A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Odd Characteristic

1 Introduction

A group algebra K[G] consists of finite sums \sum_g c_g g, with c_g\in K, multiplied by the group law and bilinearity. A unital ring is directly finite when ab=1 implies ba=1; it is stably finite when this implication holds in every finite square matrix ring. Kaplansky’s direct-finiteness conjecture asks whether K[G] is directly finite for every field K and every group G, including groups with torsion. The result here is a negative answer in one specified odd characteristic.

The word “specified” refers to finite formulas and choices in ordered finite sets. The field is a quadratic tower, the group is built from finitely many finite groups and explicit identifications, and the sums are defined without solving the word problem of that group. The construction is not claimed to have been evaluated as a practical computation. It concerns the displayed prime, not every odd prime.

The finiteness problem. For finite G, the algebra and all its finite matrix rings are finite-dimensional over K. A one-sided inverse between their left multiplication operators is therefore two-sided. This argument works even when the characteristic divides |G|. The difficulty for an infinite group is to obtain a corresponding conclusion from finite support alone.

Kaplansky proved stable finiteness in characteristic zero using finite von Neumann algebras, and asked whether the one-sided-inverse property persists in positive characteristic [Kap72, pp. 122–123]. Ara, O’Meara, and Perera proved stable finiteness for free-by-amenable groups over arbitrary division rings [AOP02]. Elek and Szabó [ES04, Corollary 4.7] established the same conclusion for all sofic groups. Such groups admit finite permutation models that approximate multiplication while keeping distinct elements separated. These theorems are constraints on the example, not hypotheses about its group. In particular, Elek–Szabó’s theorem implies, after the scalar construction is completed, that G cannot be sofic.

A separate characteristic-two construction uses incidence structures, polynomial pairings, a planar group-embedding argument, and character packing [Ope26, Theorem 1.1]. The construction here has a different local mechanism. A high-degree subspace of a modular translation representation is annihilated on one side; a weighted pairing gives a compatible annihilation on the other. The characteristic-two theorem is not an input to this odd-prime argument, and the two methods retain distinct coefficient hypotheses.

The two quantities that must be compared. A matrix F fits a simple graph when F_{ii}=1 and F_{ij}=0 for distinct nonadjacent vertices. We use a triangle-free graph on m vertices and a specified factorization

This is an explicit threshold-Kneser instance of the polynomial constructions of Haviv and Golovnev–Haviv [Hav18, Proposition 13] [GH22, Theorem 3.2 and Lemma 3.5], whose polynomial ancestry includes Alon–Babai–Suzuki [ABS91]. Low-rank fitting matrices also give storage-code recovery equations. The triangle-free constructions of Barg–Zémor, Huang–Xiang, and Barg–Schwartz–Yohananov provide related context [BZ22, HX23, BSY24]. Their results are not used to supply a group-algebra idempotent.

At every vertex we use a copy of a finite group H and an idempotent e\in K[H]. Write w=|H| and d=\dim_K eK[H], the rank of left multiplication by e on the regular representation. The second quantity is

The idempotents satisfy a directed condition: after identifying certain cyclic subgroups at adjacent vertices, e_i e_j=0 for every edge with i<j. There is no assertion about the reverse product. The asymmetry is what allows a triangular calculation to combine the two strict bounds.

The characteristic-dependent construction appears in Section 3. High augmentation degree gives the annihilated columns, in the elementary abelian setting of Jennings’ theory [Jen41]. A diagonal weighted Gram determinant and Schwartz’s grid bound [Sch80, Lemma 1 and Corollary 1], with related polynomial-identity work of Zippel [Zip79], supply enough nondegenerate pairings. Fourier character orbits turn these projections into actual elements of K[H]. The full algebra need not be semisimple; only its prime-to-p normal abelian subgroup is diagonalized, in the character-orbit setting of Clifford [Cli37].

From the comparison to scalar elements. The proof has three transitions that numerical rank inequalities alone cannot justify. First, the quotient identifying the cyclic subgroups must retain the finite vertex groups. Its injectivity proof uses the C(4)–T(4) graphical-complex mechanism described by Prytuła [Pry24, Section 8]: local short-word restrictions and triangle-freeness rule out a spherical map with exactly one nontrivial face. Ordinary graph-of-groups normal form does not by itself establish this first quotient’s injectivity. Section 4 proves it locally, including the reductions for loops and bridges.

Second, regular ranks must become identities over the same ring. A further group enlargement G_1 has a nonzero central idempotent f in R=K[G_1]. In the ring S=fR, whose identity is f, the elements P_i=fe_i admit rectangular matrices

The finite-group blocks in this step have elementary abelian p-group stabilizers, possibly nontrivial and nonnormal. Their group algebras have nilpotent augmentation ideals, and a finite change of basis proves exact integer rank identities before coordinates are paired. Section 5 gives the full construction of these matrices, with their entries in explicitly attached finite subgroups.

Third, a matrix counterexample does not yet give two scalar elements of the same group algebra. In Section 6, the fit and the ordered zero products produce a split injection

whose composite is the identity. We derive square matrices with a one-sided inverse and certify the reverse defect by a nonzero kernel vector. Section 7 then implements the finite-factor reduction of Dykema–Juschenko [DJ15, Lemma 2.1 and Theorem 2.2]. A finite Heisenberg group gives explicit matrix units, their linear independence makes the encoding injective, and a complementary idempotent restores the full scalar identity. The shift and modulation model is classical [GH09, Section 1.2]; the composite cyclic order and finite coefficient field used here are checked directly.

The dynamical consequence. A configuration over a finite alphabet A is any function G\to A. A cellular automaton has one finite memory set E\subset G and one rule applied to (x(gu))_{u\in E} at every g. A group is surjunctive when every injective such automaton, over every finite alphabet, is surjective. Gottschalk proposed this for all groups [Got73]; Gromov and Weiss proved it for sofic groups [Gro99, Wei00]. The group-ring connection is recorded by Elek–Szabó and developed in the linear theory of Ceccherini-Silberstein–Coornaert [ES04, CC06]. Phung proved stable finiteness over every field for surjunctive groups [Phu23, Theorem B]; subsequent proofs and extensions include [CCP25a, CCP25b].

Section 8 gives the elementary transfer with its actual multiplication convention. The coefficients of b define an injective automaton, those of a give a left inverse, and a nonzero coefficient of ba-1 specifies a missing point indicator. The argument excludes preimages of arbitrary support. It is a consequence of the scalar construction, not a further existence theorem.

All matrices below act on columns of free right modules. Matrix entries therefore multiply coordinates on the left. Transpose is ordinary transpose over K, with no involution. Cardinalities and dimensions in rank comparisons are integers, not elements of the coefficient field. For a positive integer n, write [n]=\{1,\ldots,n\}.

2 The field, the graph, and the local group

We first construct the finite data. This fixes every coefficient field and denominator before the modular argument begins.

Choose the least nonsquare using representatives 0,\ldots,p-1; order K by its coefficients in (1,\xi,\nu,\xi\nu). Put M=p^4-1 and choose the least element \zeta\in K^* of order M. This element exists: if b is the least common multiple of all orders in K^*, then b\mid M, while the root bound for X^b-1 gives M\le b. For each prime-power factor of b, an appropriate power of one element has that exact order. Multiplying the resulting commuting elements of coprime orders gives order b=M. In particular, M and all its powers are invertible in K.

Use natural orders on integer indices, increasing-list lexicographic orders on subsets, and lexicographic orders on tuples and finite function tables. Order characters of (\mathbb Z/M\mathbb Z)^D by their exponent lists on the ordered coordinate generators. These conventions specify the finite choices below.

Let I_1,\ldots,I_m be the 600-subsets of [1200] in order. Join distinct i,j in \Gamma exactly when |I_i\cap I_j|<200. Put

For the local algebra define the additive group P=(\F_p)^3, the function group A_0=(\mathbb Z/M\mathbb Z)^P, and

Its order is w=M^{p^3}p^3. Let c\in A_0 have value one at zero and zero elsewhere, and put

The coordinate formula v(s)=(c-s\cdot c,s) gives U\cap V=\{1\}, because c-s\cdot c\ne0 when s\ne0. We next construct an idempotent with incoming annihilation by U and outgoing annihilation by V.

3 Large idempotents with ordered annihilations

The graph has supplied the ratio t/m<1/100. Our present objective is an idempotent in the finite group algebra K[H] whose regular rank fraction exceeds 1/96. It must also have two prescribed annihilations, on opposite sides. Here

The regular rank of e\in K[H] means \dim_K eK[H], the dimension of the image of left multiplication. Although this map multiplies on the left, its image is a right ideal.

3.1 The annihilated column space

On K^P, let \mathbf e_x be the standard basis and L_s\mathbf e_x=\mathbf e_{s+x}. For the three standard generators \varepsilon_j of P set X_j=L_{\varepsilon_j}-I. The X_j commute, and characteristic p gives X_j^p=0. The vectors

are a basis. Indeed, every translation of \mathbf e_0 is a product of powers of I+X_j applied to \mathbf e_0, so the displayed family spans K^P; its cardinality is the dimension p^3.

This is the elementary abelian augmentation filtration in the setting studied by Jennings [Jen41]. Here the exact degree argument can be given directly. Let B have as columns the displayed basis vectors with n_1+n_2+n_3\ge D-\ell+1, in order. For every s, the matrix L_s-I is a polynomial in the X_j with zero constant term. Multiplication by its \ellth power raises every column’s total degree to at least D+1. Such a monomial has an exponent at least p, so vanishes. Applying the same assertion to -s and transposing yields

The transpose identity uses L_{-s}^{\mathsf T}=L_s; no scalar conjugation is present.

The columns are independent. Their number h can be counted by setting a_j=p-1-n_j: the new nonnegative indices have sum at most \ell-1. Their individual upper bounds are automatic because \ell-1<p. Stars and bars gives

For the strict inequality, the numerator minus p^3 is 3p^2-p-3>0 for p\ge3.

The same subspace is killed on the two sides in Equation (3.1), but an arbitrary projection onto it need not give the outgoing conjugate annihilation. We therefore choose the row factor by a diagonal weighted pairing.

3.2 Many nonsingular diagonal pairings

For a finite function group A_*=(\mathbb Z/M\mathbb Z)^E, its characters \chi:A_*\to K^* are all exponent lists of powers of \zeta. Put

The denominator is invertible. Reindexing the sum gives a p_\chi^{A_*}=\chi(a)p_\chi^{A_*}. A nontrivial character has sum zero, since translating its sum by an element on which it is nontrivial multiplies it by a scalar different from one. It follows that

For the last identity, sum coefficients over the independent character exponents: every nonzero coordinate of a nonidentity element is detected by a power of \zeta. Each idempotent is nonzero, there are |A_*| of them, and orthogonality makes them independent. They form a basis of K[A_*].

Specialize to A_0. Its character action is (s\lambda)(a)=\lambda((-s)\cdot a). Define

Call \lambda admissible if its stabilizer in P is trivial and N_\lambda is invertible. This condition is a finite test.

Select the first admissible character from each P-orbit containing one, and call the set of selected characters \mathscr R. Each selected orbit is free, of size p^3. These orbits cover all admissible characters, so

This argument does not need admissibility to be constant on an orbit.

3.3 Realizing the projections in the group algebra

Fix \lambda\in\mathscr R. For x,y\in P, define

Conjugation gives u(s)p_\chi^{A_0}u(s)^{-1}=p_{s\chi}^{A_0}. Fourier orthogonality and freeness of the orbit therefore imply

Indeed the Fourier product can survive only when x\lambda=(x-y+z)\lambda, which is equivalent to y=z. Products between different orbits vanish. The sum q_\lambda=\sum_x p_{x\lambda}^{A_0} is central: it commutes with A_0 and its summands are permuted by P. The basis p_{x\lambda}^{A_0}u(s) of its ideal is indexed equally by (x,y) through s=x-y. Thus these units prove

This is an explicit character-orbit block, in the normal-subgroup representation setting of Clifford [Cli37]. It does not assert semisimplicity of the entire modular algebra.

On this block, the elements u(s), c, and v(s) act respectively by

For example, left multiplication by u(s) shifts row x to row s+x, whereas c multiplies row x by (x\lambda)(c).

Define a matrix and then a finite group-algebra element by

The inverse exists by the selection criterion. Since B^{\mathsf T}D_\lambda^{-1}B=N_\lambda, we have T_\lambda B=B and T_\lambda^2=T_\lambda. Its image is exactly \im B, of dimension h. Orthogonality of the blocks proves e^2=e; on the complementary ideal its component is zero.

We now compute the regular rank, rather than the smaller rank of a single representation matrix. In \Mat_{p^3}(K), left multiplication by T_\lambda has image the matrices whose p^3 columns all lie in \im B. Its image dimension is hp^3. Summing the selected blocks gives

The incoming annihilation follows from (L_s-I)^\ell T_\lambda=0. For the outgoing annihilation, keep the factors in their actual order:

Equation (3.1) proves both assertions on every selected block. They also hold on the complementary ideal, where e is zero. This proves Proposition 3.1, including its finite prescription and the strict rank estimate.

3.4 From opposite annihilations to an ordered zero product

Take copies H_i of H. For each edge i<j, identify v(1,\bar j,\bar j^2) in H_i with u(1,\bar i,\bar i^2) in H_j, where bars denote residues in \F_p. Let G_0 be the resulting free-product quotient, and let e_i denote the image of e from H_i. The injectivity of these copies is proved in Section 4. The following ordered identity already follows from the local calculations and the quotient relations.

We have constructed the large regular rank and the edge vanishings. To compare them with the graph factorization, the remaining structural step must put exact rectangular realizations of all those ranks inside one nonzero ring.

4 Gluing the finite groups without collapsing them

The quotient G_0 of Section 3 identifies a cyclic subgroup across each edge of \Gamma. It must retain every vertex group: otherwise a later nonzero element of a finite group algebra could disappear. We first prove a general injectivity criterion and then verify it for the groups used here. The local injectivity argument belongs to the C(4)–T(4) theory of graphical complexes of groups [Pry24, Section 8]. We give a direct proof using only labelled spherical maps and Euler’s formula.

4.1 An injectivity criterion

4.2 Application to the modular construction

Recall from Section 2 the group H=A_0\rtimes P, with P=(\F_p)^3 and A_0=(\mathbb Z/M\mathbb Z)^P. Its subgroups U=u(P) and V=cUc^{-1} satisfy U\cap V=\{1\}. The quotient G_0 identifies, for every edge i<j, the elements

where \bar i denotes the residue of i in \mathbb F_p. Both elements have order p, so the corresponding maps from C_p are embeddings. We now verify the remaining local condition.

5 Realizing the local ranks in a common central sector

An idempotent in a finite group algebra has a well-defined rank as a linear operator. We now turn each of these numerical ranks into an isomorphism between modules over one common ring. The enlargement of the group is needed because the different simple or modular summands of the original group algebra need not have the same rank fraction.

Here are the precise inputs to this step. Let K be a field of characteristic p, let M be a positive integer prime to p, and assume that K contains an element \zeta of order M. In the application, M=p^4-1 and |K|=p^4. Write C_M=\mathbb Z/M\mathbb Z and put

where P acts on the functions in A_0 by translation. Suppose that a countable group G_0 contains specified copies H_1,\ldots,H_m of H. For each i, let e_i\in K[H_i] be an idempotent, and assume that the positive integer

is independent of i. All matrices below act on column vectors in free right modules, so their entries multiply coordinates on the left.

We prove the proposition in three stages. First we attach the finite groups while preserving all the given embeddings. Next we describe their f-parts as matrix algebras over group algebras of elementary abelian p-groups. Finally, nilpotence of the corresponding augmentation ideals makes the ranks exact and produces the two rectangular matrices.

5.1 Attaching the finite groups

Let z initially generate an independent copy of C_M, and write Q_{\mathrm{base}}=\langle z\rangle\times G_0. For each i, define

The groups B_i are written additively. Let \delta_i\in B_i be the constant function of value 1\in C_M. It has order M and is fixed by H_i, so it is central in L_i. Moreover, \langle\delta_i\rangle\cap H_i=1. Thus C_i=\langle z\rangle\times H_i embeds in both Q_{\mathrm{base}} and L_i, with (z^k,h) sent to (k\delta_i,h) in the latter. Define G_1 to be the quotient of the free product

that identifies these two copies of C_i for every i.

For completeness, the following action proves that all the factors embed in this quotient. Let X be a countable disjoint union of copies of the left regular Q_{\mathrm{base}}-set. Its restriction to each C_i is free and has countably infinitely many orbits. A countable disjoint union X_i of regular L_i-sets has the same property when restricted to C_i. Pair their C_i-orbits and choose one point in each paired orbit; extending those point pairings equivariantly gives a C_i-equivariant bijection X\to X_i. Transport the regular L_i-action to X using this bijection. The resulting actions of Q_{\mathrm{base}},L_1,\ldots,L_m agree on every prescribed C_i, hence define an action of G_1 on X. Each factor acts faithfully, which proves its injectivity in G_1. This argument uses the assumed embeddings H_i\hookrightarrow G_0 in defining the subgroups C_i\le Q_{\mathrm{base}}.

The element z is central in every generating factor of G_1 and therefore in G_1; its order is still M because Q_{\mathrm{base}} embeds. In R, direct multiplication gives

The coefficient of the identity in f is M^{-1}\ne0, since the elements 1,z,\ldots,z^{M-1} are distinct. Thus f is a nonzero central idempotent, and S=fR is a ring with identity f. Each P_i is an idempotent in fK[L_i]. Also, any identity e_ie_j=0 already valid in K[G_0] remains valid after multiplication by f in S.

5.2 The matrix blocks of the central sector

Fix i for the moment. A character of B_i means a homomorphism \chi:B_i\to K^*. If b_x denotes the coordinate function with value 1 at x\in H_i and 0 elsewhere, then every character is uniquely specified by exponents a_x\in\mathbb Z/M\mathbb Z through \chi(b_x)=\zeta^{a_x}. Define

Apply the Fourier calculation of Equation (3.3) to the finite index set H_i. The denominator is invertible because p\nmid M. The elements p_\chi are a basis of pairwise orthogonal idempotents whose sum is 1, and bp_\chi=\chi(b)p_\chi.

Use the action

Then hp_\chi h^{-1}=p_{h\chi}. Since z=\delta_i is fixed by H_i, the set

is H_i-invariant, and the Fourier formulas give

For each orbit in \mathcal X_i, choose one representative \theta and put T_\theta=\operatorname{Stab}_{H_i}(\theta).

Eliminating the prime-to-p part of each stabilizer is what makes its augmentation ideal nilpotent, and will permit the later change of basis. We next give the block isomorphism, including its inverse. This is the character-orbit construction already used in Section 3, now allowing nontrivial stabilizers; compare the Clifford-theoretic orbit picture in [Cli37]. It will identify the underlying K-linear rank of P_i in every block with the same original integer d. For one orbit representative \theta, abbreviate

Choose representatives r_1,\ldots,r_s of the left cosets hT. The space K[H_i] is a free right D-module with this ordered basis. Let g\in H_i act on it by left multiplication L_g, and let b\in B_i act diagonally by

Both actions commute with the right action of T. For the diagonal action this follows from (ht)\theta=h\theta for t\in T. The semidirect-product relation is also respected, because

Thus we have an algebra homomorphism \rho_\theta:K[L_i]\to\operatorname{Mat}_s(D). The element z acts as \zeta, so \rho_\theta(f)=I_s.

For t_0\in T and 1\le a,b\le s, consider

It belongs to fK[L_i] by (5.2). In the \theta-representation it sends r_bt' to r_at_0t' for every t'\in T and vanishes on every other basis coset. To check the last assertion, its left group factor sends the coset r_cT to r_at_0r_b^{-1}r_cT; the Fourier projection keeps this coset precisely when it equals r_aT, which is equivalent to c=b. In a representation associated to another orbit, the Fourier projection vanishes identically. Consequently (5.3) maps to the matrix with entry t_0 in position (a,b) of the \theta-block and zero elsewhere.

Taking the product of these representations gives a surjective map

It is injective as well. The elements p_\chi h, for \chi\in\mathcal X_i and h\in H_i, form a basis of its source. An orbit of size s contributes sw=s^2|T| basis elements, equal to the dimension of the corresponding target block. Hence the source and target have equal finite K-dimension. Formula (5.3), extended linearly in t_0 and summed over the positions and blocks, is an explicit inverse of \rho.

5.3 Turning dimensions into rectangular identities

Continue in one block and let E=\rho_\theta(P_i)\in \operatorname{Mat}_s(D). As a K-linear operator on K[H_i], E is exactly left multiplication by e_i, since f acts as the identity. Therefore

This is the link between the original rank calculation and the block decomposition. We will now obtain a basis in which E is a coordinate projection over D, not just over K. The argument is a constructive nilpotent-ideal instance of the freeness of projective modules over local rings, studied by Kaplansky [Kap58, Theorem 2]; no general projective-module theorem is needed here.

Let \varepsilon:D\to K be augmentation, and set \mathfrak a=\ker\varepsilon. By Lemma 5.2, we may choose generators t_1,\ldots,t_q of T\cong(C_p)^q with 0\le q\le3. The elements X_j=t_j-1 commute, satisfy X_j^p=0, and generate \mathfrak a. Expanding t_j=1+X_j and counting dimensions gives

It follows that

Indeed, a monomial of total degree greater than q(p-1) has an exponent at least p. This also covers q=0, when D=K and \mathfrak a=0. Matrices with entries in \mathfrak a have the same nilpotence bound under multiplication, because each entry of a product of D_{\max}+1 such matrices lies in \mathfrak a^{D_{\max}+1}.

Put \overline E=\varepsilon(E), applying augmentation entrywise, and let d_0=\operatorname{rank}_K\overline E. As \overline E is idempotent, its image and the image of I_s-\overline E form complementary subspaces of K^s. Scan the columns of \overline E in order and keep each column that increases their span; then do the same for I_s-\overline E. Build Q by taking the corresponding d_0 columns of E, followed by the corresponding s-d_0 columns of I_s-E. Its augmentation Q_0 is invertible over K. Regard Q_0 as a scalar matrix over D, and set N=Q_0^{-1}(Q-Q_0). Its entries belong to \mathfrak a, so N^{D_{\max}+1}=0. Since Q=Q_0(I_s+N),

Each of the first d_0 columns of Q is fixed by E, and every remaining column is killed by E. Thus, for \Pi=\operatorname{diag}(I_{d_0},0),

Conjugation by Q preserves K-dimension of images. Comparing with (5.5) gives the integer equalities

In particular, these are equalities of ordinary dimensions, not equalities after reduction to K.

We can now match coordinates exactly. Order pairs lexicographically and choose the increasing bijection between

the sets have equal size by (5.7). Let V be the w-by-d block matrix with s-by-s entries that sends coordinate (\mu,l) to its paired coordinate (\lambda,k). Thus V_{\lambda\mu} is the sum of standard matrix units E_{kl}^{\mathrm{std}} over all such pairings. Let W be the d-by-w block matrix reversing each pairing, with W_{\mu\lambda} the corresponding sum of E_{lk}^{\mathrm{std}}. It sends all unpaired coordinates with k>d_0 to zero. Directly on these coordinates,

Define block matrices

They have outer shapes w\times d and d\times w, respectively, with entries in \operatorname{Mat}_s(D). Their products are

Finally define the (\lambda,\mu) entry of J_i by applying \rho^{-1} to the tuple (J^{(\theta)}_{\lambda\mu})_\theta, and similarly define Y_i entrywise from the Y^{(\theta)}. The isomorphism (5.4) carries f to the tuple of identity matrices and P_i to the tuple of matrices E. The last two identities therefore pull back to (5.1). This proves Proposition 5.1.

All choices in the matrix construction are over finite sets: character orbits, coset representatives, augmentation matrices, and coordinate pairings. Taking the first element or the first independent columns in fixed orders makes the prescription definite. The inverse (5.6) is a finite sum, and the inverse (5.3) is a finite group-algebra formula. Thus the matrices furnished by the proposition are finite sums in the explicitly attached finite groups.

We now apply Proposition 5.1 to the embedded groups of Proposition 4.2 and the identical idempotents of Proposition 3.1. The group G_0 is countable because it is finitely generated. All the ranks equal the positive integer d from Equation (3.7). Hence the group G_1, the nonzero central idempotent f, and the matrices J_i,Y_i are now fixed by the constructions above. Corollary 3.3 also gives P_iP_j=0 on every edge with i<j. These are precisely the inputs to the next section.

6 Compression with a visible nonzero defect

We now use only the graph factorization and the precise common-sector output. The first lemma is independent of the group construction. Its proof explains why ordered edge annihilations, rather than symmetric orthogonality, suffice.

Apply the lemma to S=fK[G_1], whose identity is f. Distinct nonedges have F_{ij}=0 by Proposition 2.2; edges with i<j have P_iP_j=0. The rectangular identities are exactly those of Proposition 5.1. Consequently

The strict, ordinary rational inequalities are

This is more than an abstract rank paradox: it gives a nonzero vector that will certify the reverse-product defect.

Using C=C_\sharp, D=D_\sharp in Lemma 6.2 gives square matrices over the entire algebra K[G_1], not only the f-summand. The nonzero certificate is rooted in the exact order of z, which proves f\ne0, and then in D_\sharp v_*=u_*. It does not depend on identifying a coefficient by deciding equality of group words. The remaining step is to retain this defect when replacing matrices by scalar elements.

7 An injective finite-group encoding

We have a strict matrix one-sided inverse over K[G_1]. The finite-factor reduction of Dykema and Juschenko [DJ15, Lemma 2.1 and Theorem 2.2] explains the appropriate next step: put a full matrix algebra inside a finite group algebra and fill the complementary summand with the identity. We prove the actual matrix block needed here, including its injectivity. Its shift and diagonal operators are the finite Heisenberg model; the comparison with [GH09, Section 1.2] is methodological, since our cyclic order can be composite and our coefficient field is finite.

The elementary identity-completion step is worth recording separately. Its unit need not be central in the target ring.

Finite prescriptions and scope. Every choice of character, basis, field element, or finite inverse occurs in an explicitly finite set. Products of the resulting group words define finite group-algebra sums even when different words happen to coincide. No word comparisons are needed to define the elements, and the kernel certificate and injective matrix block certify the nonzero reverse defect. The countable permutation actions used to certify group embeddings are not coefficient-search inputs.

Containing torsion does not say that every element has finite order. The group must be infinite, since finite-dimensional left multiplication proves direct finiteness for every finite group algebra, in modular characteristic as well as otherwise. The coefficient field throughout remains the specified K of order p^4.

More generally, the supports of any two finite sums in a group algebra generate a finitely generated subgroup. Its group algebra embeds by the group bases, preserving both a one-sided product and its nonzero reverse defect. Thus a universal direct-finiteness theorem for finitely generated groups would already imply the assertion for all groups.

8 The cellular automaton on all configurations

The final transfer is short and characteristic-independent. We give it locally to fix the multiplication order and the meaning of a missing configuration. Its group-ring ancestry appears in [ES04, Section 1] and the linear theory of [CC06].

For any field k, group L, and finite sum c=\sum_uc_u u\in k[L], define on all functions x:L\to k

Only the coefficient sum is finite; x may have infinite support. Expanding the two finite sums gives

Thus T_cT_d=T_{cd} and T_1=\id. If \delta_h is the point indicator at h, then (T_c\delta_h)(1_L)=c_h, which also proves faithfulness.

Apply the proposition to the scalar sums in Equation (7.5). Their nonzero defect has a nonzero coefficient in the group basis, so the resulting automaton is injective and nonsurjective. The missing point indicator has no preimage even among infinitely supported configurations. If the finite formula for b displays a group element more than once, its terms simply add in the same local rule; specifying the rule requires no word-problem oracle. Thus the constructed group also gives a negative answer to Gottschalk’s surjunctivity conjecture.