Non-commutative circuits and the sum-of-squares problem
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of <italic>non-commutative</italic> arithmetic circuits and a problem about <italic>commutative</italic> degree-four polynomials, the classical sum-of-squares problem: find the smallest <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there exists an identity <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0.1 right-parenthesis left-parenthesis x 1 squared plus x 2 squared plus midline-horizontal-ellipsis plus x Subscript k Superscript 2 Baseline right-parenthesis dot left-parenthesis y 1 squared plus y 2 squared plus midline-horizontal-ellipsis plus y Subscript k Superscript 2 Baseline right-parenthesis equals f 1 squared plus f 2 squared plus midline-horizontal-ellipsis plus f Subscript n Superscript 2 Baseline comma"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0.1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="1em"/> <mml:mspace width="1em"/> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>y</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>y</mml:mi> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mspace width="1em"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} (0.1)\quad \quad (x_1^2+x_2^2+\cdots + x_k^2)\cdot (y_1^2+y_2^2+\cdots + y_k^2)= f_{1}^{2}+f_{2}^{2}+\dots +f_{n}^{2} , \quad \quad \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript i Baseline equals f Subscript i Baseline left-parenthesis upper X comma upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_{i} = f_i(X,Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a bilinear form in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals StartSet x 1 comma ellipsis comma x Subscript k Baseline EndSet"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X=\{x_{1},\dots ,x_{k}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y equals StartSet y 1 comma ellipsis comma y Subscript k Baseline EndSet"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it