TAO bounded control contracts

### PDF page 31 able to inspect or disable sensing-driven adaptation. TAO should not be used to covertly manipulate affect, intensify stress without disclosure, or target vulnerable users through hidden state inference. Extra care is required for children, trauma-exposed users, clinical populations, neurodivergent users, and high-pressure training contexts. No clinical or therapeutic claim follows from TAO without domain-specific validation, safety review, and appropriate professional oversight. Limitations Several limits should be stated clearly. First, boundedness alone does not force the five TAO base primitives. They are a curated semantic basis drawn from a projective admissibility envelope, not the only possible bounded control architecture. Higher-order, spline, learned, or task-specific flows may be used when declared as state-contract extensions and evaluated against the reference grammar. Second, state estimation is empirical. Arousal may be easier to estimate than agency, trust, or flow, and chronic low confidence can reduce the controller toward a fixed- world fallback. A biased high-confidence estimator is more dangerous than a noisy estimator; TAO can expose and gate this through contract validation, but cannot solve physiological inference by itself. Third, multidimensional control is not globally a single PSL (2,R) action on a product manifold. TAO gains interpretability by decomposing adaptive worlds into typed one-dimensional contracts, but strongly entangled variables may need joint contracts or extension models. Fourth, hysteresis, bistability, and catastrophe-like transitions are higher-order hybrid structures, not extra one-dimensional base primitives. Fifth, the synthetic benchmark verifies controller-level signatures under known dynamics; it is not a human-performance result and does not replace TAO Chamber validation. Sixth, the synthetic benchmark and BAST metrics validate controller signatures, not subjective experience or clinical benefit. Seventh, when TAO wraps a black-box policy, safety and semantics are improved only to the extent that the relevant contract variables and actuator bindings are correctly specified. No therapeutic or clinical claim follows from this architecture without domain-specific validation, safety review, and ethical oversight. These limits strengthen the paper. They prevent overclaiming and make the contribution testable. The central claim of the present paper is therefore architectural and methodological: state contracts and BAST specify how bounded adaptive systems should be declared and stress-tested. Claims about live user benefit require admissible estimators and separate empirical validation. Conclusion Computing is moving from static interfaces to generated environments. The hard problem for such systems is not only how to generate possible worlds. It is how to couple those worlds to bounded human state in a way that is stable, safe, interpretable, and empirically testable. ### PDF page 32 Thresholded Adaptive Orchestration proposes that the missing layer is a typed bounded- state interface specification, paired with boundary-stress evaluation. Human and world variables are represented as bounded observables, mapped into declared chart coordinates, acted on by TAO base primitives or extension contracts, coupled through bounded functions, protected by safety barriers, and extended with hysteretic mode switching and critical-slowing anticipation. Boundary Stress Testing, Boundary Pinning Index, Boundary Risk Integral, and Semantic Work provide a way to evaluate the failure modes that ordinary average tracking hides. The result is a concrete architecture and benchmark grammar for neuroadaptive games, XR, AI world models, training simulators, and embodied human-computer interaction. The future interactive world should not merely ask what the player did. It should ask what bounded state the player is entering, what transition the system is approaching, and which safe world deformation will keep the coupled human-world system inside its intended corridor. TAO supplies a first-principles and empirically testable candidate for that missing computation. Commutator Table for the Five Primitives Let S = (1−e)2, I = −e2, L = e(1−e), T = 1−e, and G = −e. The Lie bracket is [X,Y] = XY′−YX′. The table entries are [row,column]. S I L T G S 0 2e(e−1) (e−1)2 (e−1)2 (e−1)(e + 1) I −2e(e−1) 0 e2 −e(e−2) −e2 L −(e−1)2 −e2 0 −(e−1)2 −e2 T −(e−1)2 e(e−2) (e−1)2 0 −1 G −(e−1)(e + 1) e2 e2 1 0 Nonzero entries imply leading-order order effects for sequential interventions from different flow classes. Minimum Reporting Checklist A TAO experiment should report: 1. State contract for every controlled variable: observable, chart, primitive or extension, bounded drive, confidence rule, update schedule, safety interval, actuator binding, mode rule, and validation criterion. 2. Bounded variables and their semantic endpoints. 3. Rapidity map assigned to each world variable. ---

synthetic oracle benchmark and a boundary-stress testing protocol demonstrate controller-level signatures before human-subject validation: boundary-recovery, tuning-sweep behaviour, order effects, semantic work, and early-warning signals. Predicted empirical signatures include reduced boundary pinning, faster perturbation recovery under admissible estimates, measurable cross-class order effects, and earlier prevention of state transitions at controlled false-alarm cost. These empirical hypotheses are conditional on admissible state estimation; the present validation addresses controller-level geometry, typed interfaces, and evaluation metrics rather than live physiological inference. TAO is proposed as both a typed bounded-state interface and an evaluation grammar for living interactive worlds, not as a content generator, state- estimation solution, or mind-reading system.

4. Realistic adaptive systems require these primitives to be embedded in a hybrid architecture with bounded coupling, hysteresis, anticipation, and safety barriers. 5. This architecture produces falsifiable controller-level signatures: boundary-safe recovery, order effects, target-corridor predictions, and critical-slowing warnings. The intended contribution is therefore not another game mechanic or another generative model. It is a framework and specification for typed bounded-state interfaces in living interactive worlds: a way to make adaptive variables composable, auditable, and empirically testable. Performance claims in this paper are controller-level and conditional

The framework wins only if independently assigned flow laws generalise better than ad hoc assignment. This rule makes the taxonomy falsifiable. If designers cannot assign flow laws above chance reliability from variable descriptions alone, the taxonomy is not operational enough. Rapidity-map selection is therefore semantic first and empirical second. A chart is chosen because the endpoint meaning and composition law imply odds, inverse odds, log-odds, survival fractions, or conjunctive success. If two charts are plausible, both assignments should be pre-registered and compared on transfer, prediction error, and interpretability rather than selected after observing outcomes. TAO state contracts TAO does not operate on unnamed scalar variables. Each adaptive coordinate is specified by a state contract. A state contract declares the observable, its semantic type, the rapidity

over the nearest base contract. Extensions are therefore allowed, but they must be declared before evaluation and compared against the reference grammar. This prevents the grammar from becoming either a straitjacket or an unfalsifiable post-hoc vocabulary. State-contract calibration TAO is not parameter-free. Its claim is not that gains disappear, but that gains are calibrated inside typed bounded contracts rather than attached to untyped clipped variables. Each deployed contract should therefore include a calibration phase for the

than a Gaussian. It formalises a bounded target corridor in chart space; whether that corridor corresponds to subjective flow must be tested empirically. Hypothesis 7 is therefore evaluated with a four-cell model comparison that separates coordinate choice from tail shape:

saturating growth S(e) = (1−e)2 and log-odds alignment L(e) = e(1−e), (e) = (1−e)2. Thus a clarity intervention and a confidence/alignment intervention are predicted to show a nonzero order effect. A mode boundary produces a categorical order effect. If intervention A crosses h+ and changes the governing flow class, while intervention B does not, then A followed by B differs from B followed by A not only by a continuous Lie bracket but by a discrete mode

which measures severity as well as duration of boundary exposure. These metrics are not replacements for recovery time or tracking error; they measure the specific failure mode that bounded-state geometry predicts. Rapidity coordinates also make it possible to measure semantic work. For coordinate rj = ϕj (ej),

update_world(w) log(raw, x_hat, confidence, modes, w) Benchmarks and Falsification The benchmark must avoid strawman baselines. TAO should be compared against: 1. Fixed non-adaptive worlds. 2. Naive linear dynamic difficulty adjustment with clipping. 3. Linear PI/PID-style control with anti-windup. the same prerecorded or simulated bounded state trajectory. This isolates the controller. Replay sets should deliberately include centre-region operation, near-boundary excursions, perturbations, and mode-crossing sequences; otherwise the predicted advantages near endpoints and thresholds cannot be observed. The live-estimator benchmark uses real sensor-derived state estimates and tests the full stack.

flow class. Critical-slowing anticipation must report true-positive rate, false-positive rate, and the cost of false warnings on flow-band occupancy. The hypotheses are:

Hypothesis 1 (Boundary stability). TAO reduces overshoot near bounded endpoints relative to clipped linear controllers and should match or improve upon sigmoid-output controllers while adding explicit state-contract semantics. Hypothesis 2 (Boundary-stress signatures). Under endpoint perturbations, correctly specified TAO contracts reduce Boundary Pinning Index and Boundary Risk Integral at comparable or lower semantic work than untyped observable-space controllers after matched calibration. Hypothesis 3 (Recovery). TAO reduces recovery time after perturbation relative to fixed worlds and ordinary dynamic difficulty adjustment when the relevant state contract is correctly specified and estimator reliability is admissible. Hypothesis 4 (Flow-band occupancy). TAO increases time inside target flow bands relative to fixed and linear baselines when each live state estimate used for control satisfies the pre- specified admissibility rule ci ≥ cmin and calibration error at or below ϵi. Hypothesis 5 (Transfer). Pre-registered flow-law assignment improves transfer across related variables compared with ad hoc variable-specific heuristics. Hypothesis 6 (Order effects). Cross-class intervention sequences produce measurable order effects; same-class sequences produce smaller effects after timing and intensity are controlled. Hypothesis 7 (Critical slowing). Rising rapidity variance and autocorrelation precede some mode transitions. Controllers that act on these leading indicators reduce unwanted transition rates compared with controllers acting on instantaneous state alone, at an explicitly reported false-alarm cost. Hypothesis 8 (Logistic-distance flow band). For bounded state variables with pre- registered charts and calibration-derived width parameters, target-corridor membership is best predicted by a model comparison that separates coordinate choice from tail shape. The primary test compares Gaussian and logistic/sech-squared tails in both observable and rapidity coordinates. The framework is weakened if it does not outperform strong baselines on overshoot or recovery, if flow-law assignments fail inter-rater reliability, if estimator quality dominates all controller differences, if predicted order effects do not appear, or if black-box controllers dominate TAO simultaneously on performance, stability, transfer, interpretability, and safety. Interpretability Interpretability must not be defined circularly as “uses named flows.” It should be externally measured. Four tests are proposed. 1.Action prediction. Given controller logs up to time t, independent designers predict the next adaptation.

TAO’s interpretability claim is not that every action is obvious. It is that named bounded flow laws, targets, thresholds, modes, and barriers should make controller behaviour more predictable, explainable, transferable, and debuggable than black-box policies. Relation to Standard Bounded and Safe Control TAO should not be read as a replacement for established bounded-actuator, anti-windup, prescribed-performance, funnel-control, contraction-theoretic, or control-barrier-function methods . Those methods enforce actuator limits, transient envelopes, contraction, or