The ASA Framework
A map of how the pieces fit together.
The ASA grew in layers, each one built on the last. They are not synonyms.
| Layer | Name | What it is | Origin |
|---|---|---|---|
| 1 | MGE | The core operation: Maslov-Gibbs Einsum | Paper 201 |
| 2 | TRS | The β-deformation framework built on MGE | Paper 202 |
| 3 | ISA trilogy | Three compilers, one for each temperature | Papers 258, 419, 454 |
| 4 | H^k ladder | Complexity and cohomology classification | Papers 396–398, 420 |
| 5 | ASA | The full programme: all of the above plus all applications | Paper 219 |
Layer 1 — MGE (Maslov-Gibbs Einsum)
The single operation at the heart of everything:
\[\pi_k(\beta) = \frac{e^{-\beta E_k}}{\sum_j e^{-\beta E_j}}\]The domain of β is the full adèlic β-plane — real, imaginary, negative, complex, and p-adic values each give a different computational regime:
| β | Regime | What it computes |
|---|---|---|
| 0 | The Ambient | Uniform weights; smooth Hodge theory |
| real, finite | Forge ISA | Gibbs distribution; statistical mechanics |
| real → ∞ | Origami ISA | Tropical (min, +) semiring; hard discrete selection |
| imaginary ($it$) | Meld ISA | Quantum amplitudes; Wick rotation |
| negative | Population inversion | Lasers; anti-Boltzmann weighting |
| p-adic | Ultrametric | p-adic computation; attaches at β = 0 |
Ostrowski’s theorem closes the map: every non-trivial absolute value on ℚ is either Archimedean (giving ℝ) or p-adic (giving ℚ_p), so the adèlic β-plane is the complete parameter space for the MGE. The MGE is what makes any discrete model differentiable: replace argmin with this, and gradients flow everywhere.
→ Papers 201 (MGE), 543 (adèlic β-plane)
Layer 2 — TRS (Topological Resonance Synthesis)
TRS is the β-deformation framework built on MGE. The name describes the underlying picture: Lie groups serve as the tape of a generalised Turing machine. The Chladni resonance patterns on the group manifold encode the computational state; the topology of those nodal lines — their H⁰/H¹/H² skeleton — is the computation. The MGE makes that skeleton differentiable.
- Topological — the nodal-line topology of Chladni eigenmodes on the group manifold
- Resonance — the Chladni eigenmodes themselves; also Wigner-Racah angular momentum resonances
- Synthesis — the compiler that synthesises across classical/statistical/quantum via β
TRS is not the same as the ISA trilogy, not the same as the H^k ladder, and not the same as the ASA. It is the specific β-deformation layer.
Layer 3 — The ISA Trilogy
Four temperature regimes, three operative ISAs:
| ISA | β | Arithmetic | Paper |
|---|---|---|---|
| Ambient | 0 | Uniform / Hodge theory | 417 |
| Origami | β → ∞ | Tropical (min, +) | 258 |
| Forge | 0 < β < ∞ | Real Gibbs | 419 |
| Meld | β = it | Complex amplitudes | 454 |
Each operative ISA (Origami, Forge, Meld) uses the same five opcodes — SPLIT, SPLAT, FLIP, FLOP, TWIST — running at a different temperature. The β → it substitution (Wick rotation) turns the Forge ISA into the Meld ISA and turns statistical mechanics into quantum mechanics.
The ISAs are not the same as the H^k ladder (Layer 4). The H^k ladder classifies problem type by the cohomological obstruction that makes it hard; any ISA can in principle compute any H^k problem, but at different cost. The Origami ISA naturally handles H⁰ problems; the Forge ISA excels at H¹; the Meld ISA is required for H² (quantum) problems. But the correspondence is one of natural fit, not restriction.
Eight independent mathematical communities have each, working separately, been forced to the same five opcodes. Paper 455 explains why: Shum’s theorem (1994) identifies the free ribbon pivotal category on one self-dual object, whose generators are forced by the topology of framed tangles.
The chain complex: H^k is genuine cohomology
The H^k tiers are not merely a grading — they form a genuine chain complex. The boundary map ∂: C^k → C^{k+1} is assembled from the SPLIT and SPLAT opcodes with Koszul signs; it satisfies ∂² = 0 as a direct consequence of the Frobenius algebra axiom (SPLAT∘SPLIT = id). This is the same condition as the Pentagon identity, the MIP*=RE self-test, and Khovanov’s categorification of the Jones polynomial. The ORBIT count is the Euler characteristic of the complex; the full Poincaré polynomial is a strictly stronger invariant. At the H² level, BIND = the Kuperberg G₂ spider vertex (CMP 1996) — the complete diagrammatic axiomatisation of non-Abelian holonomy in the ISA.
→ Papers 357 (Pentagon/MIP), 571 (chain complex), 572 (G₂ spider = BIND)*
The 731-ISA: beyond the Pentagon
The three-ISA trilogy operates within the associative world — the Pentagon identity SPLAT∘SPLIT = 0 holds throughout. There is a fourth ISA that breaks this:
| ISA | Adds | Breaks | Algebraic home |
|---|---|---|---|
| Origami / Forge / Meld | — | — | ℝ, ℂ (associative) |
| 731-ISA | BIND, SPIN | Pentagon identity | 𝕆 (octonions, non-associative) |
BIND is the frog vertex — the non-abelian fusion that requires G₂ symmetry and implements the Fano associator obstruction. At j=½ it is the T-gate; at higher spin it accesses genuinely non-associative territory inaccessible to any standard quantum gate set. SPIN is the G₂ triality automorphism — the order-3 outer automorphism of Spin(8) that cyclically permutes its three 8-dimensional representations — and is the elementary injection gate for associamancy (the fourth level of the quantum resource hierarchy; see Paper 407).
A gate set is triality-complete if it contains SPIN. The 731-ISA is triality-complete; the Origami/Forge/Meld trilogy is not.
The 731-ISA is the computational realisation of the bottom-right cell of the Freudenthal-Tits magic square — the 4×4 table of exceptional Lie algebras built from pairs of division algebras (ℝ, ℂ, ℍ, 𝕆). The diagonal of that table is the division algebra ladder expressed as Lie algebras: G₂, F₄, E₆, E₈. The full non-associative frontier is described in the Non-Associative Frontier explainer.
→ Papers 207, 258, 263, 407, 405
Layer 4 — The H^k Complexity Ladder
The ISAs compute things whose complexity is classified by sheaf cohomology:
| Level | Symbol | Name | Computation | Complexity |
|---|---|---|---|---|
| 0 | H⁰ | Islands | Classical / bilateral | PSPACE |
| 1 | H¹ | Flatland | Statistical / triangular | BPP |
| 2 | H² | The Deep | Quantum / tetrahedral | BQP |
H⁰ = problems solvable by looking at each part independently. H¹ = problems requiring triangular (three-party) consistency. H² = problems requiring tetrahedral (four-party) coherence — the level at which quantum algorithms live and at which the 2008 financial crisis became unresolvable.
The snap threshold β* = (3/8)ln(1/(1−ρ)) marks the transition between H⁰ and H¹ regimes for any network with edge density ρ.
Layer 5 — The ASA (Adelic Simplicial Architecture)
The full research programme: all five layers plus their applications across quantum computing, nuclear and molecular spectroscopy, financial risk, climate economics, and pure mathematics. Organised into five portfolios:
| Portfolio | Theme |
|---|---|
| A — Core Engine | MGE, TRS, non-associative calculus |
| B — Foundations | Algebra, topology, category theory |
| C — Hardware & AI | ISA trilogy, registers, QEC |
| E — Chemistry & Physics | Bonding theory, nuclear, amplituhedron |
| F — Quantum Foundations | Magic, self-tests, paradoxes |
| G — Finance & Economics | EconIAC, gauge theory, risk |
The adelic structure — using all completions of ℚ simultaneously (real + p-adic) — is captured by the adèlic β-plane (Paper 543): the real axis governs the Origami/Forge/Meld ISA trilogy, while the p-adic completions attach at β = 0 (the Ambient). Ostrowski’s theorem guarantees this is the complete picture.
→ Paper 219 (An Adelic Invitation)
Where EconIAC fits
EconIAC is the Portfolio G application of the ASA. It uses all five layers:
- MGE — Gibbs relaxation makes every argmax differentiable; β is the rationality temperature
- TRS — the β-deformation of discrete economic models (Nash equilibria, order statistics, Leontief minimum) into smooth, calibratable functions
- Forge ISA — the compiler for statistical/probabilistic economic computation at finite β
- H^k ladder — H⁰ = bilateral risk (Basel), H¹ = triangular risk (XVA, convexity), H² = systemic risk (2008 crisis, unresolvable cascades)
- ASA — the full framework that places the Keynesian multiplier, FMO photosynthesis efficiency, and Shor’s algorithm on the same ladder
Reading order
New to the framework? Start here:
- The Maslov-Gibbs Einsum (Paper 201) — the single equation; the dodecagon of twelve unified constructs; Turing completeness as a corollary
- Planck’s Constant in Disguise (Paper 443) — the six classical dualities as β-deformations; no prerequisites
- The Adèlic β-Plane (Paper 543) — the complete parameter space: why ℏ, viscosity, volatility, and quantum amplitudes are all the same coordinate
- In Praise of Tetrahedra (Paper 386) — the geometric seed; accessible to anyone
- Eight Derivations (Paper 455) — why the five opcodes are inevitable
Then follow the Start Here routing page for your field.