FREE BROWNIAN MOTION AND EVOLUTION TOWARDS ⊞-INFINITE DIVISIBILITY FOR k-TUPLES
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Bibliographic record
Abstract
Let [Formula: see text] be the space of non-commutative distributions of k-tuples of self-adjoint elements in a C*-probability space. For every t ≥ 0 we consider the transformation [Formula: see text] defined by [Formula: see text] where ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on [Formula: see text]. We prove that 𝔹 s ◦ 𝔹 t = 𝔹 s + t , for all s, t ≥ 0. For t = 1, we prove that [Formula: see text] is precisely the set [Formula: see text] of distributions in [Formula: see text] which are infinitely divisible with respect to ⊞, and that the map [Formula: see text] coincides with the multi-variable Boolean Bercovici–Pata bijection put into evidence in our previous paper [1]. Thus for a fixed [Formula: see text], the process {𝔹 t (μ)|t ≥ 0} can be viewed as some kind of "evolution towards ⊞-infinite divisibility". On the other hand, we put into evidence a relation between the transformations ⊞ t and free Brownian motion. More precisely, we introduce a map [Formula: see text] which transforms the free Brownian motion started at an arbitrary [Formula: see text] into the process {𝔹 t (μ)|t ≥ 0} for μ = Φ(ν).
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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 | 0.001 |
| 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