A general functional equation and its stability
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Bibliographic record
Abstract
Suppose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</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 B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are vector spaces over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q comma double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q},\ \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</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="alpha 0 comma beta 0 comma ellipsis comma alpha Subscript m Baseline comma beta Subscript m Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> β </mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> β </mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha _0,\beta _0,\dots ,\alpha _m,\beta _m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are scalar such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript j Baseline beta Subscript k Baseline minus alpha Subscript k Baseline beta Subscript j Baseline not-equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:msub> <mml:mi> β </mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msub> <mml:mi> β </mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo> ≠ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha _j\beta _k-\alpha _k\beta _j\neq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to j greater-than k less-than-or-equal-to m period"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>j</mml:mi> <mml:mo>></mml:mo> <mml:mi>k</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>m</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">0\leq j>k\leq m.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript k Baseline colon upper V right-arrow upper B"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f_k:V\rightarrow B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to k less-than-or-equal-to m"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>k</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0\leq k\leq m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an
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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 | 0.002 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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