ADVERSARIAL BENCHMARK FOR
PROSPECTIVE DIMENSIONAL INFERENCE
A Blinded G₁–G₄ Computational Test of TSTOEAO Relational Source Discrimination Against Strong Conventional Baselines
John Swygert
Ivory Tower Publishing
October 2, 2026
Abstract
This paper specifies the adversarial computational benchmark required by Relational Source Discrimination and Prospective Dimensional Inference. The purpose is not to assume that the TSTOEAO protocol is methodologically original, but to determine whether it supplies measurable inferential capability beyond strong conventional pipelines assembled from nonlinear realization, observability and identifiability analysis, multi-view latent inference, Bayesian model discrimination, optimal experimental design, delay reconstruction, and explicit model-misspecification testing. Four hidden generating regimes are defined: G₁, a genuinely lower-dimensional nonlinear generator; G₂, a generator with independently certified additional registered source-state structure; G₃, exact or practical cross-dimensional observational equivalence under a registered experiment family; and G₄, generators outside every registered candidate class, including adversarial misspecification that can mimic extra dimension. The evaluator must return one of four scientific outcomes: lower-dimensional class preferred, higher-dimensional class preferred, observationally indistinguishable under the registered experiments, or none of the registered models adequate. The benchmark registers the type of dimension being tested and separates source-state, manifold, geometric-spatial, embedding, and reconstruction dimensions. Strong lower-dimensional adversaries receive delay coordinates, nonlinear state-space reconstruction, flexible latent dynamics, kernels or splines, neural state-space models, nuisance/channel models, and minimal nonlinear realizations where applicable. Three primary comparison arms are required: native TSTOEAO T, a strong conventional integrated pipeline C, and an operation-matched conventional translation C⁺. Training occurs on a subset of probes and interventions; decisive evaluation occurs on unseen probe-intervention combinations. Active experiment selection is evaluated both under native utilities and under a common externally specified utility. The benchmark is designed so that TSTOEAO can fail cleanly. A distinctive methodological claim is warranted only if the registered TSTOEAO protocol demonstrates reproducible capability or efficiency not reproduced by the operation-matched conventional control.
1. Research Objective
The preceding paper established a disciplined source-discrimination protocol but deliberately left one question unresolved: does the integrated TSTOEAO procedure add methodological value beyond an appropriately constructed conventional inference pipeline? The present work converts that question into a blinded computational benchmark.
The benchmark does not attempt to prove the physical existence of additional spatial dimensions. It tests a narrower claim: whether registered observational-equivalence reduction, relational invariants, positive and reliably absent signatures, cross-probe latent constraints, prospective intervention prediction, and active measurement selection can correctly determine when additional source structure is required, when it is unnecessary, when it is non-identifiable, and when the registered model family is wrong.
Primary test: TSTOEAO protocol T vs. strong conventional baseline C vs. operation-matched conventional translation C⁺
The benchmark must permit equivalence or defeat. If the conventional baseline produces the same decisions, calibration, and experiment choices at equal or lower cost, the proper conclusion is TSTOEAO synthesis/application rather than a distinct inference methodology.
2. Registered Objects and Decision Space
Let π = {H_(d,Ο)} denote registered source-model classes, where d is dimension and Ο registers the dimension type, π the admissible measurement/probe family, π the admissible intervention family, and π the declared equivalence and transformation rules. Register Ο ∈ {state, manifold, geometric spatial, embedding, reconstruction}. The benchmark principally tests minimum required source-state structure; geometric-spatial dimensionality is a special registered family and must not be inferred from reconstruction dimension alone. For source state X under H and experiment e=(M,I), let the predicted observation distribution be
p(O | H, M, I).
The evaluator must return exactly one of four scientific decision states:
L — a lower-dimensional registered class is preferred under the prospective criteria.
H — an additional registered source degree of freedom is required under the prospective criteria.
E — competing source classes are observationally indistinguishable under the available registered experiment family.
N — none of the registered source classes is adequate.
The decision space is intentionally not exhaustive over ontology. It is exhaustive only over the benchmark outputs. In particular, E and N prevent the evaluator from becoming a forced dimension selector.
3. Hidden Generating Regimes G₁–G₄
3.1 G₁ — Lower-Dimensional Nonlinear Generator
G₁ is generated by a nonlinear process whose observable and interventional behavior is adequately represented by a registered lower-dimensional class. Higher-dimensional candidates may fit the training data, but they are unnecessary. The primary failure mode is false dimensional discovery.
Desired outcome: G₁ → L
3.2 G₂ — Independently Certified Additional Registered Source Structure
G₂ must not be defined merely by the empirical failure of the contestants used in the benchmark. Generator construction and ground-truth certification are separated. For at least the transparent benchmark family, define a precisely bounded preregistered lower-dimensional admissible class π_d and establish independently—preferably analytically—that there exists a registered intervention I* for which no L ∈ π_d can reproduce a declared interventional consequence, while at least one constrained H ∈ π_(d+1,Ο) can. Symbolically, for the certified consequence, ∀L∈π_d, P_L(O|I*) ≠ P_G(O|I*), while ∃H∈π_(d+1,Ο) that reproduces it. The blinded evaluator is not given this certificate. In later expressive benchmarks, every claim is restricted to the strongest admissible realization in the preregistered adversarial class; finite computation does not certify failure of every conceivable lower-dimensional realization.
Desired outcome: G₂ → H
3.3 G₃ — Exact and Practical Cross-Dimensional Observational Equivalence
G₃ contains two subregimes. G₃E contains at least one pair H_a and H_b of different registered source dimensions/types that produce identical observable distributions for every experiment in an initial family π₀. G₃P contains pairs whose observable distributions are practically indistinguishable at the registered resolution, noise level, cost, and experiment budget.
G₃E: p(O | H_a,M,I) = p(O | H_b,M,I) for all registered (M,I) in π₀. G₃P: D(p_a(O|e),p_b(O|e)) ≤ Ξ΅ for every affordable e∈π₀, for a preregistered divergence D and tolerance Ξ΅.
The correct initial result is non-identifiability, not preference for the smaller class merely because it is simpler. A second experiment pool contains at least one discriminating experiment e* for G₃E and, where feasible, experiments that exceed the practical discrimination threshold for G₃P. This tests whether active selection can discover a measurement capable of breaking the equivalence when one exists without hallucinating certainty when the distinction is practically inaccessible.
Desired outcomes: before e*: G₃ → E; after informative e*: correct discrimination.
3.4 G₄ — Generator Outside the Registered Family and Adversarial Misspecification
G₄ is generated by a process not contained in any candidate class. The benchmark should include at least three subclasses: G₄a, obvious gross misspecification; G₄b, near-family misspecification; and G₄c, adversarial misspecification that allows a higher-dimensional registered model to fit substantially better than lower-dimensional candidates while retaining systematic held-out residual structure.
G₄a,G₄b,G₄c ∉ H₂ ∪ H₃ ∪ ··· ∪ H_N.
These regimes test whether the procedure can distinguish missing source structure from the wrong model family. A procedure that selects H for G₄c merely because the higher-dimensional candidate is more flexible fails the benchmark.
Desired outcome: G₄ → N
4. Strong Lower-Dimensional Adversary
The lower-dimensional competitor must be difficult enough that a higher-dimensional result cannot be obtained merely by comparing a rigid low-dimensional geometry with a flexible high-dimensional geometry. The adversarial family should include, where mathematically appropriate:
nonlinear state-space models;
Takens/Sauer-style delay-coordinate reconstructions;
minimal nonlinear realizations or approximations to them;
flexible latent dynamical models;
kernel and spline representations;
expressive neural state-space models;
explicit nuisance, receiver, and channel-transfer models;
symmetry-aware and transformation-aware lower-dimensional models.
The governing adversarial question is not whether a named lower-dimensional model fails. It is whether the strongest admissible realization in the preregistered adversarial class, under the registered data, causal restrictions, smoothness/complexity bounds, and computational budget, can reproduce the same prospective consequences.
If such a reconstruction predicts all held-out observations and interventions as well as the explicit higher-dimensional source geometry, the benchmark has not established that the geometric source dimension is required. It has established only that an adequate latent state representation exists.
5. TSTOEAO Protocol Under Test
The TSTOEAO arm applies the source-discrimination procedure specified in the preceding paper. Its registered components are:
Declare source classes, measurement operators, interventions, noise models, detection limits, symmetries, admissible transformations, and candidate relational invariants before evaluation.
Compute local observational deficit only as a regular local diagnostic, while treating observational fibers, posterior uncertainty, and candidate equivalence classes as the global objects.
Update candidate classes using positive observations, preserved invariants, calibrated negative signatures, and prospective constraints.
Require a shared latent accessibility state to constrain heterogeneous probes without probe-specific refitting when A_Ξ is used.
Permit the four outcomes L, H, E, and N.
Select the next experiment by expected reduction of registered observational equivalence, subject to cost and feasibility.
A TSTOEAO-specific relational invariant I_R may contribute only if it is preregistered relative to a transformation family π― together with its exact computation/estimator, tolerance, and failure criterion. An invariant chosen or operationally redefined after observing the benchmark cannot count as evidence of added methodology.
6. Three-Arm Conventional Controls: T, C, and C⁺
The primary comparison uses three arms. T is the native TSTOEAO formulation. C is a strong standard integrated conventional pipeline combining nonlinear realization/system identification, observability and identifiability analysis, flexible latent-state estimation, multi-view latent inference where applicable, Bayesian model comparison, calibrated non-detection likelihoods, and Bayesian model-discriminating experimental design. C⁺ is an operation-matched conventional translation of every TSTOEAO operation for which a conventional mathematical translation exists, including invariants, equivalence handling, negative signatures, shared-latent constraints, interventions, and experiment-selection logic.
All three arms receive the same training observations, probe definitions, intervention history, admissible experiment pool, structural information, and computational budget. Hyperparameter tuning and stopping rules must be symmetric enough that one arm is not privileged by search effort. The purpose of C⁺ is to distinguish an advantage due to integrated operations from an advantage unique to native TSTOEAO representation.
Interpretation is explicit: T=C=C⁺ implies no operational advantage; T>C with T=C⁺ supports useful integration/protocol architecture reproducible conventionally; T>C⁺>C motivates ablation to identify the residual TSTOEAO-specific source of advantage. If C or C⁺ outperforms T, that result must be reported directly.
7. Blinding and Data Partition
Each synthetic instance is generated with a hidden regime label and hidden generating parameters. The inference systems receive only the registered candidate families, training observations, experiment metadata, and allowed future experiment pool. Generator identity is withheld until all decisions are frozen.
Data are partitioned into three roles: fitting data, experiment-selection data available at decision time, and final untouched evaluation data. The final evaluation set contains probe-intervention combinations not used for parameter fitting or model selection.
Repeated benchmark instances should randomize admissible parameters, initial conditions, noise realizations, probe order, and nuisance effects within preregistered ranges. Randomization must not change the structural definition of the regimes. Benchmark-designer overfitting is an explicit limitation; later validation should use independently written generators, independently implemented baselines, hidden challenge instances, or third-party benchmark contributions where feasible.
8. Structural-Transport Test
Ordinary interpolation and same-channel forecasting are insufficient. The decisive test is transport across unseen combinations of observation and intervention. For example, train on probes M₁, M₂, M₃ and interventions I₁, I₂, then evaluate predictions for combinations such as
(M₄,I₁), (M₂,I₃), (M₅,I₄).
A source model earns structural credit only when the same latent/source representation predicts these new combinations without relearning a probe-specific latent state. This directly tests whether the model has captured a transportable source structure rather than a collection of channel fits.
For A_Ξ, the evidentiary hierarchy remains: shared latent fit < cross-probe prediction < cross-probe intervention prediction. Even success at the final level supports a shared causal structure only within the benchmark; it does not by itself establish a physical dimensional-expression mechanism in nature.
9. Reliable Absence and Detection Calibration
A required but absent feature can exclude a source class only when detection performance is registered. If Q is predicted by H and M is the detection process, then
P(no detection | H) = ∫ P(no detection | H,ΞΈ) p(ΞΈ | H) dΞΈ, with the simpler decomposition P(¬Q|H)+P(Q|H)P(miss|Q,M) recovered when nuisance dependence can be suppressed.
Negative signatures should therefore be generated under known sensitivity, specificity, censoring, feature-strength, nuisance, and noise conditions. The benchmark must include cases where absence is genuinely discriminating and cases where a miss is plausible. A method that treats every non-detection as exclusion should fail calibration.
10. Observational Equivalence and the Discriminating Experiment
G₃ supplies the direct test of the principle that no relational distinction exists without an experiment capable of exposing it. Under π₀, H_a and H_b are deliberately equivalent. Both arms should return E.
The available experiment pool is then expanded to include candidate probes/interventions, only some of which break the equivalence. Active selection is evaluated in two modes: a native-utility comparison, in which each arm uses its preferred criterion, and a common-utility comparison, in which T, C, and C⁺ optimize the same externally specified objective.
Native mode may use e* = argmax_e ExpectedDiscrimination_arm(current data,e). Common mode should use a preregistered shared utility such as expected reduction in Bayes classification error, expected proper-score improvement, or another common decision-theoretic objective.
This decomposition separates whole-system performance from the effect of the utility function itself. Success requires selecting or efficiently approaching a discriminating experiment without privileged knowledge of which experiment was constructed to discriminate.
11. Conditional Source-Dimension Decision
For a registered family π, experiment family π, equivalence rules π, and registered dimension type Ο, define the benchmark source-dimension result only conditionally:
d*_(π,π,π,Ο) = min{ d : a class H_(d,Ο) remains prospectively adequate and is not observationally equivalent to an adequate lower-dimensional class under the discriminating experiment set }.
This quantity is not reported for outcome E or N. It is not an intrinsic dimension of the generating universe. It is the minimum registered dimension of type Ο supported within the benchmark architecture after the strongest admissible lower-dimensional competitors in the preregistered class have been tested.
12. Primary Performance Metrics
The benchmark should report at least the following preregistered quantities:
G₁ false extra-dimension rate: P(H decision | G₁).
G₂ correct additional-structure rate: P(H decision | G₂), together with predictive calibration.
G₃ false-discrimination rate before a discriminating experiment and correct discrimination after one is obtained.
G₄ false forced-selection rate and correct N rate.
Held-out predictive log loss or another proper scoring rule.
Calibration of posterior/model confidence under noise and misspecification.
Number and cost of experiments required to reach a registered decision threshold.
Cross-probe and cross-intervention transport error.
Negative-signature calibration under known miss probabilities.
Computational cost and model complexity.
Before final evaluation, preregister one primary superiority endpoint. A recommended endpoint is the minimum expected experiment cost required to reach a calibrated correct L/H/E/N decision at fixed error tolerances across a balanced regime suite. All other metrics are secondary explanatory endpoints unless explicitly designated otherwise. Paired instances should be used where possible, with confidence intervals or other preregistered uncertainty estimates for differences in predictive loss, decision correctness, transport error, and experiment cost.
13. Candidate-Set and Information Metrics
When candidate contraction is measured geometrically, the measure must be named. For hypothesis H with measure ΞΌ_H, a normalized contraction statistic may be
Cβ = 1 − ΞΌ_H(C_H^(k)) / ΞΌ_H(C_H^(0)).
For probabilistic implementations, posterior entropy or information gain may be preferable. The benchmark must not infer stronger evidence merely from a larger raw percentage contraction in a differently parameterized model. Any TSTOEAO-specific relational-gain metric should be evaluated for invariance to permissible reparameterization or explicitly bounded to the representation in which it is defined.
14. Preregistered Success Criteria and Primary Endpoint
Before running the final blinded benchmark, numerical thresholds should be frozen. At minimum, the protocol should specify:
maximum acceptable G₁ false extra-dimension rate;
minimum G₂ detection rate at specified noise levels;
maximum G₃ false-discrimination rate before informative experiments;
minimum G₄ none-adequate detection rate;
decision thresholds for predictive scores or Bayes/model-comparison quantities;
maximum permitted probe-specific refitting in transport tests;
experiment-budget and computational-budget constraints; the primary superiority endpoint; multiplicity handling for secondary comparisons; paired-analysis and uncertainty-reporting rules.
The initial paper need not prescribe universal numerical values. The implementation study must preregister them before inspecting final benchmark labels.
15. Ablation Tests
To determine which parts of the integrated procedure actually matter, run ablations removing one component at a time: relational invariants, negative signatures, cross-probe constraints, interventions, active experiment selection, singular/global fiber analysis, and misspecification detection. If removing a TSTOEAO-specific component does not alter performance, that component has not earned empirical necessity in this benchmark.
The C⁺ arm promotes the translation ablation to a primary control. Additional ablations should remove one operation at a time from T and C⁺. If T>C but T=C⁺, the evidence supports integration/protocol architecture rather than uniquely TSTOEAO mathematics. If T>C⁺>C, ablation must identify which native representation or computation accounts for the residual difference.
16. Failure Conditions
The benchmark fails to support a distinctive TSTOEAO methodology if any of the following persists after reasonable implementation checks:
G₁ frequently produces false higher-dimensional discoveries;
G₂ is matched by a lower-dimensional observation-equivalent realization on unseen interventions;
G₃ is forced into a dimensional choice before a discriminating experiment exists;
G₄ is systematically absorbed by the most flexible/highest-dimensional candidate;
A_Ξ requires probe-specific refitting and collapses into channel modeling;
negative signatures are miscalibrated under realistic miss probabilities;
active TSTOEAO selection is no better than established model-discriminating design under native-utility and common-utility matched comparisons;
claimed relational invariants add no prospective discrimination;
the C⁺ operation-matched conventional translation reproduces all TSTOEAO decisions and efficiency, limiting the claim to synthesis/integration rather than a uniquely TSTOEAO inference method.
These outcomes do not invalidate the broader TSTOEAO architecture. They delimit the claim tested here.
17. What Would Constitute Added Methodological Value?
Added value must be measurable rather than rhetorical. The preregistered primary superiority endpoint controls the principal claim; secondary endpoints may include lower false dimensional-discovery at matched power, reliable recognition of cross-dimensional non-identifiability, better rejection of misspecified model families, fewer or cheaper experiments, improved cross-intervention transport, or a preregistered TSTOEAO relational invariant that supplies discriminating information not recovered by C⁺.
Results should be reported as paired performance profiles across generator subclasses, noise levels, experiment budgets, and adversary strengths, with confidence intervals or preregistered uncertainty summaries. Favorable secondary metrics cannot substitute for failure on the primary endpoint.
18. Interpretation of Possible Outcomes
18.1 TSTOEAO outperforms the integrated baseline
If the advantage is reproducible, survives ablation, and is traceable to a registered component of the relational protocol, there is a credible basis for investigating methodological originality and generalization beyond the synthetic benchmark.
18.2 TSTOEAO and the baseline are equivalent
The proper conclusion is that TSTOEAO provides a coherent synthesis and application of established inference machinery. This remains useful, particularly if the architecture improves conceptual discipline or transfer across domains, but it is not evidence of a new statistical method.
18.3 Conventional baseline outperforms TSTOEAO
The source-discrimination protocol should be revised or narrowed. TSTOEAO terminology should not be used to obscure a weaker inferential procedure.
18.4 Both fail
The benchmark may expose insufficient experiment families, inadequate model classes, non-identifiability, or a generator too difficult for either pipeline. The appropriate result is unresolved, followed by redesign—not a forced dimensional claim.
This creates the validation ladder: analytic truth → controlled computation → adversarial computation → real scientific inverse problem. Failure on the transparent tier blocks stronger methodological claims.
Before a large stochastic suite, the benchmark should contain a transparent four-generator tier whose correct outcomes can be established independently of either inference implementation. G₁→L must have an explicit admissible lower-dimensional realization. G₂→H must have an analytic or otherwise independent certificate that the preregistered lower-dimensional class cannot reproduce a specified intervention distribution while a registered higher-structure class can. G₃E→E must have provable observational equality under π₀ and a known withheld e* that breaks it. G₄→N must violate at least one declared held-out consequence of every registered candidate. Labels and certificates remain hidden from T, C, and C⁺ until decisions are frozen.
19. Minimal Analytic Certification Tier
20. Implementation Sequence
Construct the minimal analytic G₁–G₄ certification tier first and verify each correct outcome independently of T, C, and C⁺.
Verify exact G₃ equivalence under π₀ and verify that at least one withheld experiment can break it.
Implement strong lower-dimensional adversaries and confirm they defeat intentionally weak higher-dimensional tests.
Implement T, C, and C⁺ under matched information and budgets; run both native-utility and common-utility experiment-selection comparisons.
Freeze the primary superiority endpoint, secondary metrics, thresholds, invariant computations/tolerances, randomization ranges, analysis rules, and final benchmark seeds or hidden instances.
Run blinded evaluation and unlock generator labels only after decisions are committed.
Perform ablations, robustness analysis, and repeated trials.
Publish code, generator specifications, preregistration, failures, and null results alongside any positive claim where feasible.
21. Relationship to the Dimensional Paper Sequence
The sequence now has a clear division of labor. Substrate Relaxation and Dimensional Expression introduced the physical-accessibility hypothesis and separated observational accessibility from stronger physical interpretation. Dimensional Expression, Relational Geometry, and Inverse Reconstruction established the forward/inverse geometry and candidate-source framing. Successive Dimensional Expression and Relational Gain clarified the dimensional ladder and developed observational deficit. Relational Source Discrimination and Prospective Dimensional Inference converted those elements into a registered discrimination protocol. The present paper supplies the computational test that the protocol itself requires.
Accordingly, this benchmark is not another conceptual argument that higher dimensions exist. It is an attempt to make the preceding methodology vulnerable to controlled failure.
22. Conclusion
The next obligation of the TSTOEAO dimensional program is computational. A source-discrimination methodology earns scientific value only if it can refuse unnecessary complexity, admit independently certified necessary registered structure, preserve exact and practical non-identifiability when experiments cannot distinguish rivals, reject gross, near-family, and adversarial misspecification, and select useful new experiments without receiving the answer through benchmark design.
The revised adversarial benchmark is constructed to test those obligations through T, C, and C⁺. Its strongest lower-dimensional competitors are limited only by preregistered admissibility rather than by a convenient named model. Its decisive evaluation occurs on unseen probes and interventions. Active selection is separated into native-utility and common-utility comparisons. Its output space includes non-identifiability and model inadequacy. Its first tier requires independently transparent ground truth before scaling to stochastic and neural adversaries. Its failure conditions explicitly permit the conclusion that TSTOEAO adds no distinct methodological capability.
That possibility is essential. If TSTOEAO performs no better than C⁺, the result should be recorded as disciplined integration or synthesis even if T outperforms the less operation-matched C baseline. If it performs worse, the protocol should be revised. If T demonstrates reproducible, ablation-supported advantages over C⁺ on the preregistered primary endpoint under fair adversarial conditions, then—and only then—will there be a credible empirical basis for investigating whether a native TSTOEAO representation or relational construction contributes methodological capability in its own right.
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