Functional Analysis
9 Approaches, 7 Constraints — Mapping How Schools Model Filters
Constraint Classes = what stops doubling — Functional Analysis = how schools model it
Map: Each functional analysis tries to explain one (or more) of the 7 physical filters. Below, observed data tables show what they’re modeling.
Class 2 — Geometric / Packing — The Gap — Notations 0-64, critical at 60
What you’re seeing: 7.356° gap at Notation 60 — sphere packing fails. Data bends ε=κN^α.
| Notation | Ideal | Actual | Gap |
|---|---|---|---|
| 59 | 5.76e-18 | 5.76e-18 | ~0° |
| 60 | 1.15e-17 | 1.07e-17 | 7.356° critical |
- CDT (Ambjørn, Loll) — tetrahedra — models gap — View II_ [Class 2]
- Causal Sets (Sorkin) — discrete points — why doubling can’t be continuous — View III_
- LQG (Ashtekar, Rovelli, Smolin) — quantized area — View I_
Class 3 — Symmetry / Conservation — HOME OF LANGLANDS — 1-64 foundational
What you’re seeing: If symmetry doesn’t hold at 1-64, Notation 60 can’t be bridged. Langlands Programs (Robert Langlands, Ed Frenkel, Ngô Bảo Châu…many others) — Automorphic ↔ Galois duality preconditions spacetime. Mathematical bedrock for teaching constraints.
| Symmetry | Conserved | Dual | Data |
|---|---|---|---|
| U(1) | charge | Automorphic ↔ Galois | SUMMARY |
| SU(2) | isospin | Geometric Langlands | JSON 68KB |
- Langlands Programs — Home here at Class 3 — [Class 3 — Langlands]
- SUSY (Dimopoulos, Gates) — boson-fermion pairing would smooth gap
- Spectral Standard Model (Connes) — noncommutative geometry → Standard Model — [Class 3 → 4]
Class 4 — Field / Force — 65-134
Fields emerge and push back logarithmically ε=κ log N. Wilczek, ‘t Hooft, Witten, Vafa, Maldacena.
View IV_Logarithmic_Brake.csv (2,863)
Classes 5-7 — Entropic, Information, Biological — 90-202
| Class | Notation | Filter | Data |
|---|---|---|---|
| 5 Entropic | 90-170 | Threshold S-curve | V_ (3,006) |
| 6 Info | 130-190 | Delayed Feedback | VI_ (2,863) |
| 7 Bio/Chem | 170-202 | Stochastic | VII_ (4,033) |
Functional Analysis — Old — 9 approaches mapped to 7 filters — Class 3 = Langlands home — 81018.com