)− 2𝐶𝐴,𝐵(𝑘), 𝐶 𝐴,𝐵(𝑘)= wt(𝐴∧R 𝑘𝐵). (4.2)
This identity is exact and is also the familiar relationship between Hamming distance and overlap of
binary supports. The four complete16× 16 interference tables are retained in the supplementary source
and indata/interference_tables.json, rather than typeset in the submission manuscript. The
table for𝑘 is simply the XOR table with its second argument permuted byR𝑘; it does not by itself
establish a new additive law.
For𝑟 =[𝑅]=𝐸0⊕𝐸3, the clockwise orbit is
[𝑅]−→[ 𝐿]−→[¬ 𝑅]−→[¬ 𝐿]−→[ 𝑅]. (4.3)
Its self-overlap and XOR-weight profiles are
𝐶𝑟,𝑟 =(2, 1, 0, 1), wt(𝑟⊕R 𝑘𝑟)=(0, 2, 4, 2). (4.4)
Thus dividing the latter by four gives the four values ofsin2(𝜙/2) at 𝜙 = 0,𝜋/2,𝜋, 3𝜋/2. This is a
four-sample agreement caused by the support geometry, not a derivation of continuous sinusoidal phase
dynamics. Other carriers give different profiles:𝐸0 gives(0, 2, 2, 2).
4.2 What can be derived about Hamming measurement?
The original LM formalism gives coefficient extraction,
⟨𝑋𝑖|𝑀𝑋Θ𝑌|𝑌𝑗⟩=Θ 𝑖𝑗 (4.5)
[1, Section 4.1.4]. Counting the four positive elementary results therefore giveswt(Θ). The count is an
integer-valued aggregate of Boolean observations, not their XOR.
Theorem 4.1(Conditional uniqueness of normalized support measure). Suppose𝜇 is a nonnegative
real-valuedfunctiononthesixteenCMswith 𝜇(0)= 0,𝜇(T)= 1,rotationinvariance,andfiniteadditivity
for disjoint supports:
𝐴∧𝐵= 0 =⇒ 𝜇(𝐴⊕𝐵)=𝜇(𝐴)+ 𝜇(𝐵). (4.6)
Then𝜇(𝐴)= wt(𝐴)/4 for every CM.
Proof. The rotation acts transitively on the four atoms𝐸𝑗, so their measures are equal to some𝑝. Their
disjoint sum isT, hence4𝑝= 1. Each CM is the disjoint union of its occupied atoms. □
The assumption of rotation invariance is an equal-weight assumption about the four elementary
positions. Operationally, 𝜇(𝐴) is the probability that𝑓𝐴(𝑋,𝑌) is true if the input pair is sampled
uniformly. With a nonuniform input distribution one obtains a weighted count instead. Boolean logic
alone does not select a physical sampling distribution.
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Signed-integral correspondence-matrix phase calculus Technical companion
It is valid, after embedding bit entries inR, that
wt(𝐴)=∥𝐴∥2
𝐹 = tr(𝐴𝑇𝐴). (4.7)
The products and trace in this equation are ordinary real or integer arithmetic. Under AND/XOR
arithmetic, the analogous contraction returns parity and can vanish for a nonzero vector. This distinction
prevents an apparent squared-norm formula from being mistaken for a derivation of the physical Born
rule. Ellerman’s set-based model provides a particularly close prior comparison: it also combines
symmetric difference with an external counting interpretation of overlaps [3].
The count itself is implementable using Boolean gates. For four entries𝑎,𝑏,𝑐,𝑑 , its three binary
digits are
𝑞0=𝑎⊕𝑏⊕𝑐⊕𝑑, (4.8)
𝑞1=𝑎𝑏⊕𝑎𝑐⊕𝑎𝑑⊕𝑏𝑐⊕𝑏𝑑⊕𝑐𝑑, (4.9)
𝑞2=𝑎𝑏𝑐𝑑, wt(𝐴)=𝑞0+ 2𝑞1+ 4𝑞2. (4.10)
All sixteen cases were checked. The last equality interprets a multi-bit register as an integer; it does
not make the scalar addition characteristic two. Likewisewt(𝐴⊗𝐵)= wt(𝐴) wt(𝐵) for binary support
arrays. This tensor-count identity is not a substitute for defining a quantum tensor product of amplitudes.
4.3 Integer correlation retains orientation but is not XOR-bilinear
Writing the bit coefficients as ordinary integers, define
𝐶𝐴,𝐵(𝑘)=
3∑︁
𝑗=0
𝑎𝑗𝑏𝑗−𝑘 mod 4, C(𝐴,𝐵)=
3∑︁
𝑘=0
𝐶𝐴,𝐵(𝑘)𝑔𝑘∈ Z[𝐶4]. (4.11)
With𝑔=𝑔−1 this is𝐴★ 𝐵 in the integral group algebra. Thus
C(𝐵,𝐴)=C(𝐴,𝐵), C(𝑔𝑟𝐴,𝑔𝑠𝐵)=𝑔𝑟−𝑠C(𝐴,𝐵). (4.12)
The correlation convention is linear in the first argument after integral lifting; conjugating the evaluated
formgivestheusualDiracconvention. Thesearegenuineconjugationandcovarianceidentities. However,
the embedding of Boolean arrays as0/1 integer arrays is not additive for XOR:C(𝐴⊕ 𝐴,𝐵) = 0,
whereas 2C(𝐴,𝐵) neednotvanish. Hencethisisnotan F2-sesquilinearinnerproduct. Usualpolarization
arguments relating norm preservation to inner-product preservation cannot be assumed in that mixed
arithmetic.
5 The finite operator audit and no-go results
5.1 The exact search universe
LetA2= F2[𝐶4] withtheconvolutionproduct. Atwo-branchstateis Ψ= (𝐴,𝐵)𝑇∈A 2
2. Weenumerate
all 164= 65,536 matrices
𝑈=
𝑢11 𝑢12
𝑢21 𝑢22

, 𝑈 Ψ=
𝑢11★𝐴 ⊕𝑢12★𝐵
𝑢21★𝐴 ⊕𝑢22★𝐵

. (5.1)
Every operator is tested on all256 states. These are all two-by-twoA2-linear operators, not all possible
binary circuits, nonlinear maps, measurement models, or operators in arbitrary dimensions. As binary
matrices they form a structured subset of the8× 8matrices.
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Define𝑊(𝐴,𝐵)= wt(𝐴)+ wt(𝐵) and the weight-four shell
Σ= {(𝐴,𝐵) :𝑊(𝐴,𝐵)= 4}, |Σ|=
8
4

= 70. (5.2)
Also define the full self-correlationC(𝐴,𝐴)+C( 𝐵,𝐵) and its signed quotient, whose real value is
𝐶0−𝐶2. Fullpair-correlationpreservationmeansequalityoneverypairofstates,notmerelyonself-pairs.
Recomputed property Count
All two-by-twoA2-linear operators 65,536
Invertible operators 24,576
Involutions𝑈2=𝐼 640
Involutions sending(0,𝑟) to(𝑟,𝑟) 4
Preserve𝑊 on all256 states 32
Preserve full integer self-correlation on all states 32
Preserve full integer pair-correlation on all state pairs 32
Map the weight-four shell into itself 512
Shell preservers invertible on the full space 512
Shell-preserving involutions 96
Distinct actions on the shell / kernel size 128/ 4
Preserve full self-correlation on the shell 512
Preserve signed self-form globally / on the shell 2,048/ 8,192
Preserve full signed pair-correlation globally 32
Table 4: Recomputed classifications. The two shell-preserver sets (weight and full self-correlation)
coincideexactly. Data: finite_summary.json;membershipmasks: finite_classification.npz.
5.2 Why the global answer is exactly 32
Theorem 5.1 (Global coefficient-Hamming isometries). OnA𝑑
2, everyA2-linear transformation
preserving the total Hamming weight of the4𝑑 binary coefficients on all vectors is a permutation of the
𝑑 branches followed by an independent cyclic rotation within each branch. The group is𝐶4≀𝑆𝑑, of order
4𝑑𝑑!. For𝑑= 2 it consists of the32 matrices
diag(𝑔𝑎,𝑔𝑏),
 0 𝑔𝑎
𝑔𝑏 0

, 𝑎,𝑏 ∈{ 0, 1, 2, 3}. (5.3)
Proof. Regard the map as anF2-linear map on4𝑑 binary coordinates. Each unit vector has weight one,
so its image must be a unit vector. Images of two different unit vectors cannot coincide, since then their
weight-two sum would map to zero. Thus the map is a coordinate permutation.A2-linearity means that
it commutes with simultaneous multiplication by𝑔, which is a permutation with𝑑 disjoint cycles of
length four. A commuting permutation sends a cycle to a cycle and chooses a cyclic offset within it.
There are𝑑!cycle permutations and4𝑑 offsets. Conversely these maps plainly preserve weight. □
The proof is intentionally elementary, but the general phenomenon belongs next to the classical
MacWilliams theory of Hamming isometries and its finite-ring extensions. MacWilliams-type extension
theorems state, under their respective hypotheses, that Hamming isometries extend to monomial
transformations; Wood established the finite-Frobenius-ring extension property and Greferath–Schmidt
gave a combinatorial treatment [4, 5]. The weight used here is more specific: it counts the four binary
coefficients inside eachF2[𝐶4] symbol, rather than merely whether a ring symbol is nonzero. Thus
theorem 5.1 should be read as a specialized coefficient-basis classification obtained by combining
ordinary binary Hamming isometry with the commuting𝐶4 action, not as a new general MacWilliams
theorem.
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Fullinteger-correlationpreservationincludesthezero-lagself-correlation,whichisHammingweight.
Thusitcannot enlargethisglobalgroup. Everylistedphase-permutationdoespreservethefullcorrelation,
so the classification remains exactly32. Earlier motivation suggesting that preservation of a richer form
might enlarge the isometry group was incorrect: adding preservation constraints can only retain or
remove operators.
Thelargersigned-self-formclassdoesnotcontradictthistheorem. Thatformforgetssomeinformation,
has null directions on raw arrays, and is not a quadratic form over the original XOR addition. Requiring
the complete pair form reduces it to the same32 maps in this finite test.
5.3 The shell is richer, but still does not yield the target fringe
The shell admits the involutory mixer
𝐾 =
 1 1 ⊕𝑔2
1⊕𝑔2 1

, 𝐾 2=𝐼. (5.4)
It is not a global Hamming isometry:(𝐸0, 0) has weight one, whereas its image has weight three. All
512 shell preservers are invertible, and their restrictions form a group of128 permutations ofΣ.
The four-element kernel has a transparent explanation. The binary span of the weight-four vectors in
eight coordinates is the even-parity subspace. A linear map equal to the identity on that subspace has the
form𝐼+𝑣𝜖, where𝜖 is total parity. Commutation with the two four-cycles requires𝑣 to be constant on
each cycle, giving four choices. In ring notation they are
𝐼+
𝛼T 𝛼T
𝛽T 𝛽T

, 𝛼,𝛽 ∈ F2. (5.5)
Each is invertible because𝜖(𝑣)= 0. This explains the observed ratio512/4= 128without assigning a
new physical meaning to the shell.
For every𝑈 in the shell-preserving set, everyΨ∈Σ, and every𝑆𝑘 = diag(𝑔𝑘, 1), we computed𝑊
branch by branch after𝑈𝑆𝑘Ψ. Only seven sequences of first-branch weights occurred:
(0, 0, 0, 0),(1, 1, 1, 1),(2, 2, 2, 2),(3, 3, 3, 3),(4, 4, 4, 4), (5.6)
(1, 3, 1, 3), (3, 1, 3, 1). (5.7)
There is no sequence(0, 2, 4, 2) or(4, 2, 0, 2), including cyclic changes of phase origin. The search
covers 512· 70 operator/state cases, each at four phases. This is a complete negative result for that
experiment,notaprohibitiononallfinite-fieldquantum-liketheories. Modalquantumtheorydeliberately
uses different measurement assumptions [6].
5.4 Two explicit failures that must not be hidden
The first attempted conditional rotation used the masks[𝑅]=
1 0
1 0

and[¬𝑅]=
0 1
0 1

:
𝐹(𝐴)=(𝐴∧[¬ 𝑅])⊕(R 𝐴∧[ 𝑅]). (5.8)
For𝐴=
𝑎 𝑏
𝑐 𝑑

this is
𝐹(𝐴)=
𝑐 𝑏
𝑑 𝑑

. (5.9)
It deletes𝑎, sends𝐸0 to zero, and cannot satisfy𝐹4=𝐼. It does not rotate one column as a reversible
phase gate. The earlier assertion that its order follows fromR4=𝐼 was false.
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A later toy splitter in the convolution algebra was
𝐵=
1 𝑔2
1 1

, 𝐵 2=(1⊕𝑔2)𝐼, 𝐵 4= 0. (5.10)
It can generate the desired four output weights from a specially chosen carrier, but is nilpotent rather than
unitary. Also,𝑟|0⟩ has𝑊/4= 1/2before splitting, not one. Apparent conservation at the final detector
does not establish conservation through the circuit. No scalar normalization repairs𝐵: if𝑠2(1+𝑔2)= 1,
squaring gives0= 1.
There is an independent obstruction to a familiar Hadamard identity. Over this commutative ring,
similarity preserves trace. But𝑍𝐵= diag(𝑔2, 1) has trace𝑔2+ 1≠ 0, while𝑋 =
0 1
1 0

has trace zero.
Hencenoinvertible 𝐻 cansatisfy 𝐻𝑍𝐵𝐻−1=𝑋. Thesefailuresmotivatechangingtheamplitudealgebra
explicitly, rather than relabeling the failed maps as quantum gates.
6 The integral lift: the precise bridge from CMs to𝐽
6.1 A common source, not an embedding of characteristic two
The successful construction changes the amplitude arithmetic. Retain the four atomic CM positions, but
allow an integer number of contributions at each position:
e𝐴=𝑛0𝐸0+𝑛1𝐸1+𝑛2𝐸2+𝑛3𝐸3, 𝑛 𝑗∈ Z. (6.1)
Positive counts can describe accumulated paths; negative counts are a useful reduced representation.
Addition is now ordinary integer addition. With cyclic convolution these objects formZ[𝐶4]. Reducing
their coefficients modulo two recoversA2, but this reduction discards multiplicity information.
Next impose an additional phase relation: opposite positions represent opposite amplitudes. In
algebraic terms, quotient by1+𝑔2. The construction is the diagram
Z[𝐶4]
F2[𝐶4] Z[𝑔]/(𝑔2+ 1)  Z[𝑖].
mod 2 𝑔2=−1
(6.2)
The opposite-phase relation isa new semantic choice, not a theorem of Boolean truth tables. There is no
unital ring homomorphism from the characteristic-two algebra on the left into the characteristic-zero
algebra on the right: the equality1+ 1= 0 would have to remain true. Consequently the diagram must
not be replaced with a purported inclusionF2[𝐶4]⊂ Z[𝑖].
6.2 The intertwining equation
Define
𝑄=
1 0 −1 0
0 1 0 −1

, 𝐽 =
0 −1
1 0

. (6.3)
Then𝑄e𝐴=(𝑛0−𝑛2,𝑛 1−𝑛3)𝑇. The four elementary images are
𝑄𝐸 0=(1, 0), 𝑄𝐸 1=(0, 1), 𝑄𝐸 2=(−1, 0), 𝑄𝐸 3=(0,−1). (6.4)
Here𝑄𝐸𝑗 abbreviates𝑄𝑣(𝐸𝑗).
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Theorem 6.1 (CM rotation induces the quarter-turn). The CM rotation matrix𝑃 and the signed
quarter-turn matrix𝐽 satisfy
𝑄𝑃 =𝐽𝑄. (6.5)
The kernel of𝑄 is the ideal generated by1+𝑔2, so𝐽 is the action induced by rotation on the quotient
Z[𝐶4]/( 1+𝑔2).
Proof. For𝑛=(𝑛0,𝑛 1,𝑛 2,𝑛 3)𝑇,𝑃𝑛=(𝑛3,𝑛 0,𝑛 1,𝑛 2)𝑇, hence
𝑄𝑃𝑛 =(𝑛3−𝑛1,𝑛 0−𝑛2)𝑇 =𝐽𝑄𝑛. (6.6)
The kernel consists of(𝑎,𝑏,𝑎,𝑏 ), corresponding to(𝑎+𝑏𝑔)( 1+𝑔2). This is precisely the relation
identifying𝐸2 with−𝐸0 and𝐸3 with−𝐸1. □
This is the missing bridge behind the initial analogy. It is stronger than observing that two
transformations both have period four. It also makes the limitation unambiguous:𝐽 is not one of the
sixteen binary CMs. It is a two-dimensional signed representationinduced fromtheir four-position
rotation after lifting and quotienting. The abstract quotientZ[𝐶4]/( 1+𝑔2)  Z[𝑖] is standard algebra;
the role of theorem 6.1 is to pin that quotient to the CM coordinate convention and to expose exactly
where the scalar change occurs.
6.3 How a general CM amplitude becomes an operator
For integer coefficients𝑛𝑗, let
𝐿 e𝐴=𝑛0𝐼+𝑛1𝑃+𝑛2𝑃2+𝑛3𝑃3 (6.7)
act by convolution on another lifted CM. Put𝑢=𝑛0−𝑛2 and𝑣=𝑛1−𝑛3. Iterating equation (6.5) gives
𝑄𝐿 e𝐴=
𝑢 −𝑣
𝑣 𝑢

|    {z    }
𝑢𝐼+𝑣𝐽
𝑄. (6.8)
Thus multiplication by a CM-valued phase coefficient is not merely a verbal analogy: it is a four-position
convolution operator whose quotient is multiplication by the pair(𝑢,𝑣). On pairs, the induced product is
(𝑢,𝑣)·( 𝑠,𝑡)=(𝑢𝑠−𝑣𝑡,𝑢𝑡 +𝑣𝑠). (6.9)
This is the usual Gaussian-integer product in real coordinates. Writing pairs instead of𝑖 removes a
symbol from the implementation, not the mathematical complex structure.
6.4 What is lost, and the exact correction for XOR
The map𝑄 is not injective. Among the sixteen binary arrays its image contains nine pairs in{−1, 0, 1}2.
The zero CM,𝐼2, the XOR truth-table matrix, and the all-ones CM all map to(0, 0). These are different
predicates. Thereforeone must not use𝑄 as a semantics-preserving compression of arbitrary truth-table
CMs.
Nordoes 𝑄 preservetheoriginalBooleanoperations. Forexample, 𝐸0⊕𝐸0= 0while𝑄𝐸 0+𝑄𝐸 0=
(2, 0). Similarly𝐸0∧𝐸1= 0, whereas(𝑄𝐸 0)(𝑄𝐸 1)=(0, 1). The exact additive defect is useful:
Proposition 6.2(Carry/overlap correction). For binary CMs embedded coefficientwise in the integer lift,
𝑄(𝐴⊕𝐵)=𝑄𝐴+𝑄𝐵− 2𝑄(𝐴∧𝐵). (6.10)
Proof. Apply the integer identity𝑎⊕𝑏=𝑎+𝑏− 2𝑎𝑏 at each coordinate, then the integer-linear map
𝑄. □
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This formula is one practical way to detect an invalid “Boolean-to-amplitude” rewrite. Dropping its
overlap term changes the semantics.
Transpose satisfies𝑄(𝐴𝑇)=𝑄𝐴, where pair conjugation sends(𝑢,𝑣) to(𝑢,−𝑣). For a binary array,
complement happens to satisfy𝑄(¬𝐴)=−𝑄𝐴 because𝑄(T)= 0. That last identity is a consequence of
the information-losing quotient; it does not identify Boolean NOT with complex conjugation or with a
general physical phase operation.
7 The signed phase norm and visible CM gate operations
7.1 A quotient norm, not a new Boolean Born rule
The squared length of the quotient coordinates is
𝑁𝑄(𝑛)=∥𝑄𝑛∥2=(𝑛0−𝑛2)2+(𝑛1−𝑛3)2=𝑛𝑇(𝐼−𝑃2)𝑛. (7.1)
The matrix𝑄𝑇𝑄=𝐼−𝑃2 is positive semidefinite with rank two. It is a seminorm on the four raw count
coordinates and a positive norm on their quotient. In terms of integer self-correlation,
𝑁𝑄(𝑛)=𝐶𝑛,𝑛(0)− 𝐶𝑛,𝑛(2). (7.2)
UnlikeHammingweight, itregardsoppositecontributionsascancelling. Forexample, 𝑁𝑄(𝐸0+𝐸2)= 0
but𝑁𝑄(2𝐸0)= 4. For0/1 entries it equals
wt(𝐴)− 2(𝑎0𝑎2+𝑎1𝑎3), (7.3)
not wt(𝐴) in general.
Proposition 7.1(Conditional uniqueness of the phase quadratic form). Let a real positive-semidefinite
quadratic form onR4 be invariant under𝑃, vanish onker𝑄, and assign squared length one to𝐸0. Then
it is exactly𝑁𝑄.
Proof. Positive semidefiniteness implies that null vectors lie in the kernel of the representing symmetric
matrix,sotheformdescendsto R4/ker𝑄  R2. Itssymmetricmatrix 𝐺 mustsatisfy 𝐽𝑇𝐺𝐽 =𝐺. Writing
out a symmetric two-by-two𝐺 forces equal diagonal entries and zero off-diagonal entries:𝐺=𝜆𝐼. The
value at𝑄𝐸 0 fixes𝜆= 1. □
The theorem has substantive assumptions. In particular, specifyingker𝑄 as physically null selects
the primitive quarter-phase sector. Rotation invariance alone also permits contributions from the trivial
and sign sectors of the regular representation. In Fourier language,𝑄 selects evaluation at𝑔=𝑖 (with
its conjugate); the full coefficient squared length instead averages the energies of all four characters.
Changing from one form to the other is not merely remembering more phase data. It is changing which
directions carry measurable amplitude.
For a collection of branch amplitudes𝑧𝑥 =𝑄𝑛𝑥, define
Ψ= 2−ℎ/2
∑︁
𝑥
𝑧𝑥|𝑥⟩, ∥Ψ∥2= 2−ℎ
∑︁
𝑥
|𝑧𝑥|2. (7.4)
If we identify this representation with the usual Hilbert-space quantum model, its outcome probabilities
are the usual squared moduli. This is apullback of the Born rule under an explicit representation, not a
derivation of that empirical probability law from AND, XOR, or rotational symmetry alone.
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7.2 An 𝑆 gate that really rotates a CM amplitude
Each logical branch carries an entire lifted CM, rather than one column of a shared CM. In the branch
order(| 0⟩,|1⟩), define
𝑆CM(𝐴,𝐵)=(𝐴,𝑃𝐵). (7.5)
Applying𝑄 separately to both branches gives
(𝑄𝐴,𝑄𝐵)↦−→( 𝑄𝐴,𝐽𝑄𝐵). (7.6)
A common rotation of every quotient branch multiplies the whole state by a global phase and does not
change its physical predictions. Only relative rotations between branches are observable; four phase
labels are not four distinguishable physical states of a lone occupied basis branch.
Thisistheexactoperatormeaningofmultiplyingthe |1⟩amplitudebyaquarterphase. Thecontrolled
choice concerns thebranch label; the rotation acts on all four phase coordinates of its coefficient. This
resolves the earlier masked-column confusion.
7.3 A CM-native splitter and recombiner
One can keep the geometry visible even for Hadamard. Define on raw counts
𝐻CM :(𝐴,𝐵 ;ℎ)↦−→( 𝐴+𝐵, 𝐴+𝑃2𝐵; ℎ+ 1). (7.7)
The+ signs here count paths overZ, not modulo two. By𝑄𝑃 2=−𝑄, its quotient is
𝐻 :(𝑢,𝑣 ;ℎ)↦−→( 𝑢+𝑣,𝑢−𝑣;ℎ+ 1). (7.8)
It preserves the normalized quotient norm because
|𝑢+𝑣|2+|𝑢−𝑣|2= 2(|𝑢|2+|𝑣|2). (7.9)
The raw four-phase map is not required to be invertible on directions annihilated by𝑄; the induced map
on the quotient is unitary. An alternative raw representation uses signed subtraction directly. The two are
equivalent after𝑄, not as operations on original Boolean truth tables.
Worked example with every CM visible.Start with𝐸0|0⟩. Splitting gives(𝐸0|0⟩+ 𝐸0|1⟩)/
√
2. A
quarter-turn on the second branch gives
𝐸0|0⟩+ 𝐸1|1⟩√
2
. (7.10)
Recombination by equation (7.7) gives raw coefficients𝐸0+𝐸1 and𝐸0+𝐸3, with denominator two.
Their quotient coordinates are(1, 1) and(1,−1), with squared lengths two each. Thus both outcomes
have probability1/2 in the quantum interpretation.
With no phase shift, the raw outputs are2𝐸0 and𝐸0+𝐸2. Their quotient lengths are four and zero.
With a half-turn the outputs reverse. This is the precise distinction that XOR could not supply: equal
directions reinforce by retaining multiplicity, while opposite directions cancel under the quotient.
7.4 Tensor products and the location of entanglement
For independent registers, tensor their branch labels and multiply their amplitude coefficients using
convolution before quotienting, or Gaussian multiplication afterward. The compatibility follows from
𝑄(𝐴★𝐵)=𝑄𝐴𝑄𝐵 in theintegerconvolution algebra. Consequently
∥Ψ⊗Φ∥2=∥Ψ∥2∥Φ∥2. (7.11)
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CNOT sends|𝑥,𝑦⟩ to|𝑥,𝑥 ⊕𝑦⟩ and simply permutes the carried coefficients. Applying a splitter and
CNOT to|00⟩ yields(| 00⟩+| 11⟩)/
√
2 in the quotient representation. This is an exact encoding of a
standard entangled state, not evidence that a classical array implementation has become a nonlocal
physical system.
A previous strict-ring state,[¬𝑅]| 11⟩+[ 𝑅]| 00⟩, was also nonfactorable overA2: all 164 possible
two-factor assignments failed. Its coefficient matrix has determinant[¬𝑅]★[𝑅]= 1+𝑔2≠ 0, whereas
anouterproductoveracommutativeringhaszerodeterminant. Thisalgebraicnonseparabilityiscorrectly
distinguished from the Hilbert-space and experimental senses of entanglement.
8 A direct compiler from original CM predicates to quantum operators
The preceding quotient explains a phase coordinate system, but it intentionally loses Boolean truth-table
information. There is a second bridge that retains that information and makes direct use of CMsas
specifications of logical operators. It is particularly important for avoiding the impression that the CMs
disappear as soon as Gaussian coordinates are introduced.
Definition8.1 (ReversibleCMoracleandCMphasegate) . Fortheoriginalpredicate 𝑓𝐴inequation(2.3),
define
𝑂𝐴|𝑥,𝑦,𝑧 ⟩=|𝑥,𝑦,𝑧 ⊕ 𝑓𝐴(𝑥,𝑦)⟩, (8.1)
𝐷(𝑘)
𝐴 |𝑥,𝑦⟩=𝜁𝑘𝑓 𝐴(𝑥,𝑦)
8 |𝑥,𝑦⟩, 𝜁 8=𝑒𝑖𝜋/4. (8.2)
These act on three and two qubits respectively. They are not the original2× 2truth-table array acting on
a ket of the wrong dimension.
Proposition 8.2(Typed Boolean-oracle embedding and exact phase correction). For all two-input CMs
𝐴,𝐵:
𝑂2
𝐴=𝐼, 𝑂 𝐴𝑂𝐵=𝑂𝐴⊕𝐵, (8.3)
𝐷(𝑘)
𝐴⊕𝐵=𝐷(𝑘)
𝐴 𝐷(𝑘)
𝐵 𝐷(− 2𝑘)
𝐴∧𝐵 . (8.4)
The map𝐴↦→𝑂𝐴 is injective. For𝑘 = 4, the final factor in equation(8.4) is the identity, so XOR of
predicates translates exactly into composition of their sign-phase oracles.
Proof. The oracle preserves𝑥,𝑦 and toggles𝑧 by its predicate, so two applications cancel and two
different oracles toggle by𝑓𝐴⊕ 𝑓𝐵. Distinct predicates differ on some pair and hence produce distinct
permutations. For phase gates, apply𝑓𝐴⊕ 𝑓𝐵= 𝑓𝐴+ 𝑓𝐵− 2𝑓𝐴𝑓𝐵 as an integer equality and compare
exponents modulo eight. At𝑘 = 4, the overlap correction is a multiple of eight. □
This is an operational use of the original CM composition law, not merely a relation between
two four-cycles. It also explains why one cannot transfer an XOR rewrite unchanged to arbitrary45◦
phase gates. The correction factor may be essential. The oracle and phase identities are elementary
reversible-computationandinclusion–exclusionfacts;theirrolehereistomaketheCM-to-phaseinterface
type-correct rather than to claim a new oracle theory.
8.1 All sixteen two-input sign oracles have a simple normal form
For𝐴=
𝑎 𝑏
𝑐 𝑑

, write its algebraic normal form as
𝑓𝐴(𝑥,𝑦)=𝑑⊕𝑝𝑥⊕𝑞𝑦⊕𝑟𝑥𝑦, (8.5)
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where
𝑝=𝑏⊕𝑑, 𝑞 =𝑐⊕𝑑, 𝑟 =𝑎⊕𝑏⊕𝑐⊕𝑑. (8.6)
Then
𝐷(4)
𝐴 =(−1)𝑑𝑍𝑝
𝑥𝑍𝑞
𝑦𝐶𝑍 𝑟
𝑥𝑦. (8.7)
All sixteen cases were checked independently on all four inputs. As a worked example, OR has𝑑= 0
and𝑝=𝑞=𝑟 = 1; therefore its sign oracle is𝑍𝑥𝑍𝑦𝐶𝑍. AND gives𝐶𝑍 as a phase oracle and Toffoli as
the reversible oracle𝑂𝐴.
For a general angle the integer expansion is also useful:
𝑓𝐴(𝑥,𝑦)=𝑑+( 𝑏−𝑑)𝑥+(𝑐−𝑑)𝑦+(𝑎−𝑏−𝑐+𝑑)𝑥𝑦. (8.8)
It factors a diagonal phase predicate into constant, one-variable, and two-variable phases. This is a
semanticidentity,notanassertionthateverycontrolledphaseisacost-freeprimitiveoradmitsaparticular
ancilla-free synthesis. Synthesis constraints must be checked separately [7, 8].
8.2 Why the typed distinction matters
The array𝐸0 can mean the predicate𝑥∧𝑦, or it can mean one unit of zero-phase amplitude. In the first
roleitcompilesto 𝑂𝐸0 or𝐷(𝑘)
𝐸0
;intheseconditmapsto 𝑄𝐸 0=(1, 0). Thesearedifferentinterpretations
of the same array. A compiler must distinguish at leastpredicate CM, phase-count CM, andlinear
operator. Their allowed rewrites are different. This type discipline is a practical contribution of the audit
even if the underlying oracle constructions and Boolean phase identities are established mathematics.
9 Exact gates, coefficient domains, and the𝐶8 extension
9.1 The common-denominator 𝐶4 format
For circuits built from𝐻,𝑆 , CNOT, and Boolean basis permutations, integer Gaussian numerators with a
shareddenominator 2ℎ/2 remainavalidrepresentation. EachHadamardaddsorsubtractsnumeratorsand
incrementsℎ;𝑆 rotatespairs; apermutationreassignsbranchlabels. Thisprovesclosureunderthosegate
rules, and the quotient intertwining proves agreement with the corresponding standard quantum gates.
There are two qualifications. First, the union of the common-denominator formats is not itself closed
under arbitrary scalar addition. For example1+ 1/
√
2 cannot be written as a single Gaussian integer
divided by one power of
√
2. The format is closed under the specified synchronized circuit updates, not
an unrestricted coefficient ring. Second, allowingarbitraryreversible Boolean oracles adds gates such as
Toffoli, which are not Clifford. Hadamard and Toffoli already support universal quantum computation
with the standard real-amplitude encoding [9]. A𝐶4 phase alphabet alone therefore does not characterize
the computational power of a gate set.
For precisely𝐻,𝑆 , and CNOT, the gates generate the Clifford group; efficient stabilizer descriptions
are established and generally preferable to dense state vectors in that sector [10]. Our exact group checks
generate 24 one-qubit and11,520 two-qubit elements modulo global phase. They are checks of the
representation, not discoveries of new group orders.
9.2 Four integer coordinates for exact45◦ phases
Let
A8= Z[𝑧]/(𝑧4+ 1), 𝑧 ↦→𝜁8. (9.1)
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An element is uniquely represented by(𝑎,𝑏,𝑐,𝑑 ), meaning𝑎+𝑏𝑧+𝑐𝑧 2+𝑑𝑧 3. Multiplication by𝑧 is
the signed cyclic shift
(𝑎,𝑏,𝑐,𝑑 )↦−→(− 𝑑,𝑎,𝑏,𝑐 ). (9.2)
Thus𝑧2=𝑖,𝑧4=−1, and𝑧8= 1. This 45◦ phase is not an ordinary geometric45◦ rotation of a four-cell
binaryCM:theoriginalarraydoesnothaveeightdistinctpositions. Itisanexplicitlyenlargedcoefficient
representation. One can use eight phase bins with opposite-bin reduction, or the four independent signed
coordinates above.
Conjugation is
(𝑎,𝑏,𝑐,𝑑 )=(𝑎,−𝑑,−𝑐,−𝑏). (9.3)
The squared modulus has the exact form
|𝑎+𝑏𝑧+𝑐𝑧 2+𝑑𝑧 3|2=𝑢+𝑣
√
2, (9.4)
𝑢=𝑎2+𝑏2+𝑐2+𝑑2,
𝑣=𝑎𝑏+𝑏𝑐+𝑐𝑑−𝑎𝑑.
Theseidentitiesfollowbymultiplicationwiththeconjugateandreductionusing 𝑧4=−1and
√
2=𝑧−𝑧3.
The implementation uses the ring
D= Z[𝑧, 1/2]= Z[𝑖, 1/
√
2]. (9.5)
An amplitude is a four-integer numerator divided by2𝑠. For Hadamard, multiply each sum or difference
by𝛿=𝑧−𝑧3=
√
2and increment𝑠; this implements division by
√
2without floating point. The𝑇 gate
applies the signed shift on the selected branch, while𝑆 =𝑇 2 applies two shifts. The exact synthesis
literature studies precisely this coefficient domain [7, 8].
An important correction about the hierarchy. Once 1/
√
2 has been adjoined for normalized
Hadamards, 𝜁8 =(1+𝑖)/
√
2 is already in the ambient scalar ring. Accordingly, adding𝑇 is not
necessarily a newnumber-field extension beyond the normalized𝐶4 setting. It is an additional local
gate and a richer convenient numerator format. It extends Clifford dynamics because𝑇 is not a Clifford
generator, not because the normalization-free label𝐶4 by itself forbids every eighth-root number.
9.3 Closure and equivalence theorem
Proposition 9.1(Exactness of the phase-count representation). Using bitstring-indexed branch vectors
overD, ordinary tensor products, and the integer update rules above, every finite Clifford+𝑇 circuit
has an exact phase-count representation. Evaluating𝑧 at𝑒𝑖𝜋/4 intertwines every encoded gate with the
standard quantum gate and therefore preserves all circuit amplitudes and their Born probabilities. The
same statement holds when additional Boolean permutation oracles are supplied exactly.
Proof. Evaluation is a ring homomorphism because𝜁 4
8+ 1= 0. The gate updates agree entrywise with
𝐻, 𝑆,𝑇, and CNOT; a Boolean permutation agrees by its basis action. Sums, products, and tensor
products commute with evaluation. Induction on the gate sequence proves amplitude equality. Standard
gate unitarity, or the pairwise Hadamard norm identity and phase/permutation invariance, proves norm
preservation. The probability statement uses the usual Hilbert-space measurement rule under this
identification. □
This proposition repackages the established exact coefficient domain for Clifford+𝑇 circuits [7, 8]; it
is included to certify the CM translation, not as a new exact-synthesis theorem. It is an exact encoding of
known quantum arithmetic, not a distinct physical theory. The primitive character quotient is the reason
the encoding has complex structure, even when the code uses no complex-number datatype.
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For𝐻𝑆𝑘𝐻|0⟩, one obtains
𝑃(0)=

1, 1
2, 0, 1
2

𝑘
, 𝑃 (1)=

0, 1
2, 1, 1
2

𝑘
. (9.6)
For𝐻𝑇𝐻|0⟩,
𝑃(0)= 2+
√
2
4 , 𝑃 (1)= 2−
√
2
4 . (9.7)
Unlike the earlier Hamming-only toy splitter, these circuits preserve the appropriate normalized norm at
every step.
9.4 Measurement and projective caveats
The representation covers unitary circuits and computes their measurement probabilities exactly. A
normalized state obtained by conditioning on an arbitrary outcome may require division by√𝑝, which
need not belong toD. A measurement-capable implementation should therefore keep unnormalized
branch vectors and an exact probability record, or explicitly extend its scalar domain. Closure under
unitary updates does not silently prove closure under every normalized postselection.
Global phase may also change the apparent required field of a particular matrix representation. For
maximal CHSH measurements, real eigenvectors involvingcos(𝜋/8) suggest a larger field if that real
gauge is insisted upon. However the actual observables are implemented already by Clifford+𝑇 circuits:
𝑈+=𝑆𝐻𝑇𝐻𝑆†, 𝑈 †
+𝑍𝑈+=(𝑍−𝑋)/
√
2, (9.8)
𝑈− =𝑆𝐻𝑇†𝐻𝑆†, 𝑈 †
−𝑍𝑈− =(𝑍+𝑋)/
√
2. (9.9)
Thusa 𝐶16liftisnotnecessaryforthatexperiment. Theearlierfield-degreeargumentneglectedprojective
equivalence.
10 A Boolean phase-path theorem and its computational meaning
The gate rules have a second exact implementation: enumerate Boolean paths and accumulate integer
phasecounts. Thisisdirectlyrelatedtoestablishedpolynomialandsum-over-pathsformulations[11–13].
Proposition 10.1(CM-specialized phase-path representation). Consider a circuit built from𝐻, diagonal
powers of𝑇, and reversible Boolean gates, withℎ Hadamard occurrences. For computational basis
input𝑥 there are Boolean functions𝐹𝑥 and a phase function𝑝𝑥 with values inZ/8Z such that
⟨𝑦|𝑈|𝑥⟩= 2−ℎ/2
∑︁
𝜆∈{ 0,1}ℎ:𝐹𝑥(𝜆)=𝑦
𝜁 𝑝𝑥(𝜆)
8 . (10.1)
Proof. A Hadamard acting on a bit𝑎 introduces a new output bit𝑏 and contributes the factor2−1/2𝜁 4𝑎𝑏
8 .
A𝑇𝑘 gateonbit 𝑎 contributes𝜁𝑘𝑎
8 withoutchangingthebit. AreversibleBooleangateupdatesbitvalues
deterministically; CNOT uses XOR and Toffoli uses an AND-controlled XOR. Following these rules
produces𝐹𝑥 and the sum of all phase contributions modulo eight. Multiplying gate factors along a path
and summing paths ending at𝑦 gives equation (10.1). □
This proposition is a CM-specialized restatement of established Boolean/sum-over-paths semantics
[11–13]; its purpose is validation of the translation.
The resulting program has Boolean control expressions, but its amplitudes areinteger-weighted sums
of phases. They are not obtained by XORing all paths. With eight counts𝑛0,...,𝑛 7 for each output,
opposite-phase cancellation reduces the numerator to
(𝑛0−𝑛4)+( 𝑛1−𝑛5)𝑧+(𝑛2−𝑛6)𝑧2+(𝑛3−𝑛7)𝑧3. (10.2)
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This is the𝐶8 version of the four-position quotient.
Phase kickback is an especially transparent example. Since|−⟩ =(| 0⟩−| 1⟩)/
√
2is an eigenvector of
the bit flip,
𝑈𝑓|𝑥⟩|−⟩ =(−1)𝑓(𝑥)|𝑥⟩|−⟩ =𝐽2𝑓(𝑥)|𝑥⟩|−⟩. (10.3)
The logical predicate is converted into a half-turn phase. The signed phase sum after final Hadamards is
a Walsh-type transform of(−1)𝑓(𝑥), not the original truth-table XOR sum.
Complexityisnotremovedbynotation. Directenumerationmaytake 2ℎpaths;densestatepropagation
has 2𝑛 amplitudes. Counts need growing integers, so an honest analysis must include both arithmetic-
operation count and bit complexity. Polynomial-size Boolean descriptions can encode hard counting
tasks [11]. Any claim of a faster CM method must exploit a stated structural restriction and compare
against existing path-sum, stabilizer, tensor-network, decision-diagram, or model-counting methods [10,
14–16]. The present algorithms establish semantics and reproducibility, not a new general simulation
bound.
11 Verification of the lifted calculations
11.1 Independent implementations and exactness
The supplementary code contains an integer-only cyclotomic backend and an independently organized
path enumerator. Amplitudes are tuples of integers, denominators are powers of two, and probabilities
are rational pairs𝑟+𝑠
√
2. Equality and normalization tests in this backend are exact. A separate
complex-double backend supplies a numerical cross-check; that check is explicitly not used to certify
exact identities.
Check Newly obtained result
Parallelogram identity on small integer tuples 6,561 exact cases, no failures
Exact projective Clifford enumeration 24one-qubit; 11,520 two-qubit elements
Random exact circuit propagation 240circuits, 1–4 qubits, 24 gates each
Intermediate exact normalization 5,760 checks, no failures
Independent exact paths versus exact state propagation 120circuits, no discrepancies
Numerical cross-check against complex doubles Maximum amplitude difference
1.12× 10−15
Deutsch–Jozsa, all promised functions 8/8 at𝑛= 2; 72/72at𝑛= 3
Grover, one iteration, every marked input 𝑃= 1 at𝑛= 2;𝑃= 25/32at𝑛= 3
Bell correlations 𝐸(𝑍𝑍)=𝐸(𝑋𝑋)= 1;
𝐸(𝑍𝑋)=𝐸(𝑋𝑍)= 0
Optimal CHSH via exact𝐶8 gates 2
√
2; both local marginals1/2
CM phase carry identity 8,192 input/phase/CM cases
Two-input sign-oracle normal forms All 16 CMs verified on all inputs
Table 5: Recomputed regression results. Randomized tests use a fixed seed. Gate closure and the
intertwining identities have separate proofs; finite tests do not replace them.
The Pauli-conjugation group enumeration uses exact signed bit representations rather than floating-
point matrix matching. The random circuits include phase powers, CNOT, and where possible Toffoli;
this is intentionally broader than the Clifford-only group check. The independent path tests use at most
eight Hadamards so that all paths are actually enumerated. Full definitions and commands are given in
section B.
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11.2 What the algorithm examples do and do not demonstrate
For Deutsch–Jozsa, the all-zero output amplitude is
2−𝑛
∑︁
𝑥
(−1)𝑓(𝑥). (11.1)
It has magnitude one for constant functions and vanishes for balanced ones. Exhausting the promise sets
for two and three inputs verifies the implementation against this identity. The oracle is supplied explicitly
in the test; constructing it efficiently from an arbitrary external problem remains a separate task.
For Grover with one marked item among𝑁, one iteration gives marked amplitudesin(3𝜃) with
sin𝜃= 1/
√
𝑁. Direct gate propagation gives success probabilities one for𝑁 = 4 and 25/32 for𝑁 = 8,
matching this elementary amplitude calculation. The three-qubit marked-state phase uses a nonlinear
oracle outside the Clifford-only subgroup.
ForCHSH,theBellstateandtheobservables 𝐴0=𝑍,𝐴 1=𝑋,𝐵0=(𝑍+𝑋)/
√
2,𝐵 1=(𝑍−𝑋)/
√
2
give correlations(1, 1, 1,−1)/
√
2. The exact circuit calculation also checks that changing the remote
setting leaves each local marginal equal to1/2. This is a centralized calculation of quantum probabilities.
It is not an experimental Bell test, a local-hidden-variable explanation, or evidence that Boolean
hardware can evade Bell’s assumptions. The conclusion is representational agreement, as guaranteed by
theorem 9.1.
12 Literature review: what is already known, and what is distinctive?
12.1 Search scope and evidence standard
The review was conducted through 27 September 2026. Searches covered the paper title and author;
correspondencematricesandmatrixlogic;vectorlogicandsquarerootsofNOT;finite-fieldandset-based
quantummodels; Booleanpolynomialpathsums; exactcyclotomicsynthesis; ZHandcategoricalcalculi;
and tensor, decision-diagram, model-counting, and semi-tensor-product methods. Primary author papers,
repository versions, and publisher records were preferred. The bibliography identifies the inspected
version when version changes matter.
This is a targeted deep review, not a systematic review with a provably complete citation corpus or
a patent/priority search. No absence-of-results claim is used as proof of novelty. The search did not
establish that the precise package of CM notation, audits, and translations here has appeared before;
equally, it did not establish that it is new. Several of the central mathematical ideas have close, explicit
predecessors. The recent semi-tensor-product preprint is reported as an adjacent proposal, not as
independently verified performance evidence. A final September 2026 search also found continuing
progress in Clifford+𝑇 unitary synthesis, reinforcing that exact-circuit representation and synthesis are
active, well-developed areas rather than open novelty territory for the present encoding.
12.2 Matrix logic and vector logic: direct predecessors
Stern’s matrix logic. August Stern’s 1988 bookMatrix Logicalready develops a matrix/operator
formulation of logical transformations [17]. The publisher preview establishes historical overlap with the
broad idea of representing logic in vector spaces. A full technical comparison with that book has not
been completed here, and its broader physical interpretations are not adopted. It is enough to rule out
treating “logic represented by matrices” as a novel general principle.
Mizraji’s vector logic. Mizraji’s vector logic maps truth values to orthonormal vectors, monadic
operationstosquarematrices,anddyadicoperationstorectangularmatricesactingonKroneckerproducts
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[2]. In a two-dimensional truth space, a dyadic truth-valued gate is naturally a2× 4 map. Droncheff’s
two-input CM is instead a2× 2table whose contraction with two one-hot vectors returns a scalar truth
value. The dimensions and operational semantics differ, but equation (2.10) explicitly translates every
predicate CM into such a deterministic dyadic gate. This strengthens the prior-art connection: the name
“correspondence matrix” does not by itself establish mathematical separation.
Mizraji also studies counterfactual vector logic using square roots of NOT [18]. This is a close
conceptual predecessor to extending a logical operator calculus beyond its original Boolean values.
It is not the same as the integral phase quotient here, and its logical interpretation need not be the
interpretation used in a physical quantum circuit.
A particularly close imaginary-unit analogue.The revised manuscript on generalized square roots of
NOT explicitly discusses matrix extensions of the imaginary unit, Deutsch-type reasoning, and matrix
circular functions [19]. On a two-valued truth space, with NOT represented by𝑋, one familiar root is
𝑉 = 1+𝑖
2 𝐼+ 1−𝑖
2 𝑋, 𝑉 2=𝑋, 𝑉 4=𝐼. (12.1)
Thisisclosetothemotivatingquestion,butdiffersfrom 𝐽2=−𝐼 andfromthestrictlyBooleanconvolution
model. It uses complex coefficients; it does not make𝑖 a new Boolean truth value. The distinction is
instructive: both𝑉 and𝐽 have period four, but their squares and representations differ. The present
contribution must therefore be judged by the explicit quotient and compiler identities, not by fourfold
periodicity or the generic use of matrices for phase.
12.3 Finite-field and set-based models: close to the Boolean stage
Modal quantum theory. Schumacher and Westmoreland formulate quantum-like structures over
finite fields with a modal distinction between possible and impossible outcomes [6]. This line of work
demonstratesthatsuperposition-likealgebra,tensorstructure,andnonclassical-lookinglogicalproperties
can be investigated without the usual complex Hilbert-space probability model. It is an important
counterweight to an overbroad conclusion that characteristic two forbids every useful quantum analogue.
Our no-go theorem concerns Hamming-preservingF2[𝐶4]-linear maps under a specified measurement
functional. It does not invalidate modal models with different measurement axioms.
Quantum mechanics over sets. Ellerman’s set-based program uses the correspondence between
subsets and binary vectors, symmetric difference, and an integer-valued overlap count such as|𝑆∩𝑇|
[3]. It also explicitly discusses a sets-to-vector-spaces lifting principle. This is especially close to the
Hamming/counting portion of the present investigation. The lesson is twofold: an external non-modular
count on binary data is an established construction, and its status depends on the model’s probability
assumptions. Ourintegral𝐶4liftandopposite-phasequotientspecifyaparticularcharacterrepresentation;
the cited set program has a different organization and interpretation. Neither adjacency justifies deriving
empirical quantum probabilities from truth values without additional premises.
12.4 Hamming isometries and the finite-ring context
The global coefficient-weight theorem in theorem 5.1 sits close to classical Hamming-isometry theory.
Over a finite field, the MacWilliams extension theorem identifies linear Hamming isometries with
monomial equivalences; over finite Frobenius rings, Wood proved the corresponding extension property,
and Greferath–Schmidt supplied a combinatorial treatment [4, 5]. The present weight is not the standard
ring-symbol Hamming weight: each element ofF2[𝐶4] contributes the Hamming weight of its four
binary coefficients. Consequently, the short proof of theorem 5.1 is best understood as the ordinary
binarycoordinate-permutationtheoremtogetherwiththerequirementthatthepermutationcommutewith
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the𝑑 disjoint four-cycles induced by multiplication by𝑔. This yields𝐶4≀𝑆𝑑 directly. The classification
is useful for the CM audit, but it is not presented as a new general isometry theorem.
12.5 Boolean paths and exact amplitudes: strong prior overlap
Dawson and collaborators express quantum outputs through counts of solutions to polynomial equations
over Z2 [11]. Their connection between Boolean path constraints and signed amplitude counts is a direct
predecessor of theorem 10.1. Thus “AND/XOR descriptions plus integer signed counting can represent
quantum computation” is not a new conclusion of the CM route.
Amy develops path-sum semantics for functional quantum-circuit verification, including rewrite
methods and a tractable Clifford case [12]. Later work gives complete equational theories for sum-over-
paths expressions with unbalanced amplitudes, including coefficient-domain distinctions [13]. These
are particularly relevant to any proposed CM simplifier: a new rule should be translated into existing
path-sum syntax and tested for whether it is genuinely stronger, more local, more efficient to implement,
or merely an equivalent presentation.
The useful distinction here is representational granularity. A CM can serve as a small truth-table
object for a local predicate, while a path sum expresses global amplitude contributions. Equation (8.4)
is an exact interface between the two. Its value would be demonstrated by a verified compiler or a
benchmark in which local CM decomposition reduces an actual verification or contraction cost.
12.6 Graphical and categorical calculi
The ZH calculus was developed to handle quantum computations involving classical nonlinearity, with
compact representations related to Boolean AND and non-Clifford structure [20]. It is therefore a strong
neighboring formalism for the proposed Boolean-predicate phase translation, especially when higher-
degree monomials and controlled phases are involved. A CM-oriented graphical notation would need to
compare against ZH’s semantic and equational results rather than present graphical Boolean/quantum
unification as new in itself.
Categorical quantum semantics separates compositional structure from particular matrix coordinates
and scalar choices [21]. That viewpoint clarifies why changing scalars, tensor operations, and adjoints
must be explicit. It also suggests the right architectural goal for a CM compiler: preserve composition
under a typed interpretation. It does not remove the need to state which probability model is being
interpreted.
12.7 Exact synthesis and known coefficient rings
The ringZ[1/
√
2,𝑖] is central to exact Clifford+𝑇 synthesis. Giles and Selinger characterize multiqubit
exact synthesis with local ancillas in terms of this ring [7], while Kliuchnikov, Maslov, and Mosca study
exact single-qubit synthesis [8]. Consequently the Gaussian/cyclotomic phase-count arithmetic here
is a standard exact quantum coefficient domain in a CM-motivated coordinate system. No novelty is
claimed for its closure or for implementing𝐻,𝑆,𝑇 , and CNOT within it. Recent work continues to
improve general Clifford+𝑇 synthesis bounds [22]; that progress further supports treating the present
exact arithmetic as infrastructure rather than novelty.
Aaronson and Gottesman’s stabilizer algorithms give a strong baseline whenever the permitted gate
set is Clifford [10]. A dense CM count array will generally not compete with a compact stabilizer tableau
simplybecauseitusesBooleandata. Conversely,Aharonov’sHadamard–Toffoliuniversalityresultshows
why adding nonlinear reversible gates changes the computational story even without adding a literal45◦
phase instruction [9]. Gate-set restrictions and coefficient representations must be analyzed separately.
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12.8 Existing computational bridges and a recent close neighbor
Markov and Shi show how tensor-network contraction exploits circuit structure, with complexity
controlled by a graph-width parameter [14]. Hong and collaborators combine tensor representations with
decision diagrams [15]. Both are natural baselines for a CM proposal that uses local decomposition,
sharedsubexpressions,orfactorization. Therelevantquestionsconcernintermediatetensorsize,ordering,
graph sharing, and exact arithmetic growth, rather than array dimension in isolation.
Mei, Bonsangue, and Laarman investigate quantum simulation through model counting [16]. This is
a modern practical neighbor of the Boolean path-count interpretation. Replacing Boolean formulas by
smallCMobjectscouldbeusefulasafront-endencodingorrewritelayer,butthehardweighted-counting
task does not disappear. A credible claim would compare solver size and end-to-end cost on the same
instances.
A particularly relevant recent preprint by Li and collaborators uses semi-tensor products for exact
synthesis of CNOT and phase-polynomial circuits [23]. Its approach separates topology enumeration
frommatrix-basedinstantiationandfactorization. ThisisdirectevidencethatBoolean/matrixformalisms
are actively being used as quantum compilation tools. It is not the same representation as the CM phase
quotient, and the performance claims in that preprint were not reproduced here. The correct research
response is to investigate a translation and compare synthesis kernels, not infer that either framework
automatically dominates the other.
Prior line of work Closest overlap Main distinction to preserve
Matrix / vector logic Logical operators as matrices and
tensors; roots of NOT.
A truth-valued gate matrix is not the
same object as a scalar predicate table
or phase-count coefficient.
Modal and set models Binary superposition-like structures;
counting overlaps; lifting language.
Measurement axioms differ; physical
equivalence is not automatic.
Boolean polynomial paths Boolean constraints plus signed path
counts.
The global representation is
established; CM-local compilation may
be a useful interface.
ZH / path-sum rewriting Boolean nonlinearities and quantum
phase in a formal calculus.
New CM rules need semantic proofs
and comparative evaluation.
Exact cyclotomic synthesis Gaussian and eighth-root coefficient
rings.
Pair/array notation does not create a
new scalar theory.
Tensor / decision diagrams /
model counting
Structure-aware exact simulation. Any advantage must beat appropriate
existing backends on stated instance
families.
Semi-tensor-product
synthesis
Matrix factorization applied to
quantum circuit compilation.
A close 2026 preprint, not an
independently confirmed performance
benchmark here.
Table 6: A comparison map. Similarity is not identity, but it substantially narrows defensible novelty
claims.
12.9 A conservative novelty assessment
The following are established mathematical ingredients, not discoveries claimed here: regular cyclic
representations;groupalgebras;Gaussianandcyclotomicquotients;complexarithmeticinpairs;Boolean
path-sumdescriptions;andexactClifford+ 𝑇 simulation. Thefiniteclassificationsarereproducibleresults
for a particular explicitly defined CM phase model. They may be useful to document, but priority for
such small algebraic classifications has not been established.
The most defensible present contribution is averified synthesis: it locates the precise semantic
changesinaCM-to-phaseconstruction,provesitsintertwiningandno-gostatements,correctsmisleading
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shortcuts, and gives a typed route from original predicate CMs to reversible and phase oracles. Potential
algorithmic novelty is deferred to measurable compiler or rewriting improvements. This narrower claim
is stronger scientifically than calling an isomorphic encoding a new theory of quantum mechanics.
13 A focused next step: certified local phase rewrites
The present paper is strongest as a correction-oriented translation note and finite boundary analysis.
It does not yet establish a faster simulator or synthesizer. The most direct follow-up is therefore one
falsifiable implementation question: can local CM decompositions reduce the cost of an existing exact
phase-rewrite or verification workflow after all correction terms and conversion overhead are included?
The compiler identity
𝐷(𝑘)
𝐴⊕𝐵=𝐷(𝑘)
𝐴 𝐷(𝑘)
𝐵 𝐷(− 2𝑘)
𝐴∧𝐵 (13.1)
provides a small certificate. For bounded-input predicates, a certificate can record the two truth tables,
the phase modulus, the overlap/carry table, and the resulting circuit resources. At half-turn phases the
correction disappears; at eighth-root phases it retains exactly the information lost by translating XOR as
ordinary phase multiplication. This is not a new Boolean identity. The testable claim is whether the CM
packaging is useful as a local front end to an established path-sum, phase-polynomial, ZH, tensor, or
decision-diagram backend [12–16, 20].
13.1 Benchmark contract
A credible experiment should compare the same circuit families before and after CM-local rewriting and
reportatleast: gatecount, 𝑇 countwhererelevant,logicalancillas,peakexact-expressionsize,coefficient
bit lengths, memory, wall time, and conversion overhead. It should include favorable and unfavorable
cases. Clifford-only instances need a stabilizer baseline [10]; nonlinear Boolean oracles and non-Clifford
phase-polynomial instances should be compared with established exact methods. The outcome may be
negative. A negative result would still clarify whether the typed correction law is primarily expository
rather than computational.
13.2 Representation compression has a precise safety condition
Because𝑄 is a ring homomorphism for integer convolution, a purely convolutional subexpression may
be quotiented before or after contraction with the same result. The same move is unsafe across Boolean
XOR unless the carry correction is applied. A future compiler can therefore make “quotient late, or
quotient with a proof” an explicit type rule. This is a correctness condition and at most a constant-factor
representation reduction; no asymptotic advantage follows from it alone.
14 Conclusion
The investigation yields a deliberately narrower result than the motivating analogy suggested. Strict
Boolean CMs support a faithful fourfold rotation action, but characteristic-two XOR does not preserve
multiplicity or signed reinforcement. Under the stated coefficient-Hamming measurement, the globally
preserving F2[𝐶4]-linear maps reduce to branch permutations with independent cyclic rotations, and
the exhaustive weight-four-shell search produces no target four-sample fringe. These are model-specific
boundary results, not no-go theorems for finite-field quantum analogues in general.
A signed phase model appears only after changing scalars: lifting toZ[𝐶4] and imposing the
opposite-phase relation givesZ[𝑖], with𝑄𝑃 =𝐽𝑄 and an explicit carry formula identifying the failure
of XOR to become signed addition. Original CM truth predicates can then be compiled to reversible
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and phase oracles with a typed correction law. Exact Clifford+𝑇 arithmetic and Boolean path sums are
establishedpriorartandfunctionhereasvalidationtargets,notnewquantumformalism. Ontheevidence
presently available, the publishable contribution is the corrected CM-to-phase translation, the finite
classifications and no-fringe audit, and the reproducible separation of Boolean, integral, and quantum
semantics. Algorithmic advantage remains a future benchmark question.
A Model-boundary and correction ledger
This appendix records material changes to the exploratory discussion. Keeping the corrections visible is
part of the scientific result; a polished consolidation should not quietly preserve the exciting claims and
omit their failed premises.
Table 7: Audit of the main conceptual and computational shortcuts.
Earlier claim or ambiguity Verified replacement
The symbol𝑚 is a separate
operator.
It was a text-extraction rendering of the manuscript’s
rotated-biconditional XOR glyph. This paper consistently uses⊕ for
that operation.
The right column is𝑌 = 1. Under the supplied convention,𝑌 = 1 selects the first/left column. The
column entries are whatever the CM contains there, not the ket|1⟩
itself.
A masked rotation gives an
order-four controlled phase.
The proposed map is
𝑎 𝑏
𝑐 𝑑

↦→
𝑐 𝑏
𝑑 𝑑

. It destroys information;
𝐹(𝐸0)= 0. See equation (5.9).
A half-turn is Boolean negation. Rotation permutes positions. Complement flips values. They agree on
some arrays, but not in general. In the signed quotient a half-turn
induces additive negation by an added relation.
𝐽 is one of the original sixteen
CMs.
𝐽 has a negative entry and is not a binary CM. It is the induced
operator satisfying𝑄𝑃 =𝐽𝑄.
The four CM phases are complex
scalars insideF2.
They are group elements or operators. The scalar identities ofCdo not
hold in characteristic two.
CM multiplication is unambiguous. Pointwise AND, AND/XOR matrix multiplication, and convolution
have different units and meanings.
XOR reinforcement happens when
equal paths meet.
Equal Boolean contributions cancel. Signed amplitude reinforcement
requires keeping multiplicities.
Truth uncertainty is already
quantum superposition.
An unvaluated predicate is not automatically a coherent quantum state.
A state space, amplitude rules, and measurement interpretation must be
specified.
The four sampled weights prove a
sinusoidal phase law.
They agree with four samples for a particular carrier. No continuous
phase law or general Born behavior follows.
𝑤/4 is derived from logic alone as
a physical probability.
It is unique given disjoint additivity, normalization, and equal
rotational weighting. Those statistical assumptions are additional.
An integer correlation is anF2
inner product.
Integer counting of binary coefficients is not additive for XOR;
ordinary polarization arguments cannot be used without checking the
coefficient domain.
A richer preserved form may
enlarge the global isometry group.
Preserving a form that contains the old norm adds constraints. The full
correlation preserves the zero-lag Hamming norm and gives the same
32 global maps.
The toy splitter is a normalized
reversible analogue of𝐻.
It has𝐵4= 0. Its special initial carrier has𝑊/4= 1/2, and matching
final weights do not prove stepwise normalization.
The finite search covers every
possible Boolean quantum theory.
It covers all2× 2 operators over one specified sixteen-element
convolution ring and specified measurement conditions.
The lift embeds the entire Boolean
algebra into complex amplitudes.
The two algebras share an integral source. The quotient is not
XOR-additive and loses truth-table information; equation (6.10) gives
the defect.
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Earlier claim or ambiguity Verified replacement
Hamming norm is the same norm
after the signed lift.
The lifted quotient norm is𝐶0−𝐶2, not𝐶0. Some nonzero binary
CMs become null under𝑄.
Keeping four phases means
computation is only Clifford.
That is true only with the Clifford gate restriction. Nonlinear reversible
oracles such as Toffoli change the available gate set.
Adding𝐶8 necessarily introduces a
new ambient scalar field.
Normalized Hadamards already adjoin1/
√
2, so𝜁8=(1+𝑖)/
√
2 is
present.𝑇 is a new local gate; the four-coordinate format supports it
conveniently.
The optimal CHSH measurement
requires𝐶16.
Clifford+𝑇 implements it exactly up to the relevant global phases. A
real eigenvector gauge created the apparent extra requirement.
Algorithm agreement demonstrates
a physical quantum device or a
speedup.
It verifies a classical exact encoding of standard quantum amplitudes.
No physical nonlocal device or new complexity advantage was tested.
Every normalised
post-measurement state stays in the
same integer format.
Conditioning can require division by√𝑝 outside the original domain.
Use unnormalised states or enlarge the domain explicitly.
The matrix/Boolean-to-quantum
idea is new.
The literature contains direct predecessors. Novelty of specific
CM-based software or identities requires comparison beyond a naming
distinction.
B Reproducibility, verification boundaries, and data encoding
B.1 Files and commands
The source package contains the LaTeX manuscript, bibliography, three primary verification scripts, an
independent adversarial cross-check script, generated JSON results, and finite-set membership data. No
third-party paper or font files are redistributed. The supplied original manuscript is cited rather than
bundled.
From the project directory, run:
python code/verify_finite.py
python code/verify_quantum.py
python code/verify_bridges.py
python code/independent_hostile_checks.py
python code/make_tables.py
latexmk -pdf -interaction=nonstopmode paper.tex
The exact amplitude backend inphase_calculus.py uses Python integers and rational pairs. NumPy
isneededforthefinitesearchandindependentnumericalcomparisons. LaTeXcompilationusesstandard
mathematical packages and BibLaTeX with Biber. The README records the tested environment.
Elapsed times in the JSON files are diagnostics, not comparative performance benchmarks.
Finite search details. An element is encoded by the integerÍ3
𝑗=0𝑎𝑗2𝑗; its displayed CM is
𝑎0 𝑎1
𝑎3 𝑎2

.
A matrix is a tuple(𝑎,𝑏,𝑐,𝑑 ) in row-majoroperatororder. A two-branch state is ordered(| 1⟩,|0⟩) in
this finite model. The code builds the full16× 16multiplication table directly from circular convolution.
Every candidate is applied to all256 states; the full pair forms are checked on all state pairs for every
survivingcandidate. Pre-filtersonlydiscardcandidatesthatalreadyfailnecessaryconditions. Invertibility
is tested by whether the determinant is a unit; inF2[𝑔]/((𝑔+ 1)4), this is equivalent to odd coefficient
parity.
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Signed-integral correspondence-matrix phase calculus Technical companion
Exact quantum tests. The quantum backend uses ascending bitstrings, with qubit zero the leftmost bit.
Amplitudesarepolynomialsmodulo 𝑧4+ 1dividedby 2𝑠. Afixedseed, 20260917, generatestherandom
tests. Exact integer state propagation is cross-checked against an independently structured exhaustive
path sum on120bounded-Hadamard circuits. The complex-double comparison is separately labelled; its
small numerical discrepancy is not substituted for exact equality.
Independent adversarial reimplementation. For this revision, a separate compact script reimple-
mented the critical finite identities without importing the project backend. It independently checked
𝑄𝑃 = 𝐽𝑄 on a bounded integer box, the kernel relation, all256 Boolean carry pairs, the32 global
coefficient-Hamming isometries, the512shell preservers, the absence of the target fringe, all observed
shell phase profiles, and the oracle/phase-carry truth identities. Agreement with the primary scripts
reduces the risk of a shared implementation error; it is still not a substitute for formal proof.
What was not verified.No formal proof-assistant certification was performed. The 2018 foundational
CM manuscript was not re-audited line by line; the present companion manuscript was reviewed
end-to-end. No optimizer, hardware implementation, practical speedup, or physical experiment was
benchmarked. The small-circuit tests do not by themselves prove arbitrary-size closure; that conclusion
follows from the algebraic gate-intertwining proof. The literature review does not establish priority or
exhaust all related work.
B.2 Notation and normalization checks
The zero object is0; T is the all-ones truth table;𝐼2 is the ordinary two-dimensional identity;𝐸0 is the
convolution identity. A coefficient “1” insideA2 denotes𝐸0, notT. This distinction is essential in the
finite search tables.
A phase-count state in the common Gaussian format has denominator2ℎ/2. The exact𝐶8 backend
instead uses denominator2𝑠. Thus its squared probabilities have denominator22𝑠. These are equivalent
representations when applicable; the exponent variables must not be interchanged. The code canonically
removes common factors of two from all numerator coefficients.
B.3 Additional verified bridge facts
The supplementary checks verify𝑄𝑃 =𝐽𝑄,𝑄𝑇𝑄=𝐼−𝑃2, integral convolution compatibility, the XOR
carry correction, the four-element shell-action kernel, and the raw CM Hadamard’s quotient action. They
also verify the truth-coordinate permutation under rotation and that the reversible CM oracle embeddings
are permutations and satisfy equation (8.3). These checks are small, exact regressions for identities that
also have direct proofs.
One useful corollary of the lift is worth emphasizing. Reducing the Gaussian quotient modulo two
produces
Z[𝑖]/2Z[𝑖]  F2[𝑔]/((𝑔+ 1)2). (B.1)
It does not retain four distinct complex scalar phases:𝑔2= 1 there. Moving back to parity arithmetic
after imposing the signed relation cannot restore the lost amplitude multiplicities.
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Signed-integral correspondence-matrix phase calculus Technical companion
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