ANALYTIC GROUND TRUTH FOR PROSPECTIVE DIMENSIONAL INFERENCE: A Minimal Four-Regime Calibration of the TSTOEAO Relational Source-Discrimination Protocol

ANALYTIC GROUND TRUTH FOR
PROSPECTIVE DIMENSIONAL INFERENCE

A Minimal Four-Regime Calibration of the TSTOEAO Relational Source-Discrimination Protocol

John Swygert

Ivory Tower Publishing

October 2, 2026

Research status: computational benchmark specification and analytic calibration paper. No claim of new physical dimensionality is made.

Abstract

The preceding TSTOEAO papers developed a prospective source-discrimination protocol in which relational gain is treated as contraction of registered observational equivalence and then specified an adversarial benchmark with four legitimate inference outcomes: lower-dimensional structure preferred, higher registered source structure preferred, observationally indistinguishable, and none of the registered models adequate. The next obligation is not another conceptual extension. It is to establish transparent benchmark cases whose correct outcomes can be determined independently of the inference machinery being tested. This paper therefore defines a minimal four-regime analytic calibration suite. Each regime is constructed so that the correct decision follows from an explicit property of the registered model and experiment classes rather than from post hoc failure of benchmark contestants. The suite separates state, manifold, geometric, embedding, and reconstruction dimensions; distinguishes exact from practical observational equivalence; introduces an operation-matched conventional control C+ alongside native TSTOEAO and conventional pipelines; separates native-utility from common-utility experiment selection; and preregisters a primary endpoint. The purpose is to create a falsifiable bridge from the conceptual methodology to executable computation.

1. Research Question

Can a source-discrimination protocol correctly recover four analytically certified inference states without being allowed to define those states by its own success or failure? The benchmark must demonstrate that it can refuse unnecessary structure, identify an additional registered source degree when a bounded lower-dimensional class is provably insufficient, preserve non-identifiability when registered experiments cannot distinguish competing classes, and reject the entire registered family under model misspecification.

Decision space:  L  /  H  /  E  /  N

Here L denotes that a lower registered source class is adequate and preferred under the registered decision rule; H denotes that an additional registered source degree is required relative to the bounded admissible class; E denotes observational equivalence under the available experiment family; and N denotes that none of the registered source classes is adequate.

2. Registered Objects and the Meaning of Dimension

Every benchmark instance is defined by a tuple B = (H, M, E, I, T, Q, tau), where H is the registered hypothesis family, M the admissible measurement/experiment family, E the declared observational equivalence relation, I the intervention family, T the transformation family under which relational invariants are asserted, Q the noise/detection model, and tau the registered dimension type.

tau ∈ {state, manifold, geometric-spatial, embedding, reconstruction}

A hypothesis written H_(d,tau) therefore asserts d dimensions of a declared type. The benchmark principally tests minimum required source-state structure. A physical or geometric spatial interpretation is a special registered family and cannot be inferred merely because a reconstruction requires a higher-dimensional latent state.

3. Analytic Certification Before Blinded Evaluation

Generator construction and ground-truth certification are separated. A generator is created first. Its correct benchmark outcome is then certified from an explicit mathematical property of the registered class before any blinded inference run. Certification may use an analytic proof, an exact finite enumeration, or a rigorously bounded numerical argument whose tolerance is preregistered. The evaluator never receives the regime label or the certification argument until its decision is frozen.

The benchmark never claims failure of every imaginable lower-dimensional model. Statements of necessity are always relative to a bounded preregistered adversarial class L_d^reg with declared smoothness, causal, complexity, memory, computational, and intervention-response restrictions.

4. Regime G1: Explicit Lower-Dimensional Realization

G1 calibrates refusal of unnecessary source structure. Construct observations from an explicit lower-dimensional source whose complete registered probe and intervention behavior is generated by a known admissible realization L* ∈ L_d^reg. A higher-dimensional representation may also fit, but it is not required to reproduce the registered consequences.

∃ L* ∈ L_d^reg such that  P_L*(O | M,I) = P_G1(O | M,I)  for all registered (M,I).

Ground truth is therefore L. A method fails G1 if it systematically interprets flexibility, nuisance structure, or reconstruction dimension as evidence that an additional registered source degree is required.

5. Regime G2: Certified Additional Registered Source Degree

G2 removes the principal circularity in the earlier benchmark specification. The regime is not defined as a case in which the contestants happen to fail. Instead, choose a precisely bounded lower-dimensional admissible class L_d^reg and construct a generator with at least one registered intervention consequence that no member of that class can reproduce.

∀ L ∈ L_d^reg,  P_L(O | I*) ≠ P_G2(O | I*)

∃ H ∈ H_(d+1,tau)^reg such that  P_H(O | M,I) = P_G2(O | M,I)  on the registered suite.

The first calibration family should be chosen so that the incompatibility is analytically transparent—for example, by a rank, conservation, conditional-independence, symmetry, or intervention-response constraint that every L ∈ L_d^reg must satisfy but G2 violates. The higher-dimensional class must satisfy the same declared observational and interventional consequences with fewer structural freedoms than the unrestricted adversarial alternatives. The correct outcome is H only relative to the registered class and dimension type tau.

6. Regime G3E: Exact Cross-Dimensional Observational Equivalence

G3E tests whether the method can preserve ignorance. Construct two registered source classes of different dimension that induce identical observable distributions for every experiment in the initial family M0.

P(O | H_a,e) = P(O | H_b,e)   for every e ∈ M0

The correct outcome under M0 is E. Complexity penalties may make one representation pragmatically preferable, but they do not establish that the other source dimension is absent. A withheld admissible experiment e* is then supplied for which the predicted distributions differ.

P(O | H_a,e*) ≠ P(O | H_b,e*)

The active-design stage should identify or favor e* when it is available. This single regime tests non-identifiability recognition, refusal of false dimensional discovery, and the ability of experiment selection to expose a relational distinction when one is experimentally accessible.

7. Regime G3P: Practical Equivalence

Exact equality is not the only scientifically relevant form of non-identifiability. G3P introduces practical equivalence under registered noise, resolution, and budget. Let D be a preregistered divergence or decision distance.

D(P_a(O|e), P_b(O|e)) ≤ epsilon   for every affordable e ∈ M_budget

The correct result is practical non-discrimination at the registered tolerance. The method must not convert tiny, experimentally inaccessible distinctions into confident source claims. If a later experiment exceeds the registered separation threshold, the equivalence state may be updated prospectively.

8. Regime G4: Registered-Family Misspecification

G4 tests whether unmodeled structure is mistaken for extra dimension. The generator is outside every registered candidate class and violates at least one declared held-out consequence of each candidate. Three subclasses are used.

G4a: gross misspecification that should be easy to reject.

G4b: near-family misspecification whose training behavior closely resembles a registered model.

G4c: adversarial misspecification that makes a higher-dimensional registered model fit better than lower-dimensional candidates while retaining systematic held-out residual structure.

G4a, G4b, G4c  →  N

G4c is the critical control: the procedure must distinguish 'missing source degree' from 'wrong model family.'

9. Three Inference Arms: T, C, and C+

The benchmark compares three primary arms rather than two. T is the native TSTOEAO protocol, using registered observational equivalence, relational invariants, negative signatures, cross-probe constraints, and its native representation. C is a strong integrated conventional pipeline using standard system identification, latent-state estimation, model discrimination, calibrated negative evidence, and optimal experimental design. C+ is an operation-matched conventional translation of every TSTOEAO operation for which a conventional mathematical translation exists.

T = C = C+   → no operational advantage

T > C and T = C+   → integration/protocol advantage reproducible conventionally

T > C+ > C   → investigate residual representation-specific advantage

If the third pattern occurs, ablation must identify the responsible component rather than attributing the difference to TSTOEAO as a whole.

10. Native-Utility and Common-Utility Experiment Selection

Active measurement selection is evaluated in two modes. In native-utility mode, each arm uses its preferred experiment-selection criterion. This measures whole-system performance. In common-utility mode, all arms optimize the same externally specified utility, such as expected reduction in calibrated classification error or expected proper-score improvement. This separates gains due to inferential representation from gains due merely to different objective functions.

e_next = argmax_e U(e | current evidence)

The experiment pool, cost model, stopping rule, and access to interventions are identical across arms.

11. Relational Invariants Must Be Operationally Preregistered

A claimed relational invariant I_R is not registered merely by naming a concept. Before blinded evaluation the protocol must specify I_R, the transformation family T under which it is expected to survive, the estimator or computation, tolerance, uncertainty treatment, and failure criterion.

I_R(Tx) ≈ I_R(x)   within preregistered tolerance for T ∈ T

This prevents an invariant from being retrofitted to generator structure after the benchmark is observed. If no useful invariant can be specified prospectively, the benchmark must record that fact rather than invent a bespoke relational score.

12. Calibrated Negative Evidence

A required-but-absent signature is informative only when there was a calibrated opportunity to detect it. With nuisance parameters theta, the no-detection probability is integrated rather than treated as a fixed detector constant.

P(no detection | H) = ∫ P(no detection | H,theta) p(theta | H) dtheta

Negative evidence may contract an admissible candidate class, but it cannot be counted as exclusion merely because an expected feature was not seen.

13. Structural Transport Test

Training uses a subset of probes and interventions, for example M1,M2,M3 and I1,I2. Evaluation then includes unseen combinations such as (M4,I1), (M2,I3), and (M5,I4). Same-channel interpolation is not decisive. A proposed source representation must transport consequences across changed observation and intervention contexts without probe-specific refitting.

If a registered lower-dimensional realization predicts all held-out probe-intervention consequences as well as the explicit higher-dimensional source model, the benchmark has not established that the higher source dimension is required.

14. Primary Endpoint and Secondary Metrics

To prevent a garden of favorable comparisons, one primary superiority endpoint is preregistered:

Primary endpoint = expected experiment cost to reach a calibrated correct L/H/E/N decision

The error tolerances, regime weighting, maximum budget, and treatment of practical equivalence are fixed before evaluation. Secondary endpoints include false extra-dimension rate, correct non-identifiability recognition, misspecification rejection, held-out proper predictive loss, cross-intervention transport error, calibration, and the incremental value of I_R.

Because the same hidden instances are evaluated by all arms, comparisons are paired. Confidence intervals or corresponding uncertainty intervals are reported for paired differences in cost, loss, correctness, and transport performance.

15. Minimal Analytic Four-Generator Trial

Before any large neural or stochastic benchmark, the program begins with four transparent generators whose correct outcomes can be independently established.

Regime	Certified property	Correct result	Primary failure
G1	Explicit admissible lower realization	L	Invents extra source structure
G2	Registered lower class analytically cannot reproduce I*	H	Fails to detect required registered degree
G3E	Exact equality under M0; known e* breaks it	E	Claims distinction without exposing experiment
G4	Every registered class violates held-out consequence	N	Absorbs misspecification as dimension

The labels and certification proofs are withheld from the inference implementations. Decisions and selected experiments are frozen before unblinding. Only after this calibration succeeds should the benchmark advance to noisy stochastic systems, G3P, G4b/G4c, increasingly expressive adversaries, neural state-space realizations, and real inverse problems.

16. Benchmark-Designer Overfitting

The initial suite is necessarily designed within the same research program that developed the TSTOEAO protocol. This creates a designer-overfitting risk even under label blinding. Later validation should therefore include independently written generators, independently implemented conventional baselines, hidden challenge instances, or third-party benchmark contributions. A native TSTOEAO advantage that disappears on independent challenge instances is not evidence of a general methodological advantage.

17. Failure Conditions

The analytic calibration or the broader methodology is weakened if any of the following occur: G1 is repeatedly classified H; certified G2 is reproducible by an admissible registered lower realization; G3E is forced into L or H without a discriminating experiment; G3P produces unjustified certainty; G4 is absorbed by flexible higher-dimensional candidates; C+ reproduces every TSTOEAO decision and experiment while T is described as a distinct statistical method; native-utility gains disappear under common utility and are nevertheless attributed to representation; I_R requires post hoc tuning; negative signatures are not detector-calibrated; or the primary endpoint fails to show the claimed advantage.

A null result is scientifically admissible. If T = C+ after translation, the proper conclusion is that TSTOEAO supplies a disciplined synthesis and representation of established operations rather than a distinct inference capability.

18. Relation to the Preceding TSTOEAO Papers

The substrate and dimensional-expression papers proposed the possibility that accessibility, expression, and observation should be distinguished rather than conflated. The dimensional-geometry papers then treated upward source hypotheses and downward observations as inverse problems. Relational Source Discrimination and Prospective Dimensional Inference made observational-equivalence contraction the primary object and required prospective falsification. Adversarial Benchmark for Prospective Dimensional Inference specified how that protocol should be challenged. The present paper supplies the missing first computational rung: analytically certified cases whose answers do not depend on trusting the inference system.

The progression is therefore: conceptual distinction → formal source discrimination → adversarial benchmark → analytic ground truth → controlled computation → adversarial computation → real scientific inverse problem.

19. What This Paper Does Not Establish

This paper does not establish new physics, physical dimensional compression, a physical Lambda_D, or methodological superiority of TSTOEAO. It does not prove that finite data can exclude every conceivable lower-dimensional realization. It does not identify latent-state dimension with physical spatial dimension. It does not claim originality for system identification, minimal realization, observability, Bayesian model discrimination, multi-view latent inference, optimal experimental design, negative evidence, or delay reconstruction.

Its narrower contribution is to make the next empirical obligation explicit and falsifiable: before claims of distinctive dimensional inference are entertained, the protocol must solve transparent cases whose correct decisions are independently certified and must be compared against both a strong conventional pipeline and an operation-matched conventional translation.

20. Conclusion

A dimensional-inference methodology should not be allowed to manufacture the answer it was designed to discover. The appropriate next step is therefore an analytic calibration in which necessity, equivalence, adequacy, and misspecification are established independently of the contestants. The four-regime suite developed here makes L, H, E, and N genuine scientific outcomes rather than forced labels. The T/C/C+ comparison separates representational novelty from useful integration; native/common utility comparisons separate architecture from objective choice; and the primary endpoint constrains later claims of superiority.

If the protocol cannot reliably solve these transparent cases, scaling to more elaborate dimensional hypotheses is premature. If it can, the research program earns the right to proceed to noisy stochastic systems, stronger adversaries, independent challenge instances, and eventually real scientific inverse problems. The governing principle remains simple: no relational distinction should be claimed without an experiment capable of exposing it, and no additional source degree should be claimed when an admissible lower representation transports the registered consequences just as well.

References and Literature Boundary

This calibration paper relies primarily on the formal architecture established in the immediately preceding TSTOEAO papers. Its conventional comparison class includes established work in nonlinear realization and system identification, local and global observability/identifiability, delay-coordinate reconstruction, latent-variable and multi-view inference, Bayesian model discrimination, optimal experimental design, proper scoring, and model-misspecification testing. Those fields supply established mathematical tools; the empirical question is whether the registered TSTOEAO organization contributes anything beyond an operation-matched conventional implementation.

ADVERSARIAL BENCHMARK FORPROSPECTIVE DIMENSIONAL INFERENCE: A Blinded G₁–G₄ Computational Test of TSTOEAO Relational Source Discrimination Against Strong Conventional Baselines

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 𝓗 = {H_(d,τ)} denote registered source-model classes, where d is dimension and τ registers the dimension type, 𝔐 the admissible measurement/probe family, 𝓘 the admissible intervention family, and 𝓔 the declared equivalence and transformation rules. Register τ ∈ {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 𝓛_d and establish independently—preferably analytically—that there exists a registered intervention I* for which no L ∈ 𝓛_d can reproduce a declared interventional consequence, while at least one constrained H ∈ 𝓗_(d+1,τ) can. Symbolically, for the certified consequence, ∀L∈𝓛_d, P_L(O|I*) ≠ P_G(O|I*), while ∃H∈𝓗_(d+1,τ) 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 𝔐₀. 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 𝔐₀.   G₃P:  D(p_a(O|e),p_b(O|e)) ≤ ε for every affordable e∈𝔐₀, for a preregistered divergence D and tolerance ε.

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:

1. Declare source classes, measurement operators, interventions, noise models, detection limits, symmetries, admissible transformations, and candidate relational invariants before evaluation.

2. Compute local observational deficit only as a regular local diagnostic, while treating observational fibers, posterior uncertainty, and candidate equivalence classes as the global objects.

3. Update candidate classes using positive observations, preserved invariants, calibrated negative signatures, and prospective constraints.

4. Require a shared latent accessibility state to constrain heterogeneous probes without probe-specific refitting when A_Λ is used.

5. Permit the four outcomes L, H, E, and N.

6. 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 𝒯 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_Λ, 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,θ) p(θ | H) dθ, 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 𝔐₀, 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 𝓗, experiment family 𝔐, equivalence rules 𝓔, and registered dimension type τ, define the benchmark source-dimension result only conditionally:

d*_(𝓗,𝔐,𝓔,τ) = min{ d : a class H_(d,τ) 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 τ 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 μ_H, a normalized contraction statistic may be

Cₖ = 1 − μ_H(C_H^(k)) / μ_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_Λ 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 𝔐₀ 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

7. Construct the minimal analytic G₁–G₄ certification tier first and verify each correct outcome independently of T, C, and C⁺.

8. Verify exact G₃ equivalence under 𝔐₀ and verify that at least one withheld experiment can break it.

9. Implement strong lower-dimensional adversaries and confirm they defeat intentionally weak higher-dimensional tests.

10. Implement T, C, and C⁺ under matched information and budgets; run both native-utility and common-utility experiment-selection comparisons.

11. Freeze the primary superiority endpoint, secondary metrics, thresholds, invariant computations/tolerances, randomization ranges, analysis rules, and final benchmark seeds or hidden instances.

12. Run blinded evaluation and unlock generator labels only after decisions are committed.

13. Perform ablations, robustness analysis, and repeated trials.

14. 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.

23. References

1. Hermann, R., & Krener, A. J. (1977). Nonlinear controllability and observability. IEEE Transactions on Automatic Control, 22(5), 728–740.

2. Takens, F. (1981). Detecting strange attractors in turbulence. In D. A. Rand & L.-S. Young (Eds.), Dynamical Systems and Turbulence, Warwick 1980 (Lecture Notes in Mathematics, Vol. 898, pp. 366–381). Springer.

3. Sauer, T., Yorke, J. A., & Casdagli, M. (1991). Embedology. Journal of Statistical Physics, 65, 579–616.

4. Villaverde, A. F., Barreiro, A., & Papachristodoulou, A. (2016). Structural identifiability of dynamic systems biology models. PLoS Computational Biology, 12(10), e1005153.

5. Drovandi, C. C., et al. (2022). Optimal Bayesian design for model discrimination via classification. Statistics and Computing, 32.

6. Hyvärinen, A., Sasaki, H., & Turner, R. E. (2019). Nonlinear ICA using auxiliary variables and generalized contrastive learning. Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, 859–868.

7. Schön, T. B., Lindsten, F., Dahlin, J., Wågberg, J., Naesseth, C. A., Svensson, A., & Dai, L. (2015). Sequential Monte Carlo methods for system identification. IFAC-PapersOnLine, 48(28), 775–786.

8. Thompson, W. C., & Scurich, N. (2018). When does absence of evidence constitute evidence of absence? Forensic Science International, 291, e18–e19.

9. Swygert, J. (2026, October 1). Substrate Relaxation and Dimensional Expression: A TSTOEAO Hypothesis of Compression, Expression, Relational Coordinates, and Electromagnetic Observability. Ivory Tower Publishing.

10. Swygert, J. (2026, October 2). Dimensional Expression, Relational Geometry, and Inverse Reconstruction. Ivory Tower Publishing.

11. Swygert, J. (2026, October 2). Successive Dimensional Expression and Relational Gain: A Dimension-by-Dimension TSTOEAO Construction from Point States to Higher-Dimensional Source Geometry. Ivory Tower Publishing.

12. Swygert, J. (2026, October 2). Relational Source Discrimination and Prospective Dimensional Inference: A TSTOEAO Protocol for Observational-Equivalence Reduction, Competing Source Geometries, Negative Signatures, and Active Measurement. Ivory Tower Publishing.

RELATIONAL SOURCE DISCRIMINATION ANDPROSPECTIVE DIMENSIONAL INFERENCE: A TSTOEAO Protocol for Observational-Equivalence Reduction, Competing Source Geometries, Negative Signatures, and Active Measurement

RELATIONAL SOURCE DISCRIMINATION AND
PROSPECTIVE DIMENSIONAL INFERENCE

A TSTOEAO Protocol for Observational-Equivalence Reduction, Competing Source Geometries, Negative Signatures, and Active Measurement

John Swygert

Ivory Tower Publishing

October 2, 2026

Abstract

This paper develops a prospective source-discrimination protocol from the dimensional and inverse-reconstruction framework established in the preceding TSTOEAO papers. The central problem is not whether a higher-dimensional model can be made to fit observations, but whether an additional latent source degree of freedom is required after strong lower-dimensional alternatives, observation operators, detection limits, model complexity, and model misspecification are explicitly registered. Observational equivalence is treated as the primary object: registered measurements partition admissible source states and source classes into equivalence sets, and relational gain is defined as contraction of those registered equivalence sets. Differential-kernel contraction is retained as a local regular diagnostic rather than a universal global measure. The paper distinguishes local identifiability, global ambiguity, singular fibers, and exact cross-dimensional observational equivalence; introduces an explicit non-identifiability decision state; and conditions any minimum inferred source dimension on the registered hypothesis family, experiment family, and equivalence rules. It strengthens the dimension-blind benchmark by admitting minimal nonlinear realizations, delay embeddings, flexible latent dynamics, cross-probe and interventional tests, generators outside all registered model classes, and comparison against conventional Bayesian model-discrimination design. The component mathematics remain established. The TSTOEAO contribution proposed here is a disciplined synthesis and test protocol whose methodological originality must be earned by adversarial computational comparison rather than asserted in advance.

1. Problem Statement

The previous dimensional papers established two complementary directions. The first treated lower-dimensional observations as constrained images of candidate higher-dimensional source geometries. The second developed a dimension-by-dimension construction and identified a stronger invariant beneath the dimensional ladder: a declared observation map partitions admissible source states into observational-equivalence classes. The present paper turns those results into a prospective discrimination procedure.

The target question is deliberately narrower than “What is the true dimension of reality?” For a registered family of candidate source models and registered observation operators, the question is: what is the minimum latent source structure required to reproduce the existing observations and correctly predict observations that were not used to fit the model?

Registered source question:  H₂, H₃, …, Hₙ  →  which classes remain admissible after prospective tests?

The word dimension is therefore conditional. It refers to degrees of freedom in an admissible source model, not automatically to an ontological claim about physical spacetime. A successful D₄ model does not by itself establish a fourth physical spatial dimension. It establishes only that, within the declared model family and measurement protocol, the tested lower-dimensional competitors were insufficient and the additional source degree produced predictive gain.

2. Established Foundations and the Proposed Synthesis

The mathematical components used here have substantial precedent. Nonlinear observability and structural-identifiability analysis use differential rank conditions to determine whether nearby internal states can be distinguished from outputs. Inverse-problem theory studies recovery of source states from incomplete or transformed observations. Nonlinear realization theory asks whether an input-output behavior admits a lower-dimensional state realization and what minimal state dimension is required. Multi-view latent-variable and nonlinear source-separation methods study recovery of shared latent structure from multiple sufficiently informative views. Bayesian and sequential experimental-design methods choose measurements that discriminate among rival models or maximize expected information gain. Negative evidence becomes informative when a predicted observation should have been detectable but is not observed. Delay-coordinate methods demonstrate that a low-dimensional measured time series can reconstruct higher-dimensional dynamical state under appropriate conditions.

Accordingly, this paper does not claim originality for kernels, ranks, quotient spaces, inverse reconstruction, minimal realization, shared-latent multi-view inference, model-selection penalties, Bayesian information gain, negative evidence, or delay embedding. The proposed TSTOEAO contribution is a registered synthesis in which dimensional source classes, transformation-specific invariants, positive and negative signatures, cross-probe latent accessibility, complexity control, held-out prediction, interventions, and active next-measurement selection are evaluated together. Whether that synthesis constitutes a distinct methodology is an empirical question to be answered against strong conventional integrated baselines.

This distinction matters. The framework should stand or fail on whether the integrated procedure produces disciplined discrimination that the individual components, used casually or independently, do not guarantee.

3. Registered Source Classes and Observation Operators

Let X_H denote the admissible state space under source hypothesis H. A hypothesis may differ from another by intrinsic dimension, topology, metric structure, dynamics, symmetry, coupling, or another registered structural property. For a measurement channel i, let

Mᵢ : X_H → Oᵢ

map source states into an observation space Oᵢ. The observed datum is Oᵢ = Mᵢ(X), possibly with noise and detection limits. The measurement operator is part of the evidentiary model; it cannot be omitted and then silently absorbed into the interpretation of the source.

For k registered measurements, define the joint map

M₁:ₖ = (M₁, M₂, …, Mₖ) : X_H → O₁ × O₂ × ··· × Oₖ.

The admissible candidate set after k observations is

C_H^(k) = {X ∈ X_H : Mᵢ(X) ≈ Oᵢ for i = 1,…,k}.

The approximation relation must be specified by the noise model, uncertainty bounds, and detection protocol. Candidate contraction, rather than visual resemblance alone, is the basic inverse operation.

4. Observational Equivalence and Two Levels of Relational Gain

Two admissible source states are observationally equivalent under a registered measurement map when they produce the same registered observation:

Xₐ ~_M X_b  ⇔  M(Xₐ) = M(X_b).

For k measurements, let the observational fiber through x be

Fₖ(x) = M₁:ₖ⁻¹(M₁:ₖ(x)).

Relational gain is defined primarily as contraction of registered observational equivalence. The fiber, posterior, or admissible candidate class is therefore the global object. Differential rank is one local diagnostic of that contraction, not its universal definition.

For a smooth d-dimensional admissible source manifold X and a regular point x, define the local observational deficit

δₖ(x) = dim ker(DM₁:ₖ|ₓ) = d − rank(DM₁:ₖ|ₓ).

Adding another measurement cannot increase the kernel of the concatenated differential:

δₖ₊₁(x) ≤ δₖ(x).

The local relational gain may therefore be written

g_loc,k+1(x) = rank(DM₁:ₖ₊₁|ₓ) − rank(DM₁:ₖ|ₓ),

or equivalently as the dimension of the quotient of locally invisible directions removed by the new measurement. This quantity describes regular local identifiability only.

Globally, let Φ be a preregistered fiber-complexity or uncertainty functional appropriate to the problem. Then

g_glob,k+1(x) = Φ(Fₖ(x)) − Φ(Fₖ₊₁(x)).

No single Φ is universal. Depending on the registered source class, Φ may be finite-fiber cardinality, dimension, measure under a declared μ_H, posterior entropy, number of connected components, or the number of symmetry-equivalence classes. Candidate-set volume must therefore name its measure and should normally be normalized, for example Cₖ = 1 − μ_H(C_H^(k))/μ_H(C_H^(0)). Bayesian implementations may instead report posterior entropy or information gain.

5. Local, Global, and Singular Identifiability

Local differential identifiability and global source identifiability are distinct. The map M(x)=x² has nonzero derivative away from x=0, yet M(x)=M(−x); local rank does not remove the global twofold ambiguity. Likewise M(x,y)=(x,y²) exhibits a fold: regular points can be locally distinguished while the global fiber contains reflected states, and the critical set y=0 requires separate treatment.

The protocol therefore separates three questions. Local identifiability asks whether neighboring admissible states can be distinguished. Global identifiability asks whether the joint map is injective on the admissible class, or injective modulo a declared symmetry group. Singular-fiber analysis asks whether critical subsets require stratification rather than a single smooth-manifold rank formula. Where needed, write X = ⋃_α X_α and perform rank and fiber analysis on the relevant strata.

The governing formulation is therefore: relational gain is contraction of registered observational equivalence; differential-kernel contraction measures its local regular component.

6. Conditional Source Dimension and Non-Identifiability

Suppose the registered hypothesis family 𝓗 contains source classes H₂ through H_N, the admissible experiment family is 𝔐, and the declared equivalence/admissible-transformation rules are 𝓔. The protocol does not reward a higher-dimensional class merely because it can fit more observations. Additional degrees of freedom must earn their complexity through prospective consequences.

When a minimum surviving source dimension is reportable, write it explicitly as

d*_(𝓗,𝔐,𝓔) = min { d : H_d remains non-rejected under the registered prospective criteria }.

This is not an intrinsic metaphysical dimension. It is conditional on the registered hypothesis family, experiments, equivalence rules, noise model, and admissible transformations. A stronger lower-dimensional realization introduced later may change the result.

Different-dimensional source classes can also be exactly observationally equivalent under the registered experiment family. Define

H_a ≡_𝔐 H_b

when their permitted or probabilistic observable outcomes coincide for every admissible experiment in 𝔐. In that case the correct scientific output is not a forced dimensional selection. It is: indistinguishable under the registered experiment class. A complexity penalty may pragmatically prefer a smaller representation, but it has not demonstrated that the source lacks an additional degree of freedom.

The decision space must also permit a fourth result: none of the registered models is adequate. This prevents an unmodeled process from being mislabeled as a higher-dimensional source merely because the higher-dimensional class is the most flexible candidate available.

7. Adversarial Lower-Dimensional Baselines

A weak lower-dimensional baseline would make a higher-dimensional model appear successful too easily. The adversary should therefore approximate the strongest observation-equivalent nonlinear realization permitted by the registered data, causal assumptions, and measurement restrictions. Candidate baselines may include nonlinear state-space models, delay embeddings, latent-variable models, kernels or splines, neural state-space models, explicit nuisance/channel models, and minimal-realization constructions where available.

Takens-style delay reconstruction is especially important. A scalar or low-dimensional time series can, under appropriate smoothness and genericity conditions, reconstruct the geometry of an underlying dynamical attractor in a delay-coordinate space. Consequently, low measurement dimension is not evidence that the source itself has equally low state dimension, and successful reconstruction in a higher-dimensional delay space is not by itself proof of an additional physical spatial dimension.

For this protocol, delay-coordinate reconstruction and other nonlinear realizations count as admissible competitors whenever they reproduce the registered observations and prospective consequences without requiring the proposed source geometry. If a reconstructed state predicts all held-out observations and interventions as well as an explicit higher-dimensional geometry, the experiment has not established that the geometric source dimension is required; it has established only that sufficient latent state structure is useful.

This is a severe test, and it is intentionally so. The framework should identify additional source structure only when lower-dimensional nonlinear representations have been given a serious opportunity to succeed.

8. Complexity Control and Prospective Prediction

Training fit is insufficient. Candidate models should be compared using held-out prediction and, where appropriate, information criteria or Bayesian evidence. AIC, BIC, minimum-description-length ideas, regularized likelihood, cross-validation, and out-of-sample log likelihood are possible tools depending on the model class. No single penalty is universal; the criterion must be registered before the decisive comparison.

The decisive pattern is not “higher dimension fits better.” It is: the lower-dimensional alternatives require increasing ad hoc flexibility, while a more constrained higher-dimensional source model predicts withheld structure with fewer effective adjustments. Conversely, if a flexible lower-dimensional model predicts equally well after complexity control, the dimensional extension is not required.

Nested-model simulation should be used to estimate the false higher-dimension selection rate. Synthetic data generated from known lower-dimensional models should be passed through the same pipeline. If the procedure routinely invents unnecessary dimensions, the inference protocol fails before it is applied to unknown systems.

9. Reliable Absence as an Active Constraint

An absent feature is informative only when the feature was sufficiently expected and the measurement had sufficient sensitivity to detect it. Let Q be a signature required or strongly predicted by H and M the registered detection process. A useful negative result requires a bounded probability of non-detection under the hypothesis:

P(¬Q observed | H, M) ≤ α,

for a preregistered tolerance α, or an equivalent likelihood/Bayesian criterion. More explicitly, failure to detect Q can arise because Q is genuinely absent or because Q occurred but the measurement missed it:

P(no detection | H) = P(¬Q | H) + P(Q | H) P(miss | Q, M).

The detection model therefore belongs inside the inference. Once calibrated, non-detection can contract the candidate family rather than remaining an informal gap. A higher-dimensional model that requires a detectable artifact and repeatedly fails to produce it should lose admissibility even if it fits the positive observations.

10. Probe-Independent Accessibility and the AΛ Test

The preceding TSTOEAO papers introduced an accessibility operator A_Λ as a candidate representation of altered relational accessibility. The present protocol imposes a strong identifiability condition on that proposal. For probe i,

Oᵢ = Mᵢ(A_Λ X).

If every probe is permitted its own arbitrary accessibility operator A_(Λ,i), then the composition can be rewritten as an effective channel Mᵢ′ = Mᵢ ∘ A_(Λ,i). In that case the accessibility term has no demonstrated independent physical content; it has been absorbed into instrument or channel modeling.

A stronger TSTOEAO claim therefore requires a shared latent Λ that constrains multiple independent probes. One probe or subset may estimate Λ, but the estimated state must predict consequences in other channels without probe-specific refitting. This establishes evidence for a shared latent explanatory structure, not automatically a physical common cause. The evidentiary ordering is: shared latent fit < cross-probe prediction < cross-probe intervention prediction. A stronger Λ_D interpretation ultimately requires prospective heterogeneous consequences under registered interventions, not merely successful joint fitting.

Estimate Λ from {O₁,…,O_r}; predict {O_(r+1),…,O_k} without refitting Λ.

This criterion separates observational accessibility, Λ_A, from the stronger hypothesized physical dimensional-expression state, Λ_D. Evidence for Λ_A does not establish Λ_D. A candidate Λ_D becomes scientifically interesting only when it produces localized, probe-independent, reproducible consequences that ordinary receiver, instrument, material, thermal, mechanical, and electromagnetic models do not already explain.

11. Transformation-Specific Relational Invariants

An invariant is meaningful only relative to a declared transformation class. Let T belong to an admissible transformation family 𝒯 and let I_R be a candidate relational invariant. Then the claim is

I_R(TX) = I_R(X),   T ∈ 𝒯,

within stated tolerances. The protocol does not assume that one invariant survives every projection, embedding, compression, or measurement. Instead, each source hypothesis registers which relational structures should survive which transformations and which should not.

This turns invariance into a discriminating tool. Two source classes may reproduce the same visible shape but imply different preserved relations under a second measurement or transformation. Those differences become prospective tests rather than retrospective interpretations.

12. Active Selection of the Next Measurement

Once several source classes remain admissible, the next observation should be chosen to separate them as efficiently as possible. This is an established problem in optimal experimental design and Bayesian model discrimination. The present protocol adopts that logic explicitly.

Let H denote the registered source-class variable and O_M the possible outcome under candidate measurement M. A natural criterion is

M* = argmax_M I(H ; O_M | O₁:ₖ),

where I denotes conditional mutual information. Alternative utilities may target expected candidate elimination, reduction of candidate-set volume, expected reduction of observational deficit, or a cost-adjusted combination.

The conceptual change is important: the framework does not merely analyze whatever data happen to arrive. It asks which measurement would most sharply distinguish the remaining source explanations. The inverse problem therefore becomes prospective and experimental rather than merely reconstructive.

13. Adversarial Dimension-Blind Computational Benchmark

The decisive test should be synthetic and blinded so that the true generating regime is known to the experimenter but hidden from the evaluator. Four regimes are required:

G₁: a genuinely lower-dimensional nonlinear dynamical generator.

G₂: a generator for which an additional registered source degree of freedom is genuinely required for prospective performance.

G₃: a cross-dimensional pair that is observationally equivalent under the initial experiment family.

G₄: a generator outside every registered candidate model class.

The lower-dimensional adversary should receive every legitimate advantage except access to held-out probes and interventions. It may use nonlinear state-space reconstruction, Takens/Sauer-style delay coordinates, flexible latent dynamics, kernels or splines, expressive neural state-space models, nuisance/channel models, and minimal nonlinear realization methods where applicable. The higher-dimensional source model should not win by possessing more unrestricted flexibility; wherever possible it should have fewer structural freedoms and stronger registered constraints.

Stage 1: Register sources, hypotheses, experiments, and equivalences

Register 𝓗, the admissible experiment family 𝔐, equivalence rules 𝓔, noise and detection models, complexity criteria, intervention classes, and all decisive thresholds before fitting. Preserve hidden probes and interventions.

Stage 2: Generate heterogeneous observations

Generate informative, redundant, partially informative, noisy, and deliberately ambiguous observations. Include calibrated detection limits so required-but-absent signatures can be evaluated. For G₃, construct at least one H_a ≡_𝔐 H_b pair under the initial experiment family.

Stage 3: Fit strong competing realizations

Fit the registered geometric source models and the strongest admissible lower-dimensional nonlinear realizations under comparable computational and hyperparameter budgets. No competitor may gain access to hidden test interventions during fitting.

Stage 4: Test unseen probes and interventions

Train using a subset of probes M₁,M₂,M₃ and interventions I₁,I₂. Then test unseen combinations such as (M₄,I₁), (M₂,I₃), and (M₅,I₄). The purpose is structural transport: a shared source representation should predict consequences under new probe/intervention combinations without relearning a probe-specific latent state. Ordinary interpolation or same-channel forecasting is not decisive.

Stage 5: Require four legitimate decisions

The evaluator must be able to return: (1) lower-dimensional class preferred; (2) higher-dimensional class preferred; (3) competing classes observationally indistinguishable under the registered experiments; or (4) none of the registered models is adequate. The procedure fails if it is forced to choose a dimension when the evidence supports decisions 3 or 4.

Stage 6: Break equivalence actively

For G₃, introduce or search for a candidate measurement M* for which the rival source classes predict different observable distributions. The active-selection procedure should identify such a discriminating experiment when one exists. If no admissible experiment separates the classes, non-identifiability must remain the result.

Stage 7: Compare against strong conventional design

Active TSTOEAO measurement selection must be compared not only with random or convenience measurements but with a standard Bayesian model-discrimination or optimal experimental-design baseline. Report whether the procedures choose equivalent experiments, whether either dominates, and whether any TSTOEAO-specific invariant or transport constraint produces measurable added value.

Stage 8: False-positive and misspecification calibration

Estimate P(Ĥ_H | G₁), the false extra-dimension rate on known lower-dimensional generators. For G₄, test whether the pipeline correctly returns “registered models inadequate” rather than drifting toward the highest-dimensional candidate. Repeat across source instances, noise levels, detection limits, and misspecified nuisance models.

14. Primary Quantities to Report

At each measurement step report local observational deficit at regular points; an explicit global-equivalence measure appropriate to the problem; normalized candidate contraction under a declared μ_H or posterior information gain; held-out predictive loss; complexity penalty or model evidence; calibrated negative-signature exclusions; cross-probe and cross-intervention error; false extra-dimension rate; rate of correct non-identifiability decisions; rate of correct model-inadequacy decisions; and experiments required for discrimination.

The active-design comparison should additionally report held-out log loss, model-class selection accuracy where selection is possible, calibration under misspecification, and the number/cost of experiments required relative to conventional Bayesian model-discrimination design.

15. Decision Rules and a Hard Non-Identifiability Boundary

A higher-dimensional source class is supported within the registered experiment only when it survives calibrated negative signatures; outperforms the strongest admissible lower-dimensional realizations on prospective observations and interventions after complexity control; transports a shared latent structure across heterogeneous probes without probe-specific refitting; reproduces across source instances and noise realizations; and maintains a preregistered false extra-dimension rate on G₁ controls.

If rival dimensional source classes are observationally equivalent under 𝔐, the result is non-identifiability, not dimensional selection. If no registered class predicts the observations adequately, the result is model inadequacy, not automatic escalation to a higher dimension.

Registered Dimensional Non-Identifiability Proposition. Let H_a and H_b be registered source classes and 𝔐 an admissible experiment family. If, for every M ∈ 𝔐, the observable outcome distributions permitted by the classes coincide, p(O | H_a,M)=p(O | H_b,M), then no inference procedure restricted to observations generated by 𝔐 can establish which source class or source dimension is required. Dimensional discrimination becomes possible only if there exists at least one admissible M* ∈ 𝔐 for which the predicted observable distributions differ.

The proposition is a limitation statement grounded in standard statistical decision logic, not a claim of new mathematics. Its role in TSTOEAO is to enforce a hard boundary: no relational distinction without an experiment capable of exposing it.

16. Relationship to TSTOEAO

Within TSTOEAO, G_T remains the universal coordinate domain and domain models M_D remain overlays or representations within that architecture. The number of coordinates used in G_T is not equated with physical spatial dimensionality. The source-discrimination protocol is therefore a domain-specific inferential construction operating inside the broader relational framework.

Encoded Equilibrium Y is not identified with source dimension or Λ. Relational invariance I_R is used only when tied to a declared transformation family. A_Λ remains an accessibility operator until independent evidence supports a stronger physical interpretation. These separations preserve the architecture developed in the preceding papers while preventing the dimensional program from becoming a second, competing coordinate system.

The TSTOEAO contribution proposed here is procedural: relation before result. A dimensional interpretation is not granted because a representation is suggestive. It must emerge from the contraction of competing relational explanations under registered observations and prospective tests.

17. What Would Count as a Distinctive Result?

The strongest initial result would not be recovery of a sphere from several circles. It would be a blinded procedure that correctly distinguishes G₁ through G₄: refusing unnecessary dimension, admitting additional source structure when genuinely required, returning non-identifiability for cross-dimensional observational equivalence, and rejecting the registered family under misspecification.

An especially strong demonstration would show that a constrained higher-dimensional source model predicts unseen probe/intervention combinations that strong nonlinear lower-dimensional realizations cannot match without relearning their structure, while the reverse controls succeed on G₁. The relevant comparison is not merely against random measurement selection but against an integrated conventional pipeline using minimal realization, multi-view latent identifiability where applicable, and Bayesian model-discrimination design.

If the TSTOEAO protocol and the conventional integrated baseline are mathematically or computationally equivalent after translation, that is itself an important result: the contribution is then a TSTOEAO synthesis and application of established inference machinery rather than a new statistical methodology. A methodological originality claim requires measurable capability or performance beyond that baseline.

18. Novelty Boundary

The novelty claim must remain narrow until literature comparison and the adversarial benchmark are complete. The component mathematics and major methodological principles—including minimal nonlinear realization, multi-view shared-latent inference, observability/identifiability analysis, Bayesian model discrimination, optimal experimental design, negative evidence, and delay embedding—are established. “One latent state explains multiple probes” is therefore not by itself a novel TSTOEAO method.

The TSTOEAO-specific contribution is the way these elements are organized around relational invariance, registered dimensional-expression hypotheses, A_Λ, the Λ_A/Λ_D distinction, and observational-equivalence reduction as a gatekeeper for dimensional claims. Methodological originality should be claimed only if the complete protocol demonstrates capabilities or performance not already delivered by a strong conventional integrated pipeline. Until then, the defensible claim is a coherent TSTOEAO source-discrimination synthesis.

19. Failure Conditions

The proposed program should be considered unsuccessful or in need of revision if local deficit reduction fails to track useful global discrimination; singular fibers defeat the registered global supplement; strong lower-dimensional realizations match higher-dimensional prospective and interventional performance at comparable or lower complexity; calibrated negative signatures fail; shared Λ collapses into channel modeling; the procedure forces dimensional choices under observational equivalence; model misspecification drives systematic selection of the highest-dimensional candidate; false extra-dimension rates exceed preregistered limits; or active TSTOEAO measurement selection offers no measurable advantage over a strong conventional model-discrimination design.

These are not peripheral caveats. They define the boundaries of the hypothesis and prevent dimensional language from outrunning the evidence.

20. Conclusion

The dimensional program can now be stated without relying on a mystical ladder or visual analogy. Registered experiments partition candidate source states and source classes into observational-equivalence sets. Relational gain is the contraction of those registered equivalences. Differential-kernel contraction describes the regular local component; global ambiguity requires fibers, symmetries, posterior uncertainty, or other declared global measures, and singular strata require separate treatment.

The central scientific question is not whether a higher-dimensional model can describe the data. It is whether additional latent source structure is required after the strongest admissible nonlinear realizations have been tested on observations, heterogeneous probes, and interventions that were not used for fitting. The framework must also be capable of saying that dimensions are observationally indistinguishable or that none of the registered models is adequate.

The resulting TSTOEAO protocol treats dimensional inference as constrained source discrimination under registered equivalence. Its next obligation is computational rather than rhetorical. If the blinded G₁-G₄ benchmark can admit necessary source degrees, refuse unnecessary ones, preserve non-identifiability when experiments cannot distinguish dimensions, reject misspecified model families, exploit reliable absence, transport shared latent structure across unseen probes and interventions, and compete favorably with established optimal model-discrimination design, there will be a credible basis for investigating whether the integrated protocol itself is a methodological contribution. If not, the scientifically proper result is a TSTOEAO synthesis and application of established mathematics.

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