1. Route-Dependence

Consider three jurisdictions — Pakistan (PK), the United Arab Emirates (AE), and the Seychelles (SC) — with a bilateral recognition agreement between each pair. An entity incorporated in PK wants to operate in SC.

Route 1: PK directly to SC. The corridor re-evaluates Sanctions and Tax at SC and accepts PK’s evaluations across the other corresponded domains.

Route 2: PK to AE, then AE to SC. The first corridor re-evaluates Sanctions and Tax at AE, accepting the rest; the second re-evaluates Sanctions and Securities at SC, accepting the rest.

Does the entity arrive in SC in the same state either way? Already on these masks, no. The direct route delivers PK’s Securities evidence to SC; the staged route discards it at the second hop and SC evaluates Securities afresh. An entity whose PK Securities record is favorable prefers Route 1; an entity that would fare better under a fresh SC evaluation prefers Route 2. Route choice changes a regulatory outcome, and inconsistent determinations arise with no party acting in bad faith.

The mechanism is the classical one of treaty shopping: an entity routes through an intermediate jurisdiction whose bilateral agreements treat it better than the direct agreement does. Tax authorities have named and countered conduit structures of this kind since the OECD’s report on conduit companies (OECD, 1986), most recently through BEPS Action 6 (OECD, 2015). This note formalizes the mechanism on the applicable-rule fragment and gives a decidable check for its presence in a triangle of agreements.

The cause is not the way verdicts combine — Section 3 recalls that per-domain aggregation is a semilattice meet, indifferent to order. The cause is that the three corridors are three independently negotiated bilateral agreements. Nothing in bilateral negotiation makes the PK–SC terms the composite of the PK–AE and AE–SC terms, and when they fail to be, the routes genuinely differ. The questions are what “the composite of the terms” means precisely, and what agreement of terms guarantees. This note answers both on the applicable-rule fragment, and Section 5 carries the full comparison for the triangle above.

2. Corridors

Definition 1 (Jurisdiction-corridor graph). Let \mathcal{J} be a directed graph whose vertices are jurisdictions. Each jurisdiction U has a finite set \mathcal{D}_U of compliance domains — its regulatory vocabulary — and a versioned rulebook, with w ranging over rulebook versions. An edge m : U \to V is a corridor: a bilateral instrument under which V accepts specified evidence produced in U. A bilateral agreement induces two directed edges U \to V and V \to U, not one symmetric identification; Singapore may accept Abu Dhabi’s AML evaluation without Abu Dhabi reciprocating, and each direction carries its own parameters.

Definition 2 (Corridor parameters). A corridor m : U \to V carries four parameters (R, \mu, \gamma, W).

• Domain correspondence \mu_{U,V} : \mathcal{D}_U \rightharpoonup \mathcal{D}_V, a partial injection pairing each source domain with at most one destination domain and conversely, defined exactly where the corridor instrument names a counterpart. The correspondence records that two domains regulate the same subject matter; it is independent of what is recognized, and it does not by itself move evidence.

• Re-evaluation mask R_{U,V} \subseteq \mathcal{D}_V: the destination domains V evaluates afresh regardless of source evidence, counterpart or not. Sanctions determinations are sovereign political acts with extraterritorial reach, not quality assessments one jurisdiction can perform on another’s behalf, so \mathrm{Sanctions} \in R_{U,V} for every general-purpose corridor; a subnetwork with a shared sanctions authority may omit it only by making that authority an explicit premise of the corridor instrument.

• Grade maps \gamma_{U,V,d}, one for each domain d with \mu(d) defined and \mu(d) \notin R — the carried domains of Definition 3: a monotone map from the source grade factor at d to the destination grade factor at \mu(d) (the grade factors of Section 3), fixing the strength at which V accepts U’s verdict. Recognition may downgrade — U’s Compliant arriving as V’s Pending — and never upgrades.

• Version window W_{U,V}: the set of rulebook-version pairs (w_U, w_V) for which the other three parameters are valid. A corridor negotiated against particular rulebooks is not presumed accurate after either side revises; operation outside the window requires renegotiated parameters.

Definition 3 (Carriage). For a corridor with parameters (R, \mu, \gamma, W), a source domain d \in \mathcal{D}_U is carried when \mu(d) is defined and \mu(d) \notin R. Write \mathrm{carry}_{U,V} for the partial injection d \mapsto \mu(d) restricted to carried domains. Every destination domain outside the image of \mathrm{carry}_{U,V} — masked, or without a counterpart — is freshly evaluated at V. This fail-closed default gives every destination domain exactly one status: carried into, or freshly evaluated.

3. States and Transit

Definition 4 (Compliance assignment). The compliance assignment F gives each jurisdiction U the product F(U) = \prod_{d \in \mathcal{D}_U} S_d(U), where S_d(U) is U’s grade factor at domain d: on the applicable-rule fragment treated here, the finite chain of compliance grades \mathrm{NonCompliant} < \mathrm{Pending} < \mathrm{Compliant}, in the normal form of the companion note How Compliance Composes. NotApplicable and Exempt are applicability markers, not grades; the companion note handles mixed applicability by structured outcomes, and this note stays on the fragment where every verdict carries a grade. An element of F(U) is a compliance state at U: one grade per domain.

Transit. Fix a corridor m : U \to V, a version pair in W_{U,V}, and an entity with source state s \in F(U). The arrival state t \in F(V) is defined coordinate-wise: t(d') = \gamma_{U,V,d}(s(d)) \quad \text{if } d' = \mathrm{carry}_{U,V}(d), t(d') = \mathrm{eval}_V(d') \quad \text{otherwise,} where d is the unique carried preimage and \mathrm{eval}_V(d') is V’s fresh evaluation of the entity in domain d' under V’s rulebook version. Two modeling assumptions are in force, and they delimit the fragment:

• (E1) Deterministic evaluation. Under a fixed rulebook version, \mathrm{eval}_V is a function of the entity’s evidence and the domain. Discretionary evaluation — a judgment the rule leaves to a named official — is excluded; the companion paper Lex types such judgments as discretion holes, and route coherence in their presence is open.

• (E2) Primary-evidence evaluation. Fresh evaluation consumes the entity’s primary evidence — its own documents and facts — not the verdicts other jurisdictions have issued along the way. Carried verdicts enter the destination state only through carriage. Since primary evidence is the entity’s own, \mathrm{eval}_V does not depend on the route by which the entity arrived.

Where V independently holds its own verdict on a domain carried into — the entity already present at V, the multi-harbored case — the operative verdict is the meet of the carried grade and V’s own grade in V’s factor, by the composition law of the companion note. The meet on each factor is associative, commutative, and idempotent, so the order of aggregation never affects it. Every route-dependence question in this note therefore concerns which grades arrive, not how grades combine once arrived; both routes feed the same meet.

Transit along A \to B followed by transit along B \to C is not, in general, transit along any single corridor A \to C: fail-closed re-evaluation at B replaces carried evidence with B’s own evaluation wherever carriage fails, and the second hop may then forward B’s evaluation onward. Whether a direct corridor reproduces the staged behavior is exactly the question the next section makes precise.

4. Route Coherence

Fix corridors A \to B, B \to C, and A \to C, and a version profile v = (w_A, w_B, w_C) with (w_A, w_B) \in W_{AB}, (w_B, w_C) \in W_{BC}, and (w_A, w_C) \in W_{AC}. All comparisons in this section are at v; at another profile the corridors present different parameters and the comparison is a different instance.

Definition 5 (Route snapshot). The staged route A \to B \to C and the direct corridor A \to C reduce to snapshots over the same endpoints:

• carriage map \kappa: for the staged route, \kappa_{\mathrm{st}} = \mathrm{carry}_{BC} \circ \mathrm{carry}_{AB}, the composition of partial injections, defined where both are; for the direct corridor, \kappa_{\mathrm{dir}} = \mathrm{carry}_{AC};

• grade maps on carried domains: \gamma^{\mathrm{st}}_d = \gamma_{BC,\mu_{AB}(d)} \circ \gamma_{AB,d} for d \in \mathrm{dom}\,\kappa_{\mathrm{st}}, and \gamma^{\mathrm{dir}}_d = \gamma_{AC,d} for d \in \mathrm{dom}\,\kappa_{\mathrm{dir}};

• forwarding set \Phi: the destination domains that receive the intermediate jurisdiction’s fresh evaluation rather than evidence originating at the source, \Phi_{\mathrm{st}} = \mathrm{carry}_{BC}(\mathcal{D}_B \setminus \mathrm{im}\,\mathrm{carry}_{AB}) and \Phi_{\mathrm{dir}} = \varnothing;

• clause set K: the route’s instrument clauses — governing law, dispute forum, waivers, currency of obligation, termination and dynamic-alignment clauses — in the normal form fixed by the instrument layer, the component of the corridor system that holds each bilateral instrument’s operative clauses in that form; this note takes the layer, its normal form, and decidable equality on it as given. For the direct corridor, K_{\mathrm{dir}} is the normalized clause set of the A \to C instrument; for the staged route, K_{\mathrm{st}} is the instrument layer’s normalized combination of the clause sets of the two traversed instruments, A \to B and B \to C, under the route-combination rule that layer fixes — which clauses of each traversed instrument govern the staged arrival.

The freshly evaluated set at C is determined by the rest: E = \mathcal{D}_C \setminus (\mathrm{im}\,\kappa \cup \Phi). By injectivity, \mathrm{im}\,\kappa_{\mathrm{st}} and \Phi_{\mathrm{st}} partition \mathrm{im}\,\mathrm{carry}_{BC}, so what is freshly evaluated at the destination is determined by the final edge alone; what varies with the route is the provenance of everything else.

Definition 6 (Route coherence). The direct corridor is coherent with the staged route at v when the snapshots are equal: \kappa_{\mathrm{dir}} = \kappa_{\mathrm{st}} as partial maps — equal domains of definition and equal values — \gamma^{\mathrm{dir}}_d = \gamma^{\mathrm{st}}_d for every d in that common domain, \Phi_{\mathrm{st}} = \Phi_{\mathrm{dir}} = \varnothing, and K_{\mathrm{dir}} = K_{\mathrm{st}}. When the snapshots differ, the route-coherence defect is the set of components on which they differ; each component names the domains in disagreement and, through them, the corridor agreements whose renegotiation would close the gap. Version windows are compared alongside: a profile admissible for one route and not the other is itself a defect — the routes differ in availability before they can be compared in state.

Theorem 1 (Route independence on the applicable fragment). If the direct corridor is coherent with the staged route at v, then under (E1)–(E2) the two routes deliver the same arrival state at C for every source state s \in F(A).

The arrival state here is the graded object of Definition 4 and nothing else. Where a determination also carries the set of authorities whose grades composed it — the companion The Recognition Protocol gives one design that does, so that a participant can refuse determinations composed through a jurisdiction it has declined — that set distinguishes a staged arrival from a direct one by construction. Theorem 1 is then route independence in the grade component, and it is silent on the provenance component, which is route-dependent by design and useless if it were not. Nothing below depends on which convention is chosen, because the snapshot of Definition 5 fixes the grade component alone.

Proof. Coherence gives \Phi_{\mathrm{st}} = \varnothing, so no coordinate of the staged arrival depends on \mathrm{eval}_B. Fix d'' \in \mathcal{D}_C.

If d'' \in \mathrm{im}\,\kappa_{\mathrm{st}} = \mathrm{im}\,\kappa_{\mathrm{dir}}, let d be its unique preimage under the common carriage map. Along the staged route, the coordinate \mu_{AB}(d) holds \gamma_{AB,d}(s(d)) on arrival at B, and d'' holds \gamma_{BC,\mu_{AB}(d)}(\gamma_{AB,d}(s(d))) = \gamma^{\mathrm{st}}_d(s(d)) on arrival at C. The direct corridor delivers \gamma^{\mathrm{dir}}_d(s(d)). The grade maps agree, so the values agree.

If d'' \notin \mathrm{im}\,\kappa_{\mathrm{st}}, then on the direct route d'' \notin \mathrm{im}\,\kappa_{\mathrm{dir}} and d'' is freshly evaluated. On the staged route, if d'' were in \mathrm{im}\,\mathrm{carry}_{BC}, its preimage d' would lie either in \mathrm{im}\,\mathrm{carry}_{AB}, putting d'' \in \mathrm{im}\,\kappa_{\mathrm{st}}, or outside it, putting d'' \in \Phi_{\mathrm{st}} = \varnothing; so d'' \notin \mathrm{im}\,\mathrm{carry}_{BC} and d'' is freshly evaluated. By (E1)–(E2), both routes invoke the same evaluation, at version w_C, of the same primary evidence. \blacksquare

The check is conservative. Coherence compares the agreements, not sampled outcomes. A defective route pair can still deliver equal states for a particular entity — a fresh evaluation may happen to coincide with the carried grade it replaces — but nothing in the agreements underwrites the coincidence, and a compliance system wants agreement guaranteed, not observed. None of the snapshot components is redundant: Section 5 realizes carriage and forwarding defects with arrival states that differ; if the carriage maps agree at d but \gamma^{\mathrm{dir}}_d and \gamma^{\mathrm{st}}_d differ at some grade g, a source state with s(d) = g arrives at different grades at the common coordinate \kappa_{\mathrm{dir}}(d); a clause defect changes no grade but subjects the same arrival to different governing terms.

Decidability. The carriage map, grade maps, and forwarding set are finite objects over finite domain sets and finite grade chains, so equality on each is decidable; clause sets are compared in the instrument layer’s normal form, on which equality is decidable by the standing assumption of Definition 5. Snapshot equality is therefore a finite conjunction of decidable equalities. The machine-checked artifact accompanying the companion Op paper proves that snapshot-equality checking at the normalized layer — over carried cells, fresh-evaluation domains, and instrument clause identifiers — is sound, complete, and decidable, and raises no false obstruction: a failed check refutes coherence (route_coherence_checker_sound, route_coherence_checker_complete, route_coherence_checker_false_obstruction, route_coherence_decidable). Definitions 5 and 6 supply, on paper, the reduction from corridor-triple data on this fragment to finite snapshots of that kind; mechanizing the reduction, and the clause normal form it presupposes, remains open.

5. Worked Example

Let the three vocabularies share the domain names \{\mathrm{Sanctions}, \mathrm{Tax}, \mathrm{Securities}, \mathrm{Corporate}, \mathrm{KYC}, \mathrm{DigitalAssets}\}, with every correspondence the identity on the names it lists, all grade maps the identity on the common grade chain, and equal clause sets — so any disagreement below is genuine. The instruments, at a common version profile, are those of Section 1 made precise:

• PK \to AE: correspondence on all six names; mask R_{\mathrm{PK},\mathrm{AE}} = \{\mathrm{Sanctions}, \mathrm{Tax}\}.
• AE \to SC: correspondence on all six names; mask R_{\mathrm{AE},\mathrm{SC}} = \{\mathrm{Sanctions}, \mathrm{Securities}\}.
• PK \to SC: correspondence on five names — none for DigitalAssets; mask R_{\mathrm{PK},\mathrm{SC}} = \{\mathrm{Sanctions}, \mathrm{Tax}\}.

The carriage maps (Definition 3) are identities on their domains: \mathrm{carry}_{\mathrm{PK},\mathrm{AE}} : \{\mathrm{Securities}, \mathrm{Corporate}, \mathrm{KYC}, \mathrm{DigitalAssets}\}, \quad \mathrm{carry}_{\mathrm{AE},\mathrm{SC}} : \{\mathrm{Tax}, \mathrm{Corporate}, \mathrm{KYC}, \mathrm{DigitalAssets}\}, \quad \mathrm{carry}_{\mathrm{PK},\mathrm{SC}} : \{\mathrm{Securities}, \mathrm{Corporate}, \mathrm{KYC}\}.

Both snapshots, computed in full: \kappa_{\mathrm{st}} : \{\mathrm{Corporate}, \mathrm{KYC}, \mathrm{DigitalAssets}\}, \quad \Phi_{\mathrm{st}} = \mathrm{carry}_{\mathrm{AE},\mathrm{SC}}(\{\mathrm{Sanctions}, \mathrm{Tax}\}) = \{\mathrm{Tax}\}, \quad E_{\mathrm{st}} = \{\mathrm{Sanctions}, \mathrm{Securities}\}; \kappa_{\mathrm{dir}} : \{\mathrm{Securities}, \mathrm{Corporate}, \mathrm{KYC}\}, \quad \Phi_{\mathrm{dir}} = \varnothing, \quad E_{\mathrm{dir}} = \{\mathrm{Sanctions}, \mathrm{Tax}, \mathrm{DigitalAssets}\}.

The snapshots differ in three places, one of each kind:

1. Securities \in \mathrm{dom}\,\kappa_{\mathrm{dir}} \setminus \mathrm{dom}\,\kappa_{\mathrm{st}}. The direct corridor carries PK’s Securities verdict to SC; the staged route discards it at the AE \to SC mask and SC evaluates afresh. An entity with a Compliant PK Securities verdict arrives Compliant directly, and arrives with whatever \mathrm{eval}_{\mathrm{SC}} returns — Pending, say — when staged. Route choice changes the verdict.

2. DigitalAssets \in \mathrm{dom}\,\kappa_{\mathrm{st}} \setminus \mathrm{dom}\,\kappa_{\mathrm{dir}}. The staged route carries PK’s digital-asset evidence through AE — both hops correspond and neither masks it — while the PK–SC instrument names no counterpart, so the direct route triggers fresh evaluation. The staged route delivers a portability the direct agreement does not underwrite.

3. Tax \in \Phi_{\mathrm{st}} \setminus \Phi_{\mathrm{dir}}. On the staged route, AE freshly evaluates Tax on arrival from PK and the AE \to SC corridor carries AE’s evaluation onward; on the direct route, SC evaluates Tax itself. The routes deliver different evaluators’ verdicts on the same entity.

Each defect closes only by renegotiation, because each parameter is the content of a bilateral instrument: mask Securities in PK–SC, or unmask it in AE–SC; add a DigitalAssets correspondence to PK–SC at the composed grade, or mask DigitalAssets in AE–SC; mask Tax in AE–SC, or unmask it in PK–AE. No PK–SC term closes the forwarding defect: the direct corridor’s parameters enter only \kappa_{\mathrm{dir}} and \gamma^{\mathrm{dir}}, while \Phi_{\mathrm{st}} is fixed by the two staged edges, so a forwarding defect closes at a staged edge or not at all. The two staged closures are not alike. Masking Tax in AE–SC withdraws Tax from the final edge’s carriage; SC then evaluates Tax itself on both routes, and no other snapshot component moves. Unmasking Tax in PK–AE carries PK’s Tax verdict through AE instead; Tax leaves \Phi_{\mathrm{st}} for \mathrm{dom}\,\kappa_{\mathrm{st}}, and a carriage disagreement opens at Tax because PK–SC masks it. In general, the first-edge closure moves the forwarded domain into the staged carriage and agrees with the direct corridor only where that corridor carries the same domain under the same grade map; withdrawing the forwarded domain from the final edge’s carriage is the only single amendment that closes a forwarding defect and moves no other component. Each amendment is per-edge, not per-route: masking Tax in AE \to SC re-parameterizes every route through that edge, including routes originating at AE. After three amendments — Securities masked in PK–SC, the DigitalAssets correspondence added to PK–SC, Tax masked in AE–SC — the snapshots agree: \kappa is the identity on \{\mathrm{Corporate}, \mathrm{KYC}, \mathrm{DigitalAssets}\}, \Phi = \varnothing, E = \{\mathrm{Sanctions}, \mathrm{Tax}, \mathrm{Securities}\}. The triangle also closes with AE–SC untouched: unmask Tax in PK–AE, and re-parameterize PK–SC to the mask \{\mathrm{Sanctions}, \mathrm{Securities}\} with a DigitalAssets correspondence; then \kappa is the identity on \{\mathrm{Tax}, \mathrm{Corporate}, \mathrm{KYC}, \mathrm{DigitalAssets}\}, \Phi = \varnothing, E = \{\mathrm{Sanctions}, \mathrm{Securities}\}. Both closures satisfy Definition 6 and differ in what the agreements then guarantee — SC’s own Tax evaluation on every route under the first, PK’s carried Tax verdict under the second; the choice between them is a negotiating position, not a consequence of the check. Under either, Theorem 1 applies, and the two routes deliver identical arrival states.

6. Eligibility, Sanctions, and Instrument Clauses

Not every domain is eligible for recognition; an ineligible domain must sit in the re-evaluation mask of every corridor, and Sanctions is the standing case (Definition 2). Two consequences follow, each at its exact strength.

Eligibility is necessary, not sufficient. Per-edge eligibility does not deliver route coherence: in Section 5, Securities is eligible — and carried — on the PK \to SC edge, yet the routes disagree, because coherence is a property of the triangle of agreements, not of any edge alone.

The sanctions coordinate is route-coherent by construction. Because Sanctions lies in the mask of every general-purpose corridor, it lies outside every carriage image and every forwarding set: every route into any destination ends with a fresh sanctions evaluation there. A mandatory mask entry on every edge therefore never produces a route-coherence defect at its own coordinate. Its effect is to fix which snapshots are attainable — no route carries or forwards a sanctions verdict — not to obstruct composition.

Instrument clauses enter the snapshot on the same footing as the regulatory parameters (Definition 5): a route pair is coherent only when the clause sets also agree in the instrument layer’s normal form. A clause defect changes no grade; it subjects the same arrival to different governing law, forum, or termination terms — a route-dependence of legal position rather than of compliance state.

7. What Is Proved and What Is Open

Proved here, on the applicable-rule fragment at a fixed version profile: a direct corridor and a staged route with equal snapshots deliver the same arrival state (Theorem 1), and snapshot equality is a finite, decidable comparison of corridor data. Aggregation contributes no further obstruction, because the per-factor meet is order-independent (Section 3). Route-dependence on this fragment originates in the corridor data or not at all.

Open, in order of proximity:

• Discretion and temporal effects. (E1) excludes discretionary evaluation, and the fixed version profile excludes mid-route rulebook change. Route coherence for routes crossing typed discretion holes, or spanning version-window boundaries, is open.

• Mechanized reduction. The snapshot-equality checker is machine-checked (Section 4); the reduction from corridor-triple data to snapshots, and the clause normal form it presupposes, are paper-level. Mechanizing both is the remaining step to an end-to-end checked comparison.

• Classification. The equation \mathrm{carry}_{AC} = \mathrm{carry}_{BC} \circ \mathrm{carry}_{AB}, over all triangles at once, is the cocycle identity for transition data valued in partial injections between per-jurisdiction vocabularies. Such data compose as arrows of an inverse category (Kastl, 1979; Cockett and Lack, 2002), and as an inverse monoid when the vocabularies coincide, as in Section 5 (Lawson, 1998); in neither case a group. The classical non-abelian H^1 — cocycles modulo coboundaries with invertible coefficients (Giraud, 1971) — therefore does not apply as stated, and no cohomological classification of coherence defects is claimed. Whether an inverse-category analogue classifies defect classes over larger corridor families is open.

A defective triangle is not an error state of any single jurisdiction. It is a fact about three bilateral agreements — detected by a finite comparison, closed by renegotiating at least one of them, and, until closed, a standing difference between the routes the corridor system offers and the arrivals the agreements guarantee.

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