From Dynamic Container Encoding to a Candidate Relational Inverse Operator
A TSTOEAO Mathematical Construction for Nested-Drive Observability Without Direct Use of the Native Governing Equation
John Swygert
Ivory Tower Publishing
September 12, 2026
Research Paper
Research Note
The preceding paper, From Static Fractal Resemblance to Dynamic Container Encoding, converted an early TSTOEAO intuition into a falsifiable inverse-dynamics problem: geometric history may encode selected properties of the dynamic environment in which a system develops. Independent review identified the next hard boundary correctly. The proposed relational representation remained descriptive. It did not yet define an operator capable of taking observed geometric history Γ_G and producing an estimate of a hidden outer-container state without simply reusing the native differential equation.
This paper attempts that construction. It does not claim that the resulting operator is new mathematics, optimal, or already a TSTOEAO invariant. In fact, parts of the construction deliberately overlap with established delay-coordinate, observability, system-identification, and Koopman-style methods. That overlap is a control: if the construction reduces completely to known machinery, the result is Level 2 rather than excess scientific content.
The objective is therefore sharper than advocacy. We ask whether a domain-light relational operator can be defined from geometric history itself, whether it can reconstruct an indirectly acting outer drive, whether its observability criterion survives translation to a second physical domain, and exactly what result would force us to conclude that no distinct TSTOEAO operator has yet been found.
Abstract
We construct a candidate TSTOEAO relational inverse operator for the nested-drive problem c₂(t) → c₁(t) → x(t) → Γ_G, where only geometric history Γ_G is observed and the target is a selected property of the hidden outer drive c₂. The construction avoids direct evaluation of the native governing PDE during inference. It maps geometric histories into delay-coordinate relational features, estimates a local history-evolution operator from training trajectories, separates persistent and transient modes, and reconstructs an outer-drive coordinate from those modes. A relational observability matrix and singular-value criterion determine whether the hidden coordinate is recoverable. The resulting map is written T_rel: Γ_G → ĉ₂.
The construction is explicitly provisional. Delay embeddings, data-driven inverse operators, Koopman representations, and unknown-input observability are established subjects, so success on one domain would not constitute Level 3 novelty. The decisive TSTOEAO test is cross-domain: freeze the relational construction after calibration in Domain A, translate only measurement normalization and units in Domain B, and ask whether the same structural criterion predicts outer-drive observability and a held-out consequence. Failure, equivalence to established methods, and genuine excess prediction are separated in advance.
1. Problem Statement
Consider a nested system
dc₂/dt = F₂(c₂,t), dc₁/dt = F₁(c₁,c₂,t), dx/dt = Fₓ(x,c₁,t), g(t)=H[x(t)].
The observer receives only the geometric history Γ_G = {g(t): 0≤t≤T}. The direct inner forcing c₁ and outer forcing c₂ are hidden. The target is not necessarily the complete function c₂(t); it may be a recoverable property θ₂ = P[c₂], such as dominant frequency, direction, phase class, modulation amplitude, switching time, or another preregistered descriptor.
Γ_G → T_rel → θ̂₂ → Q̂_held-out.
The construction must not call F₁, F₂, or Fₓ during the blind inference stage. Otherwise it becomes an ordinary model-based inverse calculation.
2. What 'Domain-Light' Means
No inverse operator can be literally domain-free: measurements must be defined, sampled, scaled, and compared. Here 'domain-light' means that the inference algorithm does not require the native field equation, constitutive law, force law, or adjoint of the target system. It may use observed geometric trajectories and training labels for controlled container histories.
The allowed inputs are therefore: time-ordered geometry, sampling interval, a declared geometric feature map, and—in the supervised calibration stage—known values of selected outer-drive descriptors. The forbidden inputs during blind inference are the native PDE/ODE coefficients and an explicit forward simulator.
3. Step One: Convert Geometry into Relational Observables
Let Φ transform each observed geometry into a dimensionless relational feature vector z(t). Φ should favor relations rather than absolute coordinates. Candidate components include normalized curvature statistics, orientation distributions, winding, branch-angle distributions, anisotropy tensors, topological counts, scale-normalized roughness, temporal deformation rates, and pairwise geometric distances.
z(t) = Φ[g(t)] ∈ ℝᵖ.
Normalization removes translation, rigid rotation when irrelevant, global scale when irrelevant, and units where physically legitimate. This is not claimed to create an invariant; it creates a common observational language in which an invariant can be tested.
4. Step Two: Encode History Rather Than a Snapshot
Construct a delay vector from the relational observations:
Z_k = [z_k, z_{k-1}, …, z_{k-L+1}]ᵀ.
The delay depth L is selected before the blind test using training data and then frozen. This step embodies the Dynamic Container Encoding hypothesis: if the outer drive leaves a delayed trace in local geometry, the trace may be absent from z_k alone but present in Z_k.
This step has strong precedent. Delay-coordinate and Koopman methods already use output history to recover hidden dynamical information; data-enabled inverse-operator work has specifically shown that past output can reduce hidden-state dependence. Therefore delay history by itself is not a TSTOEAO novelty.
5. Step Three: Build the Empirical History-Transfer Operator
From training trajectories, form paired delay states (Z_k, Z_{k+1}). Define the least-squares history-transfer operator
K = Z₊ Z₋†,
where Z₋ contains delay vectors at time k, Z₊ contains the corresponding vectors at k+1, and † denotes the Moore-Penrose pseudoinverse. Regularized or nonlinear lifted variants may later be tested, but the first experiment should use the simplest frozen construction.
K is not asserted to be the physical evolution operator. It is an empirical operator on relational geometric histories. Its eigenvalues, singular vectors, or invariant subspaces describe persistent temporal structures visible in the geometry.
6. Step Four: Separate Persistent Outer Modulation from Fast Inner Response
The nested-container idea predicts a possible separation of relational time scales. Let the spectral or singular decomposition of K identify modes ψ_j with characteristic persistence τ_j. Define a preregistered persistence window W_out intended to capture modulation slower than the dominant inner response.
Π_out Z = Σ_{j∈W_out} ⟨ψ_j,Z⟩ ψ_j.
The projected coordinate
r_out(t) = Π_out Z(t)
is the candidate outer-container trace. This is a hypothesis, not a theorem. If inner dynamics possess equally slow modes, or if outer forcing acts rapidly, the separation can fail. Such failure is part of the test.
7. Step Five: Define the Candidate Relational Inverse Operator
Let θ₂ denote the selected outer-drive descriptor. On calibration data only, fit a frozen map W from the projected history coordinate r_out to θ₂. In the simplest linear version:
W* = argmin_W Σ_i ||θ₂⁽ⁱ⁾ − W r_out⁽ⁱ⁾||² + λ||W||².
The candidate operator is then
T_rel[Γ_G] = W* Π_out 𝓓_L Φ[Γ_G],
where Φ is the relational geometry map, 𝓓_L is delay embedding, Π_out is the frozen persistent-mode projection learned from geometric histories, and W* maps the retained relational coordinate to the outer-drive descriptor.
Thus
T_rel = W* ∘ Π_out ∘ 𝓓_L ∘ Φ.
This is the first explicit computational object in the present branch that maps geometric history to an outer-container estimate without evaluating the native governing equation during inference.
8. Relational Observability Criterion
A fitted estimate is not enough. We need a criterion predicting when the hidden outer coordinate is actually observable. Let A_r be the reduced evolution matrix of the retained relational modes and let C_r select the measured relational coordinates. Define
O_R(q) = [C_r; C_r A_r; C_r A_r²; …; C_r A_r^{q−1}].
The outer relational coordinate is locally recoverable in the reduced model only if the subspace carrying θ₂ is not annihilated by O_R. Numerically, define
ρ_R = σ_min(O_R|_{S_out}) / σ_max(O_R|_{S_out}).
Here S_out is the preregistered candidate outer-mode subspace. ρ_R near zero indicates practical non-observability; larger ρ_R indicates a better-conditioned reconstruction. The threshold must be selected on training data and frozen before blind evaluation.
This criterion is intentionally close to classical observability. That is scientifically useful: it makes clear that TSTOEAO has not earned novelty merely by renaming observability. The possible TSTOEAO content lies only in whether the same relational construction and criterion survive cross-domain translation.
9. The First Test: Nested Reaction-Advection-Diffusion System
Domain A should be a controlled reaction-advection-diffusion pattern-forming system. Let c₁(t) represent a local advective or rotational forcing and c₂(t) modulate one selected property of c₁, such as angular speed, amplitude envelope, or direction switching. Ground truth is generated by the native model, but T_rel is denied the governing equation during inference.
Training trajectories span a preregistered range of c₂ descriptors. Blind trajectories include histories and parameter combinations not used for fitting. The endpoint container state may be held equal across selected pairs so that static final-state fitting cannot solve the task.
The primary inference is
Γ_G → θ̂₂.
The primary prediction is not θ̂₂ itself. From θ̂₂, the method must predict a withheld consequence Q, for example the phase or orientation of a future pattern response after a standardized perturbation.
θ̂₂ → Q̂(t>T).
Only after Q̂ is frozen is the true Q revealed.
10. Required Baselines
At least four baselines are required:
Endpoint-only geometry: Φ[g(T)] without delay history.
Standard data-driven regression using the same raw geometric features.
A conventional model-based inverse or adjoint method supplied with the native governing equation.
A shuffled-history/null model that destroys temporal ordering while preserving the distribution of observed geometries.
T_rel must beat endpoint and null baselines to support history encoding. Matching a conventional data-driven inverse method is Level 2. Beating an adjoint method on one benchmark would be interesting but would not itself prove a universal relational invariant.
11. The Cross-Domain Test
After Domain A, freeze the structural choices: feature categories, delay-selection rule, persistence criterion, observability statistic, regularization rule, and held-out scoring procedure. Then move to a physically different Domain B.
A suitable Domain B could be a mechanically driven growing or deforming medium, or another system whose outer modulation and inner forcing can be independently controlled. The native equations should differ materially from Domain A.
Only domain registration is allowed: units, sampling scale, and the concrete measurement implementation of the already-declared relational features. The operator architecture itself may not be redesigned after seeing Domain B outcomes.
T_rel^A structure ≟ T_rel^B structure.
The decisive question is not whether both systems can be inverted. It is whether the same relational observability criterion predicts which outer-container variables are recoverable in both systems.
12. Candidate Cross-Domain Invariant
We can now state a concrete candidate rather than an informal list. Define the relational observability profile
I_R(Γ_G) = {rank(O_R|_{S_out}), ρ_R, τ_out/τ_in, dim(S_out)}.
The candidate invariant is not that these numbers must be identical across domains. The stronger relational claim is that a common boundary in this normalized profile separates observable from non-observable outer drives across domains.
B_R(I_R) = 1 if outer drive is recoverable; B_R(I_R)=0 otherwise.
A Level 3 candidate result would be a boundary B_R calibrated in Domain A, frozen, and then prospectively predicting observability or failure in Domain B without access to Domain B's native governing equation.
This is still a conjecture. Classical systems theory may already imply an equivalent boundary once each system is properly represented. The experiment must actively test that possibility.
13. Why This May Collapse into Known Koopman or Observability Theory
The construction contains familiar components: delay coordinates, a learned linear evolution operator, spectral projection, regression, and an observability matrix. Koopman theory already studies dynamics in spaces of observables, including delay observables; published work reports system-independent representations in particular delay coordinates. Unknown-input observability theory already asks when hidden inputs or states can be reconstructed from output histories. Data-enabled inverse operators already exploit past outputs to reduce hidden-state dependence.
Therefore the default scientific expectation should be conservative: T_rel may turn out to be a particular assembly of established tools. If a mathematical equivalence is found, we should state it explicitly and classify the result as Level 2. The experiment remains valuable because it identifies exactly which part of the original TSTOEAO intuition is established mathematics and which part, if any, survives as an additional claim.
14. Pre-Registered Failure Conditions
The candidate operator fails as a useful nested-container inference method if:
θ₂ cannot be recovered above the endpoint-only and shuffled-history baselines.
The inferred θ̂₂ does not predict the held-out Q.
ρ_R does not discriminate observable from non-observable cases out of sample.
Small changes in sampling or noise destroy the result without a predictable conditioning signature.
The persistent-mode projection merely tracks an obvious directly measured component of c₁ rather than information about c₂.
The cross-domain TSTOEAO claim fails if:
The relational observability profile requires domain-specific redesign after Domain B is revealed.
A boundary calibrated in Domain A does not prospectively classify observability in Domain B.
Any successful boundary is mathematically equivalent to a known standard observability condition with no additional prediction.
The method succeeds only by incorporating native governing equations or domain-specific hidden information.
15. What Would Count as Level 3?
A credible Level 3 result would require all of the following: the operator and boundary are specified before the second-domain test; Domain B is physically distinct; no native Domain B equation is used by T_rel during inference; the frozen relational criterion predicts a nontrivial observability boundary or held-out consequence; conventional domain-specific analysis did not supply that prediction in advance; and independent analysis verifies that the result is not a disguised known theorem.
Even then, one successful result would be evidence of excess scientific content, not proof of a Theory of Everything.
16. A More Severe Test: Zero-Shot Structural Transfer
The strongest experiment removes the fitted output map W* from cross-domain transfer. Domain A is used only to identify a normalized relational boundary B_R in the observability profile. In Domain B, T_rel is allowed to compute I_R from geometric histories but is not trained on Domain B outer-drive labels before predicting which parameter regimes are observable.
I_R^B → B_R^A(I_R^B) → predicted observable / non-observable.
After the prediction is registered, Domain B labels and native-model observability analysis are revealed. This avoids the weakest form of cross-domain success, in which a new regression is simply trained in every domain.
17. Information-Theoretic Limitation
No operator can reconstruct information that the forward dynamics have erased. Dissipative systems can strongly suppress high-frequency or old forcing information. Therefore the proper claim is not universal invertibility.
Recoverable outer information ≤ information about c₂ retained in Γ_G.
The practical implication is that T_rel should predict its own failure through rank loss, small singular values, unstable mode separation, or declining held-out performance. A theory of encoding must include the conditions of forgetting.
18. Interpretation in G_T
Within TSTOEAO, the role of G_T is now more precise. It is not a replacement equation for the physical domain. It is the proposed common relational coordinate space in which the chain
outer modulation → inner forcing → response lag/memory → geometric history → observability
is represented independently of the names of the native variables. The candidate invariant is therefore not a universal shape. It is a proposed invariant boundary governing whether information about an outer relational cause remains observable after nested transformation.
This is substantially closer to the original container intuition: the geometry need not resemble the outer container. It need only retain a measurable trace of how the outer container changed the conditions under which the geometry formed.
19. Status of the Construction
The present paper has achieved one thing the preceding paper deliberately left open: it supplies an explicit candidate operator
T_rel = W* ∘ Π_out ∘ 𝓓_L ∘ Φ
and an explicit candidate relational observability profile
I_R = {rank, conditioning, time-scale ratio, outer-subspace dimension}.
It has not demonstrated that either object is novel. On present evidence, their components have close relatives in established control, inverse, and Koopman theory. The scientific value of the construction is that the TSTOEAO claim is now exposed to a direct computational comparison rather than protected by qualitative language.
20. Conclusion
The Dynamic Container Encoding hypothesis can be drafted into an executable mathematical experiment. The resulting candidate operator converts time-resolved geometry into relational observables, embeds their history, learns an empirical history-transfer operator, isolates a candidate outer-modulation subspace, and estimates a hidden outer-container descriptor. A reduced observability matrix supplies a falsifiable conditioning criterion.
The construction is intentionally vulnerable. It may fail because the outer history has been erased. It may succeed but reduce entirely to known delay-coordinate and observability theory. Or, more interestingly, a frozen normalized observability boundary may transfer prospectively between physically different systems and correctly predict where outer-container information is recoverable.
That final possibility is the one worth testing. The next step is no longer another conceptual extension. It is numerical implementation under blind conditions.
Γ_G → T_rel → θ̂₂ → Q̂_held-out → independent reveal.
If the same frozen relational criterion survives a second physical domain, investigate it as a candidate cross-domain invariant. If it does not, preserve the failure boundary. Either result advances the question more honestly than another analogy.
References
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