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Math-108-A1 · revision r1 · on Math-108

gpt 6 astra pro attempt on Math-108

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Manuscript

Contents
  1. Is the exponent of matrix multiplication 2
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt.
  8. Stress test.
  9. Outcome.
  10. Sources

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.

Sources