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Math-108 · mathematics · Combinatorics

Is the exponent of matrix multiplication 2

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

The rational bilinear rank $R_{{\mathbb{Q}}}(n)$ is the least $r$ admitting rational linear forms $\alpha_j,\beta_j$ and fixed rational matrices $C_j$ with \[ AB=\sum_{j=1}^r\alpha_j(A)\beta_j(B)C_j\qquad(A,B\in{\mathbb{Q}}^{n\times n}). \] Set $\omega_{{\mathbb{Q}}}=\inf\{\tau:R_{{\mathbb{Q}}}(n)=O(n^\tau)\}$. The target is $\omega_{{\mathbb{Q}}}=2$, or equivalently the existence, for every $\varepsilon>0$, of exact arithmetic circuits of size $O_\varepsilon(n^{2+\varepsilon})$. This counts arithmetic operations at unit cost. It neither bounds coefficient bit lengths nor asserts that the infimum is attained by an $O(n^2)$ algorithm.

Formulation and concept references: [S1].

Short English statement

Can exact rational matrix multiplication be performed with an operation count arbitrarily close to quadratic?

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Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Math-108-A3 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-108

The stated arithmetic exponent bounds, including $9/4$ over the complex numbers, remain strictly above two. The requested exponent-two multiplication over the rational field is therefore not established, and arithmetic-operation estimates alone do not assert a bit-time implementation.

Is the exponent of matrix multiplication 2

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 107; 2026-09-24), the asserted result is that characteristic-zero matrix multiplication has square exponent below $2.258$, dual exponent above $0.465$, and rectangular exponent at aspect $0.709$ below $2.092$, with the stated finite-characteristic exceptions; In manuscript 2 (family 107; 2026-10-02), the asserted result is that the complex matrix-multiplication exponent is at most $9/4$; In manuscript 3 (family 107; 2026-09-24), the asserted result is that the square multiplication exponent is below $2.371054886006746$ over every fixed field, including every positive characteristic.

Mathematical statement of the claimed result

Manuscript 1.
In exact complex arithmetic, let $\omega$ be the square matrix-multiplication exponent, $w(k)$ the exponent for multiplying an $n\times n^k$ matrix by an $n^k\times n$ matrix, and $\alpha=\sup\{k:w(k)=2\}$. The claimed bounds are \[ \omega<1129/500=2.258,\qquad\alpha>0.465,\qquad w(0.709)<2.092. \] These are arithmetic exponents, rather than bit-complexity or finite-precision bounds.
Manuscript 2.
For every $\varepsilon>0$, the claim gives an exact complex-arithmetic algorithm multiplying two $n\times n$ matrices in \[ O_{\varepsilon}(n^{9/4+\varepsilon})\quad\text{operations},\qquad\omega\le9/4. \] Its associated tensor inequality uses $T=\sum_{i=1}^sT_i$ with every $T_i\ne0$, a shared first tensor factor and matched disjoint sectors on the other two. A tensor character $\lambda$ takes nonnegative real values and is additive, multiplicative and restriction-monotone, with $\lambda(1)=1$; define $p_X$ by $\lambda(x\sum_{j=1}^my_jz_j)=m^{p_X}$. For every probability vector $q$ the claimed inequality is \[ \lambda(T)\ge e^{p_XH(q)}\prod_i\lambda(T_i)^{q_i},\qquad H(q)=-\sum_iq_i\log q_i, \] with $0\log0=0$. These are exact arithmetic and tensor-character statements, without finite-precision assumptions.
Manuscript 3.
For every fixed field $F$, the claimed arithmetic exponent of square matrix multiplication satisfies \[ \omega_F<2.371054886006746<2.371054887. \] The exponent is the infimum of asymptotic powers achievable by exact field operations, allowing an arbitrarily small additive exponent slack. Positive and zero characteristics are both included.

Outline of the argument

For manuscript 1 (family 107; 2026-09-24), an auxiliary tensor supplies a conditional label to the missing third leg while retaining a dot-product block. Recursive completion trees and finite-family entropy optimization produce certified parameters; a strict comparison argument verifies the advertised boundaries.

For manuscript 2 (family 107; 2026-10-02), tensor degenerations separate blocks that already expose two independent variable legs. Character inequalities for polynomial multiplication, a determinant filtration, and discrete growth bound the detecting character values for matrix multiplication at the claimed exponent.

For manuscript 3 (family 107; 2026-09-24), a joint extraction from several tensor depths uses entropy and a common modular hash. Sequential deletion assigns variables compatibly, and exact integer tables repair inherited masks; the resulting direct sum is converted to an exponent bound by the rank comparison.

Formal verification

For manuscript 1 (family 107; 2026-09-24), the documented Lean scope is the stated matrix-multiplication exponent estimates over $\mathbb C$, including the rectangular bounds; the declarations do not extend these estimates to all characteristic-zero fields or the claimed finite-characteristic range.

The selected statement includes the clause used for this problem.

For manuscript 3 (family 107; 2026-09-24), OpenAI lists Lean statements for this assertion: the square multiplication exponent is below $2.371054886006746$ over every fixed field, including every positive characteristic.

The selected statement includes the clause used for this problem.

No Lean formalization is listed at this snapshot for manuscript 2 (family 107; 2026-10-02).

The linked family Comparator checks are MatrixFields.lean, MatrixMultiplication.lean.

Their named declarations include OAI.MatrixAllFields.MatrixMultiplication.AllFieldMain.omega_lt_source_constant, OAI.MatrixMultiplication.complex_alpha_gt_93_div_200, OAI.MatrixMultiplication.complex_omega_le_nine_quarters, OAI.MatrixMultiplication.complex_rectangular_omega_lt_523_div_250.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

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Math-108-A2 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-108

Is the exponent of matrix multiplication 2

Definitions and mathematical statement

The rational bilinear rank $R_{{\mathbb{Q}}}(n)$ is the least $r$ admitting rational linear forms $\alpha_j,\beta_j$ and fixed rational matrices $C_j$ with \[ AB=\sum_{j=1}^r\alpha_j(A)\beta_j(B)C_j\qquad(A,B\in{\mathbb{Q}}^{n\times n}). \] Set $\omega_{{\mathbb{Q}}}=\inf\{\tau:R_{{\mathbb{Q}}}(n)=O(n^\tau)\}$. The target is $\omega_{{\mathbb{Q}}}=2$, or equivalently the existence, for every $\varepsilon>0$, of exact arithmetic circuits of size $O_\varepsilon(n^{2+\varepsilon})$. This counts arithmetic operations at unit cost. It neither bounds coefficient bit lengths nor asserts that the infimum is attained by an $O(n^2)$ algorithm.

Formulation and concept references: [S1].

Short English statement

Can exact rational matrix multiplication be performed with an operation count arbitrarily close to quadratic?

Research attempt

Scope and source check (27 September 2026).
All coefficients below lie in $\mathbb Q$, and arithmetic operations have unit cost. For every $\varepsilon>0$ the target permits a different algorithm and a different implied constant. It does not demand one quadratic algorithm, a practical implementation, or a bound on the bit length of its constants. Bläser [S1] gives the classical rank and approximation framework. The August 2026 paper [S2] reports $\omega<2.371177$ and describes a rigorous verification procedure; its numerical certificates were not rerun here. That bound is above two. The infinite-group construction [S3, Section 3] explicitly identifies dimension deficits preventing its construction from reaching two. Neither reference is used as an assertion that the target is settled.

The attempt is to turn the desired asymptotic statement into exact finite certificates, then determine whether approximation can supply them. The arguments reproduce elementary parts of the classical framework, including the principle behind Bini's approximation method [E1]. They are not claimed as new matrix multiplication algorithms. A small tensor example will test the difference between an exact identity and a limit.

Rank bounds and the block construction.
Abbreviate $R_{\mathbb Q}(n)$ to $R(n)$. The output polynomials $p_{ij}=\sum_k a_{ik}b_{kj}$ are linearly independent: the monomial $a_{i1}b_{1j}$ occurs in exactly one of them. In a rank-$r$ algorithm all these polynomials lie in the span of the $r$ products $\alpha_s(A)\beta_s(B)$. Hence \begin{equation}\label{wsref-0-6d-6d-3a-65-6c-65-6d-65-6e-74-61-72-79} n^2\le R(n)\le n^3. \end{equation} The upper bound uses the individual products $a_{ik}b_{kj}$. Padding inputs with zero rows and columns proves $R(n)\le R(N)$ for $n\le N$.

More significantly, \begin{equation}\label{wsref-0-6d-6d-3a-73-75-62-6d-75-6c-74} R(ab)\le R(a)R(b). \end{equation} Apply a rank-$R(a)$ identity to $a$ by $a$ block matrices with $b$ by $b$ blocks. Each block product is evaluated with a rank-$R(b)$ identity. The resulting scalar multiplications are products of a rational linear form in the first input and one in the second. The outer identity remains valid for matrix blocks: after expansion every term has one first-input block followed by one second-input block, in that order. Coefficient comparison, rather than an assumption that blocks commute, proves its validity. This gives the stated product of ranks.

Fix $b\ge2$ and a rank-$r$ identity for size $b$. Recursion and padding give, for $k=\lceil\log_b n\rceil$, \[ R(n)\le r^k\le r n^{\log_b r}. \] It follows that $\omega_{\mathbb Q}\le\log_b r$. Conversely, if $R(n)\le Cn^\tau$ for all sufficiently large $n$, then $\log_bR(b)\le\tau+\log_b C$ for sufficiently large $b$. Taking $b$ to infinity and then the infimum over $\tau$ proves \begin{equation}\label{wsref-0-6d-6d-3a-66-69-6e-69-74-65} \omega_{\mathbb Q}=\inf_{b\ge2}\log_b R(b). \end{equation} In particular, exponent two is equivalent to the following exact finite certificate condition: \[ \forall\delta>0\quad\exists b\ge2\quad \exists\text{ a rational rank identity with }r<b^{2+\delta}. \] Strict inequality causes no difficulty: use half the requested exponent slack before absorbing a constant. There is no asserted bound on the size of the certificate or on how rapidly its block size grows as $\delta$ decreases.

Accounting for the linear operations.
A rank count ignores additions, but the fixed-block recursion restores their cost. All linear combinations in a chosen size-$b$ identity have a finite cost $c_b$. For inputs of size $b^k$, their total cost at the top level is at most $c_b b^{2(k-1)}$. Thus the operation count satisfies \[ T(b^k)\le rT(b^{k-1})+c_b b^{2(k-1)}. \] If $r>b^2$, summing the geometric series gives $T(n)=O_b(n^{\log_b r})$. If $r=b^2$, the same calculation gives $T(n)=O_b(n^2\log n)$. Both bounds imply the required $n^{2+\varepsilon}$ cost whenever a suitable certificate from (mm:finite) has been chosen with slack. The exceptional equality case must not be silently advertised as quadratic total cost. No uniform bound on $c_b$ is needed after $b$ is fixed for the chosen $\varepsilon$.

The distinction between a fixed saving and exponent two is now precise. A certificate $r\le b^3/c$ yields the bound $3-\log_b c$ on the exponent. If $1<c<b$, this particular bound exceeds two. It does not show that the true exponent exceeds two. Likewise a single bound $r\le Cb^2$, with $C>1$, only gives $2+\log_b C$. Such bounds for unbounded $b$ with the same $C$ would suffice, as would $r\le b^2\exp(o(\log b))$ along an unbounded sequence. A table of a few improved ranks does not provide either family.

A rational degeneration can be converted exactly.
Let $M_b$ denote the bilinear tensor for size-$b$ multiplication. Suppose an explicit identity over $\mathbb Q[t]$ has the form \begin{equation}\label{wsref-0-6d-6d-3a-64-65-67-65-6e-65-72-61-74-69-6f-6e} P(t)=\sum_{s=1}^r u_s(t)\otimes v_s(t)\otimes w_s(t) =t^qM_b+\sum_{j=q+1}^{D}t^j E_j, \qquad 0\le q\le D. \end{equation} All coefficients below degree $q$ vanish. These are formal coefficient identities, not estimates obtained by taking $t$ numerically small. Tensoring $k$ copies gives a polynomial of degree at most $kD$ whose coefficient of $t^{kq}$ is $M_b^{\otimes k}=M_{b^k}$, after reindexing.

Choose $kD+1$ distinct rational numbers $a_0,\ldots,a_{kD}$. Lagrange interpolation provides rational weights $\lambda_i$ such that the coefficient of $t^{kq}$ in every polynomial of degree at most $kD$ equals $\sum_i\lambda_i F(a_i)$. This holds coordinatewise for tensors. At each $a_i$, the tensor $P(a_i)^{\otimes k}$ has rank at most $r^k$. Consequently \begin{equation}\label{wsref-0-6d-6d-3a-65-78-74-72-61-63-74} R(b^k)\le(kD+1)r^k, \qquad \omega_{\mathbb Q}\le\log_b r. \end{equation} For the second assertion, apply (mm:finite) to block size $b^k$ and let $k$ grow: $\log_{b^k}(kD+1)$ tends to zero. Alternatively, padding gives a logarithmic factor in the rank bound, absorbed by every positive exponent slack.

Here $b,r,D,q$ are fixed before the tensor power grows. A large degree $D$ can make the practical certificate enormous without changing this limit. For a family with varying $b$, first choose the certificate and then take its tensor powers. Interchanging these choices without controlling the new overhead would be unjustified. Laurent-polynomial identities can be treated by clearing powers of $t$ first; an unspecified analytic limit or a floating-point decomposition supplies neither the coefficient identities nor their rationality.

There is also a distinction in the order of operations. Extracting an exact rank bound $R(b)\le(D+1)r$ first and tensoring it produces $((D+1)r)^k$. Extracting after tensoring, as above, costs only $(kD+1)r^k$. The former loses a fixed amount in the exponent; the latter does not. This calculation explains why a degeneration can matter even when its first exact specialization is worse than a direct algorithm.

What algebraic coefficients cost over the rationals.
The field requirement can also be audited quantitatively. Let $E/\mathbb Q$ be a finite extension of degree $h$, and suppose a rational bilinear map $T$ has a rank-$r$ decomposition over $E$. Choose a rational basis $e_1,\ldots,e_h$ of $E$ and a rational-linear map $\lambda:E\to\mathbb Q$ with $\lambda(1)=1$. On rational inputs expand $\alpha_s=\sum_u e_u\alpha_{su}$ and $\beta_s=\sum_v e_v\beta_{sv}$, where the new forms are rational. Applying $\lambda$ coordinatewise to the output gives the exact identity \[ T(A,B)=\sum_{s,u,v}\alpha_{su}(A)\beta_{sv}(B) \lambda(e_ue_v C_s). \] Thus $R_{\mathbb Q}(T)\le h^2R_E(T)$. In particular, tensor powers of one fixed size-$b$ algorithm can all be descended through the same extension, yielding $R_{\mathbb Q}(b^k)\le h^2r^k$. The factor is $h^2$, not $h^{2k}$. The same argument after interpolation in (mm:extract) gives $h^2(kD+1)r^k$ for a fixed degeneration over $E$. Neither finite extension changes the resulting exponent bound.

This observation does not authorize treating a numerical complex decomposition as an exact rational algorithm. Its coefficients must be specified exactly, its polynomial identities checked, and the finite extension hypothesis established before the displayed descent applies. For any one certificate consisting of algebraic numbers, their compositum is finite, so it qualifies. The degree may depend on the certificate chosen for a particular exponent slack. The proof still fixes that certificate before increasing its tensor power. No claim about coefficient bit complexity follows from this operation count.

A small tensor exposes the limit trap.
Let $e_0,e_1$ be a basis of $\mathbb Q^2$ and put \[ W=e_1\otimes e_0\otimes e_0+ e_0\otimes e_1\otimes e_0+ e_0\otimes e_0\otimes e_1. \] The polynomial \[ (e_0+te_1)^{\otimes3}-e_0^{\otimes3} =tW+t^2V+t^3e_1^{\otimes3} \] is a sum of two decomposable tensors at every rational $t$. Nevertheless $R(W)=3$. For a direct proof, its first-factor slice space is spanned by \[ A=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad B=\begin{pmatrix}1&0\\0&0\end{pmatrix}. \] Since $\det(aA+bB)=-a^2$, all rank-one matrices in this space lie in the single line spanned by $B$. A rank-two decomposition of $W$ would force this two-dimensional slice space to be spanned by two rank-one matrices: the two first-factor vectors must be independent, or the slice space would have dimension at most one. This is impossible. The displayed three-term expression gives the reverse bound.

Thus the limiting tensor need not have the ranks of the nearby tensors. Coefficient extraction nevertheless proves $R(W^{\otimes k})\le(3k+1)2^k$. This example is not a matrix multiplication construction and does not establish the desired ranks for $M_b$. Its purpose is to verify exactly what the approximation argument supplies, and why substituting $t=0$ into a normalized expression is not an exact rank-two algorithm.

Certificates, verification, and the remaining construction.
For a proposed exact algorithm, expand its rational coefficient arrays $\alpha_{s,ik},\beta_{s,\ell j},C_{s,pq}$. It suffices to check all $b^6$ equations \[ \sum_{s=1}^r\alpha_{s,ik}\beta_{s,\ell j}C_{s,pq} =\delta_{ip}\delta_{jq}\delta_{k\ell}. \] These are equivalent to testing every pair of matrix units in the two inputs. The local check verified all 64 scalar equations for the usual seven-product size-two identity, including the signs in the four output combinations. For the tensor $W$, rational interpolation was checked on the first three tensor powers, against explicit coefficient expansion. These finite checks support the stated identities; they are not a search result for near-quadratic algorithms.

Even monotonicity, submultiplicativity, and an infimum exponent of two do not formally force an attained quadratic bound. As an abstract test, the integer sequence \[ a(n)=\left\lceil n^2\exp(\sqrt{\log n})\right\rceil \] is monotone and submultiplicative, has infimum exponent two, and is not $O(n^2)$. Submultiplicativity follows from $\sqrt{x+y}\le\sqrt x+\sqrt y$ and $\lceil uv\rceil\le\lceil u\rceil\lceil v\rceil$ for positive $u,v$. This sequence is not asserted to be a possible matrix multiplication rank function; it only disproves that proposed deduction from those abstract properties alone.

The first missing step is an unbounded family of rational exact identities, or explicit degenerations (mm:degeneration), with $\log_b r$ approaching two. Neither the block lemma nor interpolation constructs this family. Three concrete next tasks are to search for small rational degenerations with exact residual checking; to prove a structural rank saving that improves with block size; and to track the actual degrees and coefficient fields in any proposed infinite-group construction before applying (mm:extract).

UNRESOLVED. The attempt gives finite sufficient certificates, a complete rational approximation-to-exact argument, and countertests to several stronger conclusions. It supplies no new exponent bound. Written by GPT-6 Astra Ultra with source inspection and exact finite algebra checks; no proof-assistant verification is claimed.

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Math-108-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-108

Is the exponent of matrix multiplication 2

Definitions and mathematical statement

The rational bilinear rank $R_{{\mathbb{Q}}}(n)$ is the least $r$ admitting rational linear forms $\alpha_j,\beta_j$ and fixed rational matrices $C_j$ with \[ AB=\sum_{j=1}^r\alpha_j(A)\beta_j(B)C_j\qquad(A,B\in{\mathbb{Q}}^{n\times n}). \] Set $\omega_{{\mathbb{Q}}}=\inf\{\tau:R_{{\mathbb{Q}}}(n)=O(n^\tau)\}$. The target is $\omega_{{\mathbb{Q}}}=2$, or equivalently the existence, for every $\varepsilon>0$, of exact arithmetic circuits of size $O_\varepsilon(n^{2+\varepsilon})$. This counts arithmetic operations at unit cost. It neither bounds coefficient bit lengths nor asserts that the infimum is attained by an $O(n^2)$ algorithm.

Formulation and concept references: [S1].

Short English statement

Can exact rational matrix multiplication be performed with an operation count arbitrarily close to quadratic?

Research attempt

Current frontier.
Bläser's survey explains why exact rank, approximation by polynomial families, and the asymptotic exponent must be separated. [S1], Sections 2 and 6. Dupont and collaborators' August 2026 manuscript reports $\omega<2.371177$ by optimizing combination-loss analysis; its numerical optimization and full certification have not been independently reproduced here. [S2], [E1]. Blasiak--Cohn--Grochow--Pratt--Umans construct a framework for finite algorithms from infinite groups; its remaining separating-polynomial and dimension requirements are substantive, not already a proof of exponent two. [S3], Introduction. Alman's universal-method barrier for Coppersmith--Winograd tensors precludes reaching two within that stated framework, not by every possible matrix multiplication method. [E2].
Attack route.
The selected route is to combine degenerations with field descent. A candidate discovered over ${\mathbb{C}}$ or a number field is useful to this exact rational problem only after a legitimate conversion. Likewise a limiting low-rank formula must be converted into exact arithmetic before it bounds $R_{{\mathbb{Q}}}$. I make both conversions explicit and ask whether their overhead could be the missing obstacle. It is not: it can be amortized. The remaining problem is to find the low-rank family itself, rather than to round coefficients or set a small parameter equal to zero.
Attempt.
Let $T_n$ be the rational matrix multiplication tensor. Tensor rank here counts products of one linear form in each input, with an output vector; all coefficients must lie in the stated field. First, \begin{equation}\label{wsref-0-72-32-36-3a-62-61-73-69-63} R_{{\mathbb{Q}}}(n)\ge n^2,\qquad T_a\otimes T_b\cong T_{ab}, \qquad R_{{\mathbb{Q}}}(ab)\le R_{{\mathbb{Q}}}(a)R_{{\mathbb{Q}}}(b). \end{equation} For the lower bound, the output matrices in any rank decomposition must span every matrix $AB$; taking $A=I$ shows their span has dimension at least $n^2$. For the tensor identity, index a row or column by a pair of indices and multiply elementary matrices. Expanding two rank decompositions proves submultiplicativity. These arguments give $\omega_{{\mathbb{Q}}}\ge2$, not an upper bound near two.

Lemma 26.1 (exact extraction from a rational degeneration). Suppose for some fixed $n\ge2$ there is a polynomial tensor identity over ${\mathbb{Q}}[t]$ \begin{equation}\label{wsref-0-72-32-36-3a-64-65-67-65-6e-65-72-61-74-69-6f-6e} D(t)=\sum_{i=1}^{r}a_i(t)\otimes b_i(t)\otimes c_i(t) =t^aT_n+\sum_{j=a+1}^{b}t^jU_j, \qquad 0\le a\le b, \end{equation} where the vector-valued factors are polynomials. Then for every $k\ge1$, \begin{equation}\label{wsref-0-72-32-36-3a-65-78-61-63-74} R_{{\mathbb{Q}}}(n^k)\le(bk+1)r^k, \qquad \omega_{{\mathbb{Q}}}\le\log_n r. \end{equation} Proof. The coefficient of $t^{ak}$ in $D(t)^{\otimes k}$ is exactly $T_n^{\otimes k}$: every factor has lowest degree $a$, so there are no other contributions at the lowest total degree. The whole polynomial has degree at most $bk$. Choose $bk+1$ distinct rational numbers $t_0,\ldots,t_{bk}$. Polynomial interpolation expresses the coefficient of $t^{ak}$ as a rational linear combination of the values $D(t_l)^{\otimes k}$. Each such value has rank at most $r^k$. Multiplying one tensor factor by the corresponding rational scalar and summing proves the first assertion, including the case $b=0$.

For any matrix size $N$, choose $k$ with $n^{k-1}<N\le n^k$, pad with zeros to size $n^k$, and project back. The bound is $O(\log(N+1)N^{\log_n r})$, with constants depending on the fixed base identity. For every $\varepsilon>0$, the logarithm is $O_\varepsilon(N^\varepsilon)$, so the definition as an infimum gives the second assertion. No claim that the logarithm disappears in an exact attained exponent is made.$\square$

This is a direct version of the approximation-to-exact mechanism in [S1], Section 6, proved here to make the coefficient field and parameter losses explicit. The degrees $a,b$ may be very large for the chosen base tensor but are fixed before $k\to\infty$. Allowing them to grow unaccountably with $k$ would invalidate the estimate.

Lemma 26.2 (a finite extension costs only a constant after tensoring). If $L/{\mathbb{Q}}$ has degree $d$ and a rational bilinear tensor $T$ has rank at most $r$ over $L$, then \begin{equation}\label{wsref-0-72-32-36-3a-64-65-73-63-65-6e-74} R_{{\mathbb{Q}}}(T)\le d^2r, \qquad R_{{\mathbb{Q}}}(T^{\otimes k})\le d^2r^k. \end{equation} Proof. Fix a ${\mathbb{Q}}$-basis $e_1,\ldots,e_d$ of $L$ and a ${\mathbb{Q}}$-linear map $\lambda:L\to{\mathbb{Q}}$ that is the identity on ${\mathbb{Q}}$. For rational input vectors, write the two input linear forms in a rank-one summand as $\alpha=\sum_u e_u\alpha_u$ and $\beta=\sum_v e_v\beta_v$, with rational linear forms $\alpha_u,\beta_v$. Applying $\lambda$ coordinatewise to its output gives \[ \lambda(C\alpha\beta) =\sum_{u,v}\alpha_u\beta_v\,\lambda(Ce_ue_v), \] a sum of at most $d^2$ rational rank-one terms. Applying this to all $r$ summands preserves $T$ because its output on rational inputs is rational. This proves the first bound. For the second, first tensor the rank-$r$ decomposition over the same field $L$, and only then apply the first bound. The resulting overhead is $d^2$, not $d^{2k}$. $\square$

It follows from the tensor identity and padding that any one exact rank-$r$ multiplication algorithm of base size $n$ over a number field gives $\omega_{{\mathbb{Q}}}\le\log_n r$. Coefficient bit lengths are not being estimated; the target explicitly does not charge them.

Even a complex exact decomposition of fixed size can be replaced by one over some number field. Its finitely many coefficient identities form polynomial equations over ${\mathbb{Q}}$. A complex solution means their ideal is proper. The weak Nullstellensatz over ${\mathbb{Q}}$ gives a maximal ideal above it whose residue field is a finite algebraic extension of ${\mathbb{Q}}$, and hence a solution over a number field. [E3]. Equivalently, one can solve over $\overline{\mathbb{Q}}$ and collect the finitely many coordinates in one number field. This is an existence argument, not an efficient coefficient-recovery algorithm.

Thus the arithmetic-operation exponent does not gain an obstacle merely from passing from ${\mathbb{C}}$ to ${\mathbb{Q}}$. More explicitly, if a complex family has $R_{{\mathbb{C}}}(N)=O(N^\tau)$, choose a sufficiently large fixed base $n$ with $\log_n R_{{\mathbb{C}}}(n)\le\tau+\varepsilon$. Descend its one finite decomposition and tensor it as above; the extension degree is then fixed. This gives $\omega_{{\mathbb{Q}}}\le\tau+\varepsilon$. The reverse inequality follows because a rational decomposition is complex too. Taking infima establishes equality of these exponents, without claiming equality of each finite-size rank.

A concrete benchmark rather than a new upper bound. For $A=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}e&f\\g&h\end{smallmatrix}\right)$, take \[ \begin{array}{lll} M_1=(a+d)(e+h),&M_2=(c+d)e,&M_3=a(f-h),\\ M_4=d(g-e),&M_5=(a+b)h,&M_6=(c-a)(e+f),\\ M_7=(b-d)(g+h).&& \end{array} \] The four outputs \[ (M_1+M_4-M_5+M_7,\ M_3+M_5,\ M_2+M_4,\ M_1-M_2+M_3+M_6) \] are exactly $(ae+bg,af+bh,ce+dg,cf+dh)$. Expanding proves the identity, and the companion script independently verifies it symbolically. This is Strassen's known seven-product formula, discussed in [S1], not a newly found algorithm. Tensoring yields $\log_2 7>2$; repeated use of one fixed formula cannot make that exponent drift toward two.

The positive route still needs, for every $\varepsilon>0$, a base size $n\ge2$ and a certified exact or polynomial-degeneration rank $r\le n^{2+\varepsilon}$. A bounded improvement for one base size does not provide this family. Neither interpolation nor field descent reduces $r$; each only transfers an existing identity into the target model.

Stress test.
A floating-point approximate identity is not (r26:degeneration). The latter requires the low-degree coefficients to agree exactly. A small nonzero coefficient error may survive tensor powers and interpolation. Simply sending a numerical parameter to zero can also make the individual linear forms diverge. Polynomial coefficient extraction avoids that issue only after a genuine algebraic identity has been established. The exact interpolation calculation on a small polynomial tensor is included in the companion checks.

The fixed-field qualification in Lemma 26.2 is essential to its useful asymptotics. Paying for a new unrelated extension at every tensor level, or charging enormous coefficient descriptions as bit operations, would be a different complexity analysis. The statement intentionally follows the given nonuniform unit-cost rational model. No efficient procedure for discovering the decompositions is required by that target.

The universal-method barrier in [E2] shows that merely continuing to optimize a Coppersmith--Winograd-based implementation within that method cannot reach two; it does not forbid the infinite-group framework in every form or a different tensor family. The infinite-group paper identifies an explicit dimension/separating-degree trade-off still to be met. [S3], Abstract and Section 4. No unproved construction satisfying that trade-off has been inserted here. The target also would not, by itself, decide Boolean P versus NP: its unit-cost arithmetic model and its single bilinear problem are different.

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Outcome.
Outcome: Exploratory approach with unresolved gap.

The notebook gives fully explicit exactness and field-descent conversions, including tensor-power and all-size overheads, and tests them on an exact benchmark. These are standard mechanisms developed here in the precise rational model, not claimed historical novelties. They show that numerical rounding, coefficient fields, and a failure to attain literal quadratic rank are not valid substitutes for the real missing step: a sequence of certified decompositions approaching quadratic exponent. No improved rank family or proof that such a family cannot exist has been obtained.

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