The failure of diamond on a reflecting stationary set
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Bibliographic record
Abstract
1. It is shown that the failure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal ♢ Subscript upper S"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ♢ </mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\diamondsuit _S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for a set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S subset-of-or-equal-to normal alef Subscript omega plus 1"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo> ⊆ </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ω </mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">S\subseteq \aleph _{\omega +1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that reflects stationarily often, is consistent with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif GCH"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="sans-serif">GCH</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textsf {GCH}</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="normal upper A normal upper P Subscript normal alef Sub Subscript omega"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">A</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mi> ω </mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {AP}_{\aleph _\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , relative to the existence of a supercompact cardinal. By a theorem of Shelah, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif GCH"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="sans-serif">GCH</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textsf {GCH}</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="white medium square Subscript lamda Superscript asterisk Baseline"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>◻</mml:mo> </mml:mrow> <mml:mi> λ </mml:mi> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:annotation encoding="application/x-tex">\square ^*_\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> entails <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal ♢ Subscript upper S"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ♢ </mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\diamondsuit _S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S subset-of-or-equal-to lamda Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo> ⊆ </mml:mo> <mml:msup> <mml:mi> λ </mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">S\subseteq \lambda ^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that reflects stationarily often. 2. We establish the consistency of existence of a stationary subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket normal alef Subscript omega plus 1 Baseline right-bracket Superscript omega"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ω </mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mi> ω </mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">[\aleph _{\omega +1}]^\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that cannot be thinned out to a stationary set on which the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup"> <mml:semantics> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:annotation encoding="application/x-tex">\sup</mml:annotation> </mml:semantics> </mml:math> </inline-formu
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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 | 0.000 | 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 | 0.001 |
| 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