Signed and Integral Lifts of Correspondence Matrices

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 $\cdot$ Submission revision

Contents

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 by $i$, 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 check covers a precisely specified finite universe. A regression test checks representative circuits but is not a universal proof. A research proposal identifies 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.

2 Original correspondence matrices and their types

2.1 Boolean semantics and the basis convention

We use $\oplus$ for XOR. The foundational manuscript uses a rotated-biconditional glyph for XOR; some text renderings display that glyph as $m$. It is not a new operation. Products of Boolean scalar entries mean AND. In the original convention,

Consequently, the first/left column of a CM is indexed by $y=1$, and the second/right column by $y=0$. The analogous row order is $x=1,0$. For

the associated predicate is

For a fixed valuation, exactly one of the four minterms can be true. Thus the contraction reads one entry of $A$. This is the essential content of the truth-table representation in [1, Sections 2.2–3].

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.

2.2 The four atomic CMs

For phase discussions it is useful to order the atomic features clockwise:

Then

The reordered coordinates are $v(A)={(a_0,a_1,a_2,a_3)}^T={(a,b,d,c)}^T$. 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 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 “$i$ AND $i$” 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:

in the source’s ordering $(11,10,01,00)$. More generally

The outer product $|X⟩⟨Y|$ is instead a $2\times 2$ operator-shaped array. The two contain related products but have different types. A $2\times 2$ 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.

2.5 An explicit bridge to logical gate matrices

The same predicate has a truth-valued linear gate representation

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 sixteen $2\times 4$ 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

Thus $R{E}_j=E_{j+1mod4}$. The matrix has $P^4=I_4$ and $P^2\ne I_4$ over both ${𝔽}_2$ and $\mathbb{Z}$.

The new product is circular convolution: if $A=\sum <!--\; FUNCTION\; APPLICATION\; -->a_j{g}^j$ and $B=\sum <!--\; FUNCTION\; APPLICATION\; -->b_j{g}^j$, then

Although multiplication by $g$ 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 involution $g\mapsto g^{-1}$. It is a natural phase-reversal operation for convolution, not a claim that Boolean complement is complex conjugation. The Boolean ring has

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:

Equivalently, a feature indexed by $(x,y)$ moves to $(y,\neg <!--\; FUNCTION\; APPLICATION\; -->x)$. That motion is a classical basis permutation, expressible as a swap followed by a NOT. It becomes a phase operation only after an additional amplitude interpretation is specified. This provides a useful guard against reasoning from the geometry of the drawing alone.

4 XOR interference, counting, and the limits of measurement

4.1 A well-defined interference-like experiment

For two CMs, define

This is a useful Boolean cancellation experiment. It is not yet a quantum superposition. Coincident features cancel, since $1\oplus 1=0$; two contributions in different positions survive separately. Surviving distinct features should not be described as amplitude doubling.

Let $wt<!--\; FUNCTION\; APPLICATION\; -->(A)$ count the ones using ordinary integers. Directly from $a\oplus b=a+b-2\mathit{ab}$ for bits,

This identity is exact and is also the familiar relationship between Hamming distance and overlap of binary supports. The four complete $16\times 16$ interference tables are retained in the supplementary source and in data/interference_tables.json, rather than typeset in the submission manuscript. The table for $k$ is simply the XOR table with its second argument permuted by $R^k$; it does not by itself establish a new additive law.

For $r=[R]=E_0\oplus E_3$, the clockwise orbit is

Its self-overlap and XOR-weight profiles are

Thus dividing the latter by four gives the four values of ${sin<!--\; FUNCTION\; APPLICATION\; -->}^2(\varphi ∕2)$ at $\varphi =0,\pi ∕2,\pi ,3\pi ∕2$. This is a four-sample agreement caused by the support geometry, not a derivation of continuous sinusoidal phase dynamics. Other carriers give different profiles: $E_0$ gives $(0,2,2,2)$.

4.2 What can be derived about Hamming measurement?

The original LM formalism gives coefficient extraction,

[1, Section 4.1.4]. Counting the four positive elementary results therefore gives $wt<!--\; FUNCTION\; APPLICATION\; -->(\Theta )$. The count is an integer-valued aggregate of Boolean observations, not their XOR.

The assumption of rotation invariance is an equal-weight assumption about the four elementary positions. Operationally, $\mu (A)$ is the probability that $f_A(X,Y)$ 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.

It is valid, after embedding bit entries in $ℝ$, that

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 entries $a,b,c,d$, its three binary digits are

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. Likewise $wt<!--\; FUNCTION\; APPLICATION\; -->(A\otimes B)=wt<!--\; FUNCTION\; APPLICATION\; -->(A)wt<!--\; FUNCTION\; APPLICATION\; -->(B)$ 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

With $\overline{g}=g^{-1}$ this is $A\star \overline{B}$ in the integral group algebra. Thus

The correlation convention is linear in the first argument after integral lifting; conjugating the evaluated form gives the usual Dirac convention. These are genuine conjugation and covariance identities. However, the embedding of Boolean arrays as $0∕1$ integer arrays is not additive for XOR: $C(A\oplus A,B)=0$, whereas $2C(A,B)$ need not vanish. Hence this is not an ${𝔽}_2$-sesquilinear inner product. Usual polarization 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

Let $A_2={𝔽}_2[C_4]$ with the convolution product. A two-branch state is $\mathrm{\Psi }={(A,B)}^T\in A_2^2$. We enumerate all $16^4=65,536$ matrices

Every operator is tested on all $256$ states. These are all two-by-two $A_2$-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 the $8\times 8$ matrices.

Define $W(A,B)=wt<!--\; FUNCTION\; APPLICATION\; -->(A)+wt<!--\; FUNCTION\; APPLICATION\; -->(B)$ and the weight-four shell

Also define the full self-correlation $C(A,A)+C(B,B)$ and its signed quotient, whose real value is $C_0-C_2$. Full pair-correlation preservation means equality on every pair of states, not merely on self-pairs.

5.2 Why the global answer is exactly 32

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 each ${𝔽}_2[C_4]$ 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 commuting $C_4$ action, not as a new general MacWilliams theorem.

Full integer-correlation preservation includes the zero-lag self-correlation, which is Hamming weight. Thus it cannot enlarge this global group. Every listed phase-permutation does preserve the full correlation, so the classification remains exactly $32$. 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.

The larger signed-self-form class does not contradict this theorem. That form forgets some information, 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 same $32$ 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

It is not a global Hamming isometry: $(E_0,0)$ has weight one, whereas its image has weight three. All $512$ shell preservers are invertible, and their restrictions form a group of $128$ permutations of $\mathrm{\Sigma }$.

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 form $I+\mathit{v𝜖}$, where $𝜖$ is total parity. Commutation with the two four-cycles requires $v$ to be constant on each cycle, giving four choices. In ring notation they are

Each is invertible because $𝜖(v)=0$. This explains the observed ratio $512∕4=128$ without assigning a new physical meaning to the shell.

For every $U$ in the shell-preserving set, every $\mathrm{\Psi }\in \mathrm{\Sigma }$, and every $S_k=diag<!--\; FUNCTION\; APPLICATION\; -->(g^k,1)$, we computed $W$ branch by branch after $U{S}_k\mathrm{\Psi }$. Only seven sequences of first-branch weights occurred:

There is no sequence $(0,2,4,2)$ or $(4,2,0,2)$, including cyclic changes of phase origin. The search covers $512\cdot 70$ operator/state cases, each at four phases. This is a complete negative result for that experiment, not a prohibition on all finite-field quantum-like theories. Modal quantum theory deliberately uses different measurement assumptions [6].

5.4 Two explicit failures that must not be hidden

The first attempted conditional rotation used the masks $[R]=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$and $[\neg <!--\; FUNCTION\; APPLICATION\; -->R]=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$:

For $A=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$ this is

It deletes $a$, sends $E_0$ to zero, and cannot satisfy $F^4=I$. It does not rotate one column as a reversible phase gate. The earlier assertion that its order follows from $R^4=I$ was false.

A later toy splitter in the convolution algebra was

It can generate the desired four output weights from a specially chosen carrier, but is nilpotent rather than unitary. Also, $r|0⟩$ has $W∕4=1∕2$ before splitting, not one. Apparent conservation at the final detector does not establish conservation through the circuit. No scalar normalization repairs $B$: if $s^2(1+g^2)=1$, squaring gives $0=1$.

There is an independent obstruction to a familiar Hadamard identity. Over this commutative ring, similarity preserves trace. But $Z_B=diag<!--\; FUNCTION\; APPLICATION\; -->(g^2,1)$ has trace $g^2+1\ne 0$, while $X=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$has trace zero. Hence no invertible $H$ can satisfy $H{Z}_B{H}^{-1}=X$. These failures motivate changing the amplitude algebra explicitly, rather than relabeling the failed maps as quantum gates.

6 The integral lift: the precise bridge from CMs to $J$

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:

Positive counts can describe accumulated paths; negative counts are a useful reduced representation. Addition is now ordinary integer addition. With cyclic convolution these objects form $\mathbb{Z}[C_4]$. Reducing their coefficients modulo two recovers $A_2$, but this reduction discards multiplicity information.

Next impose an additional phase relation: opposite positions represent opposite amplitudes. In algebraic terms, quotient by $1+g^2$. The construction is the diagram

$$$$

(6.2)

The opposite-phase relation is a 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 equality $1+1=0$ would have to remain true. Consequently the diagram must not be replaced with a purported inclusion ${𝔽}_2[C_4]\subset \mathbb{Z}[i]$.

6.2 The intertwining equation

Define

Then $Q\tilde{A}={(n_0-n_2,n_1-n_3)}^T$. The four elementary images are

Here $Q{E}_j$ abbreviates $\mathit{Qv}(E_j)$.

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 representation induced from their four-position rotation after lifting and quotienting. The abstract quotient $\mathbb{Z}[C_4]∕(1+g^2)\mathit{\cong \mathbb{Z}}[i]$ 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 coefficients $n_j$, let

act by convolution on another lifted CM. Put $u=n_0-n_2$ and $v=n_1-n_3$. Iterating equation (6.5) gives

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

This is the usual Gaussian-integer product in real coordinates. Writing pairs instead of $i$ removes a symbol from the implementation, not the mathematical complex structure.

6.4 What is lost, and the exact correction for XOR

The map $Q$ is not injective. Among the sixteen binary arrays its image contains nine pairs in ${\{-1,0,1\}}^2$. The zero CM, $I_2$, the XOR truth-table matrix, and the all-ones CM all map to $(0,0)$. These are different predicates. Therefore one must not use $Q$ as a semantics-preserving compression of arbitrary truth-table CMs.

Nor does $Q$ preserve the original Boolean operations. For example, $E_0\oplus E_0=0$ while $Q{E}_0+Q{E}_0=(2,0)$. Similarly $E_0\wedge E_1=0$, whereas $(Q{E}_0)(Q{E}_1)=(0,1)$. The exact additive defect is useful:

This formula is one practical way to detect an invalid “Boolean-to-amplitude” rewrite. Dropping its overlap term changes the semantics.

Transpose satisfies $Q(A^T)=\overline{\mathit{QA}}$, where pair conjugation sends $(u,v)$ to $(u,-v)$. For a binary array, complement happens to satisfy $Q(\neg <!--\; FUNCTION\; APPLICATION\; -->A)=-\mathit{QA}$ because $Q(T)=0$. 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

The matrix $Q^{T}Q=I-P^2$ 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,

Unlike Hamming weight, it regards opposite contributions as cancelling. For example, $N_Q(E_0+E_2)=0$ but $N_Q(2E_0)=4$. For $0∕1$ entries it equals

not $wt<!--\; FUNCTION\; APPLICATION\; -->(A)$ in general.

The theorem has substantive assumptions. In particular, specifying $ker<!--\; FUNCTION\; APPLICATION\; -->Q$ 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 at $g=i$ (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 amplitudes $z_x=Q{n}_x$, define

If we identify this representation with the usual Hilbert-space quantum model, its outcome probabilities are the usual squared moduli. This is a pullback of the Born rule under an explicit representation, not a derivation of that empirical probability law from AND, XOR, or rotational symmetry alone.

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

Applying $Q$ separately to both branches gives

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.

This is the exact operator meaning of multiplying the $|1⟩$ amplitude by a quarter phase. The controlled choice concerns the branch 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

The $+$ signs here count paths over $\mathbb{Z}$, not modulo two. By $Q{P}^2=-Q$, its quotient is

It preserves the normalized quotient norm because

The raw four-phase map is not required to be invertible on directions annihilated by $Q$; the induced map on the quotient is unitary. An alternative raw representation uses signed subtraction directly. The two are equivalent after $Q$, not as operations on original Boolean truth tables.

Worked example with every CM visible. Start with $E_0|0⟩$. Splitting gives $(E_0|0⟩+E_0|1⟩)∕\sqrt{2}$. A quarter-turn on the second branch gives

Recombination by equation (7.7) gives raw coefficients $E_0+E_1$ and $E_0+E_3$, with denominator two. Their quotient coordinates are $(1,1)$ and $(1,-1)$, with squared lengths two each. Thus both outcomes have probability $1∕2$ in the quantum interpretation.

With no phase shift, the raw outputs are $2E_0$ and $E_0+E_2$. 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 $Q(A\star B)=\mathit{QA}\mathit{QB}$ in the integer convolution algebra. Consequently

CNOT sends $|x,y⟩$ to $|x,x\oplus y⟩$ and simply permutes the carried coefficients. Applying a splitter and CNOT to $|00⟩$ yields $(|00⟩+|11⟩)∕\sqrt{2}$ 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, $[\neg <!--\; FUNCTION\; APPLICATION\; -->R]|11⟩+[R]|00⟩$, was also nonfactorable over $A_2$: all $16^4$ possible two-factor assignments failed. Its coefficient matrix has determinant $[\neg <!--\; FUNCTION\; APPLICATION\; -->R]\star [R]=1+g^2\ne 0$, whereas an outer product over a commutative ring has zero determinant. This algebraic nonseparability is correctly 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 CMs as specifications of logical operators. It is particularly important for avoiding the impression that the CMs disappear as soon as Gaussian coordinates are introduced.

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 arbitrary $45^{\circ }$ phase gates. The correction factor may be essential. The oracle and phase identities are elementary reversible-computation and inclusion–exclusion facts; their role here is to make the CM-to-phase interface type-correct rather than to claim a new oracle theory.

8.1 All sixteen two-input sign oracles have a simple normal form

For $A=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$, write its algebraic normal form as

where

Then

All sixteen cases were checked independently on all four inputs. As a worked example, OR has $d=0$ and $p=q=r=1$; therefore its sign oracle is $Z_x{Z}_y\mathit{CZ}$. AND gives $\mathit{CZ}$ as a phase oracle and Toffoli as the reversible oracle $O_A$.

For a general angle the integer expansion is also useful:

It factors a diagonal phase predicate into constant, one-variable, and two-variable phases. This is a semantic identity, not an assertion that every controlled phase is a cost-free primitive or admits a particular ancilla-free synthesis. Synthesis constraints must be checked separately [7, 8].

8.2 Why the typed distinction matters

The array $E_0$ can mean the predicate $x\wedge y$, or it can mean one unit of zero-phase amplitude. In the first role it compiles to $O_{E_0}$ or $D_{E_0}^{(k)}$; in the second it maps to $Q{E}_0=(1,0)$. These are different interpretations of the same array. A compiler must distinguish at least predicate CM, phase-count CM, and linear 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 the $C_8$ extension

9.1 The common-denominator $C_4$ format

For circuits built from $H,S$, CNOT, and Boolean basis permutations, integer Gaussian numerators with a shared denominator $2^{h∕2}$ remain a valid representation. Each Hadamard adds or subtracts numerators and increments $h$; $S$ rotates pairs; a permutation reassigns branch labels. This proves closure under those gate 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 example $1+1∕\sqrt{2}$ cannot be written as a single Gaussian integer divided by one power of $\sqrt{2}$. The format is closed under the specified synchronized circuit updates, not an unrestricted coefficient ring. Second, allowing arbitrary reversible 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]. A $C_4$ phase alphabet alone therefore does not characterize the computational power of a gate set.

For precisely $H,S$, 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 and $11,520$ two-qubit elements modulo global phase. They are checks of the representation, not discoveries of new group orders.

9.2 Four integer coordinates for exact $45^{\circ }$ phases

Let

An element is uniquely represented by $(a,b,c,d)$, meaning $a+\mathit{bz}+c{z}^2+d{z}^3$. Multiplication by $z$ is the signed cyclic shift

Thus $z^2=i$, $z^4=-1$, and $z^8=1$. This $45^{\circ }$ phase is not an ordinary geometric $45^{\circ }$ rotation of a four-cell binary CM: the original array does not have eight distinct positions. It is an explicitly enlarged coefficient representation. One can use eight phase bins with opposite-bin reduction, or the four independent signed coordinates above.

Conjugation is

The squared modulus has the exact form

These identities follow by multiplication with the conjugate and reduction using $z^4=-1$ and $\sqrt{2}=z-z^3$.

The implementation uses the ring

An amplitude is a four-integer numerator divided by $2^s$. For Hadamard, multiply each sum or difference by $\delta =z-z^3=\sqrt{2}$ and increment $s$; this implements division by $\sqrt{2}$ without floating point. The $T$ gate applies the signed shift on the selected branch, while $S=T^2$ applies two shifts. The exact synthesis literature studies precisely this coefficient domain [7, 8].

An important correction about the hierarchy. Once $1∕\sqrt{2}$ has been adjoined for normalized Hadamards, ${\zeta }_8=(1+i)∕\sqrt{2}$ is already in the ambient scalar ring. Accordingly, adding $T$ is not necessarily a new number-field extension beyond the normalized $C_4$ setting. It is an additional local gate and a richer convenient numerator format. It extends Clifford dynamics because $T$ is not a Clifford generator, not because the normalization-free label $C_4$ by itself forbids every eighth-root number.

9.3 Closure and equivalence theorem

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.

For $H{S}^{k}H|0⟩$, one obtains

For $\mathit{HTH}|0⟩$,

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 $\sqrt{p}$, which need not belong to $D$. 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 involving $cos<!--\; FUNCTION\; APPLICATION\; -->(\pi ∕8)$ suggest a larger field if that real gauge is insisted upon. However the actual observables are implemented already by Clifford+$T$ circuits:

Thus a $C_{16}$ lift is not necessary for that experiment. The earlier field-degree argument neglected projective 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 phase counts. This is directly related to established polynomial and sum-over-paths formulations [11, 12, 13].

This proposition is a CM-specialized restatement of established Boolean/sum-over-paths semantics [11, 12, 13]; its purpose is validation of the translation.

The resulting program has Boolean control expressions, but its amplitudes are integer-weighted sums of phases. They are not obtained by XORing all paths. With eight counts $n_0,\dots ,n_7$ for each output, opposite-phase cancellation reduces the numerator to

This is the $C_8$ version of the four-position quotient.

Phase kickback is an especially transparent example. Since $|-⟩=(|0⟩-|1⟩)∕\sqrt{2}$ is an eigenvector of the bit flip,

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.

Complexity is not removed by notation. Direct enumeration may take $2^h$ paths; dense state propagation has $2^n$ 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, 15, 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 pairs $r+s\sqrt{2}$. 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.