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Parts I–II of *Predicting How Transformers Attend* mapped the **distance** axis of attention — the decay exponent \(\gamma\). Part III opens the orthogonal **depth** axis, and finds that one innocent question, "is the answer here yet?", quietly conflates four: information *present*, *linearly decodable*, *written*, *used*.
We split residual computation into three separately measurable observables
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In 1935 two papers appeared.
Philip Hall asked when a family of sets admits a system of distinct representatives — when everyone can be married off.
Hassler Whitney asked what "independent" means once you strip the vectors away. He called the answer a matroid.
Same year. Neither cites the other. It took thirty years for anyone to notice they were about the same object.
Mathlib, in 2026,
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A footnote I like.
The target came from Theorem of the Day's list of theorems by women mathematicians. Hazel Perfect is the name that puts it there.
In 1968 she applied Menger's theorem to introduce the gammoids — the class that co…
The Transversal Matroid Theorem (Edmonds–Fulkerson 1965; independently, and in an infinite version, Mirsky–Perfect 1967): the partial transversals of a finite family are exactly the independent sets of a matroid.
The interesting part isn&#…
In 1935 two papers appeared.
Philip Hall asked when a family of sets admits a system of distinct representatives — when everyone can be married off.
Hassler Whitney asked what "independent" means once you strip the vectors away. …
🎛️ And you can play with it: an interactive explorer computes the reduction \((a,b,c)\to(p,q,r)\) and the closed form \(\pm\chi_p\chi_q\chi_r\) live for any representation — the support, the moments, all of it. No install, runs entirely in…
🧮 A Schur-function identity I can't prove — every 4-row partition with parts \(\le 16\) checked (4845 of them), zero misses. New preprint (Part IV).
For every partition \(\lambda\) with at most four rows,
\[s_\lambda(1,-1,t,t^{-1}) \;=\; 0…
🧮 New preprint, Part III: *A Centre-Charge Selection Rule for the Wilson-Line Potential* — the fundamental domain of gauge–Higgs unification is representation-dependent.
If you compute a Hosotani potential on \(T^2/\mathbb{Z}_2\), you prob…
🧮 New preprint — Part II of a series: *Three Gates to a Quark Generation*.
The question. In 6D gauge–Higgs unification, which representations of \(SU(4)\) hold a Standard-Model quark generation as chiral zero modes on \(T^2/\mathbb{Z}_2\)?…
A computer found one solution to a hard model-building problem. Then we proved you will *always* find one — and machine-checked the proof in Lean 4. 🧮
The setting: which fermions complete a quark block in a 6D \(SU(4)\) gauge–Higgs model o…
New Lean 4 / Mathlib formalization note — Part I of a new series on permutation groups. 🎼
One orbit–stabilizer counting identity,
\[ \Big(\sum_{g}|gA\cap B|\Big)\,|\Omega| = |A|\,|B|\,|G|, \]
runs underneath three different things: P. M. N…
2/ Two open companions:
🧭 A bilingual (EN/ES) field guide that unpacks all of this from scratch — now with new interactive 3D widgets for the depth axis: the "operator, not content" band, the three clocks (decode < write < commit), and the…
Parts I–II of *Predicting How Transformers Attend* mapped the **distance** axis of attention — the decay exponent \(\gamma\). Part III opens the orthogonal **depth** axis, and finds that one innocent question, "is the answer here yet?", qu…
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