1. What can another jurisdiction reuse?

A company may remain one legal person while operating through registrations in several jurisdictions. Applicable law determines which acts require an authority’s decision. An identity investigation completed for one registration may answer a question elsewhere, while the receiving authority retains any sanctions or activity decision reserved to it. The problem is to reuse the investigation without changing who decides, what they decide, or which act their decision permits.

Consider the synthetic case developed at the end of this paper. Three jurisdictions require four common evidence questions and two local questions each. Independent collection answers eighteen questions. If each authority recognizes the common observations, one collection answers the four common questions and the authorities answer their six local questions. The total is ten. The saving comes from recognition of particular evidence under current rules. A reserved refusal remains effective, and an unanswered local question still prevents an act that requires its answer.

Even complete identity evidence leaves several different questions. The same person can hold shares, owe money, and operate a service. Another person can hold a contractual right to some of the share proceeds, while a creditor has priority over those proceeds. Permission to hold the shares need not permit a public offer, a pledge, or a payment from a particular account. The record must therefore identify both the legal person and the right and action being considered. Section 2 develops these distinctions through a shareholding, software delivery, and separate funding pools.

Existing law already distinguishes one person with foreign establishments from a group of separate persons. UK guidance distinguishes an overseas company’s branch from its separately incorporated subsidiary [27]. Foreign registration, tax identification, banking access, and regulated permissions present separate questions under applicable law and provider requirements. Recognition of an earlier evaluation depends on the receiving authority’s law or an applicable instrument. A common record organizes the evidence and decisions within those relationships.

Continuation raises a related question about identity. Eligible entities can continue under another jurisdiction’s company law while preserving legal personality, subject to the applicable statutory conditions.¹ Preserving the person leaves a further question: which earlier evaluations have effect after the move? Applicable law and recognition instruments determine that effect. For example, the Cayman continuation provisions separately require any activity licence needed for the applicant’s business. Continuity and permission therefore remain distinct questions even within one statutory procedure.

The construction draws on three established practices. Mutual-recognition instruments specify when an outside evaluation has local effect. Catala and L4 represent legislative and regulatory rules as executable programs [18, 25]. Content-addressed attestations bind an evaluation to an identity, authority, rule version, and time. Here, content-addressed means that a digest identifies the exact recorded content. These parts support a record of which evidence can answer which question for a specified act.

Jurisdictional competition supplies the wider economic setting. Tiebout studied it in 1956. Romano analyzed corporate law as a market, and O'Hara, Ribstein, and Bruner developed related accounts of legal choice [24, 23, 21, 5]. The question here is what those choices imply for one person’s evaluated standing and an instrument’s lawful holders. The central economic result separates improved standing from added restrictions. Its price implications depend on a declared market mechanism and fixed buyer inputs.

Faster evidence exchange affects only part of an institutional process. Compliance review, dispute handling, service availability, beneficiary credit, and settlement finality can still determine its duration. Du, Huang, and Scharfstein distinguish those steps from payment-message speed [8]. The construction specifies reusable work and retained decisions. End-to-end time and cost remain measured outcomes.

2. Persons, rights, and permitted actions

An evaluation needs a stable subject before another authority can reuse it. The first record identifies the person and its local legal relationship. Rights and action records then specify what that person holds and what it proposes to do. A local registration is called a harbor below. The term names this relationship and carries no separate grant of authority.

Definition 1 (Harbor). A harbor is a jurisdiction-specific legal relationship of an identified entity. Its record names the home-law person, registration type, local identifier, competent authority, and active operation classes. The harbor set contains the jurisdictions with active records. Several registrations can belong to one person.

The registration, the tax position, and the permission to perform an operation are separate facts. Each has its own authority and effective period.

Definition 2 (Multi-harbored entity). Fix one legal person E, identified by its home law and registry identifier. A multi-harbored entity has a nonempty set H(E) of active harbor records tied to that person. Each record binds a governance and beneficial-ownership history, local evaluations in their declared action contexts, and the instruments giving recognized evidence its local effect. A compliance passport carries attributable evaluations and their identity, rule, time, and dependency bindings.

We index the relevant jurisdiction-specific records by their jurisdictions. Where one jurisdiction requires several records, its local evaluation uses those records before the cross-jurisdiction comparison. Thus a jurisdiction label in H(E) refers to the identified person’s active local relationship. The beneficial-ownership history identifies the persons who ultimately own or control the entity. Dependency bindings identify earlier records on which an evaluation relies.

A branch registration can already preserve one legal person across borders. UK government guidance distinguishes an overseas company’s branch from its separately incorporated subsidiary  [27]. The construction adds a common evaluation and execution contract to such legal relationships. Its contribution is reusable, versioned recognition under each receiving authority’s rules.

Table 1. Legal personality and local registration.

The UK rows illustrate that distinction. They do not determine another jurisdiction’s law. For every row, tax presence and regulated permissions require their own records. A passport supplies evidence according to its recognition contract. The relevant authority grants the operation’s legal effect.

Operation-specific recognition. For each pair (J,a) of jurisdiction and operation class, the record contains the following fields.

Table 2. What a recognition record must identify.

In Table 2, scope means the assertions and uses covered by the instrument. A freshness policy determines how current the evidence must be. These fields let the receiver distinguish an available historical record from one it may use for today’s action.

An authenticated instrument establishes the recognized scope. Its signature must come from the competent authority or a valid delegate. Registration alone supplies no omitted scope. The local evaluator returns \mathrm{Allow} with its evidence, \mathrm{Await}(q) for a specified missing premise, or \mathrm{Refuse}(r) for a controlling prohibition. A completed foreign investigation can discharge an admitted evidence question while a reserved local decision remains necessary. This permits useful recognition without transferring the receiving authority’s power.

A continuation or domestication changes the identity history under its governing law. A subsidiary introduces another person. Each transition records those consequences before any harbor record is reused. Local counsel and the competent authority must establish the matrix for a concrete deployment. The construction below uses the operation classes admitted by that matrix.

3. The composed state

The earlier sections fixed who acts, which rights are involved, and which authorities must decide. We now combine their evaluations of that same act. A domain is a category of requirements. Let \mathcal D be a fixed finite domain set. The illustrated profile takes |\mathcal{D}| = 23, including sanctions, know-your-customer checks, securities, custody, tax, employment, and Sharia compliance. The results depend on finiteness and a fixed domain set, rather than the chosen number.

For each domain d, the applicable grades form a finite distributive lattice D_d. This is a finite ordered set with a greatest common lower grade and a least common upper grade for every pair. These operations are called meet and join; distributivity means that meet distributes over join, and conversely. The usual example is the ordered chain

\mathrm{NonCompliant} < \mathrm{Pending} < \mathrm{Compliant},

with a richer factor where the domain’s internal structure requires one. A higher grade records stronger standing on the same applicable requirements. A lower combined grade retains the more restrictive judgment. Separately, each evaluation records Applicable, NotApplicable, or Exempt. Those markings answer whether a requirement applies, rather than how well it is satisfied. Section 4 treats that distinction.

Tax grades state whether registrations and filings are in order. They compose like other applicable grades. Withholding, credits, and treaty positions have quantities and interactions recorded in a separate tax ledger. The meet does not compute those amounts. Problem 7 of Section 8 identifies that remaining question.

Write \mathcal{L}^{\mathrm{App}} = \prod_{d\in\mathcal{D}} D_d. An element of this product is a vector with one applicable grade per domain. The product order compares two vectors in every coordinate: the first is lower only when each of its grades is lower or equal. The superscript \mathrm{App} restricts this space to evaluations on the fixed applicable requirements. Write T_J(E;q) for authority J’s evaluation vector of person E in action context q.

For example, suppose two authorities evaluate sanctions and custody for the same act. The first returns Compliant in both domains. The second returns Pending for sanctions and NonCompliant for custody. Their meet returns Pending and NonCompliant. Each coordinate retains its required restriction. No improvement in sanctions can compensate for the custody refusal. The formula below generalizes this coordinate-wise calculation.

How Compliance Composes proves that an idempotent, monotone operation that never weakens either input is the meet [17]. Here idempotent means that repeating an evaluation leaves its result unchanged. Monotone means that improving an input cannot lower the combined grade. Never weakens means that the output cannot exceed either input in this grade order. On the product order, the meet is pointwise: it computes each domain’s grade from that domain’s inputs alone.

Fix the action context q of Section 2. For a multi-harbored entity E, when every harbor supplies an authoritative grade on the same Applicable fragment, define

T^{\mathrm{App}}(E;q)=\bigwedge_{J\in H(E)}T_J(E;q).

More generally, let \mathcal J_d(q) be the finite nonempty set of authorities whose current rules require a judgment for applicable domain d. The action’s coordinate is T_d^{\mathrm{App}}(E;q)= \bigwedge_{J\in\mathcal J_d(q)}T_{J,d}(E;q). The scope record determines this set. The entity cannot choose it to exclude a controlling requirement. Inapplicability and unresolved scope retain their separate treatments in Section 4. Below, T_J(E) abbreviates the evaluation in the fixed context when no comparison changes that context.

The symbol \bigwedge applies meet to all the displayed inputs. The first formula covers the special case where the entity’s harbor authorities supply exactly those inputs. The second allows the applicable requirements to reach beyond the registered jurisdictions.

These meets combine authoritative judgments for the specified action. An imported observation remains evidence for a local question; it does not replace that authority’s decision. A reserved refusal survives its import. A pending required decision remains pending until its own conditions hold. Recognition may preserve or discount transported evidence. Section 5 states that transport condition. Holding, offering, pledging, and paying can have different applicability, decisions, and resource requirements for the same entity.

The imported composition theorem determines the combined evaluation. To use it for a holding decision, we still need applicability, recognition, and any reserved human judgment. Sections 4–6 specify these conditions before Section 7 derives the lawful holder set and its conditional price effects.

4. Where jurisdictions disagree about applicability

Composition is well-behaved when the harbors agree that a domain applies. They do not always agree. One jurisdiction may treat a domain as Applicable and grade it; another may treat it as NotApplicable or Exempt.

For an assumed reporting rule, a person outside its covered class is NotApplicable. A person within its class but covered by a valid exemption is Exempt. A person subject to the rule who filed the required report can be Applicable and Compliant. Each outcome answers a different question, even though none establishes a reporting failure in this example.

Combining these markings with grades on a single chain would identify legal positions that the record must distinguish. Inserting NotApplicable and Exempt between Pending and Compliant cannot retain all the original information. Being outside a requirement’s scope and satisfying that requirement remain different legal positions, even when both permit the contemplated activity.

Accordingly, composition on the full state space returns the meet where all harbors mark a domain Applicable, the shared marking where they agree it is NotApplicable or Exempt, and an explicit record of the disagreement where they disagree. We call the last an obstruction: it names which applicability markings appeared and in which harbors. The result retains the question that requires institutional action.

For example, a record can retain authority A’s Applicable/Pending judgment and authority B’s NotApplicable marking for the same domain. The obstruction identifies both authorities and markings. It asks for the relevant scope determination without treating B’s marking as proof that A’s question vanished. A typed record identifies which kind of result it contains and the information required for that kind.

Two consequences follow. Every unqualified claim here about meets or product-lattice structure is scoped to the Applicable fragment. An obstruction must also survive to the pricing layer rather than being coerced into a number. Section 7 makes that precise.

5. Movement: corridors and the passport

A combined record must retain the conditions under which a foreign evaluation can be used. Reuse across several authorities also raises a further question: does passing through an intermediate authority preserve the same evidence and local obligations?

A corridor binds a source authority, destination authority, recognition instrument, and current-use context. It specifies which attributable assertions may travel and which questions the destination must answer afresh. The carried and fresh question sets can overlap. Reusing an investigation does not remove a reserved local decision.

A typed transport specifies the kind of assertion carried and preserves its origin, subject, meaning, validity period and conditions, and permitted purpose. Where both sides declare a shared grade carrier, a transported summary may use a monotone, bottom-preserving, deflationary map \phi_{A\to B}. The shared grade carrier is the common ordered set of grades understood by both authorities. The arrow in \phi_{A\to B} names transport from A to B. Monotonicity preserves comparisons between source grades. Bottom-preservation keeps the lowest grade fixed, and deflationary means that a transported grade cannot exceed its source grade. For example, a permitted summary can reduce Compliant to Pending while retaining Pending and NonCompliant. It discounts the evaluation without converting a refusal into approval. Those properties constrain the summary. The destination’s current action-specific \mathrm{Allow} requires its own authority and completed local gate. An imported assertion cannot manufacture that decision.

The destination’s fresh questions include every decision its governing instrument reserves locally. Sanctions obligations are evaluated against the applicable current rules and subjects. A foreign result can inform that evaluation when recognized. It cannot bypass an operative local prohibition.

Route composition. A route records its ordered corridor steps, transported assertions, and remaining local questions at each step. Composition concatenates compatible steps. It retains assertion identity and every intermediate fresh obligation. This gives an executable staged route even when a direct route accepts different evidence. Two routes are equivalent for an operation only when their admitted evidence, obligations, and resulting decision agree in the same context. A path name alone does not establish that equality.

The compliance passport is the attributable evidence carried along a route. Historical signatures can be checked offline when their evidence is available. Current execution also applies the destination’s freshness and availability policy. Revocation changes current use while preserving the historical assertion. Each authority retains control of its own record and decisions.

For example, one authority can recognize a completed identity investigation while requiring its own current sanctions decision. The holder supplies the investigation once, and the receiving authority performs the reserved decision. The workload at the end of this paper measures this distinction through explicit question counts. Total operating savings additionally depend on verification, local review, maintenance, and institutional coordination.

6. Where computation stops

Statutes say fit and proper, material adverse change, good cause. These require human judgment, and a system that inserts a fixed true-or-false answer at those points would replace the required legal determination.

The rule representation records the required judgment explicitly. A typed discretion hole is an unanswered question whose specified answer must come from a competent decision-maker. It names the decision required, the officer class authorized to make it, and its scope. Supplying that answer takes a signed, revocable attestation binding the signer, the exact value, and the rule version. Evaluation of an act that requires the answer stops until it arrives. A delegated program operating the entity may propose an action. The required competent decision-maker supplies the answer.

Two companion papers define the formal systems used here. Lex types jurisdictional rules, exceptions, legal time, authority-indexed verdicts, and discretion holes. Op executes operations against those rules and records typed effects, scoped compensation, and evidence of which cross-jurisdiction operations have durably completed.

The construction does not determine what fraction of institutional work falls into discretion holes. That fraction determines how much of the maintenance layer can be automated. Its value remains an empirical question.

7. From composed standing to funded demand

For a cash-flow claim, valuation starts from its specified payments, contingencies, and discounting assumptions. Equity, debt, asset-backed distributions, and insurance supply different payment structures. A use right additionally needs a service or consumption valuation. Legal admissibility restricts who can demand the instrument under either representation.

Here an instrument is the specified equity, debt, use right, or other claim whose holding and price we study. This section reuses X for that instrument; it no longer denotes the position-vector set of Section 2. The current holding context determines which instrument-wide restrictions apply. Their intersection determines the lawful holder classes. The pricing mechanism below evaluates the funded bids of admitted participants.

Fix a multi-harbored entity E, an instrument X, and a context q for holding that instrument. Its nonempty harbor set H(X)\subseteq H(E) names registrations of the same instrument under one set of terms. Each relevant instrument-wide condition applies to that composed form. An offer in one jurisdiction can require another context.

The pricing comparison concerns instrument-wide restrictions in that fixed context. Each buyer’s actual purchase or holding has its own exact action context and current admission. The common holding map does not replace those individual decisions. The buyer inputs below retain their current requirements.

For an applicable domain, the holding gate uses \mathcal J_d(q) from Section 3. This includes every authority whose rule reaches the act or its subjects, even when the instrument has no local registration there. It excludes requirements that do not apply to that act. When that set is exactly H(X) in each domain, the gate is T^{\mathrm{App}}_{H(X)}(E;q)=\bigwedge_{J\in H(X)}T_J(E;q). For fixed comparable judgments, a meet over fewer authorities is weakly higher. Removing an authority’s label does not remove its applicable rule. The theorem below fixes q and quantifies over a generic authoritative state c and instrument-wide constraint set H. A change of action, subject, applicability, or rule roots supplies a new comparison context.

The comparison changes either grades or restrictions. We first hold the restrictions fixed and ask what an improved grade permits. We then hold the grade fixed and add an instrument-wide restriction. These are different comparisons, and they can move the lawful holder set in opposite directions.

Fix a finite set K of holder classes, such as retail, professional, and institutional investors, with further distinctions required by the relevant regimes. Write S_J : \mathcal{L}^{\mathrm{App}} \to \mathcal{P}(K) for jurisdiction J’s admissibility surface. This is a map: S_J(c) lists the classes permitted by its conditions at composed state c. The powerset \mathcal{P}(K) is the set of all subsets of K, ordered by inclusion. The one-class case K = \{\ast\} records only admitted or not admitted. Nothing below requires more structure than inclusion of holder classes.

The state order. Fix the jurisdiction’s rule version, action, subjects, applicability, and current-use context. On a declared grade fragment, require each S_J to be monotone. Thus an improvement within that fragment retains every previously admitted class. Each class’s admitting set is upward-closed: once a state admits it, every higher state does too. This is an assumption about authoritative judgments under fixed rules. Accumulating evidence need not improve a grade. New evidence can reveal a prohibition, change applicability, or require another judgment. Such a change requires another evaluation.

The harbor order. For a harbor set H write S_H = \bigcap_{J \in H} S_J. This intersection is pointwise: S_H(c)=\bigcap_{J\in H}S_J(c). A class may hold the composed instrument exactly when every condition represented in this intersection admits it. The model includes only conditions that reach the instrument wherever it is held. Examples include a controlling sanctions designation, an ownership cap in the charter, and a transfer restriction in its terms. Another harbor’s admission leaves such a restriction in force.

A condition confined to local offers and sales answers a different question. A local notification or retail distribution requirement can restrict marketing in that jurisdiction while leaving holding elsewhere unaffected. For example, the Directive on undertakings for collective investment in transferable securities (UCITS) separates home authorization from notification for marketing in another member state [11]. Home authorization is valid across member states. Article 93 governs host marketing notification, while Articles 97 and 108 retain specified host supervision and protective powers. We therefore put instrument-wide holding restrictions in S_J, and local distribution access in the bidder inputs below. A particular holder’s residency, suitability, and accreditation also enter the buyer context.

Under the stated common terms, intersecting the S_J is the meet in the inclusion order. It is never wider than an input, repeats consistently, and preserves inclusion when an input widens. The same never-wider, idempotent, and monotone conditions used for grades therefore determine this intersection. The instrument has one set of terms in every harbor. An instrument registered in one harbor alone is a different comparison object with fewer conditions. Clause (ii) below compares these objects while fixing everything except the instrument-wide restrictions. Changes in distribution or buyers require the later population comparison.

Consequently S_H is antitone in the harbor set: it can shrink when the set grows. Formally, H \subseteq H' gives S_{H'}(c) \subseteq S_H(c) for every c. An intersection of monotone maps is monotone, so S_H retains the state-order property as well. The theorem uses these two elementary inclusions. An added registration can also change applicable grades and attract funded buyers, but neither effect belongs to this fixed-state restriction comparison.

The economic channel rests on established results, used for their monotone direction only. Errunza and Losq (1985) price legally restricted holding directly: a security some class of investors may not hold trades at a discount against an otherwise identical unrestricted one. Merton (1987) prices the breadth of the investor base. Amihud and Mendelson (1986) show illiquidity is priced: wider spreads carry higher gross returns. Pástor and Stambaugh (2003) show aggregate liquidity is a priced state variable. Duffie, Gârleanu and Pedersen (2005) supply the search-theoretic foundation: in over-the-counter markets, prices and spreads respond to search frictions and bargaining power [10, 19, 1, 22, 7].

Market liquidity is the ability to trade without a large price change. Funding liquidity is the ability to obtain settlement assets when payment is due. A narrower lawful holder set can reduce market liquidity. It does not by itself show a funding shortage. Brunnermeier and Pedersen show feedback between the two forms of liquidity. The pricing theorem here signs only the holder-admissibility channel [4].

A price calculation can encounter an unresolved legal input. Reporting a number at that point would conceal the unresolved question. The result therefore has two forms: Price(p) contains a numeric price, and PriceObstruction(o) contains a reason the calculation cannot return one. The obstruction records applicability disagreement, an out-of-scope state, an empty admitted set, or a binding sanctions block, with their authorities, rule versions, and domains. Their order in the result format carries no order of legal standing.

The numeric interface below requires a supplied state in the declared Applicable fragment. A full record from Section 4 can instead contain shared NotApplicable or Exempt markings. Those markings retain their own meaning. This paper supplies no general conversion, or lift, from such a full record to an applicable-grade vector. If the required vector cannot be supplied, the out-of-scope obstruction records that gap. It does not silently assign Compliant to an exemption or an inapplicable requirement.

For the first comparison, fix potential buyers, class membership, distribution access, participation, valuations, and funded budgets. The operator changes only the admitted classes within that buyer context.

Let w map an intrinsic valuation \pi, an admitted class set S \subseteq K, and a counterparty-belief model \beta to either \mathrm{Price}(p) or a typed obstruction. Here \pi describes the instrument’s valuation under specified payment or use assumptions. The model \beta describes beliefs of the counterparties whose demand enters the chosen mechanism. Both stay fixed in the theorem, which imposes a condition on that mechanism. A composition result is either \mathrm{Applicable}(c) or \mathrm{Obstruction}(o). Define the pricing interface by \widehat w_H(\pi,z,\beta)= \begin{cases} \mathrm{PriceObstruction}(o), & z=\mathrm{Obstruction}(o),\\ w(\pi,S_H(c),\beta), & z=\mathrm{Applicable}(c). \end{cases} In this formula, z is the composition result. An applicable result supplies c to the holding map S_H. An obstruction passes directly to the output with its source record, without calling the numeric mechanism.

Theorem (Admissibility-conditioned pricing). Fix \pi, \beta, and the declared buyer context. Assume:

(H) w is monotone on its numeric branch: if S\subseteq S', w(\pi,S,\beta)=\mathrm{Price}(p), and w(\pi,S',\beta)=\mathrm{Price}(p'), then p\leq p'. This hypothesis concerns the selected pricing mechanism. The auction below supplies one construction satisfying it.

Then \widehat w_H has the following properties:

(i) Monotone in compliance. Fix the harbor set H. Suppose c \le c', both states compose without an applicability obstruction, and both calls to \widehat w_H return numeric prices p and p'. Then p\leq p'. A more compliant entity has a weakly higher numeric price on this branch.

(ii) Antitone in harbors. Fix the compliance state c and harbor sets H \subseteq H'. If the corresponding interface calls return numeric prices p_H and p_{H'}, then p_{H'}\leq p_H through the admissibility channel alone. Each added harbor can only remove classes, because the composed surface is the intersection of its harbors' surfaces.

(iii) Obstruction-faithfulness. On any input where composition returns an applicability obstruction, \widehat w_H returns a typed price obstruction carrying that provenance, never a number.

Proof. (i) S_H is an intersection of state-monotone surfaces, hence state-monotone: c \le c' gives S_H(c) \subseteq S_H(c'). Hypothesis (H) gives p\leq p' on the stated numeric branch. (ii) S_{H'}(c) = \bigcap_{J \in H'} S_J(c) \subseteq \bigcap_{J \in H} S_J(c) = S_H(c) for every c, since H \subseteq H'. Hypothesis (H) gives p_{H'}\leq p_H on the stated numeric branch. (iii) The first case of \widehat w_H returns the obstruction and its provenance directly. It makes no call to w. \square

The hypothesis is substantive: the inclusions of lawful classes alone do not prove a price inequality for every market. The following mechanism satisfies it because the bids supporting an old price remain available when more classes become admissible.

From classes to funded bidders. Here B denotes buyers, separately from the issuance matrix used in Section 2. Let B be a finite population of distinct potential buyers. A map \chi:B\to K assigns each buyer a legal class. Fix willingness u_i, distribution access d_i, submitted value v_i\ge0, and committed funding f_i\ge0. The first two are binary inputs. A value of one for u_i means willing to participate. A value of one for d_i requires distribution access and the individual buyer conditions in this context, including applicable residency, suitability, and accreditation conditions. The latter two inputs use the same currency unit. These variables are inputs, not consequences of class membership. The funded bid is b_i=\min\{v_i,f_i\}. For an admitted class set S, define

D(S)=\{i\in B:\chi(i)\in S,\ u_i=d_i=1,\ b_i>0\}.

An admitted class may contain no participating buyer. Adding a class therefore need not add a bid. When comparing harbor sets through admissibility alone, hold B,\chi,u,d,v,f fixed. Access and participation changes belong to the separate population comparison.

Worked mechanism (one-unit second-price auction). A second-price auction assigns the unit to a highest bidder at the second-highest bid. Suppress the fixed \pi, \beta, and buyer inputs in the notation. For |D(S)|\ge2, let w(S) return \mathrm{Price}(p), where p is the second-highest funded bid in D(S). Fewer funded bidders produce a typed market-depth obstruction in this example. If S\subseteq S', then D(S)\subseteq D(S'). The two bids supporting w(S) remain present in D(S'), so (H) follows. This is a statement about the specified auction. Strategic bidding and other market mechanisms require their own model.

For bids 100,80,30, removing the bidder at 30 leaves price 80. Removing the bidder at 80 instead gives price 30. Adding a legal class with no participating buyers leaves price unchanged. Starting from bids 100,80, a new arrival outside the original potential-buyer population submits a funded bid of 120, raising the price to 100. The last comparison changes the population. It therefore does not contradict the fixed-population harbor inequality.

Here accession means adding a jurisdiction to the relevant arrangement. For the net comparison, use the actual post-accession bids in D', including changes in eligibility, participation, and funding. The proposition’s phrase “new bids” refers to this whole post-accession set, including retained buyers.

Proposition (Exact accession comparison for the auction). Let D and D' be the funded bidder sets before and after accession. Both contain at least two bids. Write p for the old second-highest bid and p' for the new one. Then

\begin{aligned}p'>p&\iff |\{i\in D':b_i>p\}|\ge2,\\p'<p&\iff |\{i\in D':b_i\ge p\}|\le1.\end{aligned}

Equality holds exactly when at least two new bids are at least p, and at most one is greater than p. These conditions allow departing buyers, newly eligible buyers, and new arrivals together.

Proof. The second-highest bid exceeds a threshold exactly when at least two bids exceed it. It reaches the threshold exactly when at least two bids reach it. Apply both observations at p. \square

A harbor addition can change restrictions, authoritative grades, distribution access, and the funded bidder population. The proposition computes the net effect when the resulting bids are specified. Predicting those bids requires a demand and participation model. Thus the construction supplies an exact mechanism-level comparison without treating legal classes as demand.

The price theorem uses the state and holder-set inclusions through hypothesis (H). In particular, clause (ii) uses S_{H'} \subseteq S_H. Clause (iii) specifies the output on an obstruction. The paper proofs above establish these claims for the stated interfaces. No pricing operator is mechanized, meaning verified by a proof-assistant program. The mechanized core covers meet, applicability provenance, and the impossibility of collapsing grades and applicability onto one chain without losing information.

A corridor changes price through its effects on current admissibility and funded demand. The theorem applies when the resulting authoritative states and bidder inputs satisfy its fixed-context hypotheses. Recognition depth alone does not order local permission. Fresh questions, legal changes, or new participants can change the comparison. The route contract records those changes so the relevant economic inputs remain explicit.

The empirical comparison. The auction gives an exact answer conditional on bids. An accession study must separately estimate participation, funding, and valuation responses. It must record which restrictions and local distribution permissions changed. A zero price change is compatible with weak monotonicity. A strict reversal under fixed premises would contradict the proposed monotone price functional.

8. What is proved, what is assumed, what is open

Imported result. How Compliance Composes proves that idempotent, monotone, never-weaker composition on the Applicable fragment is the pointwise meet. Section 3 applies that result to the harbor set.

Proof-assistant results. A mechanization is a formal proof checked by a proof-assistant program. The grade-meet laws, the n-ary composition operator, exact applicability flags, the two provenance-loss witnesses, and the impossibility core are in TensorAlignment.v. It imports a five-element ordered grade model and its Heyting operations, which include a logical implication compatible with that model’s meet. Those definitions and results are in VerdictHeyting.v. The two files contain 43 and 48 named results, respectively. These results establish the finite algebraic cores. The five-element model is part of the cited verification, while the three-chain above illustrates how the present paper combines applicable grades. The pricing implications are proved in Section 7 under hypothesis (H). The rights, funding, action-admission, and earning results in Section 2 have direct paper proofs. Their finite reference examples have independent checks. The specified cases are the proceeds allocation, split and merger identities, estate transfer with existing reservations, and milestone performance and credit rules. Each case and its expected result appear in Section 2. Those checks establish the stated cases and symbolic matrix identities. They do not prove that every implementation conforms to the general paper definitions.

Paper proofs using that core. Section 7 proves that the composed surface is monotone in the compliance state and antitone in the harbor set. Clauses (i) and (ii) lift these facts through hypothesis (H) on the numeric branch. Clause (iii) specifies the interface’s output by definition.

Economic premise. Hypothesis (H) belongs to the chosen price mechanism. The holder-set and liquidity literature in Section 7 motivates its direction under those models’ conditions. The auction proves it for the specified funded bids. Applying the result to another market requires establishing (H) for that market’s mechanism and comparison context.

Open, and not claimed.

1. The price map over the full state space, where applicability disagreement is present. Only the obstruction is defined there; the number is not.

2. The counterparty-belief layer. The price a claim clears at under heterogeneous beliefs is a functional of the cross-counterparty belief distribution, not an objective price. This is an identification problem and we have not solved it.

3. The demand and participation response to an actual harbor addition. Section 7 gives the exact sign for specified auction bids.

4. Route transitivity on the full system, with applicability and temporal effects (§5).

5. Whether the objects on which the construction rests — the fixed domain set, the harbor set, the per-domain grade lattices with their applicability axis, the passport, and typed discretion holes — span institutional practice: whether every act a registrar, licensing authority or court takes on an entity is a coordinate of this object or a marked hole in it. We have not proved sufficiency and do not claim it.

6. What fraction of institutional work falls into discretion holes (§6).

7. Multi-jurisdictional tax interaction. Tax facts, filings, withholding evidence and treaty positions enter a separate ledger; deductions, credits and Pillar-2 positions are normalisation steps in that ledger, not automatic consequences of the meet.

9. Prior art

The paper combines evaluated standing with action-specific rights and demand. The following bodies of work supply different parts of that problem.

Jurisdictional competition. Tiebout, Romano, Bebchuk–Hamdani, O'Hara–Ribstein, and Bruner analyze the choice of jurisdiction and competition over legal arrangements [24, 23, 3, 21, 5]. The present construction specifies an additional input to such choices: which evaluated facts can be reused and which restrictions remain effective for one entity’s particular acts. Whether easier compliance exit improves institutional quality remains an economic question.

Recognition regimes. The UCITS Directive provides home authorization and a host marketing notification procedure, together with allocated supervisory powers [11]. It gives a concrete example of legal effects defined by a recognition instrument. The Financial Action Task Force’s mutual-evaluation methodology assesses a country’s technical compliance and system effectiveness [12]. Its unit of assessment differs from the entity and action evaluated here. The present model asks how to retain each applicable decision and its authority when combining evaluated results.

Registries and identifiers. X-Road provides data exchange between registries. The Legal Entity Identifier (LEI) identifies entities, and the OECD’s Base Erosion and Profit Shifting (BEPS) project coordinates tax measures [26, 14, 20]. The construction uses a stable identity and attributable observations, then separately determines their recognized use and the resulting admissibility. Neither identification nor data exchange alone supplies that legal effect.

Interoperability protocols. Cosmos Inter-Blockchain Communication (IBC) transports packets between chains using verification by light clients, which check the relevant chain evidence [6]. A corridor here must also record which assertions a competent authority recognizes, how their grades translate, and which local decisions remain. Successful packet delivery leaves those legal recognition questions to their governing instrument.

Rules as code and rule composition. Catala and L4 demonstrate executable representations of legislative and regulatory rules [18, 25]. Regulations Expressed as Logical Models (REALM) and LegalRuleML provide representations for legal rules and their interactions [13, 2]. This paper begins after the relevant rules have produced evaluations. Its inputs are judgments about the same act, reached under identified authorities. The composition question is which combined judgment retains all their restrictions. The holding maps then connect that evaluated state to a lawful holder set.

Asset pricing. Errunza–Losq model legally restricted holding, and Merton models the breadth of the investor base. Amihud–Mendelson, Pástor–Stambaugh, and Duffie–Gârleanu–Pedersen study the price effects of liquidity and search frictions [10, 19, 1, 22, 7]. The construction computes a holder set from the declared regulatory standing and restrictions. A chosen price model still determines how that holder set affects demand and price. Section 7 gives a theorem under an explicit monotonicity condition and an auction that satisfies it.

Legal interpretation and exit. Hart’s open texture of law and Dworkin’s account of interpretation explain why some required decisions cannot be supplied by a fixed rule evaluation [15, 9]. Section 6 records those decisions and their competent sources. Hirschman’s exit and voice provide a framework for asking what easier jurisdictional movement changes [16]. This paper establishes no theorem that cheaper exit improves regulatory quality.

10. What recognition saves

Testable implications. The declared composition law fails if its output exceeds an input on the same fixed authoritative-grade fragment. For S\subseteq S', a numeric result p>p' contradicts (H) only when valuation, beliefs, population, funding, participation, and access are held fixed. Equality satisfies (H). Observing a changed population tests a different comparison. Useful recognition occurs when an admitted foreign evaluation discharges a destination question under the current governing authority. The workload below isolates that reduction in repeated collection from the decisions retained locally.

A reproducible workload and its limits. Consider three jurisdictions, four reusable evidence questions, and two distinct local questions per jurisdiction. Independent collection performs 3(4+2)=18 question evaluations. A permitted shared collection performs 4+3\cdot2=10, while retaining all six local questions. The reference workflow gives this exact operation count. It also rejects stale recognition and preserves a local refusal. These are synthetic executions, not measured institutional deployments.

To reproduce the count, name four reusable questions and give each of three jurisdictions two distinct local questions. In the independent case, collect six answers for each jurisdiction. In the shared case, collect the four reusable answers once and all six local answers. Accept a reused answer only when its identity, question, recognition scope, and freshness conditions match. For the stale-recognition case, mark one imported answer expired: it must remain unresolved or be collected afresh. For the refusal case, retain a local Refuse answer while all imported answers satisfy their questions: the act requiring that local answer remains refused. These cases test the recognition rule’s stated boundaries and do not establish field performance.

Let c_q be the collection cost of reusable question q, and m_q the number of jurisdictions requiring it. When every recipient recognizes one common current observation, avoided collection costs equal

\sum_q(m_q-1)c_q.

Let V denote verification, coordination, and maintenance costs added by recognition. With identical local review costs, net savings equal the displayed amount minus V. Positive savings require that difference to be positive. A revocation or changed question creates a new collection obligation under the relevant dependencies. It does not preserve the old saving by assuming stale evidence remains usable.

A proposed empirical study would compare the same eligible cases under independent and recognized-evidence workflows. It would fix the case cohort, outcome definitions, and observation horizon before measuring outcomes. Independent adjudicators would assess case completeness without knowing the workflow assignment where practicable. The study would record false acceptance, false refusal, elapsed time, human minutes, counsel time, maintenance, and fully loaded cost. It would report paired differences and uncertainty intervals, including failed and withdrawn cases. Synthetic question counts establish the mechanism’s work reduction. Actual adoption, operating savings, and legal effectiveness require the corresponding institutional observations.

References

[1] Y. Amihud and H. Mendelson. Asset pricing and the bid-ask spread. Journal of Financial Economics, 17(2):223–249, 1986.

[2] T. Athan, H. Boley, G. Governatori, M. Palmirani, A. Paschke, and A. Wyner. OASIS LegalRuleML. In Proceedings of ICAIL, pages 3–12, 2013.

[3] L. A. Bebchuk and A. Hamdani. Vigorous race or leisurely walk: Reconsidering the competition over corporate charters. Yale Law Journal, 112(3):553–615, 2002.

[4] M. K. Brunnermeier and L. H. Pedersen. Market liquidity and funding liquidity. Review of Financial Studies, 22(6):2201–2238, 2009.

[5] C. M. Bruner. Re-Imagining Offshore Finance: Market-Dominant Small Jurisdictions in a Globalizing Financial World. Oxford University Press, 2016.

[6] J. Kwon and E. Buchman. Cosmos whitepaper. Interchain Foundation, 2019.

[7] D. Duffie, N. Gârleanu, and L. H. Pedersen. Over-the-counter markets. Econometrica, 73(6):1815–1847, 2005.

[8] W. Du, C. Huang, and D. Scharfstein. Competing rails for cross-border payments: Banks, fintechs, and stablecoins. Harvard Business School working paper, 15 February 2026.

https://www.hbs.edu/ris/Publication%20Files/Du_Huang_Scharfstein_14Feb2016_66992079-2e6b-4584-95cf-014441c77485.pdf.

[9] R. Dworkin. Taking Rights Seriously. Harvard University Press, 1977.

[10] V. Errunza and E. Losq. International asset pricing under mild segmentation: Theory and test. Journal of Finance, 40(1):105–124, 1985.

[11] European Parliament and Council. Directive 2009/65/EC on undertakings for collective investment in transferable securities. Official Journal of the European Union, L 302, 2009.

[12] Financial Action Task Force. Methodology for Assessing Technical Compliance with the FATF Recommendations and the Effectiveness of AML/CFT/CPF Systems. Updated June 2025.

[13] C. Giblin, A. Y. Liu, S. Müller, B. Pfitzmann, and X. Zhou. Regulations expressed as logical models (REALM). In Proceedings of JURIX, pages 37–48, 2005.

[14] Global Legal Entity Identifier Foundation. The Legal Entity Identifier and the Global LEI System. Official reference material, accessed September 2026.

[15] H. L. A. Hart. The Concept of Law. Oxford University Press, 1961.

[16] A. O. Hirschman. Exit, Voice, and Loyalty: Responses to Decline in Firms, Organizations, and States. Harvard University Press, 1970.

[17] R. Lorgat. How Compliance Composes. Companion paper, September 2026.

[18] D. Merigoux, N. Chataing, and J. Protzenko. Catala: A programming language for the law. Proceedings of the ACM on Programming Languages, 5(ICFP), Article 77, 2021.

[19] R. C. Merton. A simple model of capital market equilibrium with incomplete information. Journal of Finance, 42(3):483–510, 1987.

[20] Organisation for Economic Co-operation and Development. OECD/G20 Base Erosion and Profit Shifting Project: Final Reports. OECD Publishing, 2015.

[21] E. A. O'Hara and L. E. Ribstein. The Law Market. Oxford University Press, 2009.

[22] L. Pástor and R. F. Stambaugh. Liquidity risk and expected stock returns. Journal of Political Economy, 111(3):642–685, 2003.

[23] R. Romano. Law as a product: Some pieces of the incorporation puzzle. Journal of Law, Economics, and Organization, 1(2):225–283, 1985.

[24] C. M. Tiebout. A pure theory of local expenditures. Journal of Political Economy, 64(5):416–424, 1956.

[25] M. W. Wong et al. Overview of the CCLAW L4 project. In Workshop on Programming Languages and the Law, 2022.

[26] Nordic Institute for Interoperability Solutions. X-Road Architecture. Official documentation, accessed September 2026.

[27] Department for Business and Trade. Set up and register your business in the UK. https://www.business.gov.uk/invest-in-uk/support-topics/set-up-and-register-your-business-in-the-uk/, accessed 5 September 2026.