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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