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Boolean CM Phase Algebra:
Native Binary Realizations and Exact Detector Bounds
Prepared for Brian Droncheff
28 September 2026
Abstract
The four cyclic coordinates of a Boolean correspondence matrix give a concrete realiza-
tion of the classical modular group algebraA = F2[C4]∼= F2[u]/(u4). We separate three
questions: how to implement its operations without a primitive cyclic-convolution symbol,
what information its nilpotent phase tags retain, and what a specified linear detector can
read. The native binary implementation is exactly the centralizer of a four-cycle; allowing
arbitrary binary gates is a strictly larger model.
For the half-turn tagz = 1 +u2, we close the finite detector examples with an exact width
formula. OnN = 2n addresses withn≥ 1, c constant-annihilatingA-linear output channels
distinguish constant from balanced truth vectors by literal zero/nonzero pattern inspection if
and only ifc≥⌈N/4⌉. More generally, the threshold is⌈N/(2r)⌉ for a tag whose principal
ideal has binary dimensionr, when 4 divides N. The lower bound is an application of the
classical forbidden-weight theorem of Enomoto, Frankl, Ito and Nomura; it is not claimed
as a new coding theorem. We give an explicit optimal construction and a self-contained
power-of-two proof of the needed coding bound.
A second boundary is exact: square-zero-tag interactions vanish across a binary linear map
precisely when that map preserves the tag ideal. A reversible coordinate permutation witnesses
failure outside this class. We also provide a normal form for all512 intrinsic two-branch
unitaries, including analytic proofs of the96 involutions and72 nonmonomial involutions,
and retain reproducible finite classifications. The familiar Boolean Möbius transform explains
the phase-to-ANF identity. These are algebraic and pattern-readout results, not a Born rule,
an exact modal Deutsch–Jozsa algorithm, or a quantum query advantage.
1 Purpose and contribution boundary
A correspondence matrix (CM) records a two-input Boolean predicate in four bits. Putting those
bits on a cycle and letting a quarter-turn act on them is harmless bookkeeping. Treating all
four-bit patterns as phase coefficients requires an additional multiplication. That multiplication
is mathematically useful, but it must not be confused with pointwise Boolean conjunction or
with the ordinary product of the original2× 2 matrices.
This companion studies that additional algebra. Its purpose is not to introduce a new number
system: modular group algebras, unitary subgroups of group algebras, and Boolean Möbius
transforms have substantial prior theory [3, 5]. Its value is an explicit translation between CM
coordinates and binary operators, a closed detector-width problem with its classical antecedent
identified, and finite certificates that keep incompatible operational interpretations apart.
The principal results are Theorem 3.1 (the exact native gate algebra), Theorem 6.3 (optimal
detector width), Theorem 7.1 (the exact gate boundary for square-zero interactions), and
Theorem 4.2 (a complete two-branch unitary normal form). The detector theorem packages a
known coding bound into this coefficient model; its construction and the ideal-boundary criterion
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Boolean CM phase algebra Technical companion
are elementary. No claim of historical firstness is made for these specializations or for the finite
group counts.
Readout is part of the problem.Throughout, apattern detectoris a specified linear map
followed by inspection of its output coordinates. It is not automatically a physically realizable
measurement or a complete modal instrument. Modal quantum theory assigns possible/impossible
outcomes using a declared state and measurement structure [6, 7]; it does not give free inspection
of an entire coefficient description. Sections 4 and 8 make this distinction explicit.
Scope of the revision. The finite phase algebra and its interfaces form the paper. General
compiler-performance claims, speculative applications, and attempts to recover all standard
quantum algorithms are not included as contributions. A local Boolean-polynomial identity is
retained in Appendix B only to connect the phase algebra back to the underlying CM notation.
2 Typed coordinates and the classical phase ring
2.1 CM coordinates and the additional product
For a CMC =
(a b
c d
)
, the row and column labels are(1, 0). Thus
fC(X,Y ) =aXY +bX(1 +Y ) +c(1 +X)Y +d(1 +X)(1 +Y ). (2.1)
Here + is addition inF2; when a logical connective is written explicitly,⇕ denotes XOR. The
four cyclic basis elements are
E0 =
(
1 0
0 0
)
, E 1 =
(
0 1
0 0
)
, E 2 =
(
0 0
0 1
)
, E 3 =
(
0 0
1 0
)
.
Consequently (a0,a 1,a 2,a 3) means the CM(a0 a1
a3 a2 ), not row-major order. Address-register order,
used later, is separately ascending binary order.
Define a second multiplication by bilinearity and
Ei⋆Ej =Ei+j mod 4, (a⋆b )k =
∑
i+j≡k (4)
aibj. (2.2)
Once this algebra is declared, we use juxtaposition for its multiplication. The symbol⋆ need not
be a primitive operation in an implementation.
Operation Type Identity or diagnostic
Pointwise conjunction Boolean operation on corresponding CM entries All-ones CM
Ordinary CM product Contracted 2× 2 product over F2 Numerical I2
Phase product Equation (2.2) E0
Boolean polynomial
product
F2[X1,...,X n]/(X 2
i−Xi) X 2
i =Xi
Native phase operator
product
Ordinary product of4× 4 binary blocks I4 =ρ(E0)
For example, numericalI2 = E0 +E2 is square-zero under phase multiplication. This single
example rules out identifying the second and third rows of the table.
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Boolean CM phase algebra Technical companion
2.2 Ring structure and scalar information
Put t =E1, 1 =E0, u = 1 +t, z =t2 = 1 +u2, and
δ = 1 +z =u2, Ω =u3 = 1 +t +t2 +t3.
The group basis gives, directly,
A = F2[t]/(t4− 1) = F2[u]/(u4). (2.3)
The quarter-turnt has multiplicative order four; its half-turnz is not the additive negative of1.
Indeed−1 = 1 in this algebra, whereasz̸= 1.
Proposition 2.1. The units ofA are the eight odd-weight cyclic patterns. The only ideals are
0⊂ (u3)⊂ (u2)⊂ (u)⊂A,
with cardinalities1, 2, 4, 8, 16. Ifν(a) is the first nonzero exponent in itsu-expansion andν(0) = 4,
multiplication bya has binary rank4−ν(a) and kernel size2ν(a). The only idempotents are0
and 1.
Proof. An element with constantu-coefficient one has the form1 +h, whereh4 = 0, and inverse
1 +h +h2 +h3. An element with constant coefficient zero is nilpotent and not a unit. Evaluation
at t = 1 shows that this constant coefficient is the parity of the cyclic pattern. Every nonzero
element is urv for a unitv. Choosing the least such r in a nonzero ideal proves the ideal
classification. Multiplication by a unit is invertible; multiplication byur shifts and truncates the
four u-coordinates, proving the rank and kernel assertions. Ifa2 =a, then eithera is a unit and
a = 1, or 1 +a is a unit anda(1 +a) = 0 givesa = 0.
The unit group has the explicit presentation
A× ={tj(1 + Ω)e : 0≤j <4, e∈{ 0, 1}}∼=C4×C2.
The two cyclic factors have trivial intersection, and the displayed eight elements exhaust the
units. There are four solutions ofa2 = z: t,t 3, 1 +t +t2, 1 +t2 +t3. These are elementary
specializations of group-algebra unit questions, not a new theory of complex square roots.
3 Eliminating the star primitive without changing the model
Let V = F4
2 have basise0,e 1,e 2,e 3, and define the fixed wiring permutation
R =


0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0


.
Fora =∑
iaiti, set
ρ(a) =
3∑
i=0
aiRi. (3.1)
Every displayed operation is now binary matrix addition or multiplication. In particularρ(ab) =
ρ(a)ρ(b) and ρ(a)e0 is exactly the cyclic coefficient vector ofa.
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Boolean CM phase algebra Technical companion
Theorem 3.1(Native gate algebra). The mapρ is a faithful algebra representation with
ρ(A) = CentMat4(F2)(R).
On m coefficient blocks, the binary maps commuting withRm = diag(R,...,R ) are exactly the
regular representations ofMatm(A). Thus a circuit ofA-linear maps and its native binary block
circuit have identical coefficient outputs at every stage.
Proof. The four imagese0,Re 0,R 2e0,R 3e0 are a basis. Therefore the operatorsI,R,R 2,R 3 are
linearly independent, which proves faithfulness. Their multiplication follows the cyclic law. IfG
commutes withR, writeGe0 =∑
iaiRie0. Commutation forcesGRje0 =RjGe0 =ρ(a)Rje0 for
every j, soG =ρ(a). The converse is immediate. In block form, commutation withRm says
precisely that each block commutes withR. The circuit assertion follows by composition.
This resolves two different meanings of “without star.” Compiling⋆ intoρ removes a primitive
but leaves the model unchanged. Allowing every binary linear gate, including gates outside this
centralizer, changes the model. The second move may be useful, but conclusions proved using
A-linearity must then be checked again. Section 7 identifies the exact larger gate class that still
preserves the square-zero obstruction.
An explicit star-free interferometer. Let U0 = I4 +R and B =
(
I4 0
U0 I4
)
, so B2 = I8.
Starting with (e0, 0) and applyingB, then diag(Rj,I 4), thenB, yields
(
Rje0, (I4 +R)(I4 +Rj)e0
)
. (3.2)
Forj = 0, 1, 2, 3, the second component represents0,δ, Ω,tδ, with Hamming weights(0, 2, 4, 2).
No convolution primitive, complex scalar, or trigonometric operation is needed. These four
integers are pattern statistics, not probabilities.
4 Readout, isometries, and two-branch classification
4.1 Three different readouts
Literal coefficient inspection, integer Hamming weight, and the algebraic self-productN(a) = ¯aa
are different operations. Here¯t =t−1 =t3, extended linearly, is the conjugation corresponding
to the transpose of the displayed CM. Alsoρ(¯a) =ρ(a)T.
Writingr =a0 +a2 and s =a1 +a3, direct multiplication gives
N(a) = (r +s)1 +rs(t +t3). (4.1)
Every unit has self-product1. The null-self-product set is the ideal
K = (δ) ={0,δ,tδ, Ω}, K 2 = 0.
Every element ofK is fixed by conjugation. The remaining four scalars have self-productt +t3.
The Hermitian pairing on a free module is⟨v,w⟩ =∑
i ¯viwi. It is nondegenerate: pairing with
coordinate vectors determines every component. It nevertheless has nonzero isotropic vectors
and is not a positive real norm. In particular, zero algebraic self-product does not imply zero
pattern.
Proposition 4.1(Global coefficient-weight isometries). TheA-linear automorphisms ofAm pre-
serving the total cyclic-coordinate Hamming weight on every input are exactly block permutations
with an independent cyclic rotation in each block. Their number ism!4m, hence 32 when m = 2.
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Boolean CM phase algebra Technical companion
Proof. A binary linear Hamming isometry sends every weight-one basis vector to a distinct
weight-one vector, so is a coordinate permutation. By Theorem 3.1, it must commute with
Rm. A permutation commuting withm disjoint four-cycles permutes their orbits and, within
each orbit, is fixed by the image of one coordinate, giving one of four rotations. Every such
permutation is an isometry and commutes withRm.
For clarity, a full integer cyclic pair-correlation profile can be defined by Bj(v,w ) =∑
i coef(vi)TRj coef(wi), evaluated overZ. The same permutations preserve everyBj. Conversely
preserving the whole profile includesB0(v,v ) = wt(v), so yields the same global class. No other,
unspecified meaning of “correlation” is used.
4.2 Normal form for intrinsic unitaries
Call U∈ Mat2(A) intrinsic unitarywhen U†U =I, with conjugate transpose†. This condition
is equivalent to binary orthogonality of its regular block matrix, but it does not imply integer
Hamming-weight preservation.
Theorem 4.2(Complete normal form). LetS = ( 0 1
1 0 ) and Wk =
(1 k
k 1
)
. Every intrinsic unitary
has a unique form
U = diag(a,d )WkSe, a,d ∈A×, k ∈K, e ∈{ 0, 1}. (4.2)
There are512 such matrices, of which128 are monomial and384 nonmonomial. There are96
involutions, including the identity, and72 nonmonomial involutions.
Proof. By (4.1), a column of self-product one has exactly one unit entry and one entry inK.
Orthogonality of the columns prevents their unit entries from lying in the same row: a unit plus
an element of the maximal ideal cannot vanish. After a unique column swap if needed, write
U =
(a ak
dℓ d
)
with a,d units andk,ℓ∈K. Column orthogonality is¯aak +dℓd =k +ℓ = 0, hence
k =ℓ. Conversely, ¯aa = ¯dd = 1, ¯k =k, andk2 = 0 show that every displayed matrix is unitary.
The pattern of unit entries, followed by the two unit entries themselves and either off-diagonal
quotient, proves uniqueness. The total is82· 4· 2 = 512, andk = 0 gives exactly 128 monomial
elements.
Fore = 0, squaring shows that involutivity requiresa2 = d2 = 1. These conditions also
suffice: such units have no linearu-term, soa +d∈K and all off-diagonal terms vanish. There
are four square-one units, giving42· 4 = 64 involutions.
Fore = 1, the diagonal entries ofU 2 equal ad, so the necessary condition isd = a−1. It
suffices becausea +a−1 =a−1(a2 + 1)∈K for every unita, again killing the off-diagonal terms.
This gives8· 4 = 32 more. Exactly 16 + 8 = 24of the 96 havek = 0, leaving72 nonmonomial
involutions.
Unitary subgroups of group algebras are established objects; Balogh and Laver reproduce
a general scalar-unitary order formula that already gives eight in this scalar example [3]. It is
important not to equate that scalar result with the present2× 2 normal form. Nor can arbitrary
local-ring unitary formulas be substituted: the hypotheses in Cruickshank et al. include an
elementb with b + ¯b = 1, and later odd residue characteristic [4]. Hereb + ¯b is always a multiple
of t +t3, so the first condition already fails. The explicit proof above avoids that mismatch; it
does not certify novelty of the small matrix group.
Corollary 4.3(A scoped self-product blind spot). For any intrinsic two-branch unitaryU and
j∈{ 0, 1, 2, 3}, the two components ofU† diag(tj, 1)Ue 0 have self-products(1, 0).
Proof. Substitute (4.2). The diagonal unit factors cancel. Ife = 0, the output is(tj,k (tj + 1)); if
e = 1, it is(1,k (tj + 1)). The first entry is a unit and the second lies inK, so(4.1) applies.
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Boolean CM phase algebra Technical companion
The second component can still be a nonzero pattern. The corollary concerns this preparation,
this two-branch circuit, and this self-product readout. It is not a theorem forbidding all
interference in larger binary or modal models.
4.3 What the finite enumeration adds
Reduction modulo (u) gives
| GL2(A)| =| GL2(F2)| 23·4 = 6· 4096 = 24576.
Indeed each binary entry has eight lifts, and a matrix over this local ring is invertible exactly
when its reduction is invertible. The reproducibility package enumerates all164 = 65536 matrices
and retains their flags, not only successes.
Class Count Evidence
Intrinsic unitaries 512 Theorem 4.2; exhaustive check
Unitary involutions / nonmonomial ones 96 / 72 Analytic normal-form conditions
Global Hamming isometries 32 Proposition 4.1
Weight-four-shell preservers 512 All 70 weight-four inputs
Shell-preserver/unitary intersection 256 Complete operator inventory
Invertible splitters ofe0 22528 Both output entries nonzero
Splitters with detector profile(0, 2, 4, 2) 8192 Four phases, fixed architecture
The two classes of size512 are different. For an invertibleU =
(a b
c d
)
, the second component of
U−1 diag(tj, 1)Ue 0 is
β(1 +tj), β =ac(ad +bc)−1.
This follows by the2× 2 inverse formula. Its four Hamming weights are(0, 2, 2, 2) if ν(β) = 0,
(0, 2, 4, 2) if ν(β) = 1, (0, 4, 0, 4) if ν(β) = 2, and all zero otherwise. For valuation one,βδ = Ω
and the adjacent nonzero products areδ,tδ; the other cases follow directly. These profiles do not
define a probability theory.
5 Shared half-turn tags and Boolean Möbius computation
Use the free moduleA2n
, indexed by subsets of[n]. In this section multiple logical registers are
balanced over the same coefficient ringA. Define
C =
(
1 0
1 1
)
, M n =C⊗n, (Mn)S,T = 1T⊆S.
The tensor notation denotes the Kronecker matrix overA, with the entries0, 1 inherited from
F2. SinceC2 =I, alsoM 2
n =I.
For a Boolean truth vectorf, the vectora = Mnf is precisely its algebraic normal form
coefficient vector. This is the established subset Möbius transform, not a newly introduced
transform [5]. Let
Pf = diag(zf (x)) =I +δ diag(f), δ 2 = 0.
Then
MnPfMne∅ =e∅ +δ ANF(f). (5.1)
To verify this,Mne∅ is the all-ones vector, and multiplication byPf makes it 1 +δf. Applying
Mn gives the displayed identity. The construction is entirely star-free after replacingδ byI4 +R2
in each binary block.
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Worked two-bit example. Forf(X1,X 2) =X1∨X2, ascending truth order is00, 01, 10, 11,
the truth vector is(0, 1, 1, 1), and the ANF isX1 +X2 +X1X2. Thus the output of(5.1), in
subset order∅,{2},{1},{1, 2}, is
(1,δ,δ,δ ).
The nonempty coordinates expose the three nonzero coefficients. A single modal outcome does
not expose all three coordinates.
Kernel identities. The matrixKf =Mn diag(f)Mn has entries
(Kf)S,T =
{∑
T⊆X⊆Sf(X), T ⊆S,
0, otherwise,
and KfKg =Kfg by similarity. Meanwhile(I +δKf)(I +δKg) = I +δKf +g because δ2 = 0.
Conjugated multiplication survives before tagging; its quadratic interaction disappears after using
one shared square-zero tag. This is an algebraic explanation, not a new transform algorithm.
Costs and scope. An explicit output has2n ring coordinates, or4· 2n bits. A subset butterfly
uses n2n−1 coefficient additions. The existence of(5.1) does not make the preparation of an
unknown truth vector, application of a diagonal oracle, or complete pattern inspection free.
Structured symbolic propagation may avoid an explicit table on a particular input, but no
universal compact representation or efficient arbitrary coefficient-query algorithm is established
here.
6 Optimal constant-annihilating detector width
The finite two- and three-variable examples can be closed without an unbounded search. We
first freeze the detector contract and then expose its coding-theoretic content.
Definition 6.1(Parallel linear pattern detectors). Let B be a finite-dimensional commutative
unital F2-algebra, let 0̸= τ∈ B, and setr = dimF2(τB ). For f∈ FN
2 , the tagged pattern is
pτ(f)x = 1 +τfx. Ac-channel detector consists of fixed weightswj,x∈B and outputs
Dj(f) =
N−1∑
x=0
wj,x(1 +τfx),
∑
x
wj,x = 0 (1 ≤j≤c). (6.1)
The weights do not depend onf. The answer is “constant” exactly when allc outputs vanish,
tested by literal pattern inspection. Channels are parallel and no adaptive weight choice or
nonlinear preprocessing is included. Width means the number ofB-valued channels, not oracle-
query complexity, gate count, or storage size.
The zero-sum condition makes both constants vanish. For this algebraic definition1 +τ need
not be a unit; if reversible phase tagging is intended it must be a unit. All tagsu,u 2,u 3 used in
A satisfy that requirement.
Lemma 6.2(Forbidden middle weight; classical bound). If N is a positive multiple of four and
a binary linear subspaceC≤ FN
2 contains no vector of weightN/2, then dimC≤ N/2. The
bound is sharp.
This is the theorem of Enomoto, Frankl, Ito and Nomura [1], also restated as Proposition 2
in Bassalygo, Cohen and Zémor [2]. The result is about forbidden weight, not merely minimum
distance. Appendix A gives a self-contained proof of the specialization needed for power-of-two
lengths and a kernel containing1. The general theorem, rather than a claim to rediscover it, is
used below.
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Theorem 6.3(Exact detector width). For positive evenN, under Definition 6.1, the minimum
number of channels that separates constant from balanced vectors is
cmin(N,τ ) =



1, N ≡ 2 (mod 4) ,
⌈N/(2r)⌉, N ≡ 0 (mod 4) .
(6.2)
In particular, forA = F2[u]/(u4), τ =δ =u2, andN = 2n with n≥ 1,
cmin(n) =
⌈
2n−2
⌉
. (6.3)
Proof. The zero-sum condition reduces(6.1) toDj(f) =τ∑
xwj,xfx. Thus the combined output
is a binary linear map
H : FN
2 −→(τB )c, H 1 = 0, rankH≤rc.
Exact separation means thatkerH contains no vector of weightN/2. If 4|N, Lemma 6.2 gives
N− rankH≤N/2, sorc≥N/2.
For attainment, putL = N/2 and chooseL + 1 distinct addressesx0,x 1,...,x L. Choose
a binary basisβ1,...,β r of τB, together with liftsvs∈ B such thatτvs = βs. Form the L
differences hi(f) =f(xi) +f(x0). Pack at mostr differences into each output as∑
sβshi(f). A
packed difference is implemented by giving its two addresses the same weightvs; when several
use x0, their weights atx0 add. Every row therefore has weight sum zero. Linear independence
of theβs makes all outputs zero exactly when allL differences vanish. ThenL + 1 chosen values
are identical, impossible for a balanced vector with exactlyL zeros andL ones. Constants do
vanish. This uses⌈L/r⌉ channels.
If N≡ 2 ( mod 4), give every address weight1 in one channel. The output isτ∑
xfx, since
N is even. Both constants have even weight, but a balanced vector has odd weightN/2, so
its output isτ̸= 0. Zero channels cannot separate the two nonempty promise classes. Finally
dimF2(δA) = 2, giving (6.3) including theN = 2 case.
This is a sharp application of a classical coding theorem to a declared channel contract. The
elementary packing argument also shows why dividing by the full scalar dimension four would be
wrong forδ: every detector output lies in the two-dimensional idealδA.
Explicit four- and eight-address detectors.A basis ofδA is δ,tδ, with lifts1,t. At four
addresses the single row
(1 +t, 1, t, 0)
produces δ(f1 +f0) +tδ(f2 +f0). Vanishing forces three equal bits, excluding balance. At eight
addresses use (
1 +t 1 t 0 0 0 0 0
1 +t 0 0 1 t 0 0 0
)
.
Both outputs vanish exactly whenf0,f 1,f 2,f 3,f 4 are equal, excluding four-versus-four balance.
Theorem 6.3 proves that one channel cannot suffice at eight addresses. At sixteen addresses four
are necessary and sufficient; this closes the earlier two- and three-variable examples within the
same detector model.
Corollary 6.4(Unpromised constancy). For anyN≥ 2, the minimum width in Definition 6.1
for recognizing constancy among all Boolean vectors is⌈(N− 1)/r⌉.
Proof. Exactness now requires kerH = span{1}, so rankH = N− 1≤ rc. Pack all N− 1
differences f(xi) +f(x0) by the same construction to attain the bound.
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Variablesn AddressesN u tag (r = 3) u2 tag (r = 2) u3 tag (r = 1)
1 2 1 1 1
2 4 1 1 2
3 8 2 2 4
4 16 3 4 8
5 32 6 8 16
6 64 11 16 32
These are exact promised-detector widths, not numbers of oracle calls. They quantify information
lost through the tag ideal and recovered through parallel coefficient channels. The abruptN≡ 2
(mod 4) exception in(6.2) also shows why the address-length hypothesis cannot be dropped in a
purported universal lower bound.
7 The exact gate boundary for nilpotent interactions
7.1 What A-linearity guarantees
Suppose fixedA-linear gates are interleaved with activationsI +upDj, where the entries ofDj
may depend on declared Boolean oracle parameters. Expanding the product, any term with
q activations containsupq, since that scalar commutes with every gate. It vanishes forpq≥ 4.
Thus a sharedu tag retains at most three activation factors and a sharedu2 tag at most one.
This bounds degree in the activation parameters, not degree in the input variables inside each
predicate. It is an immediate consequence of nilpotence, not a complexity lower bound.
The u2 case has a sharper characterization even after admitting some non-A-linear binary
gates. Let V =Am viewed over F2, let ∆ be multiplication byδ on every block, and let
Km =δAm.
Theorem 7.1(Exact ideal-preservation boundary). For every binary linear mapG :V →V,
∆G∆ = 0 ⇐⇒ G(Km)⊆Km. (7.1)
Consequently, a circuit of binary linear gates preservingKm, interleaved with activationsI +
bj∆Dj, has no term of activation degree at least two whenever the fixed binary mapsDj also
preserveKm. Here thebj are Boolean parameters. The converse holds at the level of permitted
gates: if a gateG violates the right side of(7.1), two uniform activations can produce a nonzero
mixed coefficient.
Proof. On each block, im ∆ = ker ∆ = K; hence on V both spaces equal Km. Therefore
∆G∆ = 0 exactly whenG maps im ∆into ker ∆, proving (7.1).
A product of ideal-preserving maps preservesKm. In any expanded term with two consecutive
selectedactivations, thesegmentbetweentheleftfactor ∆andthenextfactor ∆isideal-preserving,
including the relevantDj. Equation (7.1) kills that term. Conversely the coefficient ofbc in
(I +c∆)G(I +b∆) is ∆G∆, nonzero when ideal preservation fails.
A reversible star-free counterexample. On one cyclic block, letG swape1 and e2 and fix
e0,e 3. This is a binary coordinate permutation, withG−1 =G and GTG =I. Nevertheless
∆G∆e0 = ∆G(e0 +e2) = ∆(e0 +e1) =e0 +e1 +e2 +e3̸= 0. (7.2)
Thus even a reversible, binary-orthogonal, star-free gate can escape the shared-tag interaction
obstruction. It is notA-linear and is not an intrinsic ring-unitary map of the declared coefficient
module. Calling both models “Boolean” is not enough to transfer a theorem between them.
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For one block there are16 binary maps commuting withR, of which eight are invertible.
There are 4096 maps preservingK, of which576 are invertible. To see the latter counts, use a
binary decompositionV =K⊕L with two-dimensional summands. Ideal preservation gives a
block upper-triangular matrix with three arbitrary2× 2 blocks, hence 212 maps; invertibility
requires both diagonal blocks invertible, giving62· 16 = 576. The symbol⊕ in this paragraph is
a vector-space direct sum, not an additional Boolean connective. Exhaustive enumeration of all
65536 binary maps independently checks (7.1).
7.2 Independent tags retain different information
InA⊗ F2A, the elementδ⊗δ is nonzero. Under multiplication into one shared copy ofA it
becomes δ2 = 0. Thus independent coefficient registers retain an interaction discarded by shared
coefficients. This is a change of tensor base and state space, not an alternative notation for the
same circuit. A full tensor ofm independent phase algebras has binary dimension4m; compressed
representations may behave differently on restricted families, but no universal compression result
follows.
Pointwisemaskingoffers anotherescape, witha differentcost. Fora taggedBoolean zb = 1+δb,
selecting itst2 coordinate and rotating that selected coordinate back extractsb. Decoding two
tags, applying Boolean AND, and retagging computes their conjunction, but the joint map is
nonlinear in the two inputs. The linear no-go theorem does not apply to such a decode–AND–retag
procedure.
8 Interfaces to modal computation and signed lifts
The detector theorem has a different contract from exact modal oracle computation. A zero
detector output can be useful information when inspecting a stored pattern, but a zero branch
is impossible in the usual modal convention; its absence is not a recorded outcome by itself.
A complete linear instrument instead requires the combined mapv↦→(Lbv)b to be injective,
so every nonzero input has at least one possible branch [7]. One must explicitly build and
analyze such an instrument before converting a pattern identity into an exact all-branches-correct
algorithm.
This distinction keeps the present paper separate from the companion exact Bernstein–
Vazirani query analysis [10]. That work concerns characteristic-twofields, recorded complete
instruments, and charged pointwise oracle calls. The present work concerns a ring and literal
linear detector patterns. A pattern that encodes all affine coefficients after one diagonal phase
layer does not refute ann-query lower bound under the former readout contract. Nor does a
constant-annihilating detector refute a one-query Deutsch–Jozsa obstruction there.
The supplied analyzer table. The companion ring example is worth retaining only as a
boundary check. Withχ = (1,z )T and X = ( 0 1
1 0 ), one hasXχ =zχ. However
Hz =
(
1 1
1 z
)
∼ diag(1,δ )
by invertible row and column additions. Itsn-fold Kronecker product over theshared ringA,
expanded over F2, has rank4 + 2n for n≥ 1: the diagonal factors with zero occurrences ofδ
contribute rank four, those with one contribute2n, and all others vanish. Its binary dimension is
4· 2n. This explains the supplied ranks6, 8, 10, 12, 14, 16 without claiming a Fourier analyzer
or importing a field theorem to a ring. The six ranks are independently reconstructed in the
revision verifier.
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Boolean CM phase algebra Technical companion
Changing scalars is not eliminating a primitive.The signed constructionZ[C4]/(1+t2)∼=
Z[i] changes the additive law:t2 becomes−1 rather than the distinct half-turn of a characteristic-
two ring. There is no unital ring homomorphism fromA into a characteristic-zero ring, since
1 + 1 = 0 would have to be preserved. A lift from binary patterns into integer coefficients is
therefore not an XOR-preserving unital embedding. The signed/integral companion [11] and the
present intrinsic algebra can share coordinate notation, but not their amplitude semantics or
probability claims.
Where each paper should own its results. The CM/LM foundations own predicate
semantics, valuation, frame transport and pointwise Boolean superposition [9]. This paper owns
the explicitly added cyclic coefficient algebra and its pattern interfaces. The exact-query paper
owns its field-valued adaptive lower bounds. The signed paper owns the integer quotient and
complex-amplitude translation. Repeating whole modal algorithm or compiler developments
across these papers would obscure, rather than strengthen, this separation.
9 Evidence, limitations, and use
The original finite verifier and its separate report checker were rerun for this revision. A new
standard-library implementation, which imports neither original checker, tests the normal form,
ideal boundary, coding subspaces, optimal constructions, and supplied analyzer ranks. The code,
complete finite operator inventory, exact output summaries and execution logs accompany the
manuscript. Appendix C separates exhaustive universes from bounded regression tests.
The exact detector width closes the original small-example agenda, but its lower-bound engine
is classical coding theory. The normal form turns finite involution counts into a transparent proof,
without claiming that this small unitary group was previously unknown. The ideal boundary
gives a precise test for which broader binary implementations retain the shared-tag truncation.
These are concrete reasons to retain the paper as a technical companion rather than discard it
as notation alone.
No practical advantage over a matched Boolean compiler or symbolic algebra system is
established. A future implementation study would need identical inputs, output contracts,
preprocessing and readout costs, including a baseline allowed the same local Boolean identities.
Adaptive sparse Möbius learning, for example, already has a substantial algorithmic literature;
the recent real-valued additive-query formulation [8] is not an XOR-valued performance baseline
without an explicit model translation.
Two focused developments are defensible beyond this note: classify useful restricted gate
families between the rotation centralizer and the ideal stabilizer, or identify a concrete retained-
information task on which the tag representation improves a matched implementation. A larger
brute-force catalog, another spelling of convolution, or an uncharged oracle layer would not
establish such an improvement.
A A self-contained power-of-two coding argument
For the detector application atN = 2n, the kernel contains1. The following proof independently
establishes the needed specialization of Lemma 6.2; its statement is already contained in the
stronger classical theorem [1].
Proposition A.1. LetN = 2L with L a power of two. IfC≤ FN
2 contains 1 and no vector of
weight L, then dimC≤L.
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Boolean CM phase algebra Technical companion
Proof. Let eL(X) be the degree-L elementary symmetric polynomial inN variables and set
Q(X) =eL(X) +eL(X + 1).
Its homogeneous degree-L terms cancel, sodegQ≤L− 1. On a binary vector of weightw, eL
has value
(w
L
)
mod 2. ForL = 2a, the identity(1 +T )w =∏
j:wj =1(1 +T 2j
) over F2 shows that
this value is theath binary digit ofw. For 0≤ w≤ 2L, it is one exactly whenL≤ w <2L.
Hence
Q(x) =
{
0, wt(x)∈{ 0,L, 2L},
1, otherwise.
Choose C = span{1}⊕W, and writem = dimW = dimC− 1. Every nonzerox∈W has weight
different from 0,L, 2L. ThereforeQ restricted toW∼= Fm
2 is zero only at the origin. Ifm = 0
the conclusion is immediate. Otherwise its unique multilinear representation is
1 +
m∏
j=1
(1 +Yj),
which has degreem. Linear substitution into Q and reduction byY 2
j = Yj cannot increase
degree, som≤ L− 1. Thus dimC = m + 1≤ L. Uniqueness of multilinear representation
follows, for example, by induction onm, separating a polynomial intoP0 +YmP1 and evaluating
at Ym = 0, 1.
The complementary-space hypothesis is essential to this particular proof presentation. The
stronger classical theorem needs no assumption that1 belongs toC and covers every length
divisible by four. The sharp construction used in Theorem 6.3 has kernel dimensionN/2, so no
asymptotic estimate is being substituted for the exact finite bound.
B Local CM nonlinearity and Boolean polynomial compilation
For (2.1), write
α =d, β =b +d, γ =c +d, κ =a +b +c +d.
Expansion gives
fC(X,Y ) =α +βX +γY +κXY. (B.1)
The parityκ is both the quadratic ANF coefficient of this two-input truth table and its augmen-
tation as a cyclic coefficient. Proposition 2.1 therefore says that the corresponding phase scalar
is a unit exactly when thislocal two-input predicateis nonaffine. This does not imply that every
substituted expression is nonlinear: substituting the same Boolean variable into both inputs of
AND givesX 2 =X.
ReplacingX,Y by Boolean polynomialsPX,PY in (B.1) is a correct local compiler rule in the
quotientX 2
i =Xi. Likewise, two CMs on the same ordered operands admit pointwise Boolean
folding before expansion. These are exact Boolean identities, not unique compiler optimizations.
The finite verifier checks all 64 elementary CM/input cases, 1500 internal nodes across 50 seeded
seven-variable DAGs, and 16384 folding valuations. The node count is not a count of 1500
independent programs. Synthetic savings against an intentionally naive expansion backend do
not establish practical superiority over optimizers allowed the same identities.
C Finite verification and reproducibility
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Boolean CM phase algebra Technical companion
Check Cases Scope
Scalar multiplication 256 All ordered pairs
Associativity/distributivity 4096 All triples
Two-branch operators 65536 All Mat2(A)
Binary orthogonality cross-check 65536 Separate inherited report
checker
Phase/ANF identities 65812 Every Boolean function,
1≤n≤ 4
Kickback branch checks 2120 Original finite branch
universe
Subset-kernel multiplication 65808 All function pairs,
1≤n≤ 3
Nilpotent regression circuits 200 Bounded seeded tests
New binary ideal-boundary test 65536 All Mat4(F2)
New unitary normal-form comparison 65536 All Mat2(A)
New subspace enumeration,N = 2, 4, 8 29230 All subspaces containing 1
New optimal promised detectors 52776 Every promised input at
evenN≤ 16, three tags
New unpromised-constancy detectors 262140 Every input at even
N≤ 16, three tags
Companion analyzer ranks 6 n = 1,..., 6
The three new tags areu,u 2,u 3; repeated input checks with different tags are counted
separately. For subspace enumeration the totals are2, 16, 29212, respectively, and maximal
balanced-avoiding dimensions are1, 2, 4. These finite universes do not replace the general proofs.
The old two-bit marked-coordinate-peak scan remains bounded evidence for its specified local
architecture, not a general search impossibility theorem.
From the package root, the new checks need only a recent Python 3 standard library:
python reproducibility/code/verify_revision.py
The inherited full inventory uses NumPy and is reproduced by
cd reproducibility/legacy
python code/verify_intrinsic.py
python code/check_report.py
The canonical revised manuscript is this single file; its bibliography is embedded below. Compile
twice with pdflatex -halt-on-error CM_Phase_Algebra_Revised.tex . The audit directory
contains review responses, overlap and priority records, and a claim crosswalk; it is not part of
the submission manuscript. Original inputs and hashes are preserved separately. The verification
programs make no network or account calls. Reported execution time is a reproducibility
observation, not a benchmark.
Provenance. This is an AI-assisted technical revision prepared for author review. Specialist
and adversarial assessments in the accompanying report are distinct analytical perspectives from
one assistant, not independent external referee reports. Historical firstness, external peer review,
and empirical performance superiority are not certified by the supplied tests.
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Boolean CM phase algebra Technical companion
References
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torics 3 (1987), 25–38. https://doi.org/10.1007/BF01788526.
[2] L. Bassalygo, G. Cohen and G. Zémor, “Codes with forbidden distances,”Discrete Mathematics213
(2000), 3–11. https://doi.org/10.1016/S0012-365X(99)00161-2.
[3] Z. Balogh and V. Laver, “Unitary subgroups of commutative group algebras of characteristic two,”
arXiv:2007.05819 (2020). https://arxiv.org/abs/2007.05819.
[4] J. Cruickshank, A. Herman, R. Quinlan and F. Szechtman, “Unitary groups over local rings,”
arXiv:1303.5106 (2013). https://arxiv.org/abs/1303.5106.
[5] M. Barbier, H. Cheballah and J.-M. Le Bars, “On the computation of the Möbius transform,”
arXiv:2004.11146 (2020). https://arxiv.org/abs/2004.11146.
[6] B. Schumacher and M. D. Westmoreland, “Modal quantum theory,”Foundations of Physics42
(2012), 918–925. https://doi.org/10.1007/s10701-012-9650-z.
[7] B. Schumacher and M. D. Westmoreland, “Almost quantum theory,” inQuantum Theory: Informa-
tional Foundations and Foils, Springer (2016), 45–81; arXiv:1204.0701.https://arxiv.org/abs/
1204.0701.
[8] Y. E. Erginbas, J. S. Kang, E. Polito and K. Ramchandran, “Adaptive sparse Möbius transforms for
learning polynomials,” arXiv:2602.06246 (2026).https://arxiv.org/abs/2602.06246.
[9] B. Droncheff,Correspondence and Logical Matrices: A Boolean Operator Calculus. Supplied revised
manuscript, September 2026. Cited for the project-specific CM/LM definitions and scope; not asserted
to be published.
[10] Exact Bernstein–Vazirani Query Complexity in Characteristic-Two Modal Computation. Supplied
companion manuscript, revised 27 September 2026. Cited by title as an unpublished project compan-
ion.
[11] B. Droncheff,Signed and Integral Lifts of Correspondence Matrices. Unpublished project companion,
27 September 2026. The comparison here concerns the scalar construction announced in its abstract.
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