Core concepts, axioms, mathematical foundations, and the boundaries of the framework.
Models, simulations, null tests, computational experiments, and reproducible tools.
What happened when the ideas met data: successful tests, failed tests, cross-market experiments, and findings that survived further checks.
| Research-program work | Empirical finding | Support level | Boundary / untested component |
|---|---|---|---|
| Structural Depth: A Containment-Based Measure of Organization | Containment/path-depth structure yields computable Θ and Ψ; Ψ>0 states are observed and separated in subsequent behavior. | DIRECT EMPIRICAL SUPPORT | Limited to the selected domain projection; does not establish a representation-independent universal measure. |
| Structural Determination: From Causality to Containment | Ψ is computed from containment/execution structure before reference to outcome variables; Ψ>0 and Ψ=0 states are compared with forward outcomes. | DIRECT OPERATIONAL SUPPORT | Universal replacement of causality by containment as a scientific explanatory principle is not established; this is not a causal-mechanism test. |
| Structural Determination Axioms (SDA) | A containment-primitive architecture is converted into a computable quantity on a selected domain projection. | PARTIAL / EMPIRICAL BRIDGE | Representation-independent validity, categorical/ontological scope, and necessity across all domains are not tested. |
| Structural Depth Theory I: Foundations of Structural Constraint Geometry via Grothendieck Sites | Market data are modeled as an admissible relational representation; containment paths, closure, and path depth are operationally tested through T1–T15. | PARTIAL SUPPORT | Grothendieck-site structure and sheafification/local-to-global consistency theorems are not directly tested. |
| Structural Depth Theory II: Structural Determination and Sheaf-Theoretic Coherence | Containment path depth, Θ, and Ψ are computed; T1–T15 and live campaigns support producibility of Ψ as a structural quantity. | PARTIAL — STRONG OPERATIONAL SUPPORT | Sheaf-theoretic coherence, invariant-residue/obstruction results, and categorical generality are not directly tested. |
| Structural Depth Theory III: Spectral Accessibility, Structural Depth, and Sheaf-Theoretic Constraint Geometry | The containment-path organization layer preceding the spectral layer is tested as an empirical counterpart to an early observable part of the chain. | PARTIAL / EARLY LAYER | Accessibility spectrum, spectral dimension, Laplacian, and spectral-accessibility results are not directly measured or validated. |
| Structural Depth Theory IV: Operational Foundations, Spectral Accessibility, and Computational Demonstrations | T1–T15, live telemetry generation of Ψ, event detection, strict matching, and reproducible audit chain reduce the theoretical quantity to a computable/falsifiable operational layer. | DIRECT OPERATIONAL SUPPORT | Not an independent representation-independent validation of all spectral and operational results in SDT IV. |
| Structural Depth and Markets: Price Formation as Structural Fixation | Ψ as a market-microstructure structural state is systematically related to subsequent ΔR and execution/microstructure variables. | PARTIAL EMPIRICAL SUPPORT | The claim that all price formation is explained by structural fixation or a single mechanism is not tested. |
| The Structural Depth Law of Time | Differences in subsequent t+1/t+3/t+5 behavior are observed between Ψ>0 and Ψ=0 states, showing temporal separation. | CONCEPTUAL COMPATIBILITY — NOT DIRECT VALIDATION | Waiting time τ or a structural-depth-dependent coordination time is not directly measured; the principal quantitative claim is not tested. |
| Structural Depth Definition of Reality: Constrained Observational Functors and Invariant Core Structure | A structural quantity becomes observable and outcome-independent within a selected domain representation. | PARTIAL / METHODOLOGICAL COMPATIBILITY | Representation-independent mathematical results concerning invariant core structure and constrained observational functors are not directly tested. |
| Structural Depth as a Spectral Constraint Field | An earlier containment/path-depth layer is empirically produced. | NOT TESTED | Spectral constraint fields, anomalous diffusion, and emergent dimensional reduction are not tested. |
| Generalized Quantum Constraint Topology | The experiment produces containment-derived structural dispersion in market microstructure. | NOT TESTED | Non-commutative accessibility and quantum constraint-topology claims are not tested. |
| Reality as a Constraint-Invariant Structure | Computability of a structural state independently of outcome variables is methodologically compatible with the projection/observation framework. | METHODOLOGICAL COMPATIBILITY — NOT TESTED | The general ontological claim that reality is a constraint-invariant structure is not empirically tested. |
| Phase Transitions: An Interpretation as Containment Expansion | Ψ events and forward microstructure differences provide observations compatible with structural reorganization/transition. | CONCEPTUAL COMPATIBILITY | A formal phase-transition mechanism defined as containment expansion is not directly specified and tested. |
| The Price of Waiting | Associations between structural states and different forward-horizon behaviors are measured. | NOT TESTED / PARTIAL COMPATIBILITY | Waiting cost, presence validity, and phase-dependent economic cost are not quantitatively measured. |
| Structural Depth and Entropy | Computability of Ψ supports an alternative measurement layer for structural organization. | NOT TESTED | Formal entropy–Structural Depth unification or empirical comparison is outside the study scope. |