Applications

Each entry below applies the pillars to a specific domain. These are not summaries of individual papers — they are the ideas that span families of papers.

# Claim Domain Key papers
1 Chemistry. The expensive part of simulating a molecule is deciding which electrons matter — and that is still done by hand. Chemistry 563, 596, 712
2 Where colour comes from. Ruby and emerald contain the same Cr³⁺ ion; the difference is one crystal-field parameter. Most mineral colour is d-electron spectroscopy, predicted by Racah’s recoupling algebra — whose central object, the 6j symbol, is a tetrahedron. Chemistry textbook
3 Proofreading looks like error correction. Kinetic proofreading and ribosomal decoding both spend energy to suppress errors — a suggestive parallel with quantum error correction, not an established equivalence. Biology 324
5 Correlation has an angle. θ_G, the Grassmannian angle between a correlated wavefunction and its mean-field reference, tracks the onset of strong correlation in real calculations. Universal 563
6 Broken symmetry is what lets a cycle do work. On a graph register, the Gibbs entropy profile is flat exactly when the graph is vertex-transitive — so asymmetry is necessary for positive efficiency. Biology / chemistry 325
7 The H^k stratification: what it is and is not. A useful organising principle across MCMC, chemistry and quantum computing — and an honest account of where the parallel stops. Universal 420 · (533, 557, 558 withdrawn)
8 Quantum speedup has a cohomological address. H⁰/H¹/H² classify which problems admit which speedups. Quantum computing 420, 472 · (473 withdrawn; 421 no DOI)
9 One threshold, two problems. The entanglement measure that flags where cheap simulation fails in chemistry is the same coordinate that separates classically simulable circuits from universal ones — a shared diagnostic, whether or not it is a shared cause. Chemistry / QC 595, 596, 570, 563
10 β is the softmax temperature. The same inverse-temperature parameter that governs Gibbs distributions governs attention, quantal response, and entropy-regularised learning. AI / ML 627

The five load-bearing ideas behind all of the above are on the Pillars page. For the full ISA family table, see Pillars → ISA Family.

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