Correspondence and Logical Matrices: A Typed Boolean Operator Calculus

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Correspondence and Logical Matrices:
A Typed Boolean Operator Calculus
Formula-valued logical matrices, valuation, pairing, and coherent operator superposition
Brian Theory
September 2026
Abstract
We develop a typed Boolean operator calculus centered on formula-valued logical matrices
(LMs) and their relation to numeric correspondence matrices (CMs). For a binary connectiveΘ,
the LM
[MXΘY ] =
[ XΘY X Θ¬Y
¬XΘY ¬XΘ¬Y
]
records the complete polarity orbit of the logical operandsX,Y . It admits the invertible
Boolean-frame normal form
[MXΘY ] = SX[Θ] SY, SX =
[
X ¬X
¬X X
]
, S2
X =I,
so valuation sends the symbolic operator to the corresponding numeric CM in its valued signed
frame. Logical pairing has the exact agreement-bit semantics
⟨A|[MXΘY ]|B⟩≡(A⇔X) Θ (Y⇔B),
and labelled basis pairings reconstruct the connective.
The central operation is Boolean superposition of aligned logical operators. If an outer Boolean
operation g is applied pointwise to formula-valued LMs, the LM lift, valuation, common signed
input-frame transport, and logical pairing all preserve the same operation. The corresponding
CM-level entrywise law is a valued corollary and has direct antecedents in truth-vector Boolean
calculus; the focus here is the formula-valued LM architecture and the coherence of its symbolic,
relational, and numeric semantics. We extend the construction to arbitrary-arity logical tensors
and rectangular correspondence maps, give conditional block constructions, and characterize
conjunctive/XOR separability by ordinaryF2 matrix rank.
The underlying truth-table, Boolean-algebra, group-action, and pointwise-composition in-
gredients are classical. The paper therefore claims a unified formula-valued symbolic/numeric
representation calculus and precise compatibility laws rather than a new underlying Boolean
algebra. No physical measurement interpretation, truth-table compression, or computational
speedup is assumed.
Contribution and scope. The contribution claimed here is a typed synthesis: a formula-valued
polarity lift, an explicit Boolean frame normal form, agreement-bit pairing, signed-frame transport,
and coherence identities linking symbolic LMs to numeric CMs. The constituent truth-table,
Boolean-matrix, group-action, pointwise-composition, tensor-flattening, and rank mechanisms have
substantial prior art, so the paper does not claim firstness for those ingredients. Its mathematical
claims are the stated representation identities and their integration into one frame-aware calculus.
The representation is exact rather than compressed, and no computational speedup or physical
measurement interpretation is assumed.
1

1 Introduction
Every binary Boolean connective has four output values. A truth table records those values as
data; a correspondence matrix records them in a2×2 array whose row and column axes are tied
to declared operands. The information content is unchanged, but the array can participate in an
operator calculus. True-first Boolean state vectors select entries; transposition exchanges operand
axes; row and column reversals absorb input negations; output complement negates the connective;
and compatible operators can be combined before any particular operands are evaluated.
A small alignment failure motivates making the frame part of the type. The two occurrences
in (X⇒Y )⇕(Y⇒X) both use the local implication table. XORing those two unlabelled arrays
would therefore give zero. In a common(X,Y ) frame, however, the second occurrence is[⇒]T, and
[⇒]ˆ⇕[⇒]T = [⇕], correctly yieldingX⇕Y. Section 5 carries this example through the symbolic
LM, valuation, and pairing layers.
The notation follows the author’s original manuscript [10]. A Greek letter such asΘ denotes a
logical operator variable, and[Θ] denotes the correspondence matrix representing that operator.
Thus Θ =⇔is a connective while
[⇔] =
[
1 0
0 1
]
is its CM. When arbitrary arity is required we use ordinary function notationf : Bn→B; for a binary
named connective,fΘ (x,y ) =xΘy. This separates the connective from its matrix representation
without introducing a second unrelated symbol for the binary CM.
Throughout the paper, uppercase symbols such asX,Y,A,B denote logical operands or formulas,
whilelowercasesymbolssuchas x,y,a,b,c,d denoteBooleanvalues, assignments, orscalarcoefficients.
The uppercase operands need not be atomic propositional variables: they may be arbitrary formulas.
The present theory nevertheless keeps LM entries in the Boolean formula algebra; an enrichment in
which the operands or entries are themselves matrix-valued operators is a separate extension and is
not assumed here.
Earlier correspondence-matrix work introduced the terminology and several constructions used
here, including binary CMs, formula-valued LMs, matrix transformations, logical pairing, and
preliminary higher-dimensional examples [10]. The present paper restates only the mathematically
supported parts in an explicit type system. In particular, the former arithmetic-modulo and
measurement analogies are not premises of the calculus; higher-dimensional objects are treated as
2n-entry truth tensors with declared matrix flattenings.
The central structural claim is not that truth tables, Boolean matrices, tensor flattenings, or
pointwise composition are new. The distinctive focus is the formula-valued LM layer and the
compatibility of its semantics with the numeric CM layer. The architecture organizes several levels
of the same Boolean function:
1. formula-valued logical operators: for example [MX∨Y ], whose four entries areX∨Y,
X∨¬Y,¬X∨Y, and¬X∨¬Y;
2. valuation to numeric correspondence operators:a valuation sends an LM to a four-bit
CM such as[⇒] =
[ 1 0
1 1
]
, with the valued operand polarities recorded by row/column reversals;
3. logical pairing: contraction of a formula-valued LM against formula states, producing an
exact relationship formula rather than merely selecting a stored bit;
4. LM operator superposition:aligned formula-valued logical operators themselves become
operands of another Boolean operation, producing a new LM; the familiar CM entrywise law is
its valued numeric image;
2

5. signed-frame actions:input permutations and polarity changes realized coherently as symbolic
substitutions and numeric matrix/tensor reindexings; and
6. higher-arity logical tensors and correspondence maps:the same construction extends to
2n symbolic polarity slots and to typed rectangular flattenings.
The point of the calculus is that these maps and operations are linked by commuting identities
rather than merely listed side by side.
A further goal is to make the binary construction scale cleanly. Ann-variable function has a
canonical 2×···×2 correspondence tensor. A formula-valued version is its symbolic polarity orbit.
A row/column bipartition ofp andq variables reshapes the tensor into a2p×2q correspondence map.
Square flattenings are endomorphisms; rectangular flattenings are linear maps between different
Boolean state spaces. Both still support a typed bra–matrix–ket contraction.
Matrix representations of logic have extensive antecedents. Boolean matrix methods predate
CMs [1]; matrix and vector logic treat logical connectives as operators [2, 3, 4]; semi-tensor-product
methods use logical structure matrices [5, 7]; and explicit XOR–AND Boolean matrix calculi also
exist [6]. Most importantly for the raw numeric operator-on-operator law, Cheng, Zhao, and Xu
state that if two Boolean functions of the same ordered variables have truth vectorsmf,mg, then
the truth vector off Φ g is obtained by applying the outer logical operatorΦ entrywise tomf,mg [6,
Prop. 3.3]. Under a2×2 reshape this is the numeric same-frame identity used here. In the inspected
Cheng source, however, the relevant construction is expressed in truth-vector/Boolean-matrix
language; we did not locate a formula-valued LM polarity lift, the present logical-pairing identity, or
a bra–matrix–ket semantic layer there. Accordingly, pointwise combination of aligned numeric truth
values is not claimed as new; the present paper centers the formula-valued LM architecture and the
way valuation, pairing, and typed frame transport preserve the same operation.
Bricken’s internally dated March 1997 notes are another close antecedent: they arrange the
sixteen binary Boolean relations as2×2 matrices, use Boolean matrix algebra, explicitly evaluate
logical operators in bra–matrix–ket form⟨x|B|y⟩, and treat logical matrices as operands of matrix
operations [14]. The original public-release date and peer-reviewed status of that technical note
should not be inferred from the current website alone. Thus neither Dirac-style notation,2×2
logical matrices, nor the broad idea that matrices representing logic can themselves be manipulated
is claimed as new. NPN transformations and packed truth-function manipulation are likewise
standard in logic synthesis [15, 16].
Eigenlogic supplies a second important boundary. Its published treatment represents logical
propositions by commuting projection operators whose truth values are eigenvalues and whose
canonical interpretations are eigenvectors [8, 9]. The CM/LM contribution pursued here is different
and narrower: formula-valued polarity operators, their valuation to numeric CMs, exact logical
pairing, typed frame alignment, and a coherence law showing that Boolean operations on operators
are respected by valuation, frame changes, and pairing. The abstract orbit embedding and polarity-
equivariance facts are standard group-action structure and are used for characterization rather than
historical priority.
2 Boolean scalar algebra and correspondence operators
2.1 Three distinct compositions
The same finite arrays participate in several different operations, and the paper keeps them explicitly
separate.
3

1. XOR–AND matrix productAB contracts an intermediate matrix index overF2.
2. Pointwise Boolean operator superpositionAˆΦ B applies an outer Boolean connective Φ to
corresponding entries of two operators in the same declared frame.
3. Boolean-function superpositionh =g(f1,...,fm) substitutes the outputs of Boolean functions
into an outer Boolean functiong.
The representation theorems below realize the third operation by the second when the operators
are aligned; they do not identify either of those operations with ordinary matrix multiplication. A
separate matched-intermediate-frame theorem later explains when the first operation also transports
cleanly through the LM representation.
2.2 States, one-hot selection, and binary CMs
Let B ={0, 1}and regard it asF2 when addition and multiplication are used. Addition is XOR,
written⇕, and multiplication is conjunction. The nonstandard XOR glyph is intentional: in the
declared true-first CM frame, the XOR matrix is the literal quarter-turn rotation of the XNOR
matrix,
[⇔] =
[
1 0
0 1
]
, [⇕] =
[
0 1
1 0
]
= rot([⇔]). (1)
The notation is paper-specific rather than a claim of standard usage. Logical equivalence of formulas
is written≡; the connective itself is written⇔.
For a bitx∈B, define the true-first state
|x⟩=
[
x
¬x
]
, ⟨x|=
[
x ¬x
]
. (2)
These areone-hot states: exactly one component is1. Thus|1⟩= (1, 0)T and|0⟩= (0, 1)T. Bra–ket
notation is used only as compact row/map/column notation in a Boolean algebra; no Hilbert-space
or physical interpretation is assumed.
Let Θ be a binary logical operator variable. Its correspondence matrix is
[Θ] :=
[
1Θ1 1Θ0
0Θ1 0Θ0
]
=
[
Θ 11 Θ 10
Θ 01 Θ 00
]
. (3)
The brackets are part of the notation:Θ is the logical operator, while[Θ] is its CM representation.
For general functional notation, writefΘ (x,y ) =xΘy.
Proposition 2.1(Unique binary representation). Once the operand axes and true-first order are
fixed, Θ ↦→[Θ] is a bijection between the sixteen binary Boolean operators and the sixteen matrices
in B2×2.
Proof. A binary operator is determined by its four values on(1, 1), (1, 0), (0, 1), (0, 0), exactly the
four entries of[Θ].
For anyC∈B2×2 define the XOR–AND contraction
EC(x,y ) =⟨x|C|y⟩= ⇕
r,s∈{1,0}
χx(r)∧Crs∧χy(s), (4)
where χx(r) = 1 exactly whenr =x.
4

Proposition 2.2(CM selection). For every binary operatorΘ and x,y∈B,
⟨x|[Θ]|y⟩=xΘy.
Proof. Both states are one-hot. Every term in(4) vanishes except the term indexed by(r,s) =
(x,y ).
Remark 2.3 (Why XOR–AND rather than OR–AND?). For one-hot basis states, XOR–AND and
OR–AND selection return the same selected coefficient because at most one product term can be
nonzero. For example, if[Θ] =
[a b
c d
]
, then⟨1|[Θ]|0⟩=b under either reduction. This observation
already appears in the historical manuscript [10].
The two algebras are not interchangeable in general. Withu = (1, 1) and v = (1, 1)T,
u·XOR/ANDv = 1⇕1 = 0, u ·OR/ANDv = 1∨1 = 1.
Thus arbitrary vector or matrix multiplication can differ as soon as more than one term survives.
XOR is retained because it gives theF2 coefficient algebra, coefficient-space linearity, cancellation,
and a single convention shared by the numeric and symbolic calculus. Short-circuit evaluation of
the logical connective OR is a separate implementation issue and does not justify replacing the
contraction algebra globally.
2.3 Basis decomposition and Boolean closure at the coefficient level
Let Ers =|r⟩⟨s|for r,s∈{1, 0}. These four one-feature matrices form the standard basis ofF2×2
2 .
Proposition 2.4(CM basis decomposition). Every binary CM has the unique expansion
[Θ] = ⇕
r,s∈{1,0}
ΘrsErs. (5)
This recovers one of the original CM constructions in a conventional vector-space form: a
connective is the XOR sum of exactly those minterm positions on which it is true [10].
The contraction is linear in the coefficients of[Θ] over F2 for fixed states. This does not imply
that the represented Boolean function is affine in its logical inputs.
2.4 Operand frames and the internal swap matrix
A CM occurrence is typed by its operand frame: row expression, column expression, input polarities,
and truth-state order. Define the transformed connectives by their truth functions
Θτ(x,y ) :=yΘx, Θ¬L(x,y ) := (¬x)Θy, Θ¬R(x,y ) :=xΘ(¬y).
Then
[Θτ] = [Θ]T, [Θ¬L] = [⇕][Θ], [Θ¬R] = [Θ][⇕]. (6)
The matrix products in(6) are XOR–AND products. Here[⇕] plays two compatible roles: as the
CM of the XOR connective and, under matrix multiplication, as the two-state permutation matrix
that exchanges true and false. This removes the need to introduce an unrelated symbolP or S for
the same 2×2 array.
Output complement is entrywise. IfJ is the all-ones2×2 matrix, then
[¬Θ] = J⇕[Θ]. (7)
5

2.5 Support order and Boolean difference
The original manuscript used the language of operator containment and “quotienting.” The sound
part is most cleanly stated as a support order. For CMsA,B∈B2×2 write
A⪯B ⇐⇒Ars≤Brs for everyr,s,
so supp(A)⊆supp(B). Define the Boolean support difference
B\A :=B∧¬A (8)
entrywise. For example,
[∨]\[∧] = [⇕].
This is ordinary Boolean set difference on truth support, not arithmetic remainder or modulo. It is
useful for decompositions and comparisons, but it is not a new scalar operation.
logical operator
Θ
numeric CM
[Θ]
formula-valued LM
[MXΘY ]
valued CM
v([MXΘY ])
numeric representation
symbolic polarity lift
general valuation
signed frame transport
all-true valuation
Figure 1: Binary CM/LM architecture. The connectiveΘ, its numeric matrix[Θ], and its formula-
valued LM are distinct typed objects. For free reference generators the all-true valuation returns
[Θ]; arbitrary valuations return signed-frame versions.
3 Formula-valued logical matrices
3.1 Binary symbolic lift
LetF be the Boolean algebra of formulas modulo logical equivalence. Unless stated otherwise,X
and Y in the defining LM are distinct free reference generators. The same formulas may later be
specialized to compound operands; in that case general valuation remains valid, but an “all-true”
valuation exists only when those substituted references are jointly satisfiable. For a formulaA and
polarity bitb∈B, write
A⟨b⟩=
{
A, b = 1,
¬A, b = 0. (9)
For a binary operatorΘ, define the formula-valued logical matrix
[MXΘY ] :=
[
XΘY X Θ¬Y
¬XΘY ¬XΘ¬Y
]
. (10)
6

Its entries are formulas, not numeric coefficients. For example,
[MX∨Y ] =
[
X∨Y X ∨¬Y
¬X∨Y ¬X∨¬Y
]
.
Thus the LM packages the complete polarity orbit of one reference formula.
3.2 Formula-valued frame normal form
There is a compact matrix factorization behind the four displayed formulas. Whenever juxtaposition
of formula-valued matrices is used in this paper, it denotes XOR–AND matrix multiplication over
the Boolean ring (F,⇕,∧); in particular this convention is reused in the matched-frame kernel
theorem. It is distinct from the hatted pointwise operator superposition introduced in Section 5.
Define the formula-valued frame matrix
SX :=
[
X ¬X
¬X X
]
. (11)
Because X∧¬X = 0 and X⇕¬X = 1,
ST
X = SX, S2
X =I. (12)
Theorem 3.1(Binary LM frame normal form). For every binary connectiveΘ,
[MXΘY ] = SX[Θ] SY. (13)
Consequently,
[Θ] = SX[MXΘY ]SY. (14)
The reconstruction identity holds in the formula Boolean ring and therefore does not require an
all-true valuation ofX and Y.
Proof. The upper-left entry ofSX[Θ] SY is the disjoint-minterm expansion
Θ 11XY ⇕Θ 10X¬Y⇕Θ 01¬XY ⇕Θ 00¬X¬Y,
which isXΘY. The remaining three entries are obtained by reversing theY polarity, theX polarity,
or both. Equation (14) follows fromS2
X = S2
Y =I.
The normal form will also make valuation and logical pairing transparent: valuatingSX gives
eitherI or [⇕], while multiplication bySX converts an external formula state into its agreement state
relative toX. Orthogonal Boolean expansions and Boolean matrix/inner-product constructions
provide close classical mechanisms for such partition-valued decompositions [18, 17]. In particular,
Example 2.13 of Gudder and Latrémolière cyclically extends a stochastic Boolean vector to an
orthonormal basis; forn = 2 and the stochastic vector(X,¬X) this produces exactly the two
complementary rows ofSX. Their Boolean-space multiplication uses join and meet rather than
the global XOR–AND coefficient algebra fixed here. ThusSX itself is not presented as a new
Boolean-space construction; Equation(13) uses that frame inside the specific CM/LM polarity
architecture.
7

3.3 Valuation to correspondence matrices
Let v :F→B be a Boolean valuation withv(X) =x and v(Y ) =y.
Theorem 3.2(LM-to-CM valuation architecture). For every binary operatorΘ and every valuation
v with v(X) =x, v(Y ) =y,
v([MXΘY ]) = [⇕]1−x[Θ][⇕]1−y. (15)
If X,Y are free reference generators, the all-true valuationvT (X) =vT (Y ) = 1 exists and gives
vT ([MXΘY ]) = [Θ].
For substituted reference formulasU,V , the corresponding all-true specialization is available only
when U and V are jointly satisfiable; Equation (15) remains valid for every actual valuation
regardless.
Proof. Apply v to the frame normal form. Since
v(SX) = [⇕]1−x, v (SY ) = [⇕]1−y,
Equation (15) follows immediately from Equation (13).
This map is conceptually important because it connects two different types of object without
changing the logical operator: a symbolic polarity matrix becomes the numeric CM for the same
connective. It is not a Möbius transform and does not produce ANF coefficients.
4 Logical pairing
For a formulaA∈F, define
|A⟩=
[
A
¬A
]
, ⟨A|=
[
A ¬A
]
.
The Boolean contraction of two formula states is XNOR/equivalence:
⟨A|B⟩= (A∧B)⇕(¬A∧¬B)≡A⇔B. (16)
Theorem 4.1 (Logical-pairing identity). For every binary logical operator Θ and formulas
A,B,X,Y ,
⟨A|[MXΘY ]|B⟩≡(A⇔X) Θ (Y⇔B). (17)
Proof. The frame matrices turn the external states into agreement states:
⟨A|SX =⟨A⇔X|, SY|B⟩=|Y⇔B⟩.
Using Equation (13) and the numeric/formula selection rule therefore gives
⟨A|[MXΘY ]|B⟩=⟨A⇔X|[Θ]|Y⇔B⟩≡(A⇔X)Θ(Y⇔B).
8

Corollary 4.2(Labelled basis reconstruction). Fori,j ∈B,
⟨X⟨i⟩|[MXΘY ]|Y⟨j⟩⟩≡Θij. (18)
Hence the four labelled basis pairings recover the complete truth table and uniquely determine the
binary connective. This is an elementary reconstruction statement, not a claim of new quantum
tomography.
Example 4.3(Pairing an implication LM). For Θ =⇒,
⟨A|[MX⇒Y ]|B⟩≡(A⇔X)⇒(Y⇔B).
TakingA =¬X and B =Y gives 0⇒1 = 1. Taking insteadA =X and leavingB arbitrary gives
⟨X|[MX⇒Y ]|B⟩≡Y⇔B,
so the output records whether the right paired formula agrees with the reference conclusionY.
Logical pairing and operator-on-operator Boolean superposition are different operations. Pairing
places an LM between formula states and returns a formula describing their relationship. Pointwise
operator superposition, developed next, uses one or more aligned CMs or LMs themselves as
operands of another Boolean operation and returns a new operator. Ordinary XOR–AND matrix
composition is treated separately.
Theorem 4.4(Valuation commutes with logical pairing). For every valuationv,
v(⟨A|[MXΘY ]|B⟩) =⟨v(A)|v([MXΘY ])|v(B)⟩. (19)
Proof. The pairing is built from complement, conjunction, and XOR, all preserved by Boolean
valuation.
Valuation therefore acts as a semantics-preserving bridge: one may pair formulas symbolically
and then valuate, or valuate the LM and states first and perform the corresponding numeric
contraction.
4.1 Partition selectors and preservation of Boolean operations
Logical pairing is a special case of a more general selector mechanism. LetI be a finite index set
and letwi∈Fbe selector weights. Define
qw(T) := ⇕
i∈I
wi∧Ti, T ∈FI. (20)
Theorem 4.5(Partition-of-unity selector characterization). Suppose the weights satisfy
wi∧wj = 0 ( i̸=j), ⇕
i∈I
wi = 1. (21)
Then qw is a unital Boolean-algebra homomorphism from the pointwise Boolean algebraFI toF:
it preserves⇕,∧, complement, and therefore every Boolean term operation. Conversely, among
F-linear maps of the form(20), unital Boolean-algebra homomorphisms have exactly such partition-
of-unity weights.
9

Proof. XOR preservation is immediate. For conjunction,
qw(T)∧qw(U) =⇕
i,j
(wi∧wj∧Ti∧Uj) =⇕
i
wi∧Ti∧Ui =qw(T∧U),
because the cross terms vanish. Equation(21) givesqw(1) = 1, so complement is preserved as well.
Conversely, the coordinate idempotents force pairwise orthogonality and unitality forces the weights
to sum to1.
For binary logical pairing the four weights are
A⟨r⟩∧B⟨s⟩, (r,s)∈B2.
They form a Boolean partition of unity even whenA andB are dependent formulas. This theorem is
the structural reason pairing can be moved through an arbitrary pointwise outer Boolean operation
later in the coherence theorem; no special distributive law for that outer operation is required.
5 Formula-valued operators as operands: aligned Boolean super-
position
The historical manuscript emphasized a same-operands result: once two subexpressions have the
same ordered operand frame, the represented operators may themselves be operands of another
Boolean connective [10]. The raw numeric truth-table law has direct antecedents: Cheng, Zhao, and
Xu state for truth vectors on the same ordered variables that an arbitrary binary outer operation
acts entrywise [6, Prop. 3.3]. The present paper therefore treats the CM identity as established
numerical structure and places the formula-valued LM statement first.
The reason the LM formulation is useful is that it makes the symbolic operands, their reference
frame, and their later valuation explicit. A pointwise outer operation is sound only when corre-
sponding cells describe the same logical polarity state. Frame agreement is therefore part of the
type contract rather than an informal convention.
For formula-valued matricesM1,M 2 in one declared frame and a binary Boolean connectiveΦ,
define (
M1ˆΦ M2
)
rs := (M1)rs Φ (M2)rs. (22)
The same notation is used after valuation for numeric CMs. This is not matrix multiplication; it is
the outer connectiveΦ applied cell by cell.
Theorem 5.1 (Aligned LM operator superposition and CM valuation). Let Θ 1, Θ 2 be binary
connectives represented in the same reference frame(X,Y ), and define the derived connectiveΩ by
xΩy := (xΘ 1y)Φ(xΘ 2y). (23)
Then the formula-valued operators satisfy
[MXΩY ] = [MXΘ 1Y ]ˆΦ [MXΘ 2Y ]. (24)
For every valuationv,
v([MXΩY ]) =v([MXΘ 1Y ])ˆΦ v([MXΘ 2Y ]). (25)
For free reference generators under the all-true valuation this reduces to the numeric CM identity
[Ω] = [Θ 1]ˆΦ [Θ 2]. (26)
10

Consequently, for bitsx,y∈B,
(xΘ 1y) Φ (xΘ 2y) =⟨x|
(
[Θ 1]ˆΦ [Θ 2]
)
|y⟩. (27)
Proof. Each LM cell is a formula in one fixed polarity position. ApplyingΦ to corresponding cells
therefore gives the polarity orbit of the derived connectiveΩ, proving(24). Boolean valuation is a
homomorphism, giving(25); the all-true specialization yields(26). Finally, one-hot CM selection
gives (27).
Corollary 5.2(Alignment before operator superposition). Suppose two LM or CM occurrences
arrive in signed framesF1,F 2 and typed frame transformsT1,T 2 transport them to one common
frameF. Their sound pointwise combination in frameF is
(T1M1)ˆΦ (T2M2),
or, after valuation,
(T1[Θ 1])ˆΦ (T2[Θ 2]).
The operation is generally unsound if the same array position refers to different assignments before
alignment.
Example 5.3(End-to-end frame alignment). Consider again
(X⇒Y )⇕(Y⇒X).
The local numeric arrays are both[⇒], but they are typed in frames(X,Y ) and (Y,X ). Combining
them before transport therefore gives the spurious zero matrix. Transporting the second occurrence
to the common(X,Y ) frame gives[⇒]T, and
[⇒]ˆ⇕[⇒]T =
[
0 1
1 0
]
= [⇕].
At the formula-valued level the same typed calculation is
[MX⇒Y ]ˆ⇕
(
[MY⇒X]
)T = [MX⇕Y ]. (28)
Here the transpose is not cosmetic: it changes the second occurrence from its local(Y,X ) coordinates
to the declared(X,Y ) frame. Valuation may be performed before or after Equation(28); the result
is the corresponding signed-frame valuation of the XOR CM, and the all-true valuation returns[⇕]
itself.
Logical pairing preserves the same calculation. Writingp = (A⇔X) and q = (Y⇔B),
⟨A|[MX⇒Y ]|B⟩≡p⇒q,
⣨
A
⏐⏐⏐
(
[MY⇒X]
)T⏐⏐⏐B
⟩
≡q⇒p,
so their XOR isp⇕q, exactly
⟨A|[MX⇕Y ]|B⟩≡(A⇔X)⇕(Y⇔B).
Thus one calculation exhibits the purpose of the typing discipline: a frame error that is invisible in
the unlabelled arrays is detected before superposition, while symbolic lift, valuation, and pairing all
preserve the repaired operation.
The raw numeric equality is therefore not the novelty claim. The structural focus is that
formula-valued logical operators can be combined as first-class operands in a typed common frame,
and the resulting symbolic operation is transported coherently through valuation and, as proved
later, through logical pairing and higher-arity LM lifts.
11

5.1 A distinct operation: matched-intermediate-frame matrix composition
Pointwise superposition should not be confused with XOR–AND matrix multiplication. The frame
normal form nevertheless yields a separate clean law when the output reference frame of one LM is
the input reference frame of the next.
Theorem 5.4(Matched-frame kernel composition). Let Θ, Ψ be binary connectives and defineΩ
by the XOR–AND matrix product
[Ω] := [Θ][Ψ] , x Ωz := ⇕
y∈B
(xΘy)∧(yΨz). (29)
Then, over the formula Boolean ring,
[MXΘY ][MY ΨZ] = [MXΩZ]. (30)
Proof. Using the frame normal form andS2
Y =I,
[MXΘY ][MY ΨZ] = SX[Θ] SY SY [Ψ] SZ = SX([Θ][Ψ]) SZ = [MXΩZ].
This is parity-based kernel composition through the matched intermediate frameY. It is
neither existential relational composition nor the pointwise outer operationˆΦ. Without a matched
intermediate frame, arbitrary products of formula-valued LMs need not remain in the same fixed-
frame LM family.
6 Higher-arity correspondence and logical tensors
The binary notation extends to arbitrary finite arity. The primary higher-arity object is a2n-entry
tensor; a matrix appears after a declared bipartition of the variables.
6.1 Canonical Boolean term extension
LetBn be the set of Boolean functionsf : Bn→B, and letFX denote the Boolean algebra of
formulas generated byX = (X1,...,Xn) modulo logical equivalence. Every such function has a
canonical disjoint-minterm extension to formulas:
˜f(A1,...,An) := ⇕
α∈Bn
f(α)∧
n⋀
i=1
A⟨αi⟩
i . (31)
Distinct minterms are logically disjoint, so OR could replace XOR in this one expansion without
changing its Boolean meaning. The XOR form keeps the expression inside the declaredF2 coefficient
calculus.
6.2 Correspondence and logical tensors
Index assignments byα= (α1,...,αn)∈Bn, using true-first order on every axis.
Definition 6.1(Correspondence tensor). The correspondence tensor off is
Cf[α] :=f(α). (32)
It has shape2×···×2 and exactly 2n Boolean entries. For a binary connectiveΘ, displayingCfΘ
as a 2×2 matrix gives the abbreviated notation[Θ].
12

Definition 6.2(Logical tensor and LM lift). For formal variablesX = (X1,...,Xn) define
Mf(X)[α] := ˜f
(
X⟨α1⟩
1 ,...,X⟨αn⟩
n
)
. (33)
Equivalently define the LM lift
LX :Bn−→FBn
X , LX(f) :=Mf(X). (34)
The binary object[MXΘY ] is the casen = 2 and f =fΘ.
Remark 6.3 (The lift as a polarity pullback). If formulas inFX are identified with their Boolean
functions of an assignmentx∈Bn, then
Mf(x,α) =f(x⇕α⇕1). (35)
Thus the LM lift is a concrete polarity pullback of the ordinary truth function. This makes clear why
its injectivity and equivariance are standard function-algebra consequences; the additional content
developed here is the typed interaction with valuation, pairing, frame transport, and operator
superposition.
Define the higher contraction
EC(x1,...,xn) = ⇕
α∈Bn
C[α]∧
n⋀
i=1
χxi(αi). (36)
Exactly one one-hot product survives, soECf (x) =f(x).
Forδ∈Bn, define the one-triggered axis-reversal operator
(πδT)[α] :=T [α⇕δ], (37)
where⇕is componentwise on assignment indices. Thusδi = 1 means that axisi is reversed.
Theorem 6.4(General valuation theorem). If v(Xi) =xi and 1 = (1,..., 1), then
v(Mf(X)) =π1⇕xCf. (38)
In particular, for the free reference generators the all-true valuation satisfies
v1(LX(f)) =Cf.
Proof. At polarity indexα, the valued argumentX⟨αi⟩
i equals the bitαi⇕(1⇕xi). Hence the valued
tensor entry isCf[α⇕(1⇕x)], which is exactly Equation (38).
6.3 LM orbit lift and Boolean superposition
Definition 6.5(Pointwise lift). Forg : Bm→B and tensorsT1,...,Tm with the same index set,
define
ˆg(T1,...,Tm)[α] :=g(T1[α],...,Tm[α]).
For formula-valued tensors,g is interpreted through its canonical Boolean term extension.
13

Theorem 6.6(LM lift and superposition law). For everyn, the mapLX is injective and preserves
arbitrary Boolean superposition. Iff1,...,fm∈Bn, g : Bm→B, and
h(x) =g(f1(x),...,fm(x)),
then
Ch =ˆg(Cf1,...,Cfm), (39)
LX(h) =ˆg
(
LX(f1),...,LX(fm)
)
. (40)
Hence the image ofLX is closed under every pointwise Boolean operation.
Proof. The identities hold at each assignment/polarity index by substitution intog. For injectivity,
ifLX(f) =LX(h), apply the all-true valuation. Equation (38) givesCf =Ch, sof =h.
Corollary 6.7(The LM image is a Boolean subalgebra). For fixed arityn, Im(LX) is a Boolean
subalgebra of the pointwise formula-tensor algebraF Bn
X and is isomorphic to the Boolean algebra
Bn of n-ary Boolean functions. Under the identification of a function with its truth tensor, all-true
valuation is the inverse of the lift on this image.
Remark 6.8 (Priority boundary for the orbit lift). The mapf ↦→(τ1⇕αf)α∈(C2)n is an instance
of standard function-algebra/group-action structure associated with the polarity group [20]: an
equivariant orbit-valued object is determined by its value at the identity element. Injectivity,
orbit generation, and the resulting equivariance are therefore used here to identify the LM image
cleanly, not as standalone novelty claims. The distinctive question for the present framework is
what additional operations—valuation to CMs, logical pairing, typed frame transport, and Boolean
superposition of the operators themselves—coexist on that orbit object and satisfy common coherence
laws.
6.4 Polarity-equivariance characterization
Forδ∈Bn, letτδnegate Xi exactly whenδi = 1. Writeα⇕δfor componentwise XOR of polarity
indices.
Theorem 6.9(Polarity-equivariance characterization). Every LM lift satisfies
τδ
(
LX(f)[α]
)
≡LX(f)[α⇕δ]. (41)
Conversely, a formula tensorT∈FBn
X belongs to the image ofLX if and only if it satisfies(41) for
every α,δ.
Proof. Negating Xi flips exactly theith polarity choice, proving the forward identity. Conversely,
an equivariant tensor is determined by its all-positive cellT[1]. That formula defines ann-ary
Boolean functionf modulo logical equivalence, and equivariance reconstructs every other cell as the
corresponding polarity substitution, henceT =LX(f).
This characterization is stronger than merely observing a “negative dual” matrix, but its group-
theoretic mechanism is standard: an equivariant orbit is determined by one reference element. Its
value here is classificatory. It gives an intrinsic membership test for formula-valued LMs, shows that
the entire 2n-cell symbolic object is generated from one reference formula by polarity changes, and
makes closure under pointwise Boolean superposition transparent.
14

Remark 6.10 (Boolean clones and superposition). A Boolean clone is a family of finitary Boolean
functions containing projections and closed under superposition. Equations(39)–(40) say that
the LM lifts respect ordinary superposition whenever the displayed functions share the declared
input arity: compose functions first and lift later, or lift the constituent functions and compose the
operators pointwise. A fully internal clone representation across arities would additionally specify
the projection operators and cross-arity substitution maps. That universal-algebra formulation is a
natural extension rather than a prerequisite here [21].
Remark 6.11 (Valuation as an intertwining map). The phrase intertwining means that a map
respects two parallel actions. Here positive valuation intertwines symbolic polarity change with
numeric axis reversal:
v1(τδLX(f)) =πδCf,
and valuation also respects every pointwise superposition:
v
(ˆg(T1,...,Tm)
)
=ˆg(vT1,...,vTm).
Thus the symbolic and numeric levels are not merely analogous; the declared operations are carried
consistently from one level to the other.
Theorem 6.12(CM/LM coherence under Boolean operator superposition). LetM1,...,Mm be
formula-valued logical tensors in the same declared frame, letg : Bm→B, and letˆg act pointwise.
Then the following operations are compatible with operator superposition.
1. Lift: if Mi =LX(fi) and h =g(f1,...,fm), then
LX(h) =ˆg(M1,...,Mm).
2. Valuation: for every Boolean valuationv,
v(ˆg(M1,...,Mm)) =ˆg(vM1,...,vMm).
3. Common input-frame change:for every common signed input reindexingT (permutation and/or
input polarity reversal, excluding output complement),
Tˆg(M1,...,Mm) =ˆg(TM 1,...,TM m).
On the symbolic level the simultaneous reference/index transport is the one specified explicitly in
Equation (56); the polarity mask is not applied twice.
4. Logical pairing: in the binary case, for formula statesA,B and pairing map pA,B(M) :=
⟨A|M|B⟩,
pA,B(ˆg(M1,...,Mm))≡˜g(pA,B(M1),..., pA,B(Mm)).
Thus Boolean operations may be performed on the aligned operators themselves without creating an
inconsistency between symbolic LM semantics, numeric CM semantics, frame transport, and logical
pairing.
Proof. The lift, valuation, and common input-frame statements hold pointwise. Logical pairing
is the partition selectorqw of Equation (20) with weights supplied by the formula states. The
partition-of-unity selector theorem therefore preserves every Boolean term operation, including the
canonical term˜g, which proves the fourth identity directly.
15

Remark 6.13 (Output complement is a different symmetry). Output negation is not one of the
unchanged-g input-frame actions in the theorem. If
gd(b1,...,bm) :=¬g(¬b1,...,¬bm),
then
¬ˆg(M1,...,Mm) =ˆgd(¬M1,...,¬Mm).
In general one cannot complement every input and the output while leaving the same outer operation
g unchanged.
aligned formula-valued LMs
(M1,...,Mm)
LM result
ˆg(M1,...,Mm)
valued CMs
(vM1,...,vMm)
valued result
ˆg(vM1,...,vMm)
LM operator
superposition
valuation valuation
same outer operationˆg
Figure 2: LM-centered coherence under valuation. The same outer Boolean operation may be
applied to aligned formula-valued operators before valuation or to their valued CMs afterward.
Logical pairing and common input-frame pullbacks satisfy analogous squares because they are
Boolean homomorphisms on the pointwise operator algebra.
Remark 6.14 (Pairing preserves the running calculation). The final line of Example 5.3 is an instance
of Theorem 6.12: after the reversed implication has been transported to the common frame, pairing
may be applied before or after the pointwise XOR. The agreement-bit result(A⇔X)⇕(Y⇔B) is
therefore the relational image of the same aligned operator computation, not a separate identity
chosen for the example.
6.5 Higher-arity logical pairing
For formula tuplesA = (A1,...,An) define
Pf(A; X) = ⇕
α∈Bn
Mf(X)[α]∧
n⋀
i=1
A⟨αi⟩
i . (42)
Theorem 6.15(Higher-arity pairing and reconstruction). For everyf : Bn→B,
Pf(A; X)≡˜f(A1⇔X1,...,An⇔Xn). (43)
Valuation commutes with this contraction. Moreover, for each assignmentβ∈Bn,
Pf
(
(X⟨β1⟩
1 ,...,X⟨βn⟩
n ); X
)
≡f(β), (44)
so the complete labelled basis-pairing table reconstructs the truth tensor.
16

Proof. The weightswα:=⋀
iA⟨αi⟩
i form a Boolean partition of unity indexed byα. To make the
selected index explicit, fix an arbitrary valuationv. Exactly one weight is true, namely the one with
αi =v(Ai) for everyi. At that cell,
v(Mf(X)[α]) =f
(
v(A1)⇕v(X1)⇕1,...,v (An)⇕v(Xn)⇕1
)
,
which isf evaluated on then agreement bitsv(Ai⇔Xi). Since this holds under every valuation,
the selected formula is˜f(A1⇔X1,...,An⇔Xn), proving Equation(43). Valuation preservation
follows from the same selector homomorphism. SettingAi =X⟨βi⟩
i makes theith agreement bit the
constantβi, proving Equation (44).
Boolean function
f∈Bn
LX(f)
formula-valued polarity LM
Cf
numeric truth tensor
formula-valued LM lift
all-true valuation
truth-value representation
Figure 3: The LM embedding is faithful because all-true valuation recovers the numeric correspon-
dence tensor. Boolean superposition may be performed before or after the lift.
7 Matrix flattenings and higher-dimensional CMs
A tensor becomes a matrix after the variables are split into a row block and a column block. This is
the clean higher-dimensional generalization of the binary CM, and it also answers a terminology
question: a matrix need not be square to act as a typed map.
Let R = (i1,...,ip) and C = (j1,...,jq) be an ordered partition of{1,...,n}, withp +q =n.
For row bitsr∈Bp and column bitsc∈Bq, let
assembleR|C(r,c)∈Bn
be the unique complete assignment whose coordinatesia equalra and whose coordinatesjb equalcb.
This explicit assembly map avoids any ambiguity when the two blocks are noncontiguous or use a
nonstandard order. Empty blocks are allowed:B0 is the singleton containing the empty assignment
and an empty Kronecker product is the scalar state1. Thus the limiting shapes1×2n and 2n×1
are covered without a separate convention.
Definition 7.1(Correspondence-map flattening). The (R|C) flattening is the2p×2q array
[f]R|C[r,c] :=f
(
assembleR|C(r,c)
)
. (45)
Its formula-valued counterpart[Mf(X)]R|C is defined by the same ordered assembly of the logical
tensor. Computationally this is an axis permutation followed by a reshape ofCf.
17

Example 7.2(A noncontiguous ordered flattening). Let n = 3, takeR = (3, 1) and C = (2), and
set f(x1,x 2,x 3) =x1⇕(x2∧x3). If the row bits arer = (r1,r 2) and the column bit isc, then
assemble(3,1)|(2)(r,c) = (r2,c,r 1).
In true-first row order(11, 10, 01, 00) and column order(1, 0) this gives
[f](3,1)|(2) =


0 1
1 0
1 1
0 0

.
The example shows why the ordered assembly map is part of the type: a noncontiguous flattening
is a precise reindexing, not an appeal to a visually guessed reshape.
The notation[Θ] remains reserved as the binary shorthand. For higher arity, the subscriptR|C
is preferable to a bare size such as[Θ] 4: two different constructions can both be4×4 while having
different types. In particular, a four-variable2|2 correspondence map is not the same object as the
4×4 coefficient-permutation matrix introduced later to rotate the four cells of a binary CM.
Under XOR–AND multiplication,[f]R|C is a linear map
F 2q
2 −→F 2p
2 .
It is an endomorphism only whenp =q. We use “operator” in the broad typed sense of an acting
map; when strict endomorphism terminology matters, we saycorrespondence mapfor a rectangular
flattening.
Let
|xR⟩=
p⨂
k=1
|xik⟩, |xC⟩=
q⨂
ℓ=1
|xjℓ⟩.
These Kronecker products are one-hot states of lengths2p and 2q.
Theorem 7.3(Rectangular bra–map–ket selection). For every bipartitionR|C,
⟨xR|[f]R|C|xC⟩=f(x1,...,xn). (46)
Thus square shape is not required for a scalar bra–matrix–ket contraction; the left and right state
spaces simply have different dimensions whenp̸=q.
Proof. The grouped row and column states are one-hot and select the single matrix entry indexed
by the complete assignment.
Remark 7.4 (Comparison with quantum terminology). In ordinary linear algebra, a rectangular
matrix is a linear map between different vector spaces, whereas a square matrix is an endomorphism.
Quantum observables and POVM effects on one Hilbert space are square; more general quantum
processes can involve maps between input and output spaces of different dimensions. A tensor
reshaping does not automatically preserve a physical-observable interpretation. The present Boolean
construction requires only the typed linear-map statement in(46), not a quantum measurement
analogy.
Remark 7.5 (Exact size boundary). The tensor, a length-2n truth vector, and every2p×2q flattening
contain the same2n Boolean entries. Reshaping may change computational locality, cache behavior,
or available algorithms, but it does not compress an arbitrary truth relation. Any benchmark
advantage of rectangular rather than square flattenings is therefore an implementation result rather
than a smaller information content.
18

order-n truth tensor
Cf
2n entries
typed flattening
[f]R|C
2p×2q
selected value
f(x)
axis permutation
+ reshape ⟨xR|·|xC⟩
Figure 4: Higher-dimensional CMs are typed matrix flattenings of the same2n-entry correspondence
tensor. Rectangular maps remain valid operands in a bra–map–ket contraction.
8 Building higher-dimensional operators from lower-dimensional
ones
Let g : Bp→B and h : Bq →B act on disjoint variable blocks, and letΘ be a binary outer
connective. Iftg,th are the true-first truth vectors, define
OΘ (tg,th)[a,b ] :=tg[a] Θth[b]. (47)
For formula variables on the two blocks, letX⟨a⟩
A denote componentwise polarity substitution, so its
ith component isX⟨ai⟩
i . Define the polarity vectors
ℓg(XA)[a] :=˜g(X⟨a⟩
A ), ℓ h(XB)[b] :=˜h(X⟨b⟩
B ). (48)
Theorem 8.1(Block-lift theorem). If
f(a,b ) =g(a) Θh(b),
then
[f]A|B =OΘ (tg,th), (49)
[Mf(X)]A|B =O˜Θ
(
ℓg(XA),ℓh(XB)
)
. (50)
Valuation commutes with both constructions.
Proof. At the entry indexed by(a,b ), the numeric right side isg(a)Θh(b) =f(a,b ). The formula-
valued statement is the same identity after polarity substitution, and valuation preservesΘ.
Theorem 8.2(Rank and conjunctive/XOR separability). For a fixed bipartitionA|B:
1. f(a,b ) =g(a)∧h(b) for some Boolean functionsg,h if and only if
rankF2[f]A|B≤1.
For a nonzero function the factorization is the ordinary rank-one factorization
[f]A|B =tgtT
h, (51)
and over F2 the nonzero factors are unique.
2. More generally, the minimum integerk≥0 for which
f(a,b ) =
k
⇕
i=1
gi(a)∧hi(b) (52)
equals rankF2[f]A|B. The empty XOR fork = 0 is defined to be0, so the zero function has
minimum k = 0.
19

The formula-valued flattening of a rank-one term factorizes analogously as the outer product of the
two polarity vectors.
Proof. A conjunctive factorg(a)∧h(b) has matrixtgtT
h and therefore rank at most one. Conversely,
every nonzero rank-one matrix overF2 is uvT for nonzero Boolean vectorsu,v, which are the truth
vectors of unique Boolean functions on the two blocks; the zero matrix is obtained by taking one
factor identically zero. For the second statement, every summand in(52) has rank at most one,
so k is at least the matrix rank. A rank factorization supplies exactly that many rank-one outer
products, proving equality.
This gives a precise modern form of the separability/factorization idea already present in the
original LM manuscript [10]. The statement concerns ordinaryF2 rank and XOR decomposition,
not the different OR-based notion usually called Boolean rank. The value also depends on the
chosen bipartition: repartitioning the variables can change the flattening rank.
Example 8.3(A 4×4 correspondence map). Let
g(W,X ) =W⇕X, h (Y,Z ) =¬Y∧Z, f =g⇒h.
In true-first order(11, 10, 01, 00),
tg = (0, 1, 1, 0), t h = (0, 0, 1, 0).
The 2|2 block lift gives
[f](W,X)|(Y,Z) =


1 1 1 1
0 0 1 0
0 0 1 0
1 1 1 1

. (53)
This recovers the useful content of the historical four-variable construction without claiming that
four variables intrinsically require a unique4×4 shape.
9 Signed variable frames in arbitrary arity
For numeric truth tensors, input permutations and polarity changes form the signed-permutation
action. Letσ∈Sn and ϵ∈Bn, and define
ϕσ,ϵ(x)i =xσ(i)⇕ϵi. (54)
Define the numeric pullbackTσ,ϵby
(Tσ,ϵC)[x] :=C[ϕσ,ϵ(x)].
With this convention the pullbacks form a right action:TϕTψ=Tψ◦ϕ. This fixes the composition
direction rather than leaving it implicit.
Theorem 9.1(Signed-frame action). For everyf : Bn→B,
Cf◦ϕσ,ϵ=Tσ,ϵCf. (55)
The transformations form the standard signed-permutation group(C2)n ⋊Sn on the input frame.
Output negation is a separateC2 action on the coefficients.
20

For the symbolic level it is useful to state the transport law without leaving the mask placement
implicit. LetPσpermute tuples by(Pσz)i =zσ(i), and define the signed reference tuple by
ϕσ,ϵ(X)i :=X⟨1⇕ϵi⟩
σ(i) .
Then for everyf : Bn→B and every polarity indexα∈Bn,
LX(f◦ϕσ,ϵ)[α] =LPσX(f)[Pσα⇕ϵ] =Lϕσ,ϵ(X)(f)[Pσα]. (56)
All three formulas are the same Boolean term after substitution. Equation(56) also prevents a
common implementation error: the polarity maskϵmay be represented in the reference tuple or
in the slot index, but not independently in both. Applying the same mask twice cancels it. In
arity one withf(x) =x, X = 1, α= 1, andϵ= 1, the desired transported value is0; a double flip
incorrectly returns 1.
Under a matrix flattening, the same action appears as row/column permutations, transpositions
or block transpositions, and—when a variable moves across the chosen bipartition—a reshape. The
symbolic identity above specifies the corresponding variable renaming and polarity substitution
exactly.
Theorem 9.2(Frame changes commute with pointwise composition). For any common numeric
reindexingT and outer Boolean operationg,
Tˆg(Cf1,...,Cfm) =ˆg(TCf1,...,TCfm). (57)
The formula-valued version holds when the same frame change includes the corresponding variable
relabeling/polarity substitution.
9.1 Relation to NPN transformations
NPN stands for input Negation, input Permutation, output Negation. Two Boolean functions
are NPN-equivalent when one can be obtained from the other by these transformations. NPN
classification is standard in logic synthesis, where it is useful for canonicalizing small truth functions
and reusing optimized implementations [15, 16].
For binary CMs the transformations are visually explicit:
input permutation [Θ] ↦→[Θ]T,
negate left input [Θ] ↦→[⇕][Θ],
negate right input [Θ] ↦→[Θ][⇕],
negate output [Θ] ↦→J⇕[Θ] entrywise.
Thus the CM frame calculus realizes the binary NPN moves as simple matrix transformations. This
is useful for frame normalization and comparison, but the existence of NPN equivalence itself is not
a novelty claim.
10 Relationship to ANF and other coordinate representations
The CM/LM architecture should not be confused with an algebraic-normal-form coefficient vector.
A correspondence tensor stores truth values; ANF stores coefficients of square-free monomials. The
two are related by Boolean Möbius inversion.
21

Write
f(x1,...,xn) = ⇕
S⊆[n]
aS
⋀
i∈S
xi. (58)
If tT =f(1T ) denotes the truth value on the assignment whose support isT, then
tT = ⇕
S⊆T
aS, a S = ⇕
T⊆S
tT. (59)
Proposition 10.1(Top ANF coefficient as correspondence parity). For every Boolean function of
positive arityn≥1,
a[n] = ⇕
x∈Bn
f(x). (60)
Hence degf =n if and only if the correspondence tensor has odd Hamming weight.
Proof. The second relation in(59), applied toS = [n], XORs every truth value. The coefficienta[n]
is one exactly when the full monomial is present.
Corollary 10.2(Binary nonlinearity parity criterion). For the binary correspondence matrix
[f]X|Y =
[
f11 f10
f01 f00
]
,
the coefficient ofXY is
aXY =f11⇕f10⇕f01⇕f00.
Thus a two-variable Boolean function has algebraic degree exactly2 if and only if its CM has odd
Hamming weight.
This parity result belongs naturally in a CM/LM foundations paper because it relates an
immediately visible operator invariant to ordinary Boolean degree. A stronger statement available
in the separate cyclic-lift program says that, after the four truth coefficients are embedded into a
particular rotation-generated algebra, the same parity becomes the unit criterion. That conclusion
depends on the additional algebra and is not used here.
For two variables, one may also compare with an STP logical structure matrix. Under the
common delta encoding1↦→(1, 0)T, 0↦→(0, 1)T and column order(11, 10, 01, 00), if
t = vec([f]X|Y )T = (f11,f 10,f 01,f 00),
then one binary-output structure matrix is
M STP
f =
[
t
¬t
]
.
This is an external representation of the same truth function, not a component of the formula-valued
LM. Likewise ANF coefficients are reached by (59), not by positive valuation.
22

11 A boundary around literal CM rotation
Literal quarter-turn rotation of a displayed2×2 CM is already part of the signed frame geometry. It
should be distinguished from the later step of imposing a cyclic multiplication on the four coefficient
positions.
Define clockwise rotation by
rot
[
a b
c d
]
=
[
c a
d b
]
. (61)
If [Θ′] := rot([Θ]), then the associated truth functions satisfy
fΘ ′(x,y ) =fΘ (¬y,x ). (62)
Thus rot4 =I and the quarter turn is a signed coordinate transformation.
To make the coefficient action explicit, use true-first row-major vectorization
vec([Θ]) = (Θ 11, Θ 10, Θ 01, Θ 00)T.
Then the 4×4 permutation matrix
R⟳ :=


0 0 1 0
1 0 0 0
0 0 0 1
0 1 0 0

 (63)
satisfies
R⟳ vec([Θ]) = vec(rot([Θ])) .
For XNOR,
vec([⇔]) =


1
0
0
1

,
so
R⟳


1
0
0
1

=


0
1
1
0

= vec([⇕]).
This is the worked meaning of
R⟳ vec([⇔]) = vec([⇕]).
The symbolR⟳ is intentionally not written as[Θ] 4: it is an operator acting on the four coefficients of
a binary CM, whereas a4×4 four-variable correspondence map such as[f](W,X)|(Y,Z) acts between
grouped logical state spaces. Equal matrix size does not imply equal type.
The historical Impax observation also sits naturally here. Since[⇔] and [⇕] have complementary
supports,
[⇔]⇕[⇕] =J,
the all-ones tautological CM. The old overlay glyph may be retained as historical notation, whileJ
or [⊤] is algebraically unambiguous.
23

A separate phase construction begins only after an additional multiplication is imposed on
the four cyclic coefficient positions. Identifying those positions with1,g,g 2,g 3 and adding cyclic
convolution yields
F2[C4]∼= F2[g]/(g4−1)∼= F2[u]/(u4), u = 1 +g.
That convolution is not the original2×2 CM matrix product, not pointwise conjunction, and not
LM pairing. Its algebraic consequences therefore lie outside the present calculus. The only fact
needed here is the logical origin of the quarter-turn action as a signed coordinate transformation.
12 Computational interpretation without a performance claim
The operator calculus has an immediate compiler interpretation. If two subexpressions depend on
the same variables but use different operand order or polarity, the signed-frame action can normalize
both to one declared frame. Section 5 then permits the outer connective to be evaluated pointwise
on their operators. This can collapse repeated Boolean work before later expansion or evaluation.
The companion computational manuscript [11] develops this idea as a typed pair compiler with
pure-structural, hybrid-retabulation, whole-retabulation, and fallback paths. The mathematical
content needed here is only the invariant: fusion is sound when the operands refer to the same
assignments in the same declared frame.
This structural simplification should not be confused with a claim of unique or universal
speedup. Packed truth-function optimizers and logic-synthesis methods can perform closely related
local reductions, and the raw same-frame pointwise law itself has prior art. The computational
question is therefore empirical: whether explicit typed frame recognition and fusion provide a useful
implementation boundary on relevant workloads. Timing, memory, comparator design, and any
positive or negative performance evidence belong to the companion computational paper and are
not used as evidence for the foundations claims here.
13 Relationship to prior operator formalisms and claim boundary
The CM/LM calculus sits among several established representations. Table 1 is a type map rather
than a novelty verdict.
A recent neighboring construction is the Binary Matrix Product (BMP) representation of
Boolean functions [19]. BMPs use products of binary matrices as compressed normal forms closely
related to BDDs, with an explicit computational aim. The correspondence tensor and its flattenings
here instead retain all2n truth coefficients and make no compression claim. The comparison is
useful precisely because both use matrix language for Boolean functions while solving different
representation problems.
13.1 Boolean closure as a claim boundary
Proposition 13.1(Core Boolean closure). Starting from a formula-valued LM or logical tensor,
polarity substitution, logical-state construction, XOR–AND contraction, pointwise Boolean superpo-
sition, block outer composition, support difference, and signed-frame reindexing remain internal to
Boolean algebra. After valuation, the corresponding numeric operations remain inF2.
Proof. Each symbolic construction is formed from Boolean term operations, substitution, finite
indexing, or reindexing. Boolean valuation preserves the term operations and yields only0 or 1
coefficients. No step requires an extension of the scalar domain.
24

Family Primary object Typical operation Relation here
Boolean matrices Binary matrices Boolean products and
transformations
Broad Boolean-matrix antecedent
[1].
Matrix/vector
logic
Truth vectors and
matrix gates
Linear or multilinear
operator action
Strong antecedent for
connective-as-operator viewpoints
[2, 3, 4].
Bricken
matrix-logic
notes
2×2 Boolean relation
matrices and
two-component truth
vectors
Explicit⟨x|B|y⟩
evaluation, Boolean
matrix algebra,
complements, transforms,
and operations on logical
matrices
Internally dated March 1997; direct
antecedent for logical
bra–matrix–ket notation and binary
operator matrices. Public-release
chronology remains to be verified
[14].
STP logic 2×2n logical structure
matrices
Semi-tensor products and
logical structure
operations
External numeric representation,
not the same type as a
formula-valued LM [5, 7].
Boolean matrix
calculus
Boolean matrices and
truth vectors over
XOR/AND
Matrix calculus plus
arbitrary binary outer
operation on
corresponding
truth-vector entries
Direct antecedent for the raw
same-frame numerical superposition
law; Prop. 3.3 states
mfΦg = mfΦ mg [6].
NPN/LUT
synthesis
Packed truth functions
and shared DAGs
Input/output phase,
input permutation, cut
matching
Strong operational antecedent for
frame transformations and
small-function reuse [15, 16].
ANF Square-free polynomial
coefficients
Möbius transform and
polynomial arithmetic
Different coordinates, related by
Boolean Möbius inversion.
Eigenlogic Commuting projectors
on interpretation space
Truth values as
eigenvalues and logical
interpretations as
eigenvectors
Direct prior art for spectral logical
observables; raw2×2 CMs have a
different F2 spectrum, while DΘ
gives the diagonal truth-observable
bridge [8, 9].
CM/LM calculus Typed truth tensor
plus
polarity-equivariant
formula tensor
Valuation, pairing, frame
transport, aligned
pointwise superposition,
matched-frame kernel
product, and block lift
Integrated object studied in this
paper.
Table 1: Representation and operation types. Individual matrix, truth-table, frame, and pointwise
operations have substantial prior art; the paper studies their particular integration in the CM/LM
architecture.
No real, complex, negative, or otherwise non-Boolean scalar is needed for the core calculus. This
closure is not by itself a historical priority claim. Its value is to state exactly which transforma-
tions belong to the CM/LM system before any optional extension—for example the cyclic phase
multiplication—is added.
The central mathematical contribution of the paper is architectural and structural rather than a
claim that any one elementary operation is new. Numeric correspondence tensors and formula-valued
LMs provide two typed realizations of the same Boolean functions; aligned operators can themselves
be operands of arbitrary Boolean superposition; and Theorem 6.12 shows that valuation, common
frame transport, and logical pairing all respect that superposition. Higher-dimensional block
construction and Boolean closure complete the calculus. The orbit-lift and polarity-equivariance
theorems identify the symbolic object cleanly, but their underlying group-action mechanism is
treated as standard.
25

14 Discussion
14.1 What makes this a calculus rather than a list of identities
The formula-valued LM is the organizing symbolic object of the calculus. Boolean superposition
acts directly on aligned LMs; valuation sends the result to numeric correspondence tensors; signed
input-frame changes reindex the symbolic and numeric levels consistently; and logical pairing is
itself a Boolean homomorphism for the same pointwise operation. The raw cellwise truth-vector
law is classical, but the LM layer records which formulas and polarities each cell means, so the
frame condition becomes an explicit semantic type rather than an implicit ordering convention.
Theorem 6.12 collects these compatibility laws.
Two structural results explain why the pieces fit. The frame normal form[MXΘY ] = SX[Θ] SY
identifies the symbolic polarity matrix as an invertible change of Boolean frame, while the partition-
selector theorem explains why pairing preserves arbitrary outer Boolean operations. Ordinary
matrix multiplication then has its own matched-intermediate-frame law, separate from pointwise
superposition. Together these distinguish a typed calculus from a mere list of matrix-decorated
truth identities without claiming a new underlying Boolean algebra.
Higher-dimensional matrices then arise from variable grouping rather than from a new truth
domain. The order-n tensor is primary; a2p×2q matrix is one flattening. This removes ambiguity
from historical higher-dimensional descriptions and makes recursive block structure explicit.
14.2 Scope beyond the present calculus
Literal CM rotation is included because it is already a signed-frame action. A cyclic convolution
on the four coefficient positions would introduce a second multiplication and a different algebraic
object; none of the results in this paper depends on that extension. Questions about such a phase
algebra, modal interpretations, or operator-valued enrichments are therefore deferred rather than
used to enlarge the claims of the present calculus.
14.3 Open structural questions
Three directions are especially natural.
1. Abstract formulation.The faithful preservation of Boolean superposition suggests a clone-
theoretic or equivariant description of the LM lift and its frame action.
2. Canonicalization. Signed permutations and NPN equivalence provide natural orbit relations;
their interaction with formula-level equivalence and efficient canonical forms deserves separate
study.
3. Empirical utility.The typed frame contract is exact, but whether it improves a compiler or
symbolic workflow is an empirical question to be tested in the computational companion rather
than assumed here.
15 Conclusion
The binary notation Θ versus [Θ] distinguishes a logical operator from its compact numeric
representation, while the formula-valued LM[MXΘY ] supplies the symbolic polarity layer that
organizes the present calculus. Its valuation recovers[Θ] in the appropriate signed operand frame,
and its logical pairing returns an exact relationship formula.
26

The central structural statement is therefore LM-centered coherence. Aligned formula-valued
logical operators may themselves be combined as first-class operands of arbitrary Boolean outer
operations; valuation transports that computation to numeric CMs, logical pairing preserves it
relationally, and signed input-frame actions preserve its typing. The frame normal form and selector
characterization explain this coherence structurally; the orbit/equivariance results classify the LM
image; Boolean closure states the scalar boundary. The CM-level pointwise law remains important
computationally, but it is treated as the valued numeric image of the broader symbolic construction
rather than as the principal novelty claim.
An n-variable correspondence tensor may be flattened into a2p×2q correspondence map for
any declared bipartition. Rectangular maps remain genuine typed linear maps fromF2q
2 to F2p
2 and
still participate in bra–map–ket selection. Block lifts and the rank/separability theorem recover
useful higher-dimensional structure from lower-dimensional subfunctions without changing the2n
information bound: the flattening rank is exactly the minimum number of conjunctive factors in an
XOR decomposition across the declared bipartition.
The paper also clarifies several boundaries. OR–AND and XOR–AND agree on one-hot selection
but differ for general linear combinations; XOR is retained to preserve theF2 coefficient algebra.
NPN transformations are standard and are realized here as explicit frame operations rather than
claimed as new. ANF is a different coordinate system. OrdinaryF2 eigenanalysis supplies algebraic
signatures of square CMs, but those eigenvalues are not the truth values of the connective; an
Eigenlogic-style diagonal truth observable lives instead on the four-dimensional interpretation space.
Finally, the4×4 quarter-turn permutationR⟳ acts on the four coefficients of a binary CM and
must not be confused with a4×4 four-variable correspondence map. Adding cyclic convolution
to that coefficient rotation produces the separate phase algebra rather than another axiom of the
CM/LM foundations.
A Spectral boundary and representative compact-CM examples
Because a binary CM is a square matrix overF2, ordinary finite-field eigenanalysis is available. It
should, however, be distinguished both from the older eigenproblem for matrices over a Boolean
algebra and from the quantum-mechanical notion of an observable. Boolean-matrix eigenproblems
have been studied since at least Rutherford and Blyth [12, 13]; those works use Boolean-algebra
matrix operations rather than necessarily the XOR–AND field algebra fixed here.
ForA∈M2(F2), define the characteristic polynomial in the ordinary field sense,
χA(λ) = det(λI−A) =λ2 + tr(A)λ+ det(A),
where subtraction and addition coincide in characteristic two. The resulting spectrum is an algebraic
signature of the CM, but it is not the truth table of the connective.
Table 2 gives a decisive boundary. If raw CM eigenvalues were to be interpreted directly as
logical measurement outcomes, OR would already be problematic: it has no eigenvalue inF2, while
XOR has only the repeated eigenvalue1. The four truth values of a binary connective are therefore
not, in general, the eigenvalues of its2×2 CM.
There is nevertheless a small projector-like subclass. Under the declared XOR–AND matrix
product, exactly eight of the sixteen binary CMs satisfyA2 =A. Exactly four are also symmetric:
the four diagonal CMs
[0], [¬∨], [∧], [⇔].
These are algebraic projection matrices overF2 (and the same0–1 arrays are orthogonal projectors
over ordinary real scalars when viewed in the standard basis). The fact that only a proper subset of
27

CM characteristic
polynomial over F2
F2 eigenstructure interpretation of the example
[∧] = [ 1 0
0 0 ] λ(λ+ 1) |1⟩has eigenvalue1;|0⟩
has eigenvalue0
idempotent projection in theF2
linear-algebra sense.
[⇔] =I (λ+ 1)2 every nonzero vector
has eigenvalue1
identity operator.
[⇕] = [ 0 1
1 0 ] ( λ+ 1)2 only the line spanned
by (1, 1)T is an
eigenspace
involutory but not diagonalizable
over F2; the true/false basis states
are exchanged rather than observed
as eigenstates.
[∨] = [ 1 1
1 0 ] λ2 +λ+ 1 no eigenvalue inF2;
roots lie inF4
a valid logical connective despite
having no F2 eigenvector.
Table 2: RepresentativeF2 spectra of binary CMs. The examples show why the raw CM spectrum
is useful algebraically but cannot by itself serve as the truth-value spectrum of a logical observable.
the sixteen compact CMs has this form is another reason not to identify the raw CM family wholesale
with quantum observables. This finite classification is an elementary structural observation rather
than a novelty claim. Indeed, forA =
[a b
c d
]
, the equationA2 =A is equivalent overF2 to bc = 0
together withb(a⇕d⇕1) = c(a⇕d⇕1) = 0, which has eight binary solutions. AddingAT =A
forces b =c = 0, leaving the four diagonal matrices above.
A.1 Matrix-element semantics versus spectral truth values
The native CM/LM semantics has already been established in Proposition 2.2 and Corollary 4.2:
truth coefficients are recovered as labelled bra–operator–ket matrix elements, and arbitrary logical
pairing returns a Boolean relationship formula. This is relational matrix-element semantics, not
spectral measurement; it supplies neither Born probabilities nor state collapse. Bricken’s technical
note provides direct prior art for the numeric bra–matrix–ket viewpoint [14]. The formula-valued
LM, its frame normal form, and the agreement-bit pairing architecture are the additional structures
studied here.
Eigenlogic takes a different route. On the four-dimensional interpretation space with canonical
assignment basis|11⟩,|10⟩,|01⟩,|00⟩, a binary connectiveΘ canonically determines the diagonal
operator
DΘ := diag(Θ 11, Θ 10, Θ 01, Θ 00). (64)
This 4×4 operator is a projector over ordinary real or complex scalars because its diagonal entries
are 0 or 1; its eigenvectors are the four interpretation basis states and its eigenvalues are exactly the
truth-table outputs. That is the spectral logic used by Eigenlogic in a more general operator setting
[8, 9]. Equation (64) therefore gives a usefulspectral truth-observable representationassociated with
a CM if a spectral interpretation is desired, butDΘ is a different representation from the compact
2×2 CM [Θ]. In the native CM representation, truth is read through the labelled matrix elements
of Proposition 2.2 and Corollary 4.2, not through the spectrum.
Formula-valued LMs also admit algebraic eigenvalue questions, since their entries lie in a
commutative Boolean ring. Ring-valued spectra are less rigid than field spectra, so the safest
intrinsic object at present is thevaluation spectral profile
ΣM :v↦−→SpecF2(v(M)). (65)
28

ForM = [MXΘY ], valuation produces row and column reversals of[Θ]. Independent left and right
reversals need not be similarity transformations, so theF2 spectrum can vary with the valuation;
simultaneous conjugate reversals preserve it. Whether this profile supplies a useful invariant of
formula-valued LMs is an open algebraic question. Nothing in this paper interprets it as a physical
observable without additional state, effect, and probability postulates.
B Binary LM basis expansion
Let b1 = 1, b2 = 0, X1 = X, X2 = ¬X, Y1 = Y, Y2 = ¬Y, and cij = biΘbj. Define the
formula-valued outer product
|Xi⟩⟨Yj|=
[
Xi∧Yj Xi∧¬Yj
¬Xi∧Yj ¬Xi∧¬Yj
]
.
Then
[MXΘY ] = ⇕
i,j∈{1,2}
cij|Xi⟩⟨Yj|. (66)
The upper-left entry is the disjoint-minterm expansion
c11XY ⇕c12X¬Y⇕c21¬XY ⇕c22¬X¬Y,
which is exactlyXΘY. Exchanging theY polarity gives the upper-right entry, exchanging theX
polarity gives the lower-left entry, and exchanging both gives the lower-right entry. Thus(66) is
equivalent to the direct binary definition (10).
C Compatibility identities
The principal maps of the calculus satisfy the following compatibility relations whenever the displayed
expressions are typed:
v
(ˆg(T1,...,Tm)
)
=ˆg(vT1,...,vTm),
Tˆg(Cf1,...,Cfm) =ˆg(TCf1,...,TCfm),
v1(τδLX(f)) =πδCf.
The first says that valuation commutes with Boolean superposition; the second that a common
frame reindexing commutes with pointwise composition; and the third that valuation intertwines
symbolic polarity changes with numeric axis reversals. These identities are the concise algebraic
content behind the commuting diagrams used in the main text.
29

D Notation and convention table
Symbol Meaning
Θ Binary logical operator variable/connective.
[Θ] Numeric 2×2 CM representingΘ in true-first order.
[MXΘY ] Formula-valued binary LM for the polarity orbit ofXΘY.
SX Formula-valued involutory frame matrix with[MXΘY ] = SX[Θ] SY.
Cf n-variable numeric correspondence tensor with2n entries.
Mf(X) Formula-valuedn-variable logical tensor.
LX Faithful LM liftf↦→Mf(X).
[f]R|C 2p×2q matrix flattening for a declared variable bipartition.
ˆg Pointwise lift of an outer Boolean function to aligned operators.
OΘ Block/outer lift constructing a correspondence map from disjoint
subfunctions.
Tσ,ϵ Signed input-frame reindexing.
πδ One-triggered numeric axis reversal:(πδT )[α] =T [α⇕δ].
qw Partition selectorT↦→⇕iwi∧Ti.
[⇕] XOR CM; also the2×2 true/false swap matrix under XOR–AND
multiplication.
R⟳ Quarter-turn coefficient permutation acting on the four vectorized
entries of a binary CM.
E Finite verification note
The supplementary materials include an independent standard-library checker, independent_
verify.py, together with the machine-readable run recordverification_results.json. The
checker is self-contained and does not import a separate computational implementation. The
recorded run passes 2,022,648 explicitly counted assertions. The families include complete formula-
frame and inverse checks over the sixteen-element formula algebra on two free generators; all163
binary outer-superposition triples; exhaustive two-generator formula pairings; independent signed-
frame alignment; matched-frame products; polarity-image and partition-selector characterizations;
complete ternary valuation, pairing, and signed symbolic transport checks; ordered rectangular
selection; all four-variable block lifts built from binary inner functions; all2×4 matrices for minimum
XOR rank decomposition; ANF parity through arity four; and the complete sixteen-CM finite
spectral checks. The checker also records boundary counterexamples, including the double-mask
transport error. These finite regression tests complement rather than replace the arbitrary-arity
proofs and provide no evidence of historical priority.
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