The Four Levels (Formal)
Companion to AAA-000, which defines the levels in plain language. This document states the formal requirements in mathematical notation. If you want to promote an ISO from Level 2 to Level 3, this is the checklist.
Informal similarity, possibly only linguistic or pictorial. No formal definition of either domain. No morphism. No mathematics. Useful for discovery. Never sufficient for proof.
Two fully specified structures A, B and a partial structure-preserving map between them. The map preserves at least one non-trivial operation or relation.
A bijective structure-preserving map with a proved inverse. Relations, transformations, and dynamics are preserved in both directions. This is the ceiling for most Faith Through Physics correspondences.
The isomorphism is empirically instantiated. Quantitative predictions verified by independent groups. This is what Maxwell achieved with electromagnetism.
The Three Boundary Crossings
Each boundary crossing has a precise formal trigger — a specific thing that becomes true that was not true before. Knowing the trigger tells you what work is actually required.
You must give both domains explicit mathematical form and produce a morphism that preserves at least one non-trivial operation or relation (composition, ordering, group law, topology).
Before Heaviside: "electricity flows like water" — analogy. After Heaviside: the ODE systems governing circuit elements and fluid flow in pipes were shown to be homomorphic — structural analogy. The crossing happened when the differential equations were written.
A proof that the mapping preserves the chosen structure. If you claim an order-preserving map, show monotonicity. If you claim a group homomorphism, show F(x ∗ y) = F(x) ∗ F(y). One worked-out non-trivial example is not enough — you must demonstrate closure under the relevant operations.
Treating pictorial similarity or shared terminology as if it already defined a morphism. "Energy flows like water" is not structural until you supply the differential equations and the variable dictionary. The words are not the morphism.
The morphism must be bijective AND preserve every operation, relation, and designated element required by the theory. You need an explicit inverse with verified natural isomorphisms. This is where most claims stall.
Initially: formal resemblance between Fermat's principle (light minimizes time) and Maupertuis' principle (particles minimize action) — Level 2. By late 19th century, the symplectomorphism was proved explicitly:
The crossing happened when the inverse mapping was constructed and the symplectic structure shown to be preserved — not just the differential equation form.
An explicit inverse mapping with a proof that ALL structure is preserved. In algebra: verify every axiom. In category theory: check that all relevant diagrams commute.
Demonstrating a bijection on underlying sets while ignoring higher-level structure — topology, smooth structure, monoidal product. Identifying state spaces pointwise while forgetting the symplectic form is not preserved. A bijection is necessary but not sufficient.
Empirical instantiation: a procedure that assigns operational definitions (measurement protocols) to the mathematical entities, PLUS experimental data showing quantitative agreement to within experimental error. The prediction must be novel — not used in building the model.
Maxwell began with an isomorphism between mechanical vortices/idle wheels (mathematical hydrodynamics) and electromagnetic quantities — a refined Level 3 within the field equations. The crossing to Level 4 occurred when Hertz measured electromagnetic waves traveling at c.
At least one novel, precise, quantitative prediction verified experimentally by an independent group. Three criteria — all required:
Declaring a mathematical model "the physics" because it is elegant or because it unifies other theories, without having produced new, uniquely corroborated predictions. Elegance is not evidence. Unification is not verification. String theorists sometimes fall prey to this charge.
Historical Crossings That Worked
Three cases from physics that successfully crossed boundaries — and what made them work. Common to all: explicit dictionaries, preservation of ALL symmetries, and checks impossible on one side but easy on the other.
Gubser-Klebanov-Polyakov/Witten filled in the correspondence of operators and correlation functions, establishing an (incomplete but compelling) equivalence of large-N limits of type-IIB string theory and N=4 SYM.
A systematic derivative expansion maps near-horizon gravitational perturbations to solutions of the Navier-Stokes equations. This is a functor between solution spaces. To prove isomorphism, invertibility must be shown — not yet done.
Built an isomorphism between categories: D-modules on Bun_G and categories of branes in the 4D gauge theory topologically twisted. All functorial requirements checked. Still mathematics, not physics — because no new empirical content.
The Ceiling for Theology-Physics
An honest assessment of where the Faith Through Physics framework can and cannot go, given current measurement capabilities.
Because soteriological stages are not (at present) operationally measurable in the laboratory, the Faith Through Physics framework lacks the operational definitions required for Level 4. You cannot measure "conviction" in Kelvin or "justification" in Joules. Without measurement protocols, Level 4 is formally unavailable.
The ceiling for theology-physics correspondences is almost certainly Level 3: mathematical isomorphism/duality.
This is exactly the status of Geometric Langlands. That program is not considered failed because it has not yet reached Level 4. A Level 3 claim — a proved, bidirectional, structure-preserving mapping — is a significant result. The Faith Through Physics framework is attempting the same thing across a harder domain boundary.
However, the ceiling is not fixed forever. If even ONE theological entity can be operationalized with a measurement protocol, that specific bridge could cross into Level 4. Three candidates:
- Conversion events measured via EEG criticality (phase transition signature at the moment of decision)
- Apostasy rates mapped to topological barrier crossing (winding number change in community structure)
- Prayer coherence measured via fMRI phase synchrony between intercessor and recipient
Formal Requirements Checklist
Before filing any ISO at a given level, work through the checklist for that level. If any item cannot be checked, the ISO cannot claim that level.
Time estimate: bottleneck is formalizing Domain B (the theological structure). Expect this to require explicit definition of objects, morphisms, and at least one non-trivial relation — not just a concept map.
This is the target level for Theophysics. Most current ISOs are on the path to Level 3 but have not yet supplied inverse functors with proofs.
Current ceiling for Faith Through Physics is Level 3. Level 4 requires operationalizing at least one theological entity. See the three candidate experiments in Section 04.