Does the Symbolic LM Calculus Canonically Induce the Modal Measurement Theory?

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  18   pdftitle={Does the Symbolic LM Calculus Canonically Induce the Modal Measurement Theory?},
  19   pdfauthor={Research synthesis prepared 18 September 2026}
  20 }
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  22 \newtheorem{theorem}{Theorem}[section]
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  30 \newcommand{\F}{\mathbb{F}}
  31 \newcommand{\B}{\mathcal{B}}
  32 \newcommand{\Sring}{\mathcal{S}_A}
  33 \newcommand{\Poss}{\mathsf{Poss}}
  34 \newcommand{\Supp}{\operatorname{Supp}_A}
  35 \newcommand{\GL}{\operatorname{GL}}
  36 \newcommand{\Id}{\operatorname{Id}}
  37 \newcommand{\vecop}{\operatorname{vec}}
  38 \newcommand{\LM}{\operatorname{LM}}
  39 \newcommand{\Bool}{\mathbf{2}}
  40 
  41 \title{\textbf{Does the Symbolic LM Calculus Canonically Induce the Modal Measurement Theory?}\\
  42 \large Adversarial bridge analysis between formula-valued Logical Matrices and the later chain-ring modal/effect framework}
  43 \author{Research synthesis and independent finite verification}
  44 \date{18 September 2026}
  45 
  46 \begin{document}
  47 \maketitle
  48 
  49 \begin{abstract}
  50 This report asks whether the formula-valued Logical Matrix (LM) calculus developed in
  51 \emph{Operator-Level Boolean Computation with Correspondence Matrices} can be regarded as the
  52 canonical symbolic source of the later modal measurement theory over the finite chain ring
  53 \(A\cong \F_2[u]/(u^4)\). The answer is mixed but mathematically useful. Paper B already proves
  54 that Boolean valuation commutes with LM pairing. That fact extends naturally to arbitrary matrix,
  55 module, and tensor expressions after an explicit scalar enrichment. However, direct Boolean
  56 valuation cannot produce the nilpotent and unit amplitudes of the chain ring: every Boolean-ring
  57 homomorphism into the local ring \(A\) lands in its idempotents \(\{0,1\}\). The correct positive
  58 bridge is therefore not direct valuation but scalar enrichment
  59 \(\B\otimes_{\F_2} A\), followed by valuation specialization
  60 \(v\otimes \Id_A\). In this enriched calculus, each symbolic amplitude has a canonical Boolean
  61 support predicate whose satisfying valuations are exactly those for which the corresponding concrete
  62 ring-valued modal amplitude is nonzero. This yields a precise symbolic-support theorem connecting
  63 formula semantics to modal possibility.
  64 
  65 The report also proves negative boundaries. The later rules that effects are rows of reversible
  66 \(A\)-bases and that a nonzero contraction denotes a possible event are additional measurement
  67 structure, not consequences of the Paper-B LM semantics. The original LM state encoding generates
  68 only coordinate effects under valuation and therefore does not generate the complete 192 reversible
  69 projective two-outcome basis family. A division-free branch update
  70 \(J_i^B=B^{-1}P_iB\) is nevertheless available for every reversible modal basis and is compatible
  71 with projective unit rescaling and valuation. Zero divisors still obstruct a closed unrestricted
  72 process theory: nonzero states need not be tensor-closed, and conditioning can take unimodular
  73 preparations to nonunimodular conditional states.
  74 
  75 Independent exhaustive checks reproduce the 16-element chain ring, its 8 units and 2 idempotents,
  76 \(|\GL_2(A)|=24576\), the 192 unordered projective reversible bases, 98,304 branch-support checks,
  77 262,144 symbolic-support fiber checks, and the failure of 144 out of 256 ordinary Boolean-ring
  78 matrix products to remain inside the restricted 16-element Paper-B LM family.
  79 \end{abstract}
  80 
  81 \tableofcontents
  82 \newpage
  83 
  84 \section{Research question and verdict}
  85 
  86 The central question is whether the schematic chain
  87 \[
  88 \text{formula-valued LM}
  89 \longrightarrow
  90 \text{Boolean valuation}
  91 \longrightarrow
  92 \text{CM/coefficient representation}
  93 \longrightarrow
  94 \text{state/resource}
  95 \longrightarrow
  96 \text{effect/measurement}
  97 \longrightarrow
  98 \text{logical or modal outcome}
  99 \]
 100 is a genuine mathematical semantics, rather than a collection of later constructions sharing matrix
 101 notation.
 102 
 103 \subsection{Executive verdict}
 104 
 105 There is a genuine bridge, but the later modal measurement theory is \emph{not} canonically derivable
 106 from Paper B as it currently stands.
 107 
 108 The strongest defensible formulation is:
 109 
 110 \begin{quote}
 111 Paper B canonically supplies a symbolic Boolean layer and a valuation-natural contraction calculus.
 112 After independently choosing the chain-ring amplitude algebra \(A\), there is a natural scalar-enriched
 113 symbolic extension in which valuation continues to commute with states, effects, contractions, matrix
 114 operations, and tensor products. In that extension, every ring-valued amplitude has a canonical Boolean
 115 support predicate describing exactly the valuations for which the corresponding modal event is nonzero.
 116 However, the rules that effects are rows of reversible \(A\)-bases and that nonzero means possible are
 117 extra measurement structure. They are not consequences of the original formula-valued LM calculus.
 118 \end{quote}
 119 
 120 Thus the clean architecture is
 121 \[
 122 \boxed{
 123 \text{Boolean symbolic LM calculus}
 124 \longrightarrow
 125 \text{\(A\)-enriched symbolic calculus}
 126 \overset{\text{valuation}}{\longrightarrow}
 127 \text{\(A\)-valued states/effects}
 128 \overset{\text{support contract}}{\longrightarrow}
 129 \text{modal outcomes}.}
 130 \]
 131 
 132 The missing structure consists of at least three independent ingredients:
 133 \begin{enumerate}[label=(\arabic*)]
 134 \item a chosen cyclic/chain-ring multiplication on the four CM coefficients;
 135 \item a chosen admissible measurement family, such as rows of matrices in \(\GL_n(A)\);
 136 \item the support interpretation \(a\neq 0\Longleftrightarrow\) ``possible.''
 137 \end{enumerate}
 138 
 139 \section{Primary sources and methodological boundary}
 140 
 141 The analysis treated the manuscript \emph{Operator-Level Boolean Computation with Correspondence
 142 Matrices} (``Paper B'') as background rather than rediscovering its results. The primary Paper-B text
 143 read for this investigation includes Section 8.1--8.2 and its proof appendix. In the parsed copy used
 144 for the review, lines 560--669 define the Boolean algebra of formulas modulo equivalence, the
 145 formula-valued LM, positive and general valuation, logical pairing, Proposition 2, and Theorem 3.
 146 Lines 780--874 contain the appendix proof and finite verification scope.
 147 
 148 The later modal source read was \texttt{PURE\_BOOLEAN\_MODAL\_CM\_ADDENDUM.tex}. In the
 149 parsed copy used for the review, lines 45--118 explicitly state that the modal nonzero-support
 150 readout is an ``additional contract,'' introduce the chain ring, define states and separability, and
 151 specify reversible effect bases. Lines 440--539 contain the complete 192-basis support classification,
 152 and lines 620--643 identify the remaining multi-round measurement/update problem.
 153 
 154 The supplied adversarial-review archive is included verbatim and unpacked in this research package.
 155 It contains the earlier novelty dossier, theorem candidates, search logs, source-audit files, and a large
 156 set of reproduced computational outputs. The current report adds a separate bridge analysis and
 157 fresh independent checks.
 158 
 159 \section{Strict type map}
 160 
 161 The investigation keeps the following object types separate.
 162 
 163 \begin{longtable}{p{0.18\textwidth}p{0.29\textwidth}p{0.23\textwidth}p{0.23\textwidth}}
 164 \toprule
 165 \textbf{Layer} & \textbf{Object} & \textbf{Map} & \textbf{Status / preserved structure} \\
 166 \midrule
 167 \endhead
 168 Binary truth function & \(C\in M_2(\F_2)\) & \(C\mapsto \LM_{X,Y}(C)\) & Injective for freely assignable \(X,Y\); preserves XOR and entrywise Boolean operations; not ordinary matrix multiplication.\\
 169 \addlinespace
 170 Formula-valued LM & \(\LM_{X,Y}(C)\in M_2(\B)\) & entrywise Boolean valuation \(v\) & Canonical for each valuation; preserves Boolean-ring operations.\\
 171 \addlinespace
 172 Valued CM & \(v(\LM)\in M_2(\F_2)\) & vectorization & Linear bijection after cell-order choice; basis/order dependent.\\
 173 \addlinespace
 174 Four coefficients & \(c\in\F_2^4\) & cyclic-basis identification & Vector-space bijection to \(A\); does not inherit ordinary \(2\times2\) matrix multiplication.\\
 175 \addlinespace
 176 Chain-ring scalar & \(a\in A\cong\F_2[u]/(u^4)\) & regular representation \(\Lambda\) & Injective algebra representation into \(M_4(\F_2)\) after multiplication on \(A\) is chosen.\\
 177 \addlinespace
 178 One-system state & \(\psi\in A^n\) & effect application & Free \(A\)-module object; one CM is only one scalar of \(A\), not a whole state.\\
 179 \addlinespace
 180 Bipartite resource & \(M\in M_{m,n}(A)\) & \(M\mapsto UMV^T\) & Local \(A\)-linear action; distinct type from original Boolean CMs.\\
 181 \addlinespace
 182 Effect & \(e\in (A^n)^*\cong A^{1\times n}\) & \(e(\psi)=e\psi\) & \(A\)-linear functional; row identification uses a free-module basis.\\
 183 \addlinespace
 184 Measurement & rows of \(B\in\GL_n(A)\) & \(B\psi\) & Reversible coordinate change; additional modal choice.\\
 185 \addlinespace
 186 Outcome scalar & \(e\psi\in A\) or \(eMf^T\in A\) & \(a\mapsto[a\neq0]\) & Support zero-test; not a ring/semiring homomorphism in general.\\
 187 \addlinespace
 188 Symbolic enriched layer & \(\Sring=\B\otimes_{\F_2}A\) & \(v_A=v\otimes\Id_A\) & Canonical once \(A\) is chosen; preserves ring, module, matrix, and tensor operations.\\
 189 \addlinespace
 190 Symbolic modal outcome & \(q\in\Sring\) & \(\Supp(q)\in\B\) & Exact valuation-wise zero/nonzero semantics; support itself is not an algebra homomorphism.\\
 191 \bottomrule
 192 \end{longtable}
 193 
 194 A key typing correction is
 195 \[
 196 \boxed{\text{one }2\times2\text{ Boolean CM}\neq\text{one modal state}.}
 197 \]
 198 After a four-feature basis is chosen, one CM becomes one scalar of \(A\). A one-bit modal state needs
 199 two such scalars:
 200 \[
 201 \psi=a_0\lvert0\rangle+a_1\lvert1\rangle\in A^2.
 202 \]
 203 A two-party resource requires four such scalars.
 204 
 205 \section{Q1: What is a formula-valued LM mathematically?}
 206 
 207 Let \(\B\) be the Lindenbaum Boolean algebra of propositional formulas modulo logical equivalence.
 208 Write it as a Boolean ring using
 209 \[
 210 f+g=f\oplus g,\qquad fg=f\land g,\qquad \neg f=1+f.
 211 \]
 212 For freely assignable symbols \(X,Y\), define
 213 \[
 214 X_1=X,\quad X_2=\neg X,\qquad
 215 Y_1=Y,\quad Y_2=\neg Y.
 216 \]
 217 For a binary truth table \(C=(c_{ij})\in M_2(\F_2)\), Paper B defines
 218 \[
 219 \LM_{X,Y}(C)
 220 =
 221 \bigoplus_{i,j}c_{ij}\lvert X_i\rangle\langle Y_j\rvert
 222 \in M_2(\B).
 223 \]
 224 Equivalently, after expansion,
 225 \[
 226 \LM_{X,Y}(\Theta)
 227 =
 228 \begin{pmatrix}
 229 X\,\Theta\,Y & X\,\Theta\,\neg Y\\
 230 \neg X\,\Theta\,Y & \neg X\,\Theta\,\neg Y
 231 \end{pmatrix}.
 232 \]
 233 
 234 \begin{theorem}[Algebraic status of the Paper-B LM family]
 235 For fixed freely assignable \(X,Y\), the map
 236 \[
 237 \ell_{X,Y}:M_2(\F_2)_{\mathrm{coeff}}\longrightarrow M_2(\B),
 238 \qquad C\longmapsto\LM_{X,Y}(C),
 239 \]
 240 is injective and \(\F_2\)-linear. It preserves entrywise Boolean operations. It is not, in general,
 241 a homomorphism from ordinary Boolean-ring matrix multiplication on \(M_2(\F_2)\) to matrix
 242 multiplication in \(M_2(\B)\).
 243 \end{theorem}
 244 
 245 \begin{proof}
 246 Linearity follows from the XOR sum defining the LM. Positive valuation sends the LM back to its
 247 coefficient CM, so equality of two LMs implies equality of their coefficient matrices. Therefore the
 248 map is injective. Entrywise Boolean operations commute with the substitution producing the four
 249 polarity positions. Ordinary matrix product is different: multiplication mixes row/column positions
 250 by XOR--AND contraction. Exhaustive evaluation of all \(16^2=256\) products of the 16 Paper-B
 251 LMs shows that 144 products leave the restricted 16-element LM family, so the family is not closed
 252 under this multiplication.
 253 \end{proof}
 254 
 255 \begin{remark}
 256 The most economical description is therefore: a Paper-B LM is a special predicate-valued matrix
 257 over a free/Lindenbaum Boolean ring. The ambient space \(M_2(\B)\) is natural; the 16 matrices
 258 generated by binary truth functions are a distinguished subset/subalgebra only for the entrywise
 259 Boolean operations. Calling the construction merely a relation would lose the XOR--AND scalar
 260 algebra used by Paper B.
 261 \end{remark}
 262 
 263 \section{Q2: What is LM pairing mathematically?}
 264 
 265 For a formula \(F\in\B\), Paper B defines
 266 \[
 267 s(F)=
 268 \begin{pmatrix}F\\ \neg F\end{pmatrix},
 269 \qquad
 270 s(F)^T=(F,\neg F).
 271 \]
 272 The symbolic pairing is
 273 \[
 274 P_C(L_1,L_2)
 275 =
 276 s(L_1)^T\LM_{X,Y}(C)s(L_2),
 277 \]
 278 where scalar multiplication is conjunction and scalar addition is XOR.
 279 
 280 \begin{proposition}[LM pairing is Boolean-ring matrix contraction]
 281 For the special LM associated with a binary connective \(\Theta\),
 282 \[
 283 \langle L_1\vert\LM_{X,Y}(\Theta)\vert L_2\rangle
 284 \equiv
 285 (L_1\leftrightarrow X)\;\Theta\;(Y\leftrightarrow L_2).
 286 \]
 287 Thus the Paper-B ``logical pairing'' is ordinary matrix contraction over the Boolean ring \(\B\),
 288 with a non-linear special state embedding \(F\mapsto(F,\neg F)\).
 289 \end{proposition}
 290 
 291 The roles of the connectives are exact:
 292 \begin{itemize}
 293 \item AND is scalar multiplication;
 294 \item XOR is scalar addition and reduction;
 295 \item negation appears in the complement-paired state encoding;
 296 \item OR is not the matrix-addition operation used by the pairing.
 297 \end{itemize}
 298 
 299 The pairing may be described semantically as predicate evaluation or relationship extraction, but
 300 those descriptions do not add algebraic structure beyond the Boolean-ring contraction.
 301 
 302 \section{Q3: Valuation naturality}
 303 
 304 Let
 305 \[
 306 v:\B\longrightarrow\F_2
 307 \]
 308 be a Boolean valuation. Extend \(v\) entrywise to matrices and vectors.
 309 
 310 \begin{theorem}[Valuation naturality]
 311 For compatible Boolean-ring matrices and vectors,
 312 \[
 313 v(MN)=v(M)v(N),
 314 \]
 315 and
 316 \[
 317 v(x^TMy)=v(x)^T\,v(M)\,v(y).
 318 \]
 319 The same statement holds for finite Kronecker tensor products.
 320 \end{theorem}
 321 
 322 \begin{proof}
 323 Every output entry is a finite expression built from Boolean-ring addition and multiplication. A
 324 Boolean valuation is a homomorphism for those operations, so it may be moved through each finite
 325 sum and product.
 326 \end{proof}
 327 
 328 This theorem is already present in Paper B for the LM pairing case. Its broader matrix/tensor version
 329 is a routine homomorphism-extension fact. It is useful for organizing the bridge, but it should not be
 330 presented as deep novelty.
 331 
 332 \section{A decisive negative theorem: direct valuation cannot produce chain-ring amplitudes}
 333 
 334 The strongest possible bridge would send Boolean formulas directly into the later amplitude ring
 335 \[
 336 A=\F_2[u]/(u^4).
 337 \]
 338 That is impossible except for the embedded Boolean scalars.
 339 
 340 \begin{theorem}[Idempotent obstruction]
 341 Let \(\B\) be a Boolean ring and let \(h:\B\to A\) be a unital ring homomorphism. Then every
 342 \(h(b)\) is idempotent. In the local chain ring \(A=\F_2[u]/(u^4)\), the only idempotents are
 343 \(0\) and \(1\). Therefore
 344 \[
 345 \operatorname{im}h\subseteq\{0,1\}\cong\F_2,
 346 \]
 347 and \(h\) factors through an ordinary Boolean valuation
 348 \[
 349 \B\longrightarrow\F_2\hookrightarrow A.
 350 \]
 351 \end{theorem}
 352 
 353 \begin{proof}
 354 Every element of a Boolean ring satisfies \(b^2=b\). Hence
 355 \[
 356 h(b)^2=h(b^2)=h(b).
 357 \]
 358 A local ring has no idempotents other than \(0\) and \(1\). The stated chain ring is local, so the
 359 image is Boolean.
 360 \end{proof}
 361 
 362 \begin{corollary}
 363 Nilpotent and nontrivial unit amplitudes such as
 364 \[
 365 u,\quad u^2,\quad u^3,\quad 1+u
 366 \]
 367 can never arise as direct values of Paper-B Boolean formulas under a ring-homomorphic valuation.
 368 \end{corollary}
 369 
 370 This rules out the claim that the chain-ring amplitude theory is obtained by simply changing the
 371 codomain of Paper-B valuation.
 372 
 373 \section{The CM-to-chain-ring identification is linear before it is multiplicative}
 374 
 375 There is a natural-looking sequence
 376 \[
 377 M_2(\F_2)
 378 \xrightarrow{\vecop_{\kappa}}
 379 \F_2^4
 380 \xrightarrow{\alpha_{\beta}}
 381 A,
 382 \]
 383 where \(\kappa\) fixes a cell order and \(\beta\) fixes an ordered \(\F_2\)-basis of \(A\).
 384 Both arrows are vector-space bijections.
 385 
 386 They are not an identification of ordinary \(2\times2\) matrix multiplication with the chain-ring
 387 multiplication. Two immediate invariants separate the algebras:
 388 \begin{itemize}
 389 \item \(M_2(\F_2)\) is noncommutative, while \(A\) is commutative;
 390 \item \(|\GL_2(\F_2)|=6\), while \(A\) has 8 units.
 391 \end{itemize}
 392 
 393 Therefore the later cyclic product
 394 \[
 395 E_r\star E_s=E_{r+s\bmod4}
 396 \]
 397 is additional algebraic structure placed on the four-component coefficient space. Once supplied,
 398 \[
 399 A\cong\F_2[C_4]\cong\F_2[u]/(u^4),
 400 \]
 401 and its regular representation
 402 \[
 403 \Lambda:A\hookrightarrow M_4(\F_2)
 404 \]
 405 is an injective algebra homomorphism. The multiplication itself, however, is not forced by the
 406 original LM valuation theorem.
 407 
 408 \section{The correct positive bridge: scalar enrichment}
 409 
 410 The direct-valuation obstruction suggests the right construction.
 411 
 412 \begin{definition}[Enriched symbolic scalar algebra]
 413 Let
 414 \[
 415 \Sring=\B\otimes_{\F_2}A.
 416 \]
 417 For the current chain ring,
 418 \[
 419 \Sring\cong \B[u]/(u^4).
 420 \]
 421 \end{definition}
 422 
 423 \begin{theorem}[Canonical specialization after choosing \(A\)]
 424 For every Boolean valuation \(v:\B\to\F_2\), there is a canonical \(A\)-algebra specialization
 425 \[
 426 v_A=v\otimes\Id_A:\Sring\to A,
 427 \]
 428 defined on pure tensors by
 429 \[
 430 v_A(f\otimes a)=v(f)a.
 431 \]
 432 It preserves addition and multiplication and therefore extends entrywise to matrices, vectors,
 433 contractions, and tensor products. In particular,
 434 \[
 435 v_A(e\psi)=v_A(e)v_A(\psi),
 436 \]
 437 and
 438 \[
 439 v_A(E M F^T)=v_A(E)v_A(M)v_A(F)^T.
 440 \]
 441 \end{theorem}
 442 
 443 \begin{remark}[Universal property]
 444 The tensor product is the coproduct of the commutative \(\F_2\)-algebras \(\B\) and \(A\).
 445 Therefore this is a genuine universal property, but it is a universal property \emph{after} the amplitude
 446 algebra \(A\) has been independently selected. It does not derive \(A\) from the formula calculus.
 447 \end{remark}
 448 
 449 For finitely many Boolean generators, if
 450 \[
 451 \Omega=\operatorname{Hom}_{\mathrm{Bool}}(\B,\Bool)
 452 \]
 453 is the valuation set, then extensionally
 454 \[
 455 \B\cong\F_2^{\Omega}
 456 \quad\text{and hence}\quad
 457 \boxed{\Sring\cong A^{\Omega}.}
 458 \]
 459 So an enriched symbolic scalar may be read simply as an \(A\)-valued function of the Boolean
 460 valuation. This is the finite Boolean-power viewpoint.
 461 
 462 \section{The strongest bridge theorem: symbolic support realizes modal possibility}
 463 
 464 Let
 465 \[
 466 q\in\Sring\cong A^{\Omega}.
 467 \]
 468 Define the symbolic support predicate \(\Supp(q)\in\B\) to have exactly the valuation set
 469 \[
 470 \{v\in\Omega: v_A(q)\neq0\}.
 471 \]
 472 For
 473 \[
 474 q=f_0+f_1u+f_2u^2+f_3u^3,
 475 \qquad f_i\in\B,
 476 \]
 477 one may write
 478 \[
 479 \Supp(q)=f_0\lor f_1\lor f_2\lor f_3.
 480 \]
 481 Although this coordinate formula uses the displayed basis, the resulting Boolean predicate is
 482 basis-independent up to logical equivalence because it is defined by the intrinsic condition
 483 \(v_A(q)\neq0\).
 484 
 485 \begin{theorem}[Symbolic-support bridge]
 486 Let
 487 \[
 488 e\in\Sring^{1\times n},\qquad \psi\in\Sring^n.
 489 \]
 490 Define
 491 \[
 492 \Poss(e,\psi)=\Supp(e\psi)\in\B.
 493 \]
 494 Then for every Boolean valuation \(v\),
 495 \[
 496 \boxed{
 497 v\bigl(\Poss(e,\psi)\bigr)=1
 498 \iff
 499 v_A(e)\,v_A(\psi)\neq0.}
 500 \]
 501 For a bipartite symbolic resource \(M\),
 502 \[
 503 \boxed{
 504 v\left(\Supp(eMf^T)\right)=1
 505 \iff
 506 v_A(e)v_A(M)v_A(f)^T\neq0.}
 507 \]
 508 \end{theorem}
 509 
 510 \begin{proof}
 511 Choose an \(\F_2\)-basis \(\{\beta_k\}\) of \(A\) and write the symbolic scalar contraction as
 512 \[
 513 q=\sum_k f_k\beta_k.
 514 \]
 515 At valuation \(v\),
 516 \[
 517 v_A(q)=\sum_k v(f_k)\beta_k.
 518 \]
 519 Linear independence of the basis gives
 520 \[
 521 v_A(q)=0\iff v(f_k)=0\ \text{for all }k.
 522 \]
 523 Therefore
 524 \[
 525 v_A(q)\neq0
 526 \iff
 527 \bigvee_kv(f_k)=1
 528 \iff
 529 v\left(\bigvee_k f_k\right)=1.
 530 \]
 531 \end{proof}
 532 
 533 \begin{corollary}[Satisfiability/possibility bridge]
 534 \[
 535 \Supp(e\psi)\text{ is satisfiable}
 536 \iff
 537 \exists v:\;v_A(e)v_A(\psi)\neq0.
 538 \]
 539 The same holds for bipartite contractions \(eMf^T\).
 540 \end{corollary}
 541 
 542 This is the strongest bridge found in the investigation. It says that a single symbolic contraction can
 543 compute an ordinary Boolean formula describing exactly the parameter valuations under which the
 544 corresponding concrete ring-modal event is possible.
 545 
 546 \section{Q4: Can formula-valued LMs represent modal effects?}
 547 
 548 Two meanings of ``represent'' must be distinguished.
 549 
 550 \subsection{As measurement semantics: only a restricted subset}
 551 
 552 The Paper-B logical state map is
 553 \[
 554 s(L)=\begin{pmatrix}L\\\neg L\end{pmatrix}.
 555 \]
 556 For every valuation,
 557 \[
 558 v(s(L))\in
 559 \left\{
 560 \begin{pmatrix}1\\0\end{pmatrix},
 561 \begin{pmatrix}0\\1\end{pmatrix}
 562 \right\}.
 563 \]
 564 So the bare Paper-B bra/ket construction yields only coordinate rows after valuation. It does not
 565 directly yield the embedded-\(\F_2\) effect \((1,1)\), which already occurs in a standard modal
 566 \(X\)-type basis.
 567 
 568 \begin{proposition}[Negative measurement-family characterization]
 569 The original Paper-B LM pairing semantics does not generate the complete 192 reversible projective
 570 modal bases over \(A\). It does not even generate every reversible basis in the embedded
 571 \(\F_2\) subtheory.
 572 \end{proposition}
 573 
 574 This is a negative version of the requested measurement-family characterization.
 575 
 576 \subsection{As coefficient data: every effect can be encoded after choices}
 577 
 578 Fix a CM cell ordering and the cyclic \(\F_2\)-basis of \(A\). Every \(a\in A\) has a unique four-bit
 579 coefficient representation, hence a unique Boolean CM in that basis, and therefore a Paper-B LM
 580 representative in a chosen operand frame. An effect
 581 \[
 582 e=(a,b)\in A^2
 583 \]
 584 can consequently be encoded by two such LMs, and a \(2\times2\) reversible basis by four.
 585 
 586 Thus every modal effect is LM-encodable as coefficient data, but this encoding is:
 587 \begin{itemize}
 588 \item dependent on the chosen four-feature basis;
 589 \item dependent on CM cell order;
 590 \item dependent on the chosen LM operand frame/positive-valuation convention;
 591 \item not a derivation of effect semantics;
 592 \item not an identification of LM matrix multiplication with \(A\)-multiplication.
 593 \end{itemize}
 594 
 595 The correct statement is therefore:
 596 \begin{quote}
 597 Every modal effect is LM-encodable after the chain-ring coefficient basis is fixed, but not every modal
 598 effect is an LM effect in the original Paper-B pairing sense.
 599 \end{quote}
 600 
 601 \section{Q5: ``nonzero = possible'' is extra structure}
 602 
 603 The later modal source explicitly introduces nonzero readout as an additional contract. The algebra
 604 shows why it cannot be hidden inside ordinary Boolean valuation semantics.
 605 
 606 Define
 607 \[
 608 \sigma:A\to\{0,1\},\qquad
 609 \sigma(a)=\begin{cases}0,&a=0,\\1,&a\neq0.\end{cases}
 610 \]
 611 This map is neither additive in the \(\F_2\) sense nor multiplicative to Boolean support.
 612 For example,
 613 \[
 614 \sigma(u)=\sigma(u^2)=1,
 615 \]
 616 but
 617 \[
 618 \sigma(u+u^2)=1\neq1\oplus1=0.
 619 \]
 620 More importantly, zero divisors give
 621 \[
 622 u\neq0,\qquad u^3\neq0,\qquad uu^3=u^4=0,
 623 \]
 624 so
 625 \[
 626 \sigma(u)\land\sigma(u^3)=1
 627 \quad\text{while}\quad
 628 \sigma(uu^3)=0.
 629 \]
 630 
 631 \begin{proposition}[Support is not a semiring homomorphism]
 632 The nonzero map \(\sigma:A\to\Bool\) cannot be a homomorphism from the chain ring to either the
 633 Boolean ring \((\Bool,\oplus,\land)\) or the ordinary Boolean semiring \((\Bool,\lor,\land)\).
 634 Consequently, the support semantics is an independently selected zero-test rather than a functorial
 635 quotient of the amplitude algebra.
 636 \end{proposition}
 637 
 638 The symbolic-support theorem does not contradict this. It says that, \emph{once the zero-test is chosen},
 639 its valuation fibers can be represented exactly by Boolean formulas.
 640 
 641 \section{Q6: Symbolic observables/effects}
 642 
 643 The most useful symbolic generalization is not a new Boolean algebra of observables but a module of
 644 symbolic effects:
 645 \[
 646 \mathcal E_n=(\Sring^n)^*\cong\Sring^{1\times n}.
 647 \]
 648 An effect may carry genuine formulas in its coefficients, for example
 649 \[
 650 e=
 651 \bigl(
 652 (x\land\neg y)+u(x\oplus z),\;
 653  u^2(f\leftrightarrow g)+u^3z
 654 \bigr).
 655 \]
 656 For a symbolic state \(\psi\), the contraction
 657 \[
 658 e\psi\in\Sring
 659 \]
 660 retains the complete symbolic \(A\)-valued amplitude. Applying support gives
 661 \[
 662 \Supp(e\psi)\in\B,
 663 \]
 664 a Boolean formula describing exactly when the effect is modally possible after specialization.
 665 
 666 Thus the framework naturally exposes two levels of output:
 667 \[
 668 \boxed{e\psi\ \text{(symbolic amplitude)}}
 669 \qquad\text{and}\qquad
 670 \boxed{\Supp(e\psi)\ \text{(symbolic possibility condition)}}.
 671 \]
 672 
 673 Calling this an ``effect algebra'' would currently be premature because no partial sum, orthosupplement,
 674 or effect-algebra order has been shown to satisfy the standard axioms. ``Symbolic effect module'' or
 675 ``predicate-parametrized effect calculus'' is more accurate.
 676 
 677 \section{Q7: Sequential measurement and update}
 678 
 679 The chain ring creates a normalization problem because possible amplitudes can be nonunits. However,
 680 reversible-basis measurements admit a clean branch update that requires no division by the observed
 681 amplitude.
 682 
 683 Let
 684 \[
 685 B\in\GL_n(A)
 686 \]
 687 be a reversible measurement basis, with rows interpreted as effects. Let \(P_i\) be the diagonal
 688 coordinate projector having a single \(1\) in coordinate \(i\). Define
 689 \[
 690 \boxed{J_i^B=B^{-1}P_iB.}
 691 \]
 692 
 693 \begin{theorem}[Division-free reversible-basis update]
 694 For every commutative ring \(A\) and every \(B\in\GL_n(A)\), the maps
 695 \(J_i^B\) satisfy
 696 \[
 697 (J_i^B)^2=J_i^B,
 698 \]
 699 \[
 700 J_i^BJ_j^B=0\quad(i\neq j),
 701 \]
 702 and
 703 \[
 704 \sum_iJ_i^B=I.
 705 \]
 706 If
 707 \[
 708 c_i=(B\psi)_i,
 709 \]
 710 then
 711 \[
 712 \boxed{J_i^B\psi\neq0\iff c_i\neq0.}
 713 \]
 714 Moreover,
 715 \[
 716 B(J_i^B\psi)=P_iB\psi,
 717 \]
 718 so a repeat of the same measurement has support only on outcome \(i\).
 719 \end{theorem}
 720 
 721 \begin{proof}
 722 Idempotence, orthogonality, and resolution of the identity follow by conjugating the coordinate
 723 projectors. Let
 724 \[
 725 b_i=B^{-1}e_i.
 726 \]
 727 Then
 728 \[
 729 J_i^B\psi=c_i b_i.
 730 \]
 731 The column \(b_i\) is unimodular because it is a column of an invertible matrix. Hence its entries
 732 generate the unit ideal, so there exist \(r_j\) with
 733 \[
 734 \sum_j r_j(b_i)_j=1.
 735 \]
 736 If \(c_i b_i=0\), multiplying by the \(r_j\) combination yields \(c_i=0\). The converse is
 737 immediate. Finally,
 738 \[
 739 B J_i^B\psi=P_iB\psi.
 740 \]
 741 \end{proof}
 742 
 743 \subsection{Projective well-definedness}
 744 
 745 If the effect rows are independently rescaled by units,
 746 \[
 747 B'=DB
 748 \]
 749 with \(D\) diagonal and invertible, then
 750 \[
 751 (B')^{-1}P_iB'
 752 =B^{-1}D^{-1}P_iDB
 753 =B^{-1}P_iB.
 754 \]
 755 Therefore the branch projector is invariant under the same row-unit projectivization already used by
 756 the modal support theory. Outcome permutations only relabel \(i\).
 757 
 758 \subsection{Symbolic update}
 759 
 760 For
 761 \[
 762 \widetilde B\in\GL_n(\Sring)
 763 \]
 764 define
 765 \[
 766 \widetilde J_i=\widetilde B^{-1}P_i\widetilde B.
 767 \]
 768 Then every specialization satisfies
 769 \[
 770 v_A(\widetilde J_i)=J_i^{v_A(\widetilde B)}.
 771 \]
 772 So valuation commutes with the branch-update construction as long as the symbolic basis remains
 773 invertible under the relevant specialization.
 774 
 775 \section{Why this is not yet a closed process theory}
 776 
 777 The projector update is algebraically clean, but the zero divisors prevent the set of all nonzero
 778 states from being a well-behaved monoidal state class.
 779 
 780 \subsection{Nonzero states are not tensor-closed}
 781 
 782 Take
 783 \[
 784 \psi=(u,0),\qquad \phi=(u^3,0).
 785 \]
 786 Both are nonzero, but over \(A\)
 787 \[
 788 \psi\otimes_A\phi=0
 789 \]
 790 because \(uu^3=u^4=0\).
 791 
 792 \subsection{Unimodular preparation is not conditioning-closed}
 793 
 794 Restricting preparations to unimodular vectors repairs the elementary tensor-annihilation problem for
 795 simple tensors, but conditioning can leave that class. Consider the bipartite coefficient matrix
 796 \[
 797 M=\begin{pmatrix}1&0\\0&u\end{pmatrix}.
 798 \]
 799 As a four-component vector it is unimodular because one entry is a unit. Applying Alice's second
 800 standard effect leaves Bob in
 801 \[
 802 (0,u),
 803 \]
 804 which is nonzero but nonunimodular.
 805 
 806 Therefore a satisfactory closed sequential theory still needs an explicit decision about admissible
 807 states, branch objects, quotients, ideals, or a richer process semantics. The symbolic predicate layer
 808 can describe when a branch exists, but it does not turn a nonunit branch amplitude into an invertible
 809 one.
 810 
 811 \section{Failure results and exact boundaries}
 812 
 813 \begin{longtable}{p{0.34\textwidth}p{0.58\textwidth}}
 814 \toprule
 815 \textbf{Proposed identification} & \textbf{Result} \\
 816 \midrule
 817 \endhead
 818 ``An LM is a modal effect.'' & False as stated. Paper-B bras evaluate only to coordinate effects.\\
 819 \addlinespace
 820 ``Boolean valuation directly gives \(A\)-amplitudes.'' & False. Every Boolean-ring homomorphism into the local chain ring lands in \(\{0,1\}\).\\
 821 \addlinespace
 822 ``The ordinary \(2\times2\) CM algebra is the phase ring.'' & False. The four-component identification is linear; the cyclic product is additional structure.\\
 823 \addlinespace
 824 ``Valuation naturality derives modal possibility.'' & False. It derives concrete amplitudes. The zero/nonzero interpretation is a separate support contract.\\
 825 \addlinespace
 826 ``Nonzero support is ordinary relational semantics.'' & False compositionally. XOR cancellation and zero divisors prevent support from being a semiring homomorphism.\\
 827 \addlinespace
 828 ``Existing LM semantics generates all 192 measurements.'' & False. The full family is LM-encodable as coefficient data after adding the \(A\)-structure, but not generated by the original complement-paired effect semantics.\\
 829 \addlinespace
 830 ``Symbolic formulas solve normalization.'' & False. They can describe branch conditions but do not invert nonunits.\\
 831 \addlinespace
 832 ``All nonzero ring states form a monoidal state theory.'' & False: \(u\otimes_Au^3=0\).\\
 833 \addlinespace
 834 ``Unimodular preparations solve closure.'' & False: conditioning can yield nonunimodular nonzero states.\\
 835 \addlinespace
 836 ``The bridge extends unchanged to arbitrary finite rings.'' & False as stated. The clean \(\B\otimes_{\F_2}A\) enrichment requires an \(\F_2\)-algebra such as a characteristic-two ring; other scalar systems require a modified control/Boolean-power construction.\\
 837 \bottomrule
 838 \end{longtable}
 839 
 840 \section{Prior-art matrix}
 841 
 842 The novelty search translated each candidate theorem into conventional mathematics rather than
 843 searching only CM terminology.
 844 
 845 \begin{longtable}{p{0.31\textwidth}p{0.62\textwidth}}
 846 \toprule
 847 \textbf{Construction/result} & \textbf{Prior-art assessment} \\
 848 \midrule
 849 \endhead
 850 \(2\times2\) matrices for the 16 binary truth functions & Known. Bricken's 1997 notes explicitly discuss XOR as binary addition, AND as multiplication, all 16 binary logic matrices, truth-value vectors, and expressions of the form \(\langle x\vert B\vert y\rangle\). Edwards, Stern, and matrix-logic traditions are older.\\
 851 \addlinespace
 852 Truth vectors and matrix logical gates & Known. Mizraji's vector logic represents classical logical connectives by matrices acting on Kronecker products of truth vectors.\\
 853 \addlinespace
 854 Quantum-flavored logical observables & Known adjacent work. Eigenlogic treats logical observables in a genuine vector/Hilbert-space style distinct from Paper-B formula-valued LMs.\\
 855 \addlinespace
 856 Formula/Lindenbaum algebra modulo logical equivalence & Standard algebraic logic.\\
 857 \addlinespace
 858 Valuation commuting with matrix expressions & Standard homomorphism preservation and already proved in Paper B for LM pairing.\\
 859 \addlinespace
 860 \(\B\otimes A\) enrichment and \(A^\Omega\) interpretation & Standard base-change / Boolean-power mathematics. The contribution is its application as the exact typed bridge between the two CM layers.\\
 861 \addlinespace
 862 Boolean-to-local-ring idempotent obstruction & Elementary ring theory; not a novelty target.\\
 863 \addlinespace
 864 Modal effects with possible iff nonzero & Established in modal quantum theory over finite fields, especially Schumacher--Westmoreland.\\
 865 \addlinespace
 866 Finite-field/XOR quantum-like programming & Established adjacent work in James--Ortiz--Sabry.\\
 867 \addlinespace
 868 Semiring-valued quantum-like process theories & Established broad area; Gogioso's framework is a direct comparison point.\\
 869 \addlinespace
 870 Possibilistic contextuality as support/global-section obstruction & Established by Abramsky--Brandenburger and follow-on sheaf-theoretic work.\\
 871 \addlinespace
 872 States/effects and categorical measurement/update & Established broader categorical foundations, including effectus theory.\\
 873 \addlinespace
 874 Exact symbolic-support theorem for the CM-derived chain-ring enrichment & No exact prior formulation located in this search. Best characterized as a potentially useful new synthesis theorem rather than a new base algebraic construction.\\
 875 \addlinespace
 876 Negative theorem that original LM pairing does not generate the 192-basis family & Appears to be a useful internal comparison theorem; mathematically elementary once the types are separated.\\
 877 \addlinespace
 878 \(B^{-1}P_iB\) update & Standard conjugated-projector/change-of-basis algebra; useful as the natural division-free update for the current modal model.\\
 879 \bottomrule
 880 \end{longtable}
 881 
 882 \section{Adversarial interpretation}
 883 
 884 The project survives an adversarial reading, but in a narrower form than a claim of derivation.
 885 
 886 \subsection{What is routine}
 887 \begin{itemize}
 888 \item valuation naturality is homomorphism preservation;
 889 \item matrix semantics for logic has substantial prior art;
 890 \item semiring/ring-valued process models and modal nonzero semantics have prior art;
 891 \item conjugated coordinate projectors are standard algebra.
 892 \end{itemize}
 893 
 894 \subsection{What is genuinely informative in the synthesis}
 895 \begin{itemize}
 896 \item the idempotent obstruction proves that the strongest direct-valuation bridge is impossible;
 897 \item the tensor-product scalar enrichment gives the correct universal replacement;
 898 \item the support predicate gives an exact formula-level representation of concrete modal possibility;
 899 \item the effect-family comparison identifies which sense of ``LM measurement'' does and does not extend to the 192 modal bases;
 900 \item the update construction shows that nonunit amplitudes do not by themselves block a division-free branch map;
 901 \item the zero-divisor closure counterexamples locate the remaining process-theoretic weakness precisely.
 902 \end{itemize}
 903 
 904 \section{Computational verification}
 905 
 906 A fresh independent checker is included as
 907 \begin{center}
 908 \texttt{tests/lm\_modal\_bridge\_checks.py}.
 909 \end{center}
 910 It uses only the Python standard library. The exact machine-readable output is included as
 911 \begin{center}
 912 \texttt{tests/lm\_modal\_bridge\_checks\_output.json}.
 913 \end{center}
 914 
 915 The expanded run verified:
 916 
 917 \begin{longtable}{p{0.59\textwidth}r}
 918 \toprule
 919 \textbf{Check} & \textbf{Count/result} \\
 920 \midrule
 921 \endhead
 922 Elements of \(A=\F_2[u]/(u^4)\) & 16\\
 923 Units & 8\\
 924 Idempotents & exactly \(0,1\)\\
 925 \(|\GL_2(A)|\) & 24,576\\
 926 Unordered reversible projective two-outcome bases & 192\\
 927 Projector idempotence checks & 384\\
 928 Projector orthogonality checks & 384\\
 929 Resolution-of-identity checks & 192\\
 930 Branch nonzero iff effect coefficient nonzero & 98,304\\
 931 Same-basis repeatability checks & 98,304\\
 932 Unit row-scaling invariance checks & 24,576\\
 933 Symbolic support fiber checks & 262,144\\
 934 Paper-B LM matrix products tested & 256\\
 935 Products remaining in restricted 16-LM family & 112\\
 936 Products leaving restricted 16-LM family & 144\\
 937 Nonzero-state tensor annihilation example & confirmed\\
 938 Unimodular preparation to nonunimodular conditional example & confirmed\\
 939 \bottomrule
 940 \end{longtable}
 941 
 942 The 262,144 symbolic-support checks exhaust all
 943 \[
 944 16^4=65,536
 945 \]
 946 symbolic \(A\)-valued scalars whose four coefficient formulas are Boolean functions of two variables,
 947 and all four valuations of those variables.
 948 
 949 The package also preserves the original checker and output generated during the first pass, plus
 950 rerun logs showing that both the original and expanded scripts execute successfully in the current
 951 environment.
 952 
 953 \section{Paper implications}
 954 
 955 The bridge is strong enough to deserve a substantial theory section, but the present evidence does
 956 not support a standalone paper whose headline claim is that Paper-B LMs derive modal quantum
 957 measurement.
 958 
 959 The best placement is a central section of the broader Boolean--modal synthesis paper. A strong
 960 narrative is:
 961 \[
 962 \text{formula LM}
 963 \to
 964 \text{Boolean-ring contraction}
 965 \to
 966 \text{valuation naturality}
 967 \to
 968 \boxed{\text{idempotent obstruction}}
 969 \to
 970 \text{\(A\)-enrichment}
 971 \to
 972 \boxed{\text{symbolic-support theorem}}
 973 \to
 974 \text{modal effects/bases}
 975 \to
 976 \boxed{\text{extra support contract}}
 977 \to
 978 \text{division-free branch update}
 979 \to
 980 \boxed{\text{tensor/update closure obstruction}}.
 981 \]
 982 
 983 This structure has three advantages:
 984 \begin{enumerate}
 985 \item it answers the skeptical question of why the original logical pairing has anything to do with the
 986 later modal measurement rule;
 987 \item it clearly separates generic prior art from CM-specific typing and synthesis;
 988 \item it uses negative theorems as part of the contribution rather than forcing an overstated positive
 989 unification.
 990 \end{enumerate}
 991 
 992 \section{Strongest defensible theorem and final answer}
 993 
 994 The single best theorem to carry forward is the support-bridge statement
 995 \[
 996 \boxed{
 997 v\!\left(
 998 \Supp\bigl(\langle E,S\rangle_{\B\otimes A}\bigr)
 999 \right)=1
1000 \iff
1001 \left\langle
1002 v_A(E),v_A(S)
1003 \right\rangle_A\neq0.}
1004 \]
1005 Together with its existential form,
1006 \[
1007 \boxed{
1008 \Supp(\langle E,S\rangle)
1009 \text{ satisfiable}
1010 \iff
1011 \exists v:\langle v_A(E),v_A(S)\rangle_A\neq0.}
1012 \]
1013 
1014 It should be stated beside the negative companion theorem
1015 \[
1016 \boxed{
1017 \text{No direct Boolean-ring valuation }
1018 \B\to\F_2[u]/(u^4)
1019 \text{ can generate non-Boolean amplitudes.}}
1020 \]
1021 
1022 The pair identifies the bridge almost exactly:
1023 \begin{quote}
1024 The formula-valued LM calculus and the later modal theory do more than merely coexist, but the modal
1025 theory is an enriched realization of the symbolic Boolean semantics, not a theorem forced by the
1026 original LM semantics alone. The correct bridge is scalar enrichment plus specialization, followed by
1027 an independently declared support interpretation and measurement-family choice.
1028 \end{quote}
1029 
1030 \appendix
1031 \section{Reproducibility notes}
1032 
1033 Run the expanded checker from the root of this package with
1034 \begin{lstlisting}[language=bash]
1035 python tests/lm_modal_bridge_checks.py
1036 \end{lstlisting}
1037 The output should match \texttt{tests/lm\_modal\_bridge\_checks\_output.json} modulo the environment
1038 string. The script contains no network calls and depends only on the Python standard library.
1039 
1040 The prior adversarial-review archive is preserved under \texttt{inputs/} and unpacked under
1041 \texttt{prior\_adversarial\_review/}. The package manifest records SHA-256 hashes for every included
1042 file.
1043 
1044 \section{Search strategy}
1045 
1046 The prior-art search used conventional mathematical translations rather than CM-specific vocabulary.
1047 Queries included Boolean matrix logic, matrix logic and vector logic, formula/Lindenbaum algebra,
1048 matrices over semirings, Boolean powers, modal quantum theory, finite-field quantum computation,
1049 possibilistic contextuality, effectus theory, categorical quantum mechanics, and semiring-valued
1050 quantum-like theories. The machine-readable search ledger is included separately in
1051 \texttt{research/SEARCH\_LOG.csv}.
1052 
1053 \begin{thebibliography}{99}
1054 
1055 \bibitem{Bricken1997}
1056 William Bricken.
1057 \newblock \emph{Notes on Matrix Techniques for Logic}.
1058 \newblock March 1997.
1059 \newblock \url{https://wbricken.com/pdfs/01bm/01math/03math-supporting/math-tangential/04matrix-tech.pdf}.
1060 
1061 \bibitem{Edwards1972}
1062 C. R. Edwards.
1063 \newblock The logic of Boolean matrices.
1064 \newblock \emph{The Computer Journal}, 15(3):247--253, 1972.
1065 \newblock doi:10.1093/comjnl/15.3.247.
1066 
1067 \bibitem{Stern1988}
1068 August Stern.
1069 \newblock \emph{Matrix Logic: Theory and Applications}.
1070 \newblock North-Holland, 1988.
1071 \newblock doi:10.1016/C2009-0-14445-8.
1072 
1073 \bibitem{Mizraji1992}
1074 Eduardo Mizraji.
1075 \newblock Vector logics: The matrix-vector representation of logical calculus.
1076 \newblock \emph{Fuzzy Sets and Systems}, 50(2):179--185, 1992.
1077 \newblock doi:10.1016/0165-0114(92)90216-Q.
1078 
1079 \bibitem{Mizraji2008}
1080 Eduardo Mizraji.
1081 \newblock Vector logic: A natural algebraic representation of the fundamental logical gates.
1082 \newblock \emph{Journal of Logic and Computation}, 18(1):97--121, 2008.
1083 \newblock doi:10.1093/logcom/exm057.
1084 
1085 \bibitem{ChengQi2010}
1086 Daizhan Cheng and Hongsheng Qi.
1087 \newblock A linear representation of dynamics of Boolean networks.
1088 \newblock \emph{IEEE Transactions on Automatic Control}, 55(10):2251--2258, 2010.
1089 \newblock doi:10.1109/TAC.2010.2043294.
1090 
1091 \bibitem{Cheng2011}
1092 Daizhan Cheng, Yin Zhao, and Xiangru Xu.
1093 \newblock Matrix approach to Boolean calculus.
1094 \newblock In \emph{Proceedings of the 50th IEEE CDC-ECC}, pages 6950--6955, 2011.
1095 \newblock doi:10.1109/CDC.2011.6160289.
1096 
1097 \bibitem{DuboisToffano2016}
1098 Francois Dubois and Zeno Toffano.
1099 \newblock Eigenlogic: A quantum view for multiple-valued and fuzzy systems.
1100 \newblock arXiv:1607.03509, 2016.
1101 
1102 \bibitem{SchumacherWestmoreland2012}
1103 Benjamin Schumacher and Michael D. Westmoreland.
1104 \newblock Modal quantum theory.
1105 \newblock \emph{Foundations of Physics}, 42:918--925, 2012.
1106 \newblock doi:10.1007/s10701-012-9650-z; arXiv:1010.2929.
1107 
1108 \bibitem{JamesOrtizSabry2011}
1109 Roshan P. James, Gerardo Ortiz, and Amr Sabry.
1110 \newblock Quantum computing over finite fields: Reversible relational programming with exclusive disjunctions.
1111 \newblock arXiv:1101.3764, 2011.
1112 
1113 \bibitem{AbramskyBrandenburger2011}
1114 Samson Abramsky and Adam Brandenburger.
1115 \newblock The sheaf-theoretic structure of non-locality and contextuality.
1116 \newblock \emph{New Journal of Physics}, 13(11):113036, 2011.
1117 \newblock doi:10.1088/1367-2630/13/11/113036; arXiv:1102.0264.
1118 
1119 \bibitem{ChoEtAl2015}
1120 Kenta Cho, Bart Jacobs, Bas Westerbaan, and Abraham Westerbaan.
1121 \newblock An introduction to effectus theory.
1122 \newblock arXiv:1512.05813, 2015.
1123 
1124 \bibitem{Gogioso2017}
1125 Stefano Gogioso.
1126 \newblock Fantastic quantum theories and where to find them.
1127 \newblock arXiv:1703.10576, 2017.
1128 
1129 \bibitem{PaperB}
1130 Brian Theory.
1131 \newblock \emph{Operator-Level Boolean Computation with Correspondence Matrices}.
1132 \newblock Unpublished manuscript supplied in the ChatGPT File Library, accessed 18 September 2026.
1133 
1134 \bibitem{ModalAddendum}
1135 Brian Theory.
1136 \newblock \emph{Pure Boolean Modal CM Addendum}.
1137 \newblock Research manuscript/source file \texttt{PURE\_BOOLEAN\_MODAL\_CM\_ADDENDUM.tex}, accessed 18 September 2026.
1138 
1139 \end{thebibliography}
1140 
1141 \end{document}