Bibliographic record
Abstract
We prove that in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Linial–Meshulam stochastic process the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(d - 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> st homology group with integer coefficients vanishes exactly when the final maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(d - 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional face is covered by a top-dimensional face. This generalizes the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d equals 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> case proved recently by Łuczak and Peled [Discrete Comput. Geom. 59 (2018), pp. 131–142] and establishes that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals d log n slash n"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p =d \log n/n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the sharp threshold for homology with integer coefficients to vanish in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Subscript d Baseline left-parenthesis n comma p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Y_d(n, p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , answering a 2003 question of Linial and Meshulam [Combinatorica 26 (2006), pp. 475–487].
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How this classification was reachedexpand
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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.007 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.005 | 0.006 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".