1. System Object

ATLAS is specified as a tuple of measurable and governed objects. Every component is defined in the sections that follow with its domain, role, and admissible use.

• \(\mathcal{X}\) — observable state space (cross-asset, regime-scoring inputs)
• \(\mathcal{D}\) — governed artifact space (schema-validated, provenance-tracked outputs)
• \(\mathcal{G}\) — governance / data-quality state space (admissibility of \(\mathcal{D}\))
• \(\mathcal{S}\) — structural regime state space (finite, slow, persistent)
• \(\mathcal{T}\) — tactical instability state space \(\{\text{Stable},\, \text{Transitional},\, \text{Unstable}\}\)
• \(\mathcal{C}\) — confidence / boundary-distance space
• \(\mathcal{F}\) — flip-risk / transition-pressure space
• \(\Pi\) — admissible policy map (Section 8)
• \(\mathcal{V}\) — validation operator family (Section 10)

ATLAS is a measurement and decision-governance system. It is not a return forecaster, an unconstrained optimizer, or an execution engine. Sections 2–14 define each component formally; Sections 15–16 give the public glossary and the explicit non-claims.

1.1 Standing Assumptions

All random variables in this document are defined on a fixed probability space \((\Omega, \mathcal{H}, \mathbb{P})\) equipped with the discrete-time filtration \((\mathcal{F}_t)_{t \in \mathbb{Z}_{\ge 0}}\) introduced in Section 2. Statements of equality involving random variables are understood \(\mathbb{P}\)-almost surely; statements of equality involving deterministic objects are absolute.

The remainder of the document operates under the following labeled assumptions. Each is stated as part of the specification, not asserted as universally true of the underlying market or of the data-generating process.

• A1 (Finite latent state). The structural regime takes values in a finite set, \(|\mathcal{S}| < \infty\), and evolves as a Markov chain with transition matrix \(P\) (Section 3).
• A2 (Information adaptation). All decision-time objects — \(S_t,\, T_t,\, C_t,\, F_t,\, D_t^{\,score},\, \pi_t\) — are \(\mathcal{F}_t\)-measurable, where \(\mathcal{F}_t\) is the natural artifact filtration introduced in Section 2.
• A3 (Fail-closed tactical layer). Inputs that are NA-driven or whose governing artifacts are inadmissible cannot resolve to \(T_t = \text{Stable}\) (Section 4).
• A4 (Deterministic validation). Each validator \(V_j\) is a deterministic function on artifact contracts (Section 10).
• A5 (Closed-set authorization). The authorization predicate \(\operatorname{Auth}(o, s, r)\) is two-valued (Section 11).

Assumptions A1–A5 are referred to by label where their force is invoked.

2. Observation and Artifact Layer

Let \(t\) denote decision time and \(d\) an observation date. Let \(X_t \in \mathcal{X}\) denote the observable state vector and \(\mathcal{D}_t\) the set of artifacts available at \(t\):

For each artifact define its maximum observation date and lag:

Artifact state is a closed-set label:

Eligibility for downstream use admits labeled states (OK, STALE, DEGRADED) and blocks the two value-less states (UNAVAILABLE, INSUFFICIENT):

A STALE or DEGRADED artifact carries its label downstream rather than being silently substituted; an UNAVAILABLE or INSUFFICIENT artifact has no value to consume, and downstream computations fail closed (Section 10).

The natural artifact filtration is the increasing family of \(\sigma\)-algebras

where \(I_t\) is the set of artifact indices available at \(t\). By Assumption A2, every decision-time object defined in Sections 3–8 is \(\mathcal{F}_t\)-measurable: ATLAS cannot use information that is not in \(\mathcal{F}_t\), and any apparent use of such information is a governance violation.

In the live implementation, \(q_i\) is decomposed into three orthogonal axes — a freshness label (timeliness and calendar-completeness), a health label (input quality), and an observability label (sample sufficiency for derivative quantities) — treated as governance vocabulary rather than analytic detail. The closed set above is the public abstraction.

3. Structural Regime

The structural regime \(S_t \in \mathcal{S}\) is a finite-valued classification of the prevailing macro environment. Let \(Z_t = (Z_{1,t}, \ldots, Z_{k,t})\) denote a low-dimensional vector of governed scores along the growth, inflation, risk-appetite, and liquidity axes. Then

\(\Theta_S\) denotes the governed structural parameters; \(G_t\) is the artifact-admissibility state at time \(t\). The map \(f_S\) is stateful (the argument is the score history \(Z_{1:t}\), not \(Z_t\) alone); slowness and persistence of the resulting state sequence are formalized by the inequality below. Structural-regime estimation is in the regime-switching tradition (Hamilton, 1989), in which the latent state evolves as a Markov chain with transition matrix

The diagonal \(P_{ii}\) is the one-step self-persistence of regime \(i\); the off-diagonal mass governs reachability between regimes. \(P\) is a member of \(\Theta_S\) and is estimated on a declared calibration window (Section 14).

Structural regime is not tactical instability:

\(T_t\) does not appear as an argument of \(f_S\); equivalently, \(S_t\) is \(\sigma(Z_{1:t}, \Theta_S, G_t)\)-measurable and not \(\sigma(T_t)\)-measurable in general. A single-layer model that conflates the two horizons violates the persistence inequality above (Ang & Timmermann, 2012). Separation is monitored by the leakage diagnostic \(\mathrm{lk}(W) = I(S_t; T_t \mid Z_t)\) of Section 12 (HEDGEHOG).

4. Tactical Instability

The tactical state \(T_t \in \{\text{Stable},\, \text{Transitional},\, \text{Unstable}\}\) is derived from high-frequency diagnostics of classification certainty. Define discrete differences:

The tactical objects are:

• \(H_t\) — transitional severity (integer-valued)
• \(N_t\) — NA-driven indicator (logical)
• \(K_t\) — trigger count (integer-valued)
• \(C_t,\ \Delta C_t\) — confidence and confidence change (Section 5)
• \(F_t,\ \Delta^2 F_t\) — flip-risk and flip-risk acceleration (Section 6)

The tactical fusion is

Thresholds and parameter values within \(\Theta_T\) are not disclosed. The fail-closed invariant is stated as an implication:

Equivalently: a tactical input that is NA-driven or whose governing artifacts are inadmissible can never resolve to Stable. The structural-tactical separation invariant — \(T_t\) does not appear as an argument of \(f_S\) (Section 3) — is enforced by construction. Any structural override is the subject of a separately governed rule, never implicit in tactical motion.

5. Confidence as Boundary Distance

Confidence \(C_t \in \mathcal{C}\) is defined as proximity to a decision boundary (Bishop, 2006, Ch. 4), not as subjective belief. Let \(M_s : \mathbb{R}^k \to \mathbb{R}\) denote the score/margin function assigned to structural state \(s\). The live confidence object is the score-margin form:

Let \(\Gamma \subset \mathbb{R}^k\) denote the structural decision boundary — under the margin form, the indifference locus \(\Gamma = \{\,z \in \mathbb{R}^k \,:\, M_s(z) = M_{s'}(z) \text{ for some } s \neq s'\,\}\); under the geometric form, an estimated surface in the score space. A geometric distance formulation

is computed in a shadow geometry layer as a research-only diagnostic and does not gate live policy. In both formulations, the boundary limit

holds by construction. Both \(C_t\) and \(\Delta C_t\) are first-class \(\mathcal{F}_t\)-measurable quantities (A2); \(\Delta C_t < 0\) is information about the trajectory and is not equivalent to a regime flip.

6. Flip Risk and Transition Pressure

Flip risk \(F_t \in \mathcal{F}\) is a function of proximity and motion in the score space:

The map \(\phi\) is monotone-non-increasing in \(C_t\) (proximity to the boundary raises flip risk) and increasing in \(\|\Delta Z_t\|\) (velocity in the score space raises flip risk). Three properties hold by construction:

• Change dominates level. Static-level exceedance alone is not sufficient for elevated transition pressure (Diebold & Rudebusch, 1996).
• Acceleration emphasis. \(\Delta^2 F_t\) can be informative even when \(F_t\) has not crossed a static threshold.
• No silent stabilization. Short or incomplete histories are labeled through the observability axis of Section 2; they are never imputed to Stable.

7. Disagreement

Score disagreement is the range across the structural-score axes (the live implementation applies the max–min functional on a designated axis subset):

\(D_t^{\,score}\) is a live object and is one of the families that contributes to admissibility of an abstention (Section 8).

Surface disagreement is the indicator that eligible classification surfaces disagree on the structural label:

\(D_t^{\,surface}\) is research-only; it is measured but not wired into the live admissible-action map. Any promotion to live status would require an explicit, declarative governance pass (Section 10).

8. Abstention as an Admissible Decision

The policy map is

where \(\mathcal{U}\) is the set of admissible non-null exposure actions and \(\varnothing\) denotes no action — abstention. In the current implementation:

The admissible decision set at time \(t\) is the gated subset

where \(u\) is the admissibility map and \(\pi_t\) is \(\mathcal{F}_t\)-measurable by A2. Gating reduces the action set; it does not select within it.

Abstention is not failure. It is a first-class output with reason codes, an episode lifecycle, and governance attribution. ATLAS manages exposure to uncertainty; it does not maximize return. This formulation follows the reject-option framework of Chow (1970) and its extension to selective classification in El-Yaniv & Wiener (2010).

An abstention episode is a maximal time interval over which \(\pi_t = \varnothing\), paired with a post-hoc outcome label. Formally:

where the outcome label \(\omega\) is assigned post-hoc at a pre-registered forward horizon:

• \(\omega = \mathrm{JC}\) (Justified Caution) — forward drawdown of sufficient magnitude (abstaining avoided realized loss);
• \(\omega = \mathrm{FC}\) (False Caution) — forward drawdown within tolerance (abstaining was unnecessary);
• \(\omega = \mathrm{IN}\) (Inconclusive) — insufficient forward data or intermediate outcome.

Horizons and drawdown thresholds are declared in advance to prevent horizon-selection bias (Harvey, Liu & Zhu, 2016). \(\mathcal{E}^{\,\mathrm{abst}}\) is the input set to the LANTERN confusion matrix (Section 12).

9. Policy Separation

ATLAS enforces four disjoint layers. The boundaries are architectural, not stylistic; each is stated as a measurability or domain identity.

• Measurement \(\neq\) policy. \(C_t,\, F_t,\, D_t^{\,score}\) are \(\mathcal{F}_t\)-measurable inputs to the admissibility map \(u\). They are not elements of \(\mathcal{U}_t\); knowing a measurement does not constitute taking an action.
• Policy \(\neq\) execution. \(\pi_t \in \mathcal{U}_t\) is a posture. The execution map is a separately scoped function on \(\mathcal{F}_t\) and is gated by operator authorization (Section 11); a posture is not a trade.
• Reporting \(\neq\) allocation authorization. A reported state object is \(\mathcal{F}_t\)-measurable; allocation authority requires \(\operatorname{Auth}(o, s, r) = 1\) (A5) and is not implied by the reportability of the underlying state.
• Research \(\neq\) live governance. Shadow and research artifacts are \(\mathcal{F}_t\)-measurable but do not enter the domain of \(u\). Promotion to live governance requires passing the gates of Section 10.

10. Validation and Governance

Let \(\mathcal{V} = \{V_1, V_2, \ldots, V_n\}\) be the family of validators, each producing a closed-set verdict:

Fail-closed semantics:

The validator family covers, at minimum:

• schema validation (column presence, type, closed-set membership);
• key uniqueness;
• date class consistency;
• freshness (\(\ell_i(t)\) within tolerance);
• stale-state propagation: if \(D_j\) is computed from \(D_i\), then \(q_i(t) \neq \mathrm{OK} \Rightarrow q_j(t) \neq \mathrm{OK}\) (downstream artifacts inherit upstream non-OK states; they do not mask them);
• shadow / governed-track separation (no mutation of governed roots from shadow surfaces);
• promotion gates (explicit, declarative, pre-registered);
• operator authorization (Section 11).

A \(\mathrm{FAIL}\) verdict is non-recoverable: governance refuses to publish. \(\mathrm{INSUFFICIENT}\) expresses a well-defined computation against a sample too thin to commit a value — it is labeled, not silently coerced to Stable or to a default. Governed roots are never written from shadow tracks. Hypotheses that fail empirical validation are recorded and disabled (Popper, 1959).

11. Multitenancy and Operator Scope

Let \(o \in \mathcal{O}\) denote an operator and \(s \in \mathcal{S}_o\) a sleeve in that operator’s scope. Authorization is a closed-set predicate over \((o, s, r)\) where \(r\) is the requested resource:

A request is valid only if

Missing operator context is fail-closed: a surface that requires \((o, s)\) to be safe refuses to render rather than render against a default. Let \(\mathrm{scope}(D)\) return the \((o, s)\) tuple to which an artifact is scoped, with \((o, \varnothing)\) marking org-wide artifacts. The per-operator-sleeve filtration is

Cross-sleeve leakage is structurally prohibited: any value rendered for \((o, s)\) is \(\mathcal{F}_t^{(o,s)}\)-measurable, and \(\mathcal{F}_t^{(o,s)} \cap \mathcal{F}_t^{(o,s')}\) for \(s \neq s'\) contains only org-wide \((o, \varnothing)\) state. Reports, exports, overrides, governed artifacts, and promotion gates are all scoped to \((o, s)\). The audit trail records \(\mathrm{scope}(\cdot)\) for any action taken, so post-hoc review can reconstruct what was observable when a decision was made.

12. Measurement Programs

The programs below are measurement and evaluation tracks, each with an explicit status tag. None of them are allocation engines on their own; promotion to live policy requires the gates of Section 10.

HEDGEHOG   STATUS: LIVE. A forensic-measurement program for the regime-and-tactical stack. Its scope is the tuple of measurement axes

Each axis admits a formal definition over a discrete window \(W = [t_0, t_1] \cap \mathbb{Z}_{\ge 0}\) and a scope \(R\) (a subset of artifacts, regimes, or sleeves):

Reading the axes:

• Coverage \(\mathrm{cov}_R(W)\): fraction of decision times at which every artifact in scope \(R\) was eligible (Section 2).
• Latency \(\mathrm{lat}_i(W)\): median artifact lag \(\ell_i(t)\) (Section 2) over the window; tail-percentile variants are reported alongside the median.
• Persistence \(\mathrm{per}(s)\): expected dwell time in structural state \(s\); the architectural prior \(\mathbb{P}(S_{t+h} = S_t) \gg \mathbb{P}(T_{t+h} = T_t)\) (Section 3) is checked against \(\mathrm{per}(s)\).
• Calibration \(\mathrm{cal}(W)\): expected calibration error over confidence bins \(\{B_k\}\) partitioning the unit interval, with \(\overline{\hat{p}}^{(B_k)}\) the bin-mean predicted probability and \(\overline{y}^{(B_k)}\) the realized rate.
• Leakage \(\mathrm{lk}(W)\): conditional dependence between structural and tactical states given the observable score vector; \(I(\,\cdot\,;\,\cdot\mid\cdot\,)\) is conditional mutual information (or an equivalent conditional-dependence functional). The architectural invariant is \(\mathrm{lk}(W) \approx 0\); persistent positive deviations are escalated for review.

HEDGEHOG is a measurement program, not an allocation surface. None of \(\mathrm{cov}\), \(\mathrm{lat}\), \(\mathrm{per}\), \(\mathrm{cal}\), \(\mathrm{lk}\) enters \(\mathcal{U}_t\). Instrumentation depth along each axis is itself a HEDGEHOG-reported metric, not a uniform claim of completeness.

SENTINEL   STATUS: RESEARCH. Transition-forecasting evaluation. Let \(\hat{p}^{(M)}_{ij}(t; h)\) denote model \(M\)'s estimate of \(\mathbb{P}(S_{t+h} = j \mid S_t = i)\). For a horizon \(h\), a scoring rule \(\ell\), and an out-of-sample index set \(\mathcal{I}_h\):

SENTINEL reports \(\Delta\mathcal{L}(h)\) over a held-out window and across horizons \(h\), comparing a baseline \(B\) to a candidate \(C\). There is no map from \(\Delta\mathcal{L}\) into \(\mathcal{U}_t\): SENTINEL is measurement of forecastability, not policy.

LANTERN   STATUS: SHADOW. Caution-policy calibration. Let \(\pi_t^{\star}\) denote a post-hoc reference action evaluated against forward outcome at a pre-registered horizon (see Section 8). LANTERN computes the policy-state confusion matrix over the realized policy \(\pi_t\) and the reference \(\pi_t^{\star}\):

\(\Lambda\) is reviewed by operators in a shadow learning surface. LANTERN does not write into \(\Pi\); there is no \(\Lambda\)-derived map into \(\mathcal{U}_t\).

TICM   STATUS: LIVE labeling. Stress-event window labeling. TICM defines a family of event windows and a corresponding indicator:

\(\tau_t^{\,\mathrm{TICM}}\) participates as a conditioning variable in the evaluation programs above. TICM labels never enter \(\mathcal{U}_t\) directly and carry no policy authority of their own.

WS7   STATUS: RESEARCH. Surface-disagreement track. Building on \(D_t^{\,surface}\) (Section 7), WS7 reports the windowed disagreement rate

and stratifications of it by structural regime \(S_t\) and by TICM label \(\tau_t^{\,\mathrm{TICM}}\). WS7 is measured but not wired into \(\mathcal{U}_t\); there is no \(\rho^{\,\mathrm{WS7}}\)-derived gate on \(\pi_t\).

13. Optionality / Protection Measurement

A protection bundle at decision time \(t\) is a finite set of contracts

with strike grid \(\{k_j\}\), contract counts \(\{n_j\}\), and per-contract proxy premia \(\{c_j\}\). The protection-cost proxy is

where \(c_j^{(t_0)}\) is the contract-proxy premium taken at a snapshot date \(t_0\) (\texttt{STATIC\_PROXY\_AT\_T0}) — not the live mid-market price at \(t\). The budget constraint is

The Slice F successor would replace each \(c_j^{(t_0)}\) by a live mid-market \(c_j(t)\) drawn from an option-chain pricing surface; this remains DEFERRED. Until promotion, \(P_t^{\,\mathrm{proxy}}\) carries no allocation, abstention, or gross-reduction authority and does not equal the realizable cost of acquiring \(\mathcal{P}_t\) at \(t\). It is a reporting estimate, labeled as such at the row level.

14. Estimation, Calibration, and Uncertainty

Only methods that are implemented or explicitly evaluated against ATLAS data are described here. ATLAS does not claim methods it does not use.

Structural-regime estimation is in the regime-switching tradition (Hamilton, 1989) with a finite latent state and a transition matrix \(P\) (Section 3). Parameters \(\Theta_S\) and \(\Theta_T\) are estimated on a calibration window declared in advance and held out from out-of-sample evaluation:

For a generic parameter \(\theta \in \Theta\):

Sample-size guardrail. Estimators carry an explicit minimum-sample requirement \(n_{\min}(\theta)\); below threshold, the estimator returns a labeled non-value rather than committing one:

The following properties are encoded in the formal objects above and in Sections 2–8:

• \(W_{\mathrm{cal}} \cap W_{\mathrm{oos}} = \varnothing\): calibration and out-of-sample windows are disjoint, declared before evaluation begins.
• \(|W_{\mathrm{cal}}| < n_{\min}(\theta) \Rightarrow \hat{\theta} = \mathrm{INSUFFICIENT}\): the sample-size guardrail commits no value below threshold.
• Forward evaluation horizons \(\{h_1, \ldots, h_K\} \subset \mathbb{Z}_{>0}\) are declared before \(W_{\mathrm{oos}}\) is entered, and outcomes for \(\mathcal{E}^{\,\mathrm{abst}}\) are aggregated across this fixed set (Harvey, Liu & Zhu, 2016; Tetlock & Gardner, 2015).
• Composite uncertainty is the vector \((C_t,\, F_t,\, D_t^{\,score})\) feeding the admissibility map \(u\); it is not collapsed into a single continuous probability, since such a collapse would itself require a calibration model subject to the same instability (Taleb, 2007). Admissibility \(u(\cdot) \in 2^{\,\mathcal{U} \cup \{\varnothing\}}\) is a closed-set output.

Parametric confidence intervals, bootstrap-based interval estimation, and spline-based smoothing are not claimed as components of the live estimation stack. Parameter uncertainty is real and is not collapsed into false precision.

15. Public Mathematical Glossary

Every cross-section ATLAS symbol introduced in Sections 1–14 and 16 appears below with its domain, interpretation, implementation status, and a public-safety note. Standing assumptions A1–A5 are stated in Section 1.1. Standard set-theoretic and probability notation is not duplicated. Program-local symbols defined inline within a single subsection — the abstention-episode set \(\mathcal{E}^{\,\mathrm{abst}}\) and outcome labels \(\{\mathrm{JC}, \mathrm{FC}, \mathrm{IN}\}\) (Section 8); the scope predicate \(\mathrm{scope}(\cdot)\) and per-operator-sleeve filtration \(\mathcal{F}_t^{(o,s)}\) (Section 11); the structural decision boundary \(\Gamma\) (Section 5); the HEDGEHOG axes \(\mathrm{cov},\, \mathrm{lat},\, \mathrm{per},\, \mathrm{cal},\, \mathrm{lk}\) and supporting \(W,\, R,\, B_k,\, \tau_s\); the SENTINEL objects \(\hat{p}^{(M)},\, \mathcal{L}^{(M)},\, \Delta\mathcal{L},\, \mathcal{I}_h\); the LANTERN objects \(\pi_t^{\star},\, \Lambda,\, M_{\pi,\,\pi^{\star}}\); the TICM objects \(\mathcal{E}^{\,\mathrm{TICM}},\, W_e,\, \tau_t^{\,\mathrm{TICM}}\); the WS7 statistic \(\rho^{\,\mathrm{WS7}}\); the optionality objects \(\mathcal{P}_t,\, k_j,\, n_j,\, c_j,\, B_t\); the calibration objects \(W_{\mathrm{cal}},\, W_{\mathrm{oos}},\, n_{\min}\); and the admissibility map \(u\) — are not duplicated here. Implementation status is one of LIVE, SHADOW, RESEARCH, or DEFERRED.

Programs (Section 12) carry their own tags: HEDGEHOG LIVE, TICM LIVE labeling, LANTERN SHADOW, SENTINEL RESEARCH, WS7 RESEARCH. The optionality proxy \(P_t^{\,\mathrm{proxy}}\) (Section 13) is ESTIMATE_PROXY; real option-chain pricing is DEFERRED.

16. Non-Claims

Three statements, made explicitly to prevent inferential drift.

ATLAS does not assert

as a universal trading claim.

ATLAS does not define

as an unconstrained return-maximization problem.

ATLAS does define

The objective ATLAS is governing is exposure to uncertainty under regime instability, subject to validation \(\mathcal{V}\), operator scope \((o, s)\), and the layer separation of Section 9. It is not portfolio-return maximization under regime stability. The two problems are complementary, not interchangeable.