Core Ideas

Each page here states one significant claim — something we believe is new to the literature — and lays out the papers, experiments, and open questions behind it. These are not summaries of individual papers; they are the ideas that span families of papers.

The TRS/ASA programme has produced a number of results we believe are genuinely novel. The pages below present the most significant of them, one claim per page, written for a reader who wants the idea first and the machinery second.

Each page follows the same structure:

  1. The claim — one or two sentences, as precisely as possible
  2. Why it matters — what changes if the claim is true
  3. The evidence — the papers and experiments that support it
  4. What would falsify it — the prediction that would prove us wrong
  5. Open questions — what we do not yet know

The claims, in brief

# Claim Key papers
1 Magic has a periodic table. Quantum non-classicality is not binary — it has orbits, valences, and dark states invisible to standard measures. 365, 366, 467, 469, 470
2 Every molecule is already running a programme. Enzyme catalysis, nitrogen fixation, and spin-state switching are ISA programmes readable in the group-orbit language. 488, 489, 490, 491, 509
3 The Fano crystal is universal. Whether an orbit is closed or open — the same binary — governs stability from quark confinement to financial contagion to photosynthetic efficiency. 317, 319, 325, 357, 545
4 Quantum speedup has a cohomological address. H⁰, H¹, and H² classify which computational problems admit which kinds of speedup — and why some problems are hard at every level. 420, 421, 472, 473
5 Biology runs quantum error correction. Kinetic proofreading, protein chaperones, and the ribosome implement H⁰ × H¹ × H² QEC — the same three-tier structure as fault-tolerant quantum computing. 510, 511, 515, 324
6 β is a coordinate. Planck’s constant, viscosity, volatility, and temperature are all the same object — coordinates on the adèlic β-plane — and Ostrowski’s theorem proves the map is complete. 201, 443, 419, 454, 543
7 The H^k stratification is not an analogy. The three-tier structure appears across MCMC, EVT, causal inference, chemistry, and quantum computing — some mappings are exact theorems, some make quantitative predictions, some are useful taxonomy. 420, 533, 557, 558, 559
8 Every molecule runs a Galois programme. The spin state, valence, and catalytic barrier of any transition-metal complex are determined by G-orbit walks on its molecular G-set — computable in O(1), without DFT, at 300 K. 488, 489, 490, 491, 492, 562
9 The Grassmannian is the universal space for correlated systems. A single angle θ_G = arccos(σ₀) — the geodesic distance from the classical fixed point — diagnoses multi-reference chemistry, QEC threshold failure, nuclear bonding, and financial contagion, with the same β* snap at θ_G ≈ 20° across all four domains. 563, 568, 570, 574, 577, 578, 580

More claims are in preparation. The ISA semiring-polymorphism claim (same programme, different arithmetic, different physics) is treated in The Operative and Harmonic ISAs and The ISA Opcodes rather than here, since those pages already give it a full treatment.

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