A one-sided inverse that cannot be reversed in a group algebra

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Family 197 · Algebra
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

For square matrices over a field, $AB=I$ forces $BA=I$. A left inverse and a right inverse cannot differ. On an infinite-dimensional space, this principle fails: a shift can have a one-sided inverse with a missing direction.

A group algebra lies between these settings. The group may be infinite, but each algebra element is a finite sum. Kaplansky’s direct-finiteness conjecture asks whether that finite support is enough to recover the finite-dimensional implication.

Family 197 claims that it is not. Its strongest group-theoretic version uses a finitely presented torsion-free group over $\mathbb F_2$. Other manuscripts give a torsion-containing characteristic-two construction, one specified odd-characteristic construction, and a dependent counterexample to a determinant inequality. Their hypotheses and dependencies are different and need to be kept separate.

2. Background and essential definitions

One-sided inverses and stable finiteness

The group algebra $K[G]$ consists of finite formal sums $\sum a_g g$, with multiplication extended from the group law. Its identity is the group identity with coefficient one.

A unital ring is directly finite if

$$ab=1\quad\Longrightarrow\quad ba=1.$$

It is stably finite if every finite square matrix ring over it is directly finite. A matrix counterexample shows failure of stable finiteness, but does not automatically provide scalar elements $a,b$ in the original ring. Some manuscripts in this family explicitly enlarge the group to complete that passage.

If $ab=1$ but $ba\ne1$, the element $e=1-ba$ is a nonzero idempotent: $e^2=e$. Also $ae=0$. Thus this defect supplies zero divisors, but a zero product by itself does not establish failure of direct finiteness. This is a stronger phenomenon than the headline of family 196.

Finite approximations and local dynamics

A sofic group admits finite approximate permutation models for every finite part of its multiplication table, with small errors and good separation of distinct elements. Soficity is an approximation property, not a requirement that the group itself be finite.

A configuration over a finite alphabet $K$ is an arbitrary function $x:G\to K$. A cellular automaton applies the same rule at every group position using only finitely many nearby coordinates. A group is surjunctive if every injective such automaton is also surjective. Gottschalk’s conjecture proposes this property for every group.

For $a=\sum a_u u\in K[G]$, one convention for its linear local rule is

$$(T_a x)(g)=\sum_u a_u x(gu).$$

The sum is finite even when the configuration has infinite support, and $T_aT_b=T_{ab}$. A scalar one-sided inverse therefore gives an injective automaton with a local left inverse; a nonzero reverse defect can obstruct surjectivity.

An infinite-dimensional determinant

An integral group-ring matrix acts on square-summable functions on the group. Its Fuglede–Kadison determinant uses the trace of the logarithm of its positive square, extending the absolute value of an ordinary determinant.

For an invertible operator $T_A$ on $n$ copies of $\ell^2(G)$, the convention is

$$\log\det_{\mathcal N(G)}(T_A) =\frac12\tau_n\!\left(\log(T_A^*T_A)\right),$$

where $\tau_n(I)=n$. The trace is built from the identity coefficient in each diagonal entry. The group-ring Determinant Conjecture asserts an analogue of the lower bound one for nonzero integer determinants.

3. Development history and previous partial results

For a finite group, its group algebra and all finite matrix rings are finite-dimensional, so stable finiteness follows from linear algebra in every characteristic. Kaplansky’s characteristic-zero theorem uses finite von Neumann algebras, whose trace replaces ordinary dimension. Montgomery gave a concise operator-algebra proof. The result extends from complex coefficients to every characteristic-zero field.

In positive characteristic, Ara, O’Meara, and Perera proved stable finiteness for free-by-amenable groups over arbitrary division rings. Elek and Szabó extended it to all sofic groups. Consequently, any group supporting the claimed counterexamples here must be nonsofic; this is a deduction from their theorem, not an independently assumed property of the construction.

Dykema, Heister, and Juschenko organized the finite cancellation patterns of one-sided inverses and established positive results for several small supports. Dykema and Juschenko also explained how to convert a matrix defect into a scalar defect after taking a suitable finite-group factor. That distinction is central to the torsion-containing constructions.

The dynamical background begins with Gottschalk’s surjunctivity question. Gromov and Weiss proved surjunctivity for sofic groups, and the linear cellular-automaton theory developed the relation to group-algebra finiteness. Later work of Phung and others established further general connections between surjunctivity and stable finiteness.

For determinants, approximation methods give the lower bound for large classes including amenable and sofic groups. A failure of a determinant approximation formula is a different matter: the manuscript discusses Kammeyer’s example where finite-quotient determinants do not converge to the group determinant, although the relevant determinants remain above one. The claim here violates the lower bound itself.

The torsion-free manuscript develops the parity and coned-graph strategy of family 196. The determinant manuscript instead takes the torsion-containing characteristic-two scalar construction as its exact input. The odd-characteristic construction is presented independently of that binary theorem.

4. What the new manuscripts claim

Source	Main algebraic setting	Claimed conclusion
M1, 4 October 2026	$\mathbb F_2[G]$, finitely presented torsion-free $G$	$ab=1$, $ac=0$, $c\ne0$, hence $ba\ne1$
M2, 23 September 2026	A finite characteristic-two field and finitely presented $G$ with odd-order torsion	Scalar $ab=1\ne ba$
M4, 26 September 2026	A specified field of order $p^4$, one specified odd prime $p$, and finitely generated torsion-containing $G$	Scalar $ab=1\ne ba$
M3, 23 September 2026	An integral square matrix over another finitely generated group	Rational group-ring inverse and determinant strictly between zero and one

M1 additionally gives a finite two-dimensional classifying complex, whose universal cover is contractible. Its graph matchings are selected by an existence argument, rather than listed as a practical concrete example.

M2 gives a terminating finite prescription for its field, presentation, and sums. The manuscript states that this prescription has not been executed and need not be computationally practical. Its field is not asserted to be $\mathbb F_2$, and its group contains torsion.

M4 specifies the prime by

$$m=\binom{1200}{600},\qquad p=\min\{h>1:h\text{ divides }(m!)^2+1\}.$$

It claims the result for this prime, not every odd prime. Its field has order $p^4$, and its finite formulas likewise are not presented as an executed practical computation. The theorem states finite generation of its group; the stronger finite-presentation conclusion of M1 and M2 should not be silently transferred to it.

M2 and M4 explicitly derive injective nonsurjective cellular automata over their finite fields, giving claimed counterexamples to Gottschalk’s conjecture. The established sofic finiteness theorem makes the direct-finiteness example groups nonsofic.

M3 depends on M2’s scalar characteristic-two defect. It claims an integral matrix invertible over the rational group ring with

$$0<\det_{\mathcal N(G_D)}(T_A)<1.$$

Its logarithmic integral is finite and negative, with the spectrum bounded away from zero. This is not an example where a divergent integral or a zero kernel convention creates the failure.

5. Why the advance matters

Provisional importance: 5/5. If correct, the family overturns central finiteness conjectures connecting algebra, geometric group theory, probability-preserving approximation, and local dynamics.

The torsion-free prime-field result is especially sharp: finite-order elements and a larger coefficient field are not needed for that version. The torsion-containing constructions contribute different explicit prescriptions and, in the odd-characteristic case, a separate coefficient regime.

The consequences extend beyond rearranging products. A nonsurjunctive cellular automaton is locally defined and injective, yet some global configurations never occur. A nonsofic group lacks the finite approximate multiplication models guaranteed for many familiar groups. The determinant construction would show that integral coefficients alone do not enforce the proposed infinite-dimensional volume lower bound.

6. The main idea

Modify parity so that exactly one contribution survives

M1 uses finite labelled graphs attached by cones to a rose of generator loops. Closed graph paths become group relations. Root-to-vertex labels define finite sums, and simultaneous equal-label steps preserve the group element represented by each product pair.

For a zero product, every vertex of a simultaneous-step graph has odd degree. Each component then has even size, causing cancellation over $\mathbb F_2$. To obtain product one, M1 arranges a single even-degree exception at the root pair. Its component has odd size and contributes exactly the identity; all other components cancel.

Seven extra labels organized by the Fano plane create this parity pattern while preserving balanced inverse-label counts. Pairing with a second graph gives $ac=0$, and root protection ensures $c\ne0$. If $ba=1$, then $c=b(ac)=0$, a contradiction.

Random graph estimates, planar separators, and cone surgeries provide asphericity and root protection, adapting the strategy of family 196. The estimates are reproved for the changed vertex types.

From finite incidence data to scalar inverses

M2 starts with finite incidence data and vectors whose outer products sum to zero along specified lines but to a nonzero identity over the whole set. Low-degree interpolation produces these pairing identities. A probabilistic construction with local trades supplies the required incidence regularity and large girth.

Finite groups are glued according to that incidence geometry. A planar argument ensures that the relevant local groups embed, so a later algebraic witness does not collapse in the quotient. Character idempotents and prescribed conjugations pack a rectangular factorization into a matrix one-sided inverse.

The reverse defect is detected inside an embedded subgroup algebra by a rank obstruction. A suitable finite-group algebra then supplies matrix units, allowing an injective encoding of the matrix pair as scalar elements. Adding the complementary identity restores the full scalar identity. The final step is necessary: a matrix defect alone would only prove failure of stable finiteness.

M4 uses a different modular mechanism. A triangle-free graph has a low-rank fitting matrix, while finite-group idempotents have a larger normalized regular rank. Directed zero products along graph edges make these inequalities compatible. Group embeddings, exact rectangular identities, and a finite Heisenberg-group encoding again turn them into a scalar defect. The construction uses the specified odd prime throughout.

Turn a binary defect into a determinant below one

M3 represents the finite characteristic-two field by matrices over $\mathbb F_2$, giving $XY=I_d$ and a nonzero idempotent $C=I_d-YX$. It enlarges the group by finitely supported copies of the cyclic group of order three, indexed by the original group.

Averages over those cyclic coordinates isolate one coefficient of $C$. The resulting rational orthogonal projection $P$ has trace $d+\theta$, where $\theta>0$, but its reduction modulo two factors through only $d$ columns. Odd denominators make that reduction meaningful.

A modular lift and block-matrix construction turn this discrepancy into an integral matrix with rational inverse. Trace calculations remove block shears and give the exact determinant

$$\det_{\mathcal N(G_D)}(T_A)=q^{-\theta}<1$$

for a suitable power $q$ of two. The determinant calculation transfers the binary defect; it does not independently construct the original one-sided inverse.

7. Scope, limitations, and open questions

The four sources do not provide one universal counterexample simultaneously satisfying every field and group condition. M1 is torsion-free over $\mathbb F_2$; M2 and M4 use torsion; M4 concerns one odd prime; and M3 changes the group again and depends specifically on a characteristic-two input.

No claim contradicts direct finiteness in characteristic zero. The determinant bound is a different invariant, so an integral matrix with determinant below one can coexist with the characteristic-zero direct-finiteness theorem.

Existence, a terminating prescription, and an executed small example are different levels of explicitness. The manuscripts state the first two in different places and do not claim practical computations of the full constructions.

Questions about small presentations, minimal supports, other odd primes, and how broadly the determinant transfer applies remain distinct from the stated counterexamples. The explanation reports the constructions and their consequences as claimed, without auditing their proofs.

8. Sources and further reading

Kaplansky’s Fields and Rings, second edition (1972), formulates the finiteness question. Montgomery’s Left and right inverses in group algebras (1969), Ara–O’Meara–Perera’s Stable finiteness of group rings in arbitrary characteristic (2002), and Elek–Szabó’s Sofic groups and direct finiteness (2004) give the principal positive results.

Dykema and Juschenko’s finite-group reduction explains the scalar-versus-matrix distinction. Gottschalk’s Some general dynamical notions (1973), Weiss’s Sofic groups and dynamical systems (2000), and Ceccherini-Silberstein and Coornaert’s linear cellular-automaton work provide the dynamical context.

Fuglede and Kadison introduced the operator determinant; Lück’s L²-Invariants develops the group-ring determinant question. The parity predecessor is explained in family 196. The source list below identifies the four separate claims and their versions.

Source manuscripts

• M1. OpenAI, A Torsion-Free Group Algebra That Is Not Directly Finite (October 4, 2026).
• M2. OpenAI, A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Characteristic Two (September 23, 2026).
• M3. OpenAI, A Counterexample to the Group-Ring Determinant Conjecture (September 23, 2026).
• M4. OpenAI, A Counterexample to Kaplansky’s Direct-Finiteness Conjecture in Odd Characteristic (September 26, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.