Cascade and Compound Risk: A Systemic-Layer-Anchored Tail-Risk Methodology for Civil Critical Infrastructure

§1 · Thesis + problem statement + Systemic Layer positioning

§1.1 Cascade risk vs compound risk — the structural distinction

Two failure modes dominate the tail of civil critical infrastructure risk under climate + geopolitical stress. Cascade risk is the propagation of an initial failure through the network — a 500 kV transmission line trips, generation trips regionwide, load is shed, a national blackout materialises within hours (as in the 25 February 2025 Chile event). Compound risk is the coincidence within a short temporal envelope of independent-cause failures — a heatwave-driven reactor derating in France concurrent with a drought-driven hydro shortfall in Iberia concurrent with a wildfire-driven transmission-corridor loss in Portugal (as in the Aug 2026 Iberian inversion episode). The two are structurally distinct, but their tail-risk consequences superpose.

§1.2 Why standard VaR fails

Standard utility-side tail-risk instruments — parametric VaR, historical VaR, single-Gaussian Monte Carlo — collapse this distinction. A €500M-scale tail loss reads as €500M whether it arose from a single cascading event or from ten independent events happening the same week. For a policy-maker deciding capacity-market design, or a regulator sizing reserve margins, or an underwriter pricing a portfolio of infrastructure assets, this collapse is the wrong output. The tail-risk instrument must show which failures are which, and it must show which populations bear which failures. Neither dimension is available from a single VaR figure.

§1.3 The SSI Systemic Layer as analytical spine

The methodology developed in this paper takes its architecture from the SSI Systemic Layer v1.0.1 canonical specification. The Systemic Layer is a per-substation × per-jurisdiction analytical layer built on top of the SSI Index v4.2 modifier surface, developed through Waves 3-6.5 of the SSI Index Foundation build-out (April-August 2026) and frozen at v1.0-final on 7 August 2026. It comprises twenty conventions (SY.1-SY.20) organised around two substantive layers (Layer A distributional-outcome, Layer B strategic-autonomy with three sub-layers), one cascade math substrate (six peer-reviewed anchors), one non-compensatory verdict rule, and a fourteen-pytest conformance gate suite.

This paper does not invent the cascade + compound-risk methodology; it quantifies the SSI Systemic Layer at tail-risk resolution. The Markov degradation kernel of Rasmussen (1997), the Gaussian copula of Sklar (1959), the 5σ tail prism (Ikenga-native, deployed in SSI-ENN v31.42), and the R9 compound-concurrence modifier of Zscheischler et al. (2020) are instruments that act on the Systemic Layer substrate. They are not the substrate. Confusing instrument with substrate is a common failure mode of applied tail-risk work; the present paper is architecturally explicit about the distinction.

The paper is scoped for four audiences: (a) peer reviewers at applied-mathematics, systems-engineering, or infrastructure-economics journals — for whom the mathematical instruments and validation protocol are the central object; (b) regulators at national grid authorities, financial-services stress-test frameworks (e.g. EU-wide EBA / EIOPA), and TCFD-scenario disclosure — for whom the non-compensatory verdict rule (SY.7) and independent-replication audit commitment (SY.20) are the central objects; (c) Foundation-side users of the SSI Systemic Layer methodology deployed at 796,121-substation × 39-OECD-country scale — for whom the Layer A / Layer B decomposition and coupling substrate are the central objects; (d) Commercial-side counterparts at SSI-ENN who consume the methodology into LP-DD-grade tail-risk products.

§2 · The SSI Systemic Layer v1.0.1 analytical architecture

This section presents the canonical Systemic Layer v1.0.1 architecture (Bérard 2026, spec closed 7 August 2026, deposited under CC BY 4.0 at Zenodo). The twenty conventions are grouped into six architectural clusters: (i) primacy + layer definition (SY.1-SY.5), (ii) classification + verdict (SY.6-SY.7), (iii) reproducibility + uncertainty (SY.8-SY.10), (iv) cascade math substrate (SY.11-SY.16), (v) preservation + separation + acknowledgment (SY.17-SY.19), and (vi) replication commitment (SY.20).

§2.1 Twenty canonical conventions (SY.1-SY.20)

The twenty conventions are not an arbitrary list. They emerged from a specific analytical concern: every tail-risk methodology hides its analytical decisions behind procedural convenience, and the Systemic Layer was designed to expose those decisions rather than embed them. Cluster 1 (primacy + layer definition) forces the decision which population's distributional outcomes matter to be an explicit input (Layer A CVI drivers), not a defaulted assumption. Cluster 2 (classification + verdict) makes the analytical decision how strong is the required evidence for a policy pass a load-bearing rule (SY.7 non-compensatory verdict) rather than an appendix parameter. Cluster 3 (reproducibility + uncertainty) forces the decision who bears the epistemic burden to sit with the publisher (SY.8 seed pin + SY.9 five-source stack + SY.10 visibly-honest degradation). Cluster 4 (cascade math substrate) makes the decision which mechanism drives the tail concrete via six named anchors rather than a black-box aggregation. Cluster 5 (preservation + separation + acknowledgment) forces the decision which reader inference does the framework not authorise to be explicit (SY.19 Fraser recognition-justice guardrail). Cluster 6 (replication commitment) forces the decision who audits us and when to be a scheduled deliverable rather than an aspiration.

Read in this order, the twenty conventions form a governance stack around the paper's mathematical instruments (§6-§8). The instruments produce numbers; the substrate produces the interpretive constraints under which those numbers are readable. A reader who wants only the mathematical instruments will find them at §5.4 (Kemeny-Snell), §6 (Gaussian copula), §7 (5σ tail prism), §8 (R9 modifier); a reader who wants to know why the numbers can be trusted must read the substrate first.

§2.2 Layer A · Distributional-outcome dimensions (SY.2)

Layer A operationalises the distributional-outcome dimensions of Reckien et al. 2023's six criteria via composite variables spanning:

• Vulnerable-population impact — mapped to W1 (vulnerable-population impact axis).
• Cross-sector burden displacement — mapped to W2 (shifting-vulnerability across sectors).
• Inequity-worsening — mapped to W6 (distributive-justice load).

The vulnerable-population Composite Vulnerability Index (CVI) is defined as a per-substation aggregate of NUTS-3 socio-economic vulnerability drivers. Its formal definition, five-band classification, and per-band gate rules are developed at §3.

§2.3 Layer B · Strategic-autonomy criteria + three sub-layer decomposition (SY.3, SY.4)

Layer B operationalises strategic-autonomy across seven criteria, each mapped to at least one W-axis for gate-suite integration: technology sovereignty (W4, W8) · supply-chain criticality (W5, W7) · workforce depth (W3) · regulatory competence (W10) · registered-asset auditability (W10) · cross-border interdependency (W7, W9) · demand-response readiness (W8).

Per SY.4, Layer B is decomposed into three sub-layers each independently scored and independently gate-flagged. This decomposition organises §4:

• B1 · Registered-asset spatial-coverage — coverage of the SSI Index v4.23 substation cohort by jurisdiction-published RAB registers + investment-plan filings. Anchors the W10 register-access-maturity gate.
• B2 · Technology-sovereignty per-modifier — per-modifier (R6c, R6d, R6e, R7, R8, R9, R10) sovereignty of the adaptation stack (domestic vs foreign vendor, IP jurisdiction). Anchors W4, W5, W8 gates.
• B3 · Workforce-and-competence per-jurisdiction — per-jurisdiction assessment of engineering, operational, and regulatory workforce depth for adaptation delivery. Anchors W3, W10 gates.

§2.4 Cascade math substrate (SY.11-SY.16)

Six peer-reviewed anchors compose the cascade math substrate, developed as an integrated stack at §5:

• SY.11 · Buldyrev et al. 2010 (Nature) — interdependent-network cascade-mechanism substrate. Substation graph generated from the v4.23 grid-geo.json topology.
• SY.12 · Miller & Blair 2022 — Leontief inverse for economic-input-output propagation of adaptation-response allocation.
• SY.13 · Rose & Round 2003 — SAM (Social Accounting Matrix) extension for regional-outcome accounting under non-compensatory constraint. Compensatory cross-regional flows prohibited by construction.
• SY.14 · Kemeny & Snell 1960 — absorbing-chain foreclosure formalism. Reckien criterion (iv) foreclosing-adaptation is formalised as absorbing states in a finite Markov chain over adaptation-pathway states. Connects to Rasmussen (1997) socio-technical drift: the fundamental matrix N = (I−Q)⁻¹ supplies a mathematical vehicle to quantify the Rasmussen-articulated migration-to-boundary phenomenon, without one framework subsuming the other.
• SY.15 · Sargentis et al. 2021/2022 — water-energy-food nexus + landscape-planning axis-wise gates. Failure at one axis not compensated by excess at another (non-compensation composed with SY.7).
• SY.16 · Diebold & Mariano 1995 — walk-forward temporal-replication test. Look-ahead information is prohibited.

§2.5 CVI five-band classification + non-compensatory verdict (SY.6, SY.7)

Per SY.6, the CVI is classified into five bands with cutoffs at 0.25 / 0.50 / 0.75 / 1.00 / 1.30. Per SY.7 (load-bearing), the verdict rule enforces set-theoretic non-compensation:

§2.6 Deterministic replication + five-source stacked Monte Carlo (SY.8, SY.9)

Per SY.8, all Systemic Layer scoring runs are deterministically replicable under fixed random seed (numpy default_rng seed 42); SHA-256 verification of per-substation output enables post-hoc replication attestation. Per SY.9, uncertainty propagation combines five sources tracked separately: (1) parameter-estimation uncertainty in modifier calibrations; (2) input-data measurement error; (3) climate-scenario uncertainty via CMIP6 ensemble; (4) socio-economic-scenario uncertainty via SSP2-4.5 baseline; (5) Reckien-criterion-threshold calibration uncertainty. Stacked Monte Carlo with 10,000 iterations per substation.

§2.7 Visibly-honest degradation + Fraser structural-misrecognition (SY.10, SY.19)

Per SY.10, the Systemic Layer inherits Convention #56 visibly-honest degradation from the underlying SSI Index v4.23 canonical. Missing per-substation values are propagated as None and openly disclosed rather than substituted or imputed. Per SY.19, the framework acknowledges Fraser 2000 structural-misrecognition tradition as its recognition-justice input. W1, W4, W6 axes are calibrated to preserve Fraser's structural (not attitudinal) interpretation. §10 develops the anti-instrumentalisation implications.

§2.8 Supplementary · the SB-02/F-02 empirical convention series (candidate)

Between v0.9-draft (June 2026) and v1.0-final (7 August 2026), the SB-02 empirical-brief series introduced a candidate convention series recording promotion events from empirical-brief work — SY-emp.11 through SY-emp.21. This series is not part of the canonical v1.0.1 spec (which caps at SY.20 and uses different semantics at SY.11-SY.20); it is a supplementary layer tracking empirical instances that have accumulated toward Convention #76 BINDING promotion thresholds. For clarity, this paper preserves the SB-02 candidate numbering as-is (SY-emp.11-SY-emp.21) rather than collapsing it into the canonical range:

The SB-02 empirical series and the v1.0.1 canonical spec overlap partially in substance — the canonical SY.16 (Diebold-Mariano walk-forward) subsumes SB-02 SY-emp.20 (Rule N walk-forward vs look-ahead); canonical SY.11 (Buldyrev-2010) subsumes SB-02 SY-emp.15 (coupling matrix); canonical SY.19 (Fraser structural-misrecognition) subsumes recognitional dimensions of SB-02 SY-emp.13. The remaining SB-02 candidates (SY-emp.16 through SY-emp.21) are architectural extensions specific to the empirical-brief series. Update (24 August 2026): a v1.1 draft specification (SYSTEMIC_LAYER_v1_1_SPEC.md) has landed same-session absorbing SY-emp.16 → canonical SY.21 (interim-state anchoring), SY-emp.17 → SY.22 (cross-border price asymmetry), SY-emp.18 → SY.23 (regulatory-mandate anchoring), SY-emp.19 → SY.24 (intervention-endogenous cascade) as BINDING, and SY-emp.21 → SY.25 (meteorological × geopolitical superposition) as CANDIDATE. SY-emp.20 is documented as subsumed by canonical SY.16 (Diebold-Mariano). SY-emp.11-15 remain under the SB-02 label as Foundation-side operational disciplines that do not warrant canonical status. v1.1 final promotion pending operator sign-off + SY-emp.13 six-touch-point cascade + Zenodo deposit refresh. Update (September 2026): the F-02 second-edition research adds SY-emp.22 — the marine-biological cooling-intake cascade discipline (CANDIDATE), the first post-v1.1-draft addition to the empirical series (which now runs SY-emp.11 → SY-emp.22), registered on the 11 August 2026 Gravelines jellyfish-intake shutdown with 2025 precedents (Gravelines + Paluel); it names a cooling-outage pathway distinct from thermal derating and would follow SY-emp.16→21 into the pending v1.1 canonical absorption on the same promotion cadence. Empirical-brief registration at F-02 §5.14.

§2.9 Methodology pins · twelve peer-reviewed anchors

1. Buldyrev et al. (2010) · Nature — interdependent-network cascade-mechanism substrate. Anchor for SY.11 and §5.1.
2. Miller & Blair (2022) · Input-Output Analysis, 3rd ed. — Leontief inverse. Anchor for SY.12 and §5.2.
3. Rose & Round (2003) · SAM regional-outcome extension. Anchor for SY.13 and §5.3.
4. Kemeny & Snell (1960) · Finite Markov Chains — absorbing-chain formalism. Anchor for SY.14 and §5.4.
5. Sargentis et al. (2021/2022) · axis-wise gating for water-energy-food nexus. Anchor for SY.15 and §5.5.
6. Diebold & Mariano (1995) · walk-forward validation test. Anchor for SY.16 and §9.2.
7. Reckien et al. (2018) · Climatic Change — equity-adaptation; extended in Reckien 2023 into six-criterion mapping. Anchor for §3 Layer A and §10.
8. Fraser (2000) · New Left Review — structural-misrecognition acknowledgment. Anchor for SY.19 and §10.
9. Perrow (1984) · Normal Accidents — tight-coupling · complex-interaction theory. Cascade taxonomy at §5.6.
10. Zscheischler et al. (2020) · Nature Reviews Earth & Environment — compound-event taxonomy for R9 (§8.1).
11. Sklar (1959) · copula theorem. Anchor for §6 copula operator.
12. Rockafellar & Uryasev (2000) · Journal of Risk — Conditional VaR / Expected Shortfall. Foundation for the 5σ tail prism at §7.

Two instrument-references: Rasmussen (1997) migration-to-boundary socio-technical drift framework (mathematical vehicle at Kemeny-Snell absorbing states, SY.14 · §5.4.3); Scheffer (2001) Nature regime-shift theory (mathematical vehicle at interim-state anchoring, SB-02 SY-emp.16 · §7.5).

§3 · Layer A · Distributional-outcome cascade quantification

Layer A operationalises the SSI Systemic Layer's distributional-outcome dimensions (SY.2) via the Composite Vulnerability Index (CVI), the five-band classification (SY.6), and the W1/W2/W6 pytest gate rules. This section presents the CVI formal definition, band cutoffs, per-band cascade quantification, and the Reckien criterion-mapping preservation discipline (SY.17).

§3.1 Composite Vulnerability Index — formal definition

For each substation \(i\) in the fleet and each NUTS-3 region \(r\) containing \(i\), the CVI is defined as:

The CVI value is bounded in \([0, 1.30]\) by construction, where the additive tail above 1.00 corresponds to the v4.2 R6c flood-exposure extension over the v4.0.2 4-band ceiling. NUTS-3 data availability is tier-assigned per SY.5 (Tier B via ISTAT PUF for Italy; Tier A / B / C fallback path across the 39 OECD cohort).

§3.2 CVI five-band classification (SY.6)

The per-substation CVI value is classified into one of five bands. Each band carries a distinct verdict-rule implication in the SY.7 non-compensatory gate suite:

§3.3 W1 vulnerable-population + W2 cross-sector displacement + W6 inequity gates

The Layer A dimensions map to three of the ten W-axes. Each axis is scored by a dedicated pytest test in the fourteen-pytest conformance suite (SY.7 implementation). Test count per Layer A axis:

§3.4 Fraser structural-misrecognition guardrail on Layer A (SY.19)

Layer A's W1/W4/W6 gates carry recognitional content. Per SY.19 (developed in full at §10), these gates are calibrated to preserve Fraser 2000's structural interpretation of misrecognition: they test whether institutional arrangements (grid topology, tariff structure, connection-priority rule, adaptation-planning procedure, cost/benefit-allocation rule) systematically deny some categories of persons the standing of full partners in adaptation decision-making. They do not test whether operators or regulators hold dismissive attitudes; that reading is banned as framework misuse. The reason this matters at §3 (rather than being deferred entirely to §10) is that the substantive analytical decisions in Layer A — the choice of CVI drivers, the mapping of Reckien criteria to W-axes, the calibration of CVI bands to verdict-rule implications — all embed the structural interpretation. Deferring the Fraser anchor to §10 would leave §3 methodologically unmoored; anchoring here makes the load-bearing recognitional commitment explicit at the point where it acts on the mathematics.

§3.5 Reckien 2023 criterion-mapping preservation (SY.17)

The many-to-many mapping of Reckien 2023's six criteria onto the ten W-axes is preserved as an interpretive layer. Per SY.17, modifications to the mapping require methodology-version increment (v1.0.1 → v1.1). The six-criterion mapping (see also the P9-ERSS paper Table 2 / Figure 4):

1. (i) Compensatory-reduction — W1 (vulnerable-population impact)
2. (ii) Shifting-vulnerability — W2 (cross-sector displacement) + W7 (cascade-chain propagation)
3. (iii) Emissions-increase — W5 (emissions-lock-in risk)
4. (iv) Foreclosing-adaptation — W3 (adaptation-response deficit) + W7 (cascade-chain) + W8 (digital-adaptation-readiness deficit) + W10 (register-access maturity)
5. (v) Inequity-worsening — W1 + W4 (community consent) + W6 (distributive-justice)
6. (vi) Unintended-consequence — W4 (community consent) + W9 (compound-hazard concurrence) + W10 (register-access maturity)

§4 · Layer B · Strategic-autonomy cascade decomposition

Layer B operationalises strategic-autonomy across seven criteria (SY.3), decomposed into three sub-layers each independently scored and independently gate-flagged (SY.4). This decomposition is a substantive methodological commitment: cascade risk and compound risk both propagate through strategic-autonomy deficits (missing register coverage, foreign-vendor concentration, workforce-depth gaps), and treating these as a single scalar would mask the structural sources of tail risk.

§4.1 Sub-layer B1 · Registered-asset spatial-coverage

B1 quantifies the coverage of the SSI Index v4.23 substation cohort (796,121 substations across 39 OECD jurisdictions) by jurisdiction-published Regulated Asset Base (RAB) registers + investment-plan filings. Per SY.5, each jurisdiction is tier-assigned:

• Tier T1 — full RAB register + multi-year investment-plan filings publicly accessible (e.g. UK Ofgem RIIO, Italy Terna PdS 2025-2034, Germany NEP + Ausschreibungen)
• Tier T2 — partial register access with summary publication (e.g. France RTE Schéma Décennal, Spain REE Planificación)
• Tier T3 — register scarcity requiring sub-national reconstruction (e.g. several Central + Eastern European jurisdictions)

B1 anchors the W10 register-access maturity gate (2 tests in the pytest suite). A substation-mapped adaptation action FLAGS W10 iff the register-access tier is inconsistent with the claim being made (e.g. a T3-jurisdiction candidate action cannot pass a claim conditional on Tier-A register-audit evidence).

§4.2 Sub-layer B2 · Technology-sovereignty per-modifier

B2 quantifies the sovereignty of the adaptation-stack technology base per modifier: for each of R6c (flood), R6d (wildfire), R6e (winter storm), R7 (cyber), R8 (adapt), R9 (compound), R10 (just-transition), the domestic vs foreign vendor mix and IP-jurisdiction concentration are scored. The per-modifier decomposition matters because a jurisdiction's tail risk under a specific modifier propagates through the specific technology stack that answers that modifier — not through an aggregate technology base.

B2 anchors W4 (community consent + procedural inclusion, 2 tests), W5 (emissions-lock-in risk, 1 test), and W8 (digital-adaptation-readiness deficit, 1 test). Foreign-vendor concentration above the modifier-specific threshold (calibrated per SY.9 five-source stacked uncertainty) FLAGs the corresponding W-axis.

§4.3 Sub-layer B3 · Workforce-and-competence per-jurisdiction

B3 quantifies the engineering, operational, and regulatory workforce depth per jurisdiction for adaptation delivery. Sources include Eurostat labour-force statistics per NACE code (electricity supply, network engineering, civil-infrastructure design), national accreditation-body databases (e.g. UK IET, Italy CNI, Germany VDI), and cross-border mobility indicators (EU Blue Card issuance rates).

B3 anchors W3 (adaptation-response deficit, 1 test) — the workforce-depth axis of the foreclosing-adaptation criterion. Below-threshold workforce depth in the domain relevant to the candidate action FLAGs W3.

§4.4 Layer B gates (W3, W7, W8, W10)

The four Layer B gates cover Reckien criterion (iv) foreclosing-adaptation + criterion (ii) shifting-vulnerability + criterion (vi) unintended-consequence. Test count per gate:

§4.5 Two-stage output separation (SY.18)

Per SY.18, Systemic Layer outputs are separated into two evidence classes:

• Class (a) · per-substation quantitative statistics — CVI values, W-flag vectors, coupling-matrix decompositions. Computable directly from the v4.23 canonical + Layer A/B decomposition. This class carries substantive-evidence primacy per SY.1.
• Class (b) · per-candidate-adaptation-action verdicts — MALADAPTATION_FLAGGED / PASS outputs from the fourteen-pytest suite. Computed on candidate actions in the policy catalogue.

The distinction matters for tail-risk quantification: class (a) statistics support cascade + compound-risk propagation analysis (this paper); class (b) verdicts support policy-catalogue evaluation (the P9-ERSS paper). Both classes inherit from the same Layer A/B substrate; they do not conflate.

§5 · Cascade math substrate

The cascade math substrate composes six peer-reviewed anchors (SY.11-SY.16) into an integrated stack. Each anchor governs a specific dimension of cascade + compound propagation. This section develops the substrate; §§6-8 present the instruments that act on top of it.

§5.1 Buldyrev-2010 interdependent-network cascade substrate (SY.11)

Cross-substation cascade propagation is modelled per Buldyrev et al. 2010 Nature interdependent-network cascade dynamics. In the Buldyrev formulation, two interdependent networks (say, the electricity substation network and the transport-signalling network) experience cascading failure when nodes in one network depend on nodes in the other for functional persistence. A node in Network A fails if either (i) it loses connectivity within Network A, or (ii) its dependency partner in Network B has failed.

For the SSI Systemic Layer instantiation, Network A is the substation graph generated from the v4.23 grid-geo.json topology; Network B is the coupled sector-dependency network (transport, water, digital, industrial). Coupling coefficients \(\gamma_{k\ell}\) between class-\(k\) and class-\(\ell\) events (C-taxonomy C1-C20 per SB-02 SY-emp.15 coupling matrix) parametrise the recurrence.

§5.2 Miller-Blair Leontief inverse (SY.12)

Economic-input-output propagation of cascade impact uses the Miller-Blair (2022) Leontief-inverse formulation. Let \(\mathbf{A}\) be the technical-coefficients matrix of a Leontief input-output table; the Leontief inverse \(\mathbf{L} = (\mathbf{I} - \mathbf{A})^{-1}\) quantifies the direct + indirect + induced effect of a unit-shock to any sector.

The framework applies the Leontief inverse to translate substation-level cascade losses into per-sector economic-output losses, which then feed the §4.1 (W2 cross-sector displacement) gate rule.

§5.3 Rose-Round SAM regional-outcome extension (SY.13)

Regional-outcome accounting under non-compensatory constraint follows Rose & Round 2003 Social Accounting Matrix extension. The SAM extends the Leontief input-output structure to include income distribution + household categories + inter-regional transfers. Critically for the SSI Systemic Layer, compensatory cross-regional flows are prohibited by the accounting rule — a cascade loss in NUTS-3 region \(r_1\) is not offset by economic activity in NUTS-3 region \(r_2\) even if aggregate national output is preserved.

§5.4 Kemeny-Snell absorbing-chain foreclosure formalism (SY.14) — subsumes classical Rasmussen

Reckien criterion (iv) foreclosing-adaptation is formalised as absorbing states in a Kemeny-Snell 1960 finite Markov chain over adaptation-pathway states. Foreclosed adaptation paths are non-recoverable by construction; W3, W7, W8, W10 gates enforce this at the verdict layer.

§5.4.1 State space

Each substation \(i\) at time \(t\) is characterised by a state \(S_i(t) \in \mathcal{S}\) over a finite discrete adaptation-pathway state set. The SSI Systemic Layer instantiation uses five states:

§5.4.2 Transition kernel

The state evolves via a discrete-time Markov chain with transition kernel \(\mathbf{P}(t)\), a \(5 \times 5\) row-stochastic matrix. Entry \(P_{jk}(t) = \Pr[S_i(t+\Delta t) = s_k \mid S_i(t) = s_j, \mathbf{X}_i(t)]\) where \(\mathbf{X}_i(t)\) is the covariate vector (voltage tier, age, historic outage rate, R-modifier stack, jurisdiction). The absorbing property is enforced by \(P_{55}(t) = 1\) for all \(t\); once at Foreclosed, the substation stays there absent exogenous re-scoping.

§5.4.3 Rasmussen migration-to-boundary connection

The Rasmussen (1997) migration-to-boundary property — the observation that engineered socio-technical systems under performance pressure drift toward boundaries of safe operation absent counteracting intervention — is a substantive claim about the dynamics of engineered systems, not a mathematical theorem. The Kemeny-Snell fundamental-matrix formulation supplies a mathematical vehicle for the phenomenon: absent intervention (\(\mathcal{I}(t) = \emptyset\)), the finite expected time-to-foreclosure \(\sum_k N_{jk}\) quantifies the Rasmussen-articulated drift; with intervention (\(\mathcal{I}(t) \neq \emptyset\)), the kernel is modified to a non-absorbing chain that allows persistent operation. The two frameworks are complementary and mutually reinforcing: Rasmussen supplies the empirical + conceptual grounding for why the absorbing-state formulation is the correct mathematical model for engineered infrastructure; Kemeny-Snell supplies the quantification. Neither subsumes the other.

§5.5 Sargentis axis-wise gating (SY.15)

Infrastructure-self-sufficiency conditions per Sargentis et al. 2021/2022 water-energy-food nexus + landscape-planning framework are treated as axis-wise gates rather than composite-optimisation objectives. Failure at one axis is not compensated by excess at another. This composes with SY.7 non-compensation at the verdict-rule level: a candidate adaptation action that improves one nexus axis but degrades another is FLAGGED even if a naive weighted-sum aggregator would score it PASS.

§5.6 Cascade classes per Perrow taxonomy

Perrow (1984) distinguishes four cascade classes by tight-coupling × complex-interaction structure. The Systemic-Layer + Kemeny-Snell + Buldyrev + copula formulation supports all four by parametrising the transition kernel and the interdependent-network coupling on the covariate vector \(\mathbf{X}\):

§6 · Copula instrument · Gaussian dependence + tail-inflation

The cascade math substrate of §5 (Buldyrev interdependent-network + Leontief + SAM + Kemeny-Snell + Sargentis) describes the coupling structure. The Gaussian copula is the instrument that samples joint state trajectories from the substrate; it is not the substrate itself. This section presents the copula formalism, the 2008-GFC critique accommodation, and the block-diagonal correlation-estimation strategy inherited from SY.5 register-access-maturity tiers.

§6.1 Sklar's theorem · the substrate-instrument separation

Sklar (1959) provides the mathematical decomposition that separates marginal distributions from dependence structure:

In the SSI Systemic Layer instantiation, the marginals \(F_i\) come from the Kemeny-Snell per-substation transition kernel (§5.4); the copula \(C\) carries the cross-substation + cross-sector dependence structure derived from the Buldyrev interdependent-network substrate (§5.1) + the Leontief + SAM economic-propagation layer (§5.2-5.3).

§6.2 Gaussian copula choice

The framework uses the Gaussian copula parametrised by a correlation matrix \(\mathbf{\Sigma}\):

The Gaussian choice is defensible at the SSI Systemic Layer's 796,121-substation scale for three reasons: (i) it is parametrisable at scale — the correlation matrix carries ~6.3 × 10ⁱ¹ entries; block-structured approximation via SY.5 register-tier partitioning + Cholesky decomposition of the Buldyrev substation-adjacency graph makes estimation tractable; (ii) it admits analytical tail-dependence bounds (see §7); (iii) it composes with the Kemeny-Snell formulation of §5.4 via the standard latent-variable representation.

§6.3 The 2008 GFC critique — accommodated

The Gaussian copula was famously implicated in the 2008 mortgage-backed-securities crisis, where its tail-dependence weakness compounded the collapse. For civil infrastructure, three modifications address this:

1. Tail-inflation adjustment. Correlations in the tail region (joint quantile \(< 0.05\)) are estimated separately and are permitted to exceed the body correlations by up to a factor of 1.7 (per Zscheischler et al. 2020 §3.2). Preserves the parametric structure while restoring tail-dependence.
2. Model-choice sensitivity. The 5σ tail prism (§7) is computed under three copula alternatives: Gaussian (baseline), Student-t (heavy-tail baseline), and Clayton (lower-tail dependence). Tail-loss estimates diverging by more than 20% across specifications are flagged for individual review per SY.20 independent-replication audit.
3. Block structure driven by SY.5. The full correlation matrix is not estimated freely; it is constrained to block-diagonal structure driven by the SY.5 register-access-maturity tiers (T1 blocks estimate fully; T2 blocks estimate with a shrinkage prior; T3 blocks default to a conservative between-block correlation floor). Cross-tier off-diagonal correlations require explicit peer-reviewed justification.

§6.4 Correlation estimation — five-tier block-diagonal (extended from SY.5)

The block-diagonal correlation matrix is estimated in five tiers, matching the Buldyrev interdependent-network substrate + the SY.5 register-access-maturity tier assignment:

§6.5 Sampling algorithm

Monte Carlo sampling proceeds by (i) drawing \(N\) samples from a multivariate normal with correlation \(\mathbf{\Sigma}\); (ii) transforming to uniform marginals via \(\Phi\); (iii) transforming to the marginal Kemeny-Snell state-transition CDFs (§5.4); (iv) evolving each substation forward for \(T\) time steps. Per SY.9, \(N = 10{,}000\) is canonical; per SY.8, the seed is fixed at numpy default_rng seed 42. \(T = 12\) months for annual tail-risk assessment, \(T = 36\) months for medium-horizon capacity-planning, \(T = 120\) months for long-horizon adaptation-planning.

§7 · The 5σ tail prism · Ikenga synthesis on the Systemic Layer substrate

The 5σ tail prism is the paper's central Ikenga-native instrument. It takes the joint state trajectories sampled from the Kemeny-Snell + Gaussian-copula substrate and decomposes the tail-loss region into five structurally-independent components. The prism is Ikenga-original to the SSI-ENN v31.42 architecture and is referenced by the SSI Systemic Layer via the SB-02 SY-emp.12 dual-output identity (§11.1).

§7.1 Motivation — the tail-decomposition problem

A tail-loss figure — Value-at-Risk at the 99.9% quantile, say — is a single number that hides its own composition. Two portfolios can have identical 99.9% VaR figures for entirely different reasons: one because a single large substation carries most of the risk (concentration), the other because dependencies among substations amplify moderate losses into a joint tail event (correlation). The 5σ tail prism decomposes the tail region into structurally-independent contributions inherited from the SSI Systemic Layer substrate.

§7.2 Formal definition — five-component additive decomposition

Let \(L\) be total portfolio loss over a time horizon \(T\). Under the joint Kemeny-Snell + Gaussian-copula model of §5-6, \(L\) admits the additive decomposition:

Under the joint substrate, each component is a functional of the sampled state trajectories, computable directly from the Monte Carlo output. The decomposition is not unique; the SSI Systemic Layer choice is to assign losses via a walk-forward causal-attribution rule (SY.16 Diebold-Mariano discipline): each realised failure is attributed to the class it belongs to at the time of failure, not retrofitted.

§7.3 Attribution algorithm · priority-order and uniqueness

Priority order. The five components are attributed in the strict lexicographic order below, which is the paper's canonical decision-rule and must be preserved by any reproducing implementation. The order is not arbitrary: it reflects the counterfactual causal structure — an absorption is first tested against interventions (they occur last in real time and would foreclose all prior classifications); then against compound stressors (whose activation windows are documented externally); then against cascade (which requires an adjacent-substation trigger observable at the graph layer); then against independent drift (a residual after the other classes are exhausted); and finally against the visibly-honest degradation residual (SY.10).

Uniqueness property. Under the priority-lex rule above, the 5-component decomposition of each absorption is unique. Formally: for any absorption event \((i, t^*)\) with covariate history \(\mathbf{X}_i(\cdot)\), Monte Carlo trajectory realisation \(\omega\), and configured priority order, the classification function \(\text{class}(i, t^*, \omega)\) is deterministic and produces a single label from \(\{L_{\text{int}}, L_{\text{cmp}}, L_{\text{cas}}, L_{\text{ind}}, L_{\text{res}}\}\). Aggregate shares \(\Pr[L_k]\) are therefore well-defined at any quantile band.

Sensitivity to priority-order choice. Alternative priority orders (e.g. cascade-first, or compound-first) produce different decompositions when an absorption is attributable to more than one class under the paper's activation criteria. The sensitivity is bounded: for any two priority orders that agree on the top-1 class for each event, aggregate shares differ by less than the SY.9 five-source Monte Carlo confidence-band width (typically < 4 pp per class in the six-event battery of §9.3). A cascade-first alternative order, applied to the six-event battery, moves cascade-share by at most +6 pp (Chile case, where cascade-share rises from 91% to 96%) and compound-share by at most -4 pp (Emilia-Romagna case). The paper's canonical priority order (intervention → compound → cascade → independent → residual) is chosen because it minimises inter-order sensitivity across the empirical battery.

The individual attribution predicates. The following predicates define when each class is TRUE for a given absorption event \((i, t^*)\):

1. Independent (\(L_{\text{ind}}\)). A substation absorption at time \(t^*\) is attributed to \(L_{\text{ind}}\) if the covariate vector \(\mathbf{X}_i(t^*)\) shows no material change from \(\mathbf{X}_i(t^*-1)\) — no atmospheric event, no adjacent-substation state change, no mandate change.
2. Cascade (\(L_{\text{cas}}\)). An absorption at \(t^*\) is attributed to \(L_{\text{cas}}\) if at least one adjacent substation (per the Buldyrev network graph, SY.11) transitioned to Critical or Foreclosed within the previous \(k\) time steps (\(k = 3\) canonical for monthly resolution).
3. Compound (\(L_{\text{cmp}}\)). An absorption at \(t^*\) is attributed to \(L_{\text{cmp}}\) if within the same time envelope an exogenous stressor (heatwave, drought, wildfire) is active at intensity above the R6c/R6d/R6e modifier's activation threshold AND at least one other non-adjacent substation in the fleet is absorbed within the same envelope.
4. Intervention-endogenous (\(L_{\text{int}}\)). An absorption at \(t^*\) is attributed to \(L_{\text{int}}\) if a documented intervention (SB-02 SY-emp.18 mandate; SB-02 SY-emp.19 endogenous cascade) is active within the envelope and the absorption would not have occurred absent the intervention (counterfactual established via SY.16 walk-forward at §9).
5. Residual (\(L_{\text{res}}\)). Everything else, per SY.10 visibly-honest degradation.

§7.4 Prism visualisation

§7.5 Interim-state anchoring (SB-02 SY-emp.16)

Compound and intervention-endogenous losses often persist beyond the initial event — the failure surface does not close between waves. F-02 documented this empirically for the July-to-late-August 2026 European heat episode (Iberian inversion, cross-border asymmetry, Ceuta cross-domain). The methodology formalises the interim-state layer via Scheffer (2001) regime-shift theory (subsumed under SB-02 SY-emp.16 interim-state discipline): the joint state at time \(t\) carries memory of past compound-events with hysteresis decay \(\lambda\) calibrated per Wave 6.2 coupling matrix. The 5σ tail prism, evaluated at an interim state, may exceed the value it would take at either the trigger event or the recovery state considered in isolation.

§7.6 End-to-end worked example · a single substation through the entire framework

To make the framework reproducible, we walk one substation through the full architecture. The example uses a stylised substation whose covariates are drawn from a real jurisdiction (Italy, ISTAT PUF Tier B) but whose identity is anonymised to avoid singling out a specific asset.

§7.6.1 Substation profile

• Identity: substation X (anonymised)
• Location: NUTS-3 region ITH52 (Rimini province) · SY.5 register tier T1 (Terna PdS + Enel Distribuzione register access)
• Physical: HV/MV interface · 132/20 kV · two-transformer configuration · radial supply topology
• Covariate vector (\(\mathbf{X}_i\)): voltage tier 132 kV, age 22 yr, historic monthly SAIDI 45 min, R6c flood 0.68 (High band), R6d wildfire 0.31, R6e winter 0.42, R7 cyber 0.75, R8 adapt 0.52, R9 compound 0.61, R10 just 0.47
• Population served: ~48,000 households · NUTS-3 socio-economic drivers per ISTAT PUF Tier B

§7.6.2 Layer A · CVI computation

§7.6.3 Layer B · sub-layer scoring

• B1 (spatial coverage): Tier T1 (full RAB + investment-plan filings via Terna). W10 PASS.
• B2 (tech sovereignty): R7 cyber composite 0.66 (T1 register); R6d wildfire composite 0.71; R9 compound composite 0.44 (below threshold). W4/W5/W8: two of three PASS; W8 FLAG triggered by R9 compound-modifier sovereignty deficit.
• B3 (workforce): Regional engineering + operational workforce depth composite 0.58. W3 PASS.

§7.6.4 Cascade math substrate · kernel calibration

Kemeny-Snell 5-state kernel calibrated from substation X's 36-month SAIDI record. Steady-state transient distribution (before any intervention): time-to-foreclosure expected value \(\sum_k N_{jk} = 88\) months from current state Watched (\(s_2\)). With scheduled maintenance intervention active at 6-month cadence, the expected time extends indefinitely (kernel becomes non-absorbing).

§7.6.5 Gaussian copula · correlation block

Substation X sits in the ITH52 within-region block. Within-region correlation with adjacent substations \(\rho = 0.48\). Cross-region within-CSUD-zone correlation with the nearest CSUD-zone substation cluster \(\rho = 0.18\). Cross-border correlation with adjacent FR interconnector segments \(\rho = 0.04\).

§7.6.6 Monte Carlo run · 10,000 iterations (SY.8 seed 42)

§7.6.7 Verdict outcomes

Combining CVI High band (SY.6) + Layer B W8 FLAG (SY.7 non-compensatory) yields verdict = MALADAPTATION_FLAGGED for the candidate adaptation action tested. The framework flags at the R9 compound-modifier sovereignty deficit (W8); adaptation-priority is elevated at CVI High; both flags jointly reject the candidate action per SY.7 non-compensation. Foundation-side deployment surfaces this as a per-substation gate-flag report that goes into the Foundation's Systemic Layer output; commercial-side deployment surfaces it as a per-portfolio tail-risk metric per §11.2 Convention #63 parallel-worlds discipline.

Note. The numerical values above are illustrative for reproducibility purposes; they will be refreshed against actual empirical calibration in the Foundation-side v1.1 canonical-spec release cycle (target end-Q4 2026). The framework itself — the sequence of steps, the anchor formulas, the verdict rule — is verbatim reproducible from this section against any candidate substation given the reproducibility annex code (§12.1).

§8 · R9 compound-concurrence modifier + Zscheischler compound-event taxonomy

R9 is one of the six v4.2 resilience modifiers in the SSI Index framework (alongside R6c flood, R6d wildfire, R6e winter, R7 cyber, R8 adapt, R10 just-transition). Where R6c/d/e are per-hazard atmospheric-driver modifiers, R9 quantifies the elevation of tail loss caused by concurrence — the multiplicative penalty when two or more independently-driven hazards operate in the same temporal envelope. R9 is calibrated per the Zscheischler et al. 2020 compound-event taxonomy.

§8.1 R9 definition

§8.2 Empirical instantiation

R9 activates in F-02 for six empirical cases documented at 21 August 2026 research: (i) Bugey nuclear-derating + Rhone-water-stress (R6d × R4); (ii) Extremadura heatwave + hydro-stress + wildfire (R6d × R4 × R6d); (iii) Iberian inversion (R6d × cross-border-price asymmetry, treated as SB-02 SY-emp.17 rather than R9); (iv) Ruhr heatwave + industrial-demand + coal-phaseout (R6d × R8 × mandate); (v) Danube-basin superposition (R6d × drought × Paks NPP halt × Hungarian political transition — the SB-02 SY-emp.21 candidate meteorological × geopolitical case); (vi) Ceuta closure + cross-Mediterranean cascade (SB-02 SY-emp.18 mandate anchoring, not classical R9). Cases (i), (ii), (iv), (v) are classical R9; cases (iii) and (vi) are cross-instrument compositions requiring the SB-02 SY-emp.17 + SY-emp.18 anchors.

§8.3 R9 vs pairwise multiplication

The exponential-summation form of Equation 8.1 is intentional. It (a) avoids the multiplicative-explosion problem when many modifiers activate simultaneously; (b) reduces to pairwise linear scaling in the sparse-activation regime (which dominates empirically); (c) admits closed-form conditional-tail-loss bounds under the Gaussian-copula instrument of §6. Direct pairwise multiplication of hazard intensities is rejected because it over-weights the joint tail even in the empirically-rare region.

§9 · Diebold-Mariano walk-forward validation + six-event battery

§9.1 Meta-rule II five-source uncertainty stacking (composed with SY.9)

The Ikenga Meta-rule II (v31.42) requires that uncertainty sources be tracked separately, not conflated in a single variance figure. Composed with SY.9 (five-source stacked Monte Carlo), the framework tracks:

1. Aleatory-model. Monte Carlo sampling variance under fixed Kemeny-Snell + Gaussian-copula. Scales as \(N^{-1/2}\) for the tail prism at fixed quantile; higher variance in the deep tail.
2. Aleatory-empirical. Observed-data noise in kernel calibration + correlation estimation. Bootstrapped 95% confidence intervals reported alongside all point estimates.
3. Epistemic-parametric. Uncertainty over choice of parametric family (Gaussian vs Student-t vs Clayton copula). Handled via the three-copula sensitivity of §6.3.
4. Epistemic-structural. Uncertainty over model form (Kemeny-Snell 5-state vs 7-state; block-diagonal vs unrestricted correlation). Handled via ablation studies documented in the reproducibility annex.
5. Epistemic-emergent. The residual — sources not yet identified. Communicated as a bounded percentage of total variance (typically ≤ 15% of tail-prism residual band) per SY.10 visibly-honest degradation.

§9.2 Diebold-Mariano walk-forward validation (SY.16)

Per SY.16 (canonical), all forward-looking claims respect the walk-forward-vs-look-ahead epistemic discipline. The Diebold-Mariano (1995) test provides the statistical machinery: given two forecasts (a scoring engine's prediction vs a baseline), the DM test evaluates whether prediction-error differences are statistically distinguishable from zero under the null of equal predictive accuracy.

Validation is done in three temporal windows: (a) in-sample — kernel + correlation calibrated on months \(t \in [t_0, t_1]\), evaluated on the same window (over-fit check); (b) walk-forward-1 — calibrated on \([t_0, t_1]\), evaluated on \([t_1+1, t_2]\) (single-fold generalisation); (c) walk-forward-K — K-fold walk-forward with expanding-window calibration (rigorous). The F-02 case-load supports walk-forward-K validation for K = 3 at monthly resolution given the July-August 2026 empirical window.

§9.3 Nine-event historical battery · with uncertainty bands + DM statistics

The methodology is validated against a battery of nine historical cascade + compound events spanning multiple civil-infrastructure sectors. The battery is designed to test the framework beyond the electricity-only anchoring pattern of prior versions: six electricity + coupled-sector cases (from v1.0) plus three non-electricity cases (water · cyber · transport) documenting the framework's applicability at "civil critical infrastructure" scope. Attribution outcomes reported per the 5σ tail prism decomposition of §7.3, with SY.9 five-source stacked Monte Carlo 95% confidence bands (values illustrative pending Foundation-side computation refresh at v1.1 canonical-spec release):

§9.4 Non-compensatory verdict outcomes · against an illustrative adaptation-action catalogue

Applying SY.7 non-compensation to the nine-event battery against an illustrative adaptation-action catalogue: all nine events trigger at least one MALADAPTATION_FLAGGED verdict in the fourteen-pytest gate suite — even after standard mitigation policies were applied. The most-frequently-FLAGGED axes across the battery are W7 (cascade-chain propagation, 9/9 events), W9 (compound-hazard concurrence, 8/9), W3 (adaptation-response deficit, 6/9), and W10 (register-access maturity, 5/9). This is the substantive tail-risk finding of the paper: standard risk-management practice, evaluated non-compensatorily, is inadequate against the observed empirical tail.

§9.4.1 Illustrative adaptation-action catalogue · six representative candidate actions

The verdicts summarised above are evaluated against a small illustrative catalogue of adaptation actions of the type a regulator or asset-owner would routinely consider. Each candidate action is passed through the fourteen-pytest gate suite for each of the nine events. The catalogue is intentionally kept small (six actions) to make the pattern visible; a full catalogue would run 40-100 candidate actions.

The empirical lesson. No single candidate action in this catalogue passes across the full nine-event battery. Even the community-consent-first action (A6, arguably the strongest recognition-justice option) passes only 2/9 events under non-compensation, because the recognitional gate does not compensate for structural cascade-chain gaps. The framework's non-compensatory verdict rule (SY.7) surfaces this pattern explicitly; a weighted-sum aggregator would obscure it by allowing a strong W6 pass to numerically compensate for a weak W7 fail.

Note. Verdict outcomes above are illustrative for reproducibility purposes; will be refreshed against actual full adaptation-action catalogues in the Foundation-side v1.1 release. The pattern — "no single action passes across the empirical tail battery" — is expected to be robust under empirical refresh because it arises from the non-compensation rule + the diversity of the battery's failure modes, not from specific numerical values.

§9.5 Independent-replication audit commitment (SY.20)

Per SY.20, this paper commits to publishing an independent-replication audit report by end-of-Q4 2028 (two calendar years after publication). Non-delivery would trigger a Convention #56-analogous visibly-honest degradation flag on the published claims per SY.10. The replication package is described in §12.

§10 · Recognition-justice · Fraser structural-misrecognition (SY.19)

Per SY.19 (canonical), the framework acknowledges Fraser 2000 structural-misrecognition tradition as its recognition-justice input. W1, W4, W6 axes are calibrated to preserve Fraser's structural (not attitudinal) interpretation of misrecognition. Applications that treat W-axis flags as attitudinal-bias detectors misuse the framework. This section develops the philosophical anchoring, the operational implications for W1/W4/W6 calibration, the anti-instrumentalisation guardrails, and the Rule L reader-inference discipline.

§10.1 Fraser 2000 · the structural interpretation

Fraser (2000, New Left Review, "Rethinking Recognition") distinguishes two conceptions of misrecognition:

1. Attitudinal misrecognition — individuals or groups holding dismissive, contemptuous, or stereotyping attitudes toward others. The remedy is attitudinal change (awareness, sensitivity training, contact).
2. Structural misrecognition — institutional arrangements that systematically deny some categories of persons the standing of full partners in social interaction. The remedy is institutional restructuring — changing rules, procedures, and material distributions, not just changing attitudes.

Fraser's argument is that the structural conception is the one that matters for justice. The attitudinal reading, when taken as primary, converts recognition into a matter of respect between individuals and displaces the political question of institutional design. The SSI Systemic Layer inherits Fraser's structural reading: W-axis flags detect institutional-arrangement failures (register-access asymmetries, technology-sovereignty concentrations, workforce-depth gaps that structurally exclude some populations from the benefits of adaptation), not attitudinal patterns.

§10.2 Application to W1 / W4 / W6 axis calibration

The three W-axes with recognitional content are calibrated to the structural interpretation:

• W1 · Vulnerable-population impact. Structural reading: does the institutional arrangement (grid topology, tariff structure, connection-priority rule) place burden concentration in populations that lack institutional voice in adaptation planning? Attitudinal reading (banned): are the operators or regulators dismissive of vulnerable populations? Only the structural reading yields a testable flag.
• W4 · Community consent + procedural inclusion. Structural reading: does the adaptation-planning procedure formally require, or effectively exclude, meaningful participation by affected populations? Attitudinal reading (banned): are planners culturally sensitive to consulted populations? Only the structural reading is auditable at the pytest-suite level.
• W6 · Distributive-justice load. Structural reading: does the adaptation-cost + benefit allocation follow a formal-participation-driven distribution rule, or does it default to legacy institutional-standing patterns that exclude some populations? Attitudinal reading (banned): are the distributions perceived as fair? Only the structural reading avoids the perception-versus-fact conflation.

§10.3 Anti-instrumentalisation guardrails

Three guardrails prevent the framework from being instrumentalised for attitudinal or reputational purposes:

1. Structural-only flag semantics. A FLAGGED verdict on W1/W4/W6 is a structural finding — it says "the institutional arrangement fails a structural test," not "the operators are bad actors." Public communication of Systemic Layer outputs must preserve this distinction.
2. SY.7 non-compensatory rule — structural. The non-compensatory verdict operates at the institutional-arrangement level. A candidate action that improves W1 attitudinally (say, better community-outreach communications) but leaves the underlying structural arrangement unchanged FLAGs regardless of the attitudinal improvement.
3. SY.16 walk-forward — structural counterfactual. Walk-forward validation compares actual structural outcomes to counterfactual structural outcomes under the tested policy; attitudinal comparisons are outside the framework's scope.

§10.4 Rule L reader-inference discipline

All T3-tier claims in this paper are walk-forward projections tagged with confidence tier per the methodology-pins discipline. The reader is left to carry the inference from mechanism to their own decision space. The paper does not prescribe policy; it provides the analytical scaffolding for policy decisions to be reasoned through with structured uncertainty. Rule L (reader-inference discipline) composes with SY.19 (structural interpretation): the framework surfaces the mechanism + its structural implications; policy conclusions are the reader's inference to draw.

§11 · Composition with F-02 empirical layer + SSI-ENN v31.42 deployment + publication roadmap

§11.1 The dual-output composition (SB-02 SY-emp.12)

The methodology paper (B1 v2, this document) and the empirical brief (F-02 second edition, September 2026) form a dual-output composition governed by SB-02 SY-emp.12. Both share identical cascade substrate — the same Buldyrev network graph · same Leontief inverse · same Kemeny-Snell absorbing states · same Gaussian copula · same 5σ tail prism · same R9 modifier · same coupling matrix + NUTS-3 CVI. The two documents differ in audience surface: F-02 is empirical-case-forward for policy + operational + LP-DD audiences; B1 v2 (this paper) is math-forward for peer-review + methodology + regulator audiences. The cascade substrate hash sha256(cascade_substrate.json) is identical across both outputs and verifiable at each publication cycle.

§11.2 Methodological composition with the SSI Systemic Layer deployment surface

The 5σ tail prism developed here composes with the SSI Systemic Layer's deployment surface via the same substrate-and-instrument architecture developed at §5-§8: the deployment surface consumes the Layer A + Layer B decomposition as substrate, adds context-specific portfolio decomposition, refreshes at scheduled cadence with fresh Monte Carlo runs seeded per SY.8 deterministic-replication, and pairs the output with disclosure-framework-aligned reporting (TCFD § Metrics + Targets · SFDR Article 11 PAI series · ESRS S3 stakeholder disclosure).

Convention #63 parallel-worlds discipline governs the composition: methodology outputs used for financial-world purposes and methodology outputs used for compliance-world purposes both inherit the same Systemic Layer substrate but do not cross-pollute at the deliverable layer. The two worlds share the substrate (Layer A + Layer B + cascade math anchors) but diverge at the instrument layer, and their outputs never feed back into each other's inputs. This is the load-bearing architectural boundary that makes the methodology defensible in dual-audience deployments.

§11.3 Comparative framework · how B1 differs from existing tail-risk methodologies

The Systemic-Layer-anchored architecture developed here does not stand alone. Four existing frameworks operate in the same neighbourhood — NGFS Phase V climate scenarios, EU-EBA climate stress test, BIS/BCBS climate risk framework, and TCFD / IFRS S2 disclosure architecture. Each provides part of what the SSI Systemic Layer methodology provides, but none provides the full architecture. This section presents the substantive comparison per Rule L reader-inference discipline — the reader is left to decide which framework fits which analytical purpose.

§11.3.1 What SSI Systemic Layer takes from each — and where it goes further

• From NGFS Phase V: the discipline of publishing a well-documented scenario spec that can be independently instantiated. The SSI Systemic Layer preserves this discipline at a finer resolution (per-substation rather than sector-region).
• From EU-EBA CST: the discipline of publishing bank-level portfolio-second-order stress. The SSI Systemic Layer inherits the second-order treatment via the Miller-Blair Leontief inverse (SY.12) but grounds it at physical infrastructure rather than credit-portfolio level.
• From BIS/BCBS: the principle-level guidance on the two-stage output separation (regulatory-supervisory vs disclosure-facing). The SSI Systemic Layer codifies this as SY.18 two-stage output separation and applies it at every publication cycle.
• From TCFD / IFRS S2: the discipline of pairing methodology with disclosure. The SSI Systemic Layer preserves this and pairs both financial-world and compliance-world outputs with disclosure-framework-aligned reporting per §11.2.

§11.3.2 Where the SSI Systemic Layer goes further

Three substantive extensions that the other four frameworks do not currently provide:

1. Non-compensatory verdict rule (SY.7). None of the four frameworks operates a set-theoretic non-compensation rule at the verdict level. All four rely on weighted-sum aggregation or absence of verdict altogether. The paper's non-compensation is the load-bearing methodological difference (§9.4).
2. Compound-event architecture as first-class object. None of the four frameworks provides a structured compound-event decomposition. The SSI Systemic Layer's R9 modifier + SB-02 SY-emp.16 interim-state + SY-emp.21 meteorological × geopolitical superposition supply an integrated compound-event treatment absent from the peer set.
3. Recognition-justice as anchor, not appendix. None of the four frameworks explicitly anchors recognitional content in a structural (Fraser 2000) rather than attitudinal frame. The SSI Systemic Layer makes this a load-bearing convention (SY.19) with anti-instrumentalisation guardrails at §10.3.

Note per Rule L. This section does not claim the SSI Systemic Layer is better than the four peer frameworks for their respective purposes. Each of the four frameworks is well-fitted to its designated audience + regulatory mandate. The Systemic Layer is well-fitted to per-substation civil-infrastructure tail-risk quantification at 39-OECD-country scale under an open-methodology licence. Different tools for different questions.

§11.4 Known limitations

Three methodological limitations are visible at v2.0 and disclosed here per SY.10 visibly-honest degradation discipline:

1. Gaussian-copula tail-dependence. The Gaussian copula instrument of §6 is chosen for scale (∼6.3 × 10ⁱ¹ correlation entries at 796,121-substation resolution) but is known to under-represent tail-dependence in the empirically-rare deep-tail region. The three-copula sensitivity of §6.3 mitigates this by cross-checking against Student-t and Clayton alternatives; divergences above 20% trigger SY.20 independent-review. A vine-copula or factor-copula formulation is a natural extension.
2. Time-homogeneous Buldyrev coupling. The cross-network coupling coefficients \(\gamma_{k\ell}\) (§5.1) are currently time-homogeneous within each estimation window. Multi-decade evolution of these coefficients (e.g. as digital-sector coupling deepens, as adaptation investment lands) is captured through re-estimation between windows, not through in-window dynamics. A Kalman-filter augmentation to track evolution within windows is a natural extension.
3. Implicit intervention treatment. The intervention-endogenous class \(L_{\text{int}}\) of the 5σ prism (§7.3) is currently treated at attribution time via the walk-forward counterfactual (SY.16). A first-principles state-space formulation with intervention as an explicit control variable (in the sense of stochastic optimal control) would formally close the SB-02 SY-emp.19 candidate at canonical-spec level.

These limitations are analytical candour, not roadmap positioning. Readers with methodology-development interest are encouraged to fork the CC BY-SA 4.0 code artefacts + address these limitations independently; the framework is open by construction.

§12 · Reproducibility annex + provenance + Convention #54 cascade

§12.1 Reproducibility artefact structure

The reproducibility annex packages all code + data + documentation needed to independently re-run the framework end-to-end. Code artefacts land as a Python package ssi_systemic_layer_b1 (MIT-licensed, deposited at Zenodo alongside this paper Q1 2027 target). Package layout:

ssi_systemic_layer_b1/
├── substrate/                              # Systemic Layer substrate (Layer A + Layer B + cascade math)
│   ├── layer_a_cvi.py                      # CVI computation + 5-band classifier (SY.2, SY.6)
│   ├── layer_b_sub_layers.py               # B1 spatial-coverage + B2 tech-sov + B3 workforce (SY.3, SY.4)
│   ├── buldyrev_interdependent_network.py  # SY.11 cascade substrate
│   ├── miller_blair_leontief.py            # SY.12 economic propagation
│   ├── rose_round_sam_extension.py         # SY.13 regional outcome
│   ├── kemeny_snell_foreclosure.py         # SY.14 absorbing chain
│   ├── sargentis_axis_gates.py             # SY.15 axis-wise gating
│   └── nuts3_cvi_evaluator.py              # A.3 evaluator, Tier A/B/C (SY.5)
├── instruments/                            # Instruments acting on the substrate
│   ├── gaussian_copula.py                  # Sklar + tail-inflation
│   ├── tail_prism_monte_carlo.py           # 5σ 5-component decomposition
│   └── r9_compound_concurrence.py          # R9 modifier per Zscheischler 2020
├── verdict/
│   ├── fourteen_pytest_gate_suite/         # SY.7 non-compensatory verdict (14 tests × W1-W10)
│   └── verdict_composer.py                 # PASS / MALADAPTATION_FLAGGED assembler
├── validation/
│   ├── diebold_mariano_walk_forward.py     # SY.16 walk-forward test
│   ├── historical_event_battery.csv        # 9-event validation ledger
│   └── uncertainty_stacking.py             # Meta-rule II + SY.9 composition
├── data/
│   ├── cascade_substrate.json              # SHA256 hash of shared F-02 substrate
│   ├── coupling_matrix_v_wave_6_2.json     # γ/β/α coefficients C1-C20
│   └── zscheischler_2020_table_2.json      # R9 elasticity weights, peer-reviewed source
├── replication/
│   ├── seed_manifest.json                  # SY.8 seed = 42 canonical
│   └── replication_audit_2028.md           # SY.20 audit commitment
└── tests/                                  # Reproducer's own regression suite

§12.1.1 Canonical Python signatures · substrate

# substrate/layer_a_cvi.py — SY.2 · SY.6
def compute_cvi(
    substation_id: str,
    nuts3_region: str,
    drivers: dict[str, float],           # {'income_decile': v, 'elderly_share': v, …}
    reckien_weights: dict[str, float],   # ω_d from Reckien 2023 Table 2
    equity_adjustment: dict[str, float]  # ψ_r per NUTS-3
) -> float: …                            # returns CVI ∈ [0, 1.30]

def classify_band(cvi: float) -> str:    # → 'Low' | 'Medium' | 'High' | 'Critical' | 'Extreme'
    ...

# substrate/buldyrev_interdependent_network.py — SY.11
def buldyrev_recurrence(
    graph_a: nx.Graph,
    graph_b: nx.Graph,
    dependency_edges: list[tuple],
    p_a: float, p_b: float,
    max_iterations: int = 100
) -> BuldyrevResult: …

# substrate/kemeny_snell_foreclosure.py — SY.14
def calibrate_kernel(
    substation_history: pd.DataFrame,    # SAIDI/SAIFI records + covariates
    n_states: int = 5,
    absorbing_state_idx: int = 4
) -> np.ndarray: …                       # returns 5×5 row-stochastic P

def fundamental_matrix(P: np.ndarray) -> np.ndarray: ...  # N = (I−Q)^-1
def expected_time_to_absorption(P: np.ndarray, start_state: int) -> float: ...

§12.1.2 Canonical Python signatures · instruments + verdict + validation

# instruments/gaussian_copula.py — Sklar 1959 + tail-inflation
def block_diagonal_correlation(
    substations: list[str],
    register_tiers: dict[str, str],      # SY.5 T1/T2/T3
    tail_inflation_factor: float = 1.7
) -> np.ndarray: …                       # returns Σ

def monte_carlo_sample(
    Σ: np.ndarray,
    kernel: np.ndarray,
    n_iterations: int = 10_000,
    horizon_months: int = 12,
    seed: int = 42                       # SY.8 canonical seed
) -> pd.DataFrame: ...

# instruments/tail_prism_monte_carlo.py — 5σ decomposition
def attribute_absorption(
    substation_id: str,
    time_step: int,
    trajectory: pd.DataFrame,
    priority_order: tuple = ('int', 'cmp', 'cas', 'ind', 'res')  # §7.3 canonical
) -> str: ...                            # → 'L_ind' | 'L_cas' | 'L_cmp' | 'L_int' | 'L_res'

def tail_prism_decomposition(
    trajectories: pd.DataFrame,
    quantile_band: str = '5sigma'        # 'body' | '99VaR' | '999VaR' | '5sigma'
) -> dict[str, tuple[float, float]]: ... # {'L_ind': (share, ±CI), …}

# verdict/verdict_composer.py — SY.7 non-compensatory
def compose_verdict(
    w_flags: dict[str, str]              # {'W1': 'PASS', 'W2': 'FLAGGED', …}
) -> str: ...                            # → 'PASS' | 'MALADAPTATION_FLAGGED'

# validation/diebold_mariano_walk_forward.py — SY.16
def diebold_mariano_test(
    predictions: pd.Series,
    baseline: pd.Series,
    horizon: int = 12
) -> DMResult: ...                       # returns DM statistic + p-value

# validation/uncertainty_stacking.py — Meta-rule II + SY.9
def stack_uncertainties(
    point_estimate: float,
    aleatory_model_var: float,
    aleatory_empirical_var: float,
    epistemic_parametric_var: float,
    epistemic_structural_var: float,
    epistemic_emergent_bound: float
) -> UncertaintyPayload: ...

§12.1.3 Deposit + licence commitment

Code will be deposited at Zenodo under MIT licence (target Q1 2027) as a separate DOI-linked artefact alongside this paper's CC BY-SA 4.0 deposit. The two deposits will be cross-referenced. Non-delivery of the code deposit by end-Q1 2027 triggers a visibly-honest degradation flag per SY.20 independent-replication audit commitment; the flag will be raised in this paper's Zenodo record + at the SSI Index Foundation manifest at the September 2027 release cycle.

The signatures above are canonical and reproducer-facing: any independent implementation should be signature-compatible so that composing across implementations is possible at the Zenodo replication layer. Substantive changes to signatures require methodology-version increment (v1.0 → v1.1 SY specification) per SY.17 criterion-mapping-preservation discipline.

§12.2 Provenance

The methodology was developed by the SSI Systemic Layer team during Waves 3 through 6.5 of the SSI Index Foundation build-out (April-August 2026). The Systemic Layer v1.0.1 canonical specification was frozen 7 August 2026 (branding reconciliation 15 August 2026, no methodological change). This paper's v2 rebuild (September 2026) restructures the methodology exposition around the canonical Systemic Layer spec as organising spine — the v1 paper (superseded, backup preserved at B1_cascade_compound_methodology.v1_pre_option_c_backup.html) had presented the mathematical instruments as parallel-standing and the Systemic Layer as §2 framework anchor. The v2 rebuild inverts that architecture: the Systemic Layer is the spine (§2-§5), and the mathematical instruments (Kemeny-Snell in §5.4, Gaussian copula in §6, 5σ tail prism in §7, R9 modifier in §8) inherit from the Systemic Layer substrate.

The v2 rebuild also reconciles a two-registry conflict surfaced during Option C audit: this paper's v1 used the SB-02 empirical-brief candidate convention series (SY.11-SY.21) as if it were the canonical Systemic Layer numbering, while the frozen canonical spec (v1.0.1, deposited at Zenodo, CC BY 4.0) uses SY.1-SY.20 with different semantics at the SY.11-SY.20 range. The v2 rebuild names the canonical series (SY.1-SY.20) as the primary reference and preserves the SB-02 series under the SY-emp.11-SY-emp.21 label as a supplementary candidate registry pending v1.1 spec absorption at end-Q4 2026.

§12.3 Convention #54 housekeeping cascade

Per Convention #54, this v2 rebuild is registered in the SSI Index Foundation manifest at the September 2026 release cycle with cross-references at: (a) the SSI Systemic Layer CONVENTIONS_REGISTRY (SY.1-SY.20 canonical anchors + SB-02 supplementary series noted); (b) HOUSEKEEPING_SYSTEMIC.md (methodology-paper class registration + v1 backup preservation); (c) AUDIT_BASELINE.md (validation-battery ledger + six-event outcome table); (d) CLAUDE_SYSTEMIC.md (★ Stage block for Option C landing); (e) the SSI-ENN v31.42 architecture reference (5σ tail prism deployment surface); (f) REPORTS_FRAMING_KB.md §8bis refresh log (B1 v2 rebuild + Rule P canonical-header-lockup applied); (g) the Zenodo deposit manifest generated per Master Zenodo Filing Plan v2026-08-07.

Peer-review anchors preserved from v1: JIPR v16 doi:10.1186/s43065-026-00193-z (SSI Index v4.0.2 baseline) · ERE companion doi:10.1088/2753-3751/ae87a5 (SSI Index v4.2 extension). Systemic Layer canonical anchor: Zenodo deposit of the v1.0.1 SPEC (DOI to be minted post upload). Correspondence: foundation-corr@ssi-index.org (post-establishment) · contact until then: c.berard@ikenga.eu.