Asymmetric Rogers–Ramanujan type identities. I. The Andrews–Uncu conjecture
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
In this work, we start an investigation of asymmetric Rogers–Ramanujan type identities. The first object is the following unexpected relation <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n greater-than-or-equal-to 0 Endscripts StartFraction left-parenthesis negative 1 right-parenthesis Superscript n Baseline q Superscript 3 StartBinomialOrMatrix n Choose 2 EndBinomialOrMatrix plus 4 n Baseline left-parenthesis q semicolon q cubed right-parenthesis Subscript n Baseline Over left-parenthesis q Superscript 9 Baseline semicolon q Superscript 9 Baseline right-parenthesis Subscript n Baseline EndFraction equals StartFraction left-parenthesis q Superscript 4 Baseline semicolon q Superscript 6 Baseline right-parenthesis Subscript normal infinity Baseline left-parenthesis q Superscript 12 Baseline semicolon q Superscript 18 Baseline right-parenthesis Subscript normal infinity Baseline Over left-parenthesis q Superscript 5 Baseline semicolon q Superscript 6 Baseline right-parenthesis Subscript normal infinity Baseline left-parenthesis q Superscript 9 Baseline semicolon q Superscript 18 Baseline right-parenthesis Subscript normal infinity Baseline EndFraction"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:munder> <mml:mfrac> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-OPEN"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mfrac linethickness="0"> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-CLOSE"> <mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>9</mml:mn> </mml:msup> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>9</mml:mn> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>18</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>9</mml:mn> </mml:mrow> </mml:msup> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>18</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi>
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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| 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