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 and accepts the rest. The second re-evaluates Sanctions and Securities at SC and accepts the rest.

These masks already produce a route difference. The direct route delivers PK’s Securities evidence to SC. The staged route discards it at the second hop, where SC evaluates Securities afresh. An entity whose PK Securities record is favorable prefers Route 1, while one that would fare better under fresh SC evaluation prefers Route 2. Route choice can therefore change a regulatory outcome without any 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 three corridors implement independently negotiated bilateral agreements. Bilateral negotiation does not require the PK–SC terms to equal the composite of the PK–AE and AE–SC terms, so the routes can differ. Per-domain aggregation adds no source of variation: it is a semilattice meet, independent of order (Section 3). We must therefore define “the composite of the terms” and establish what equality of terms guarantees. The construction below answers both questions on the applicable-rule fragment, and Section 5 compares the triangle above in full.

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 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.

For an entity already present at V, the multi-harbored case, the destination may independently hold a verdict on a carried domain. If V holds such a verdict, the composition law of the companion note gives the operative verdict as a meet. It combines the carried grade and V’s own grade in V’s factor. The meet is associative, commutative, and idempotent, so aggregation order never affects it. Route-dependence therefore concerns which grades arrive. Both routes combine those grades through the same meet.

Transit along A \to B followed by transit along B \to C need not equal transit along any single corridor A \to C. Wherever carriage fails, fail-closed re-evaluation at B replaces carried evidence with B’s own evaluation. The second hop may forward B’s evaluation to the destination. Route coherence asks whether a direct corridor reproduces this staged behavior.

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 use v. A different profile supplies different corridor parameters and defines a separate comparison.

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 in Theorem 1 is the graded object of Definition 4. A determination may also carry the set of authorities whose grades composed it, as in the companion The Recognition Protocol. That set lets a participant refuse determinations composed through a jurisdiction it has declined. It necessarily distinguishes a staged arrival from a direct one. Theorem 1 then establishes route independence only in the grade component and makes no claim about the deliberately route-dependent provenance component. The results below apply under either convention 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 requires agreement between the instruments. A defective route pair can still deliver equal states for a particular entity if a fresh evaluation happens to equal the carried grade it replaces. Such a coincidence does not give the guarantee a compliance system needs from the agreements.

Each snapshot component can expose a distinct disagreement. Section 5 exhibits carriage and forwarding defects with different arrival states. If the carriage maps agree at d but \gamma^{\mathrm{dir}}_d and \gamma^{\mathrm{st}}_d differ at some grade g, choose a source state with s(d) = g. It arrives at different grades at the common coordinate \kappa_{\mathrm{dir}}(d). A clause defect leaves grades unchanged 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. Equality on each is therefore decidable. Clause sets use the instrument layer’s normal form, whose equality is decidable by the standing assumption of Definition 5. Snapshot equality is a finite conjunction of these decidable equalities.

The machine-checked artifact accompanying Op proves sound, complete, decidable snapshot-equality checking at the normalized layer of carried cells, fresh-evaluation domains, and instrument clause identifiers. It 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 give the paper-level reduction from this fragment’s corridor-triple data to such finite snapshots. Mechanizing the reduction and its assumed clause normal form 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 identity correspondences on the listed names, identity grade maps on the common grade chain, and equal clause sets. These choices ensure that differences below reflect corridor disagreement. At a common version profile, the instruments of Section 1 are:

• 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 because both hops correspond and neither masks it. The PK–SC instrument names no counterpart, so the direct route triggers fresh evaluation. Only the staged agreements support this portability.

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 requires renegotiation because each parameter is part of a bilateral instrument. The alternatives are:

• 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 the two staged edges fix \Phi_{\mathrm{st}}. A forwarding defect therefore requires an amendment to a staged edge.

Masking Tax in AE–SC withdraws Tax from the final edge’s carriage. SC then evaluates Tax itself on both routes, with every other snapshot component unchanged. Unmasking Tax in PK–AE instead carries PK’s Tax verdict through AE. Tax leaves \Phi_{\mathrm{st}} for \mathrm{dom}\,\kappa_{\mathrm{st}}, creating a carriage disagreement because PK–SC masks it.

In general, the first-edge closure moves the forwarded domain into staged carriage. It 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 without changing another component. Each amendment affects the whole edge: masking Tax in AE \to SC changes 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. The first guarantees SC’s own Tax evaluation on every route, while the second guarantees PK’s carried Tax verdict. The parties must negotiate which guarantee to adopt. The check does not choose between them. Under either closure, Theorem 1 gives identical arrival states on the two routes.

6. Eligibility, Sanctions, and Instrument Clauses

Every corridor must mask domains ineligible for recognition, with Sanctions the standing case (Definition 2). This requirement has two consequences.

Eligibility is necessary, not sufficient. In Section 5, Securities is eligible and carried on the PK \to SC edge, yet the routes disagree. Coherence is a property of the triangle of agreements, so per-edge eligibility alone cannot ensure it.

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. It fixes which snapshots are attainable: no route carries or forwards a sanctions verdict. This restriction does not obstruct composition.

Instrument clauses explain why equal grades alone cannot close every defect. They enter the snapshot alongside regulatory parameters (Definition 5), and coherence requires equal clause sets in the instrument layer’s normal form. A clause defect leaves grades unchanged but subjects the same arrival to different governing law, forum, or termination terms. Closing this difference in legal position also requires renegotiating the instruments.

7. What Is Proved and What Is Open

On the applicable-rule fragment at a fixed version profile, Theorem 1 proves equal arrival states for direct and staged routes with equal snapshots. Comparing those snapshots is finite and decidable. Since the per-factor meet is independent of order (Section 3), all route-dependence on this fragment originates in corridor data.

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 records a mismatch between three bilateral agreements, without implying an error state in any single jurisdiction. A finite comparison detects it. Closure requires renegotiating at least one agreement. Until then, the routes remain subject to different arrival guarantees.

References

• Cockett, J. R. B., and Lack, S. (2002). Restriction categories I: Categories of partial maps. Theoretical Computer Science 270(1–2), 223–259.
• Giraud, J. (1971). Cohomologie non abélienne. Grundlehren der mathematischen Wissenschaften 179, Springer.
• Kastl, J. (1979). Inverse categories. In H.-J. Hoehnke (ed.), Algebraische Modelle, Kategorien und Gruppoide, Studien zur Algebra und ihre Anwendungen 7, 51–60. Akademie-Verlag, Berlin.
• Lawson, M. V. (1998). Inverse Semigroups: The Theory of Partial Symmetries. World Scientific.
• OECD. (1986). Double Taxation Conventions and the Use of Conduit Companies. Committee on Fiscal Affairs, OECD, Paris.
• OECD. (2015). Preventing the Granting of Treaty Benefits in Inappropriate Circumstances, Action 6 – 2015 Final Report. OECD/G20 Base Erosion and Profit Shifting Project, OECD Publishing, Paris.