PROSE AID ONLY — use PDF/page images and exact source for mathematics. PDF PAGE 1 Process Closure and Restricted Conversion in a Chain-Ring Modal Model Technical companion to the CM/LM publication project Internal research companion ∗ 20 September 2026 Abstract Let A = F2[u]/(u4). Declaring every nonzeroA-amplitude possible is incompatible with independently composed events, sinceu3u = 0. Restricting deterministic states to primitive vectors does not repair selective conditioning. This companion separates the resulting static support model from a closed process theory. It gives a field-modal completion, a Frobenius realization of the original bipartite support tables by coarse binary effects, and exact restricted conversion results for the markedA-linear subtheory. The one-copy filter preorder is compo- nentwise order on padded Smith exponents. For collective filters retaining all phase registers, extraction of a free target is possible exactly when the residue matrix has sufficient rank. This criterion is elementary local-ring algebra and is not presented as a new distillation principle. If resolved coefficient measurements are allowed before phase disposal, an explicit two-copy protocol produces a free single-register target; forgetting the outcomes instead produces a mixed modal state. The comparison is technically useful, but current priority evidence supports a technical companion rather than a separate fourth research paper. Status. All universal statements below have written proofs. Exhaustive computations are corrobo- ration over stated finite universes, not substitutes for those proofs. The document has not received independent human refereeing or proof-assistant certification. It asserts no Born rule, physical implementation, algorithmic speedup, or matched-contract communication advantage. 1 Question, scope, and contribution The chain-ring model used in the CM/LM project associates bipartite resources with matrices over A = F2[u]/(u4). Its static support tables and Smith-type resource labels are mathematically well defined. The difficulty is operational: if nonzero amplitudes are interpreted as possible events and independent events multiply, then two locally possible nilpotent events can have an impossible joint event. A process theory must also survive selective measurements, discarded outcomes, references, and classical communication. Field-modal theories already supply generalized states, effects, conditioning, and mixtures [9, 8]. Module types over finite chain rings are classical [4]; matrix factorization comparison and its stabilized variants also have established algebraic antecedents [2, 1, 6]. Accordingly, the contribution claimed here is a careful operational comparison and boundary analysis: ∗Prepared for mathematical review within the CM/LM publication project. Author metadata and publication status are not assigned. 1 PDF PAGE 2 1. a concise obstruction to faithful nonzero-support independence and to primitive-only repair; 2. a closed literal binary-modal contract and an exact realization of the declared ring support tables; 3. the linear normalizer of the markedA-operation algebra and the associated coarse measurement family; 4. exact one-copy conversion underA-linear filters, with its relation to established algebraic com- parison orders; 5. an exact retained-phase free-target criterion and a resolved-disposal activation example. These results do not identify the ring model with its binary realization as tensor theories. The original CM matrices,A-modules, literal binary systems, full binary gates, coarse ring effects, and physical quantum measurements remain distinct mathematical objects. 2 Why faithful nonzero support does not close Write 0 for an impossible scalar event. Suppose tensoring scalar events agrees with multiplication in A. Theorem 2.1(Independent-success obstruction). There is no Boolean possibility predicatep :A→ {0, 1} such thatp(0) = 0, every nonzero scalar has value one, andp(ab) =p(a)p(b). Proof. Both u3 and u are nonzero, whileu3u = u4 = 0. Multiplicativity would give0 = p(0) = p(u3u) =p(u3)p(u) = 1. A vector inAn is primitive when its coordinates generate the unit ideal. Primitive vectors close under invertibleA-linear maps and balanced tensor products. That observation alone does not produce a process theory. Proposition 2.2(Primitive states fail after selection). Primitive preparations with the original nonzero-amplitude outcome rule do not have independently factorizing outcome sets, and selected outcomes need not be primitive. Proof. The vectors(1,u 3) and (1,u ) are primitive. Their second coordinate outcomes are locally nonzero, but the joint second–second amplitude isu3u = 0. Selecting the second coordinate of (1,u 3) also produces the nonprimitive scalaru3. Typed quotient normalization does not fix the problem naturally. PutBℓ =A/(uℓ). Dividing a nonzero vector of minimum valuationa gives a primitive vector overBℓ−a. For (1,u 3)∈ B2 4, selecting the second coordinate and normalizing gives1∈B1. Tensoring that branch with1∈B1 remains nonzero. If one first tensors the original state with1∈ B1, the image is(1, 0) and the second branch is zero. Thus conditioning and tensoring do not form the required natural square. There is a unique multiplicative Boolean alternative. Proposition 2.3(Unit predicate). If p :A→{ 0, 1} is multiplicative withp(0) = 0 and p(1) = 1, then p(a) = 1 exactly for the units ofA. Proof. If a is a unit,1 =p(a)p(a−1), sop(a) = 1. Ifa is nilpotent, thenp(a)n =p(an) =p(0) = 0. Units and nilpotents exhaust this finite local ring. The unit predicate reduces primitive states operationally to their nonzero residue classes over A/(u)∼= F2. It closes, but deletes every nilpotent support event and all higher Smith information. No preservation of coherent addition as Boolean disjunction is asserted. 2 PDF PAGE 3 3 A closed literal modal contract The minimal closed completion used here is ordinary generalized modal theory overF2 [8]. Definition 3.1(Literal contract). A system is a finite-dimensionalF2-vector spaceV. A possible state is a nonzero subspaceS≤V; the zero subspace denotes an impossible branch. A selective process is a subspaceK≤ HomF2(V,W ) acting by K(S) = span{k(s) :k∈K, s∈S}. An instrument is a finite family{Ko} whose constituent maps have common kernel zero. Effects are subspaces ofV ∗. A destructive measurement is complete when its effect subspaces spanV ∗. Discard uses all ofV ∗. Classical data are incoherent families of subspaces, composed by blockwise spans. Independent systems use⊗F2. Proposition 3.2 (Closure). The contract in definition 3.1 closes under sequential and parallel composition, selective conditioning, untouched references, classical feed-forward, outcome erasure, and discard. Proof. Stacking every constituent map of a complete instrument gives an injection. Composing injections and tensoring them over a field preserve injectivity, including after tensoring with an identity on a reference. A possible selected branch is nonzero by definition, and two nonzero independent subspaces contain a nonzero simple tensor. Associativity and interchange follow from their linear-map counterparts. For outcome erasure, letk range over every constituent map andf over the corresponding output dual. The span offk has annihilator equal to the common kernel, hence zero. Finite-dimensional duality makes that span the entire input dual. Thus forgetting a complete instrument produces the same remote reduced subspace as discard. The forgetting operation is a span of labelled branches, not a coherent XOR, so branches do not cancel. Full binary effects can distinguish some vectors related by multiplication by anA-unit. Therefore algebraic orbit equivalence underA-linear gates is not equality of all literal measurement statistics. The operational equivalence relation must always be declared with the effect set. 4 Exact coarse realization of the ring tables Let R(a) be regular multiplication bya∈ A in the ordered basis(1,u,u 2,u 3). Let λ : A→ F2 extract the coefficient ofu3, and letJ be the anti-diagonal matrix of the Frobenius pairingλ(ab). Then R(a)J =JR(a)T. ForM∈ Matm×n(A) define a binary bipartite coefficient matrixΦ(M) by the 4-by-4 blocks Φ(M)ij =R(Mij)J. For a primitive rowx∈Am, letEx be the four-dimensional dual subspace containing the coefficient covectors ofz↦→xz∈A. Theorem 4.1(Coarse support realization). An invertibleA-basis of primitive rows gives a complete coarse measurement in the literal contract. For primitive rowsx,y, (Ex,E y) is possible onΦ(M) ⇐⇒ xMy T̸= 0. 3 PDF PAGE 4 Moreover, for compatibleA-matricesP,Q, R(P )Φ(M)R(Q)T = Φ(PMQ T ). Proof. Block multiplication and the Frobenius identity give R(x)Φ(M)R(y)T =R(xMy T )J. The right side is nonzero exactly whenxMy T is nonzero. Expanding an invertibleA-basis in the polynomial basis gives an invertible binary matrix, so the four-row effect blocks span the full binary dual. The local intertwining formula is the same calculation with matrix-valuedP,Q. This preservation statement is deliberately narrow. The forgetful assignment fromA-modules to binary spaces has natural surjections U(M)⊗F2U(N)−→U(M⊗AN), but it is not strong monoidal: forM =N =A the domain has binary dimension16 and the target dimension 4. The scalar Frobenius lift ∆(a) = 3∑ i=0 ui⊗u3−ia satisfiesµ∆(a) = 4u3a = 0. Thus the balanced comparison kills every lifted scalar. Ring separability is also not preserved:diag(1, 0) factors overA but its binary lift has Schmidt rank four. 5 The marked linear symmetry group Set V =U(A2), C ={R(a)I2 :a∈A}, B =R(Mat2(A))⊆ EndF2(V ). Theorem 5.1(Algebra normalizer). The binary linear normalizer of the marked operation algebra is NGL8(F2)(B) = GL2(A) ⋊ AutF2(A), |N| = 98,304. It is also the state-transformation group preserving the family of coarseA-projective measurements. Its induced permutation group on the24 projective points has order12,288. Proof. The centre ofB isC, and the centralizer ofC isB. A normalizer ofB therefore normalizesC and induces anF2-algebra automorphismσ of A. Removing coefficientwiseσ leaves an invertible A-linear map. Conversely both factors normalizeB. Every automorphism sends u↦−→u +cu2 +du3, c,d ∈ F2, and all four choices occur. Reduction modulou gives| GL2(A)| =| GL2(F2)|212 = 6· 4096. For the measurement family, realize its column counterpart as the24 free rank-one submodules Av≤ A2. A binary map fixing every such point fixes both coordinate axes and has block form diag(P,Q ). FixingA(1, 1) givesP =Q, and fixing every graphA(1,a ) forces P to commute with all R(a). Hence the pointwise stabilizer is the eight scalar units. These units spanC, so any setwise stabilizer normalizesC and belongs toN. Conversely semilinear maps preserve the points and their complementary pairs. Finally putJ2 = diag(J,J ). Under the Frobenius pairing, effect columns areJ2 times projective state columns. Pullback by a state map introduces transpose and conjugation byJ2; because J2CTJ2 =C, the resulting condition is again membership inN. The kernel on the point set is the eight scalar units, giving98,304/8 = 12,288. 4 PDF PAGE 5 This is not the abstract automorphism group of the distant graph. Projective-line geometry supplies related semilinear constructions and radical parallelism [3]; the exact historical priority of this ambient-binary stabilizer specialization remains unresolved. 6 One-copy conversion under local filters Every matrix over the chain ringA admits a Smith form. Write its sorted exponents as 0≤a1≤···≤ aD≤ 4, where exponent 4 denotes a zero diagonal entry. When dimensions differ, pad both lists by4 to the maximum relevant diagonal length. Theorem 6.1(Exact filter preorder). LetM and N be nonzero rectangularA-matrices with padded Smith exponentsai and bi. The following are equivalent: 1. N =LMQ T for compatibleA-matricesL,Q; 2. Φ(N) is a possible pure target under localA-linear filters, finite classical feed-forward, and local logical product ancillas; 3. bi≥ai for everyi. Proof. Let H = im(M). The image ofMQ T is a submodule ofH, and applyingL makes im(N) a quotient of that submodule. For a finiteA-module T define fj(T ) = dimF2 ( uj−1T∩ soc(T ) ) , 1≤j≤ 4. If T has decreasing cyclic lengthsℓi, thenfj(T ) = #{i : ℓi≥ j}. An injection preserves these subspaces, so their dimensions cannot increase. A surjection dualizes to an injection under the exact binary dual; transposed nilpotent Jordan blocks preserve cyclic lengths. Hence every target cyclic length is bounded by its source counterpart. Since the lengths are4−bi and 4−ai, necessity follows. For sufficiency, reduce both matrices to rectangular Smith forms. Select the required source coordinates and multiply coordinatei byubi−ai when bi< 4; set all remaining target coordinates to zero. Undo the target basis changes. This constructsL,Q. Along a fixed adaptive transcript, local maps compose toL,Q. If a coarse outcome has a one-dimensional nonzero final state, expanding every process subspace into spanning maps yields at least one nonzero constituent trajectory producing that same target. Thus classical control and coarse-graining do not enlarge pure-state reachability. The complete monotones are the threshold counts cj(M) = #{i :ai1, so no general Smith theorem overBk is used. The retained-phase contract allowsBk-linear local rectangular filters, finite adaptive instruments, local logical product ancillas, and classical selected transcripts. Every phase register remains in the target; there is no free correlated phase ancilla and no phase measurement or partial phase disposal. Theorem 8.1(Free-target extraction over a local ring). Let (B, m) be a commutative local ring with residue fieldF. For a matrixN over B, there exist rectangularL,Q such that LNQ =Id if and only ifrankF (N mod m)≥d. Proof. Reduction modulo m proves necessity. Conversely, choose ad-by-d minor whose residue determinant is nonzero. Its determinant is outside the maximal ideal and is therefore a unit. IfS,T select the corresponding rows and columns andH =SNT , thenH is invertible. TakingL =H −1S and Q =T givesLNQ =Id. This unit-minor proof is standard local-ring matrix algebra; compare the behavior of minor ideals under multiplication in [10]. 6 PDF PAGE 7 Corollary 8.2(Exact retained-phase criterion). Letr0 be the number of unit Smith factors of an A-matrix M. Under the retained-phase contract,k copies of ΦA(M) can produce the free structured target ΦBk(Id) on a possible selected branch if and only if rk 0≥d. Proof. Regrouping thek independent binary copies gives the Frobenius lift of N =M(u1)⊗···⊗ M(uk). Residue commutes with the Kronecker product, so its rank isrk 0. Apply theorem 8.1 and the local intertwining identity from theorem 4.1. The earlier socle-rank obstruction is precisely this residue rank in binary form. Ifs =u3 1··· u3 k, then sm = 0 and sN =s ¯N. Regular multiplication bys has binary rank one, so rankF2RBk(sN) = rankF2( ¯N). ForM = diag(1,u 2),r0 = 1; no number of retained-phase copies produces a free target of coordinate rank two. Atk = 2 the ordinary binary input rank is36, while the free target rank is32. Ordinary rank permits the conversion, but the retained-phase residue rank forbids it. 9 Resolved phase disposal activates the example We now enlarge the operation class. After aB2-linear filter, each party may measure the second phase register in the binary coefficient dual basis, retain the outcome label, and keep only the first phase register. This is a valid selected effect in the literal modal theory, but it is not aB2-linear quotient map. Therefore corollary 8.2 no longer applies after the measurement. Writev =u2 and order the two-copy logical coordinates as(0, 0), (0, 1), (1, 0), (1, 1). For two copies of diag(1,u 2), the regrouped coefficient matrix is N = diag(1,v 2,u 2 1,u 2 1v2). Proposition 9.1(Two-copy resolved-disposal witness). Two copies ofΦA(diag(1,u 2)) can produce ΦA(I2) on a possible selected branch when coefficient-resolved disposal of the second phase register is allowed. Proof. Choose the 2-by-4 filters L = ( 1 0 0 0 v 0 0 0 ) , Q = ( v3 0 0 0 v2 0 0 0 ) . Then LNQ T = ( v3 v2 0 v3 ) =W. Let ℓ extract the coefficient ofv3 from F2[v]/(v4). On the scalar Frobenius lift ∆v(a) = 3∑ i=0 vi⊗v3−ia, one has (ℓ⊗ℓ)∆v(a) =ℓ(a). Both parties select this effect on their second phase register. Applying ℓ entrywise toW givesI2, so the retained state is exactlyΦA(I2). 7 PDF PAGE 8 This witness uses only theN00 = 1 sector. The same construction works with two copies of ΦA(diag(1, 0)), so it does not isolate a power unique to nilpotent amplitudes. It transfers part of the phase entanglement already present in the Frobenius lift into a logical coordinate. Outcome resolution matters. In the displayed protocol, forgetting the phase labels joins two independent output vectors: span{ΦA(I2), ΦA(E12)}. The unconditional output is therefore a two-dimensional mixed modal state, not the desired pure target. This calculation does not classify every protocol with unobserved phase disposal; it only shows why selected measurement and blind discard are different contracts. 10 Priority assessment and publication disposition Table 1: Disposition of the main restricted-conversion results. Result Mathematical status Priority assessment Coarse binary realiza- tion Proved exactly for the declared bipartite support tables Frobenius machinery is classical; exact CM/LM comparison is a synthesis claim Linear normalizer Proved by centre, commutant, and projective-point stabilizer Exact ambient-binary specialization not certified as first One-copy filter preorder Proved for rectangular systems and adaptive pure branches Standard chain-ring module and matrix-factorization consequence Filtered ranks Proved monotone, not complete Normalized functions are established quotient Sylvester ranks Retained-phase free ex- traction Necessary and sufficient residue-rank criterion Elementary unit-minor result; not a new distillation principle Resolved-disposal acti- vation Exact two-copy witness Useful operation-boundary example; also works on unit sectors Contextuality increase Explicit selected-filter counterexample Direct consequence of P01’s static classification Cao’s work on matrix monoids over Artinian chain rings explicitly studies Green relations [5], but only the publisher abstract was accessible during this review. No theorem number or exact equivalence is attributed to that source. A bounded search and an inaccessible full text cannot establish novelty. The appropriate disposition is therefore a technical companion, not a separate fourth full paper. Its useful role is to prevent category mistakes: static ring support is not a closed process theory; the Frobenius realization is not monoidal; conversion claims depend on the operation set; resolved conditioning differs from discard; and several apparent monotones are standard algebra in operational dress. 11 Computational verification The analytic proofs above cover arbitrary declared dimensions. Independent finite checks provide regression evidence: • all 65,536 two-by-two matrices overA were classified independently; • 1,966,080 left and right one-sided filter cases matched the exponent downsets; 8 PDF PAGE 9 • 1,225 ordered comparisons of three-entry profiles matched threshold-count order; • all 65,536 two-by-two matrices overF2[x,y ]/(x2,y 2) passed the residue/socle and explicit free- extraction checks; • a declared 4,096-matrix sample overB2 passed the same checks; • the resolved-disposal witness was verified by full binary matrix multiplication, with output rows identical to ΦA(I2); • the frozen baseline exhaustive checker and its twelve unit tests were rerun from byte-identical copies. The baseline normalizer enumeration constructs98,304 distinct semilinear maps; its upper bound is the analytic proof in theorem 5.1, not an exhaustive scan ofGL8(F2). Exact scripts, inputs, raw outputs, hashes, and environment records accompany the review package. The larger canonical audit’s previously reported provenance mismatches are separate and were not rewritten here. 12 Conclusion The chain-ring support tables have a faithful static realization inside a closed binary-modal theory, but the realization does not preserve tensor composition or separability. Once operations are fixed, one-copy conversion is governed by the full Smith exponent order. Collective retained- phase extraction of a free target is governed exactly by residue rank. Allowing a resolved phase measurement changes the resource theory and activates the two-copy example, while blind outcome erasure produces a mixed state. The principal lesson is methodological: every resource claim must name the systems retained, the scalar action respected by each process, the available effects, and whether outcomes are selected or forgotten. Within those boundaries, the proofs are closed. Their algebraic core is largely established machinery, so the appropriate publication vehicle is this technical companion. References [1] Ramon Antoine, Pere Ara, Joan Bosa, Francesc Perera, and Eduard Vilalta. The cuntz semigroup of a ring, 2023. [2] Gonzalo Aranda Pino, K. R. Goodearl, Francesc Perera, and Mercedes Siles Molina. Non-simple purely infinite rings. American Journal of Mathematics, 132(3):563–610, 2010. [3] Andrea Blunck and Hans Havlicek. On distant-isomorphisms of projective lines.Aequationes Mathemat- icae, 69:146–163, 2005. [4] Eimear Byrne, Anna-Lena Horlemann, Karan Khathuria, and Violetta Weger. Density of free modules over finite chain rings.Linear Algebra and its Applications, 651:1–25, 2022. [5] Yonglin Cao. On the multiplicative monoid ofn×n matrices over artinian chain rings.Communications in Algebra, 38(9):3404–3416, 2010. [6] Tsz Fun Hung and Hanfeng Li. Malcolmson semigroups.Journal of Algebra, 623:193–233, 2023. [7] Andrei Jaikin-Zapirain and Diego López-Álvarez. On the space of sylvester matrix rank functions, 2020. [8] Benjamin Schumacher and Michael D. Westmoreland. Almost quantum theory, 2012. [9] Benjamin Schumacher and Michael D. Westmoreland. Modal quantum theory.Foundations of Physics, 42:918–925, 2012. [10] The Stacks Project Authors. The stacks project: Fitting ideals, tag 07z6, 2026. Accessed 20 September 2026. 9