Abstract
A holder sells five grams of pledged gold. The buyer can acquire the existing right without acquiring an unencumbered right or another five grams. The transaction must preserve the lender’s interest, any physical shortage, and unfinished payment instructions. Similar problems arise when delivery is disputed, an award remains unpaid, or a successor institution takes over unfinished work. This paper explains constructions that preserve rights, quantities, authority records, and remaining duties through these changes. Their results concern specified records and transitions under explicit assumptions. Legal effect, custody, and provider performance require separate evidence. An award supplies no positive lower bound on cash recovery.
1 Five grams, one existing right
Alice owns five grams in identified metal held by a custodian. She has pledged that holding to a lender. Bob agrees to buy it subject to the pledge. Assume the governing instruments permit this transfer, the necessary parties consent, and the custodian gives the required acknowledgment. The transaction changes the holder. It leaves the same metal, debt, security interest, and priority in place.
Recording a second five-gram allocation for Bob would duplicate the physical support. Deleting the pledge would enlarge the right without the lender’s release. Keeping Alice as the holder would leave the agreed transfer incomplete. A correct record changes the entitled person while preserving the right’s continuing conditions.
Now suppose only three of the five grams remain supported by custody evidence. The recorded face quantity is still five grams, with three supported and two deficient. Transfer does not repair that shortage. Bob receives the specified impaired right, with its deficit and associated claims, if the applicable instruments permit that transaction. A new holder’s name cannot turn missing metal into present metal.
This case follows the transfer construction in Allocated Title, § 7 [1]. It introduces the paper’s central question: which parts of an obligation may change, and which must survive? The answer requires more than a balance. It requires the exact right, its physical support, the authority for changing it, and everything already done toward performance. Authority is permission held by the named institution for the particular action at the relevant time.
2 What the register must distinguish
Allocated title means ownership tied to identified metal under the applicable law. A promise to deliver equivalent metal is a different right. The first depends on effective property arrangements and actual custody. The second depends on an obligor’s performance. A display showing “five grams” does not identify which right the holder has.
The construction identifies each physical lot and its quantity. Different descriptions of the same lot cannot create additional capacity. It assigns non-overlapping quantity portions to holdings. These portions are accounting coordinates within identified metal, not necessarily separately cut pieces of a bar. The governing law must give the recorded allocation its stated proprietary effect.
A pledge adds a security interest without adding metal. A permitted lease can instead replace present ownership with a claim for equivalent metal’s return. The register then records the redelivery obligation and no longer presents that quantity as the holder’s allocated metal. A shortage reduces support without silently deleting the entitlement. Restored support first follows the recorded restoration rule for the affected holders.
Money needs comparable care. A single receipt of 100 units can perform two obligations through allocations of 60 and 40. It cannot perform both for 100. The payment’s enduring identity prevents another report of the same movement from creating new quantity. Cash available for a new payment is separate: the institution may already have spent money received earlier.
Across record systems, authenticated correspondence must distinguish reports of the same movement from distinct causal legs. Matching field names alone cannot decide this. Corrections and alias reconciliation must preserve consumed quantities and expose unsupported credits.
An invariant is a condition preserved by every permitted operation. An election is a holder’s authorized choice among the permitted uses of the right. Allocated Title, § 5.6, proves its per-election invariant and conserved provider attribution under the stated transition rules and evidence assumptions. These results preserve the construction’s accounting distinctions. They do not establish that the vault holds the reported metal. Independent reconciliation and access to the underlying custody evidence remain required premises.
Even aggregate equality can mislead. Seventy grams of owned metal and thirty grams of authorized redelivery claims can account for a hundred-gram customer position. The same totals could conceal thirty grams lent without authorization. Quantity checks must therefore accompany the legal state and each holder’s actual election.
3 Four changes of holder
The transfer construction separates four operations in Allocated Title, § 7.1. Each changes an existing right rather than issuing another allocation.
A registered-owner change replaces the person in whose name allocated or encumbered title is held. The beneficial owner remains the same. A properly authorized trustee replacement is the motivating case.
A beneficial-owner change replaces the person entitled to the economic benefit. The registered owner remains in place. The metal and continuing encumbrances retain their identities.
A combined transfer changes both registered and beneficial ownership to the acquirer. Alice’s sale can use this mode when both interests pass to Bob.
A redelivery assignment changes the creditor of a contractual metal-return obligation. It transfers no present allocation of metal. The assignment must carry the effective transfer of each supporting collateral or guarantee right and any required obligor acknowledgment.
The admission decision determines whether the proposed operation is permitted. It must bind the precise instructions, parties, right, legal act, consents, and current authority. A continuing lien needs its full schedule, including priority and release conditions. The custody or obligation route must support the successor’s right. An unsupported assertion that a person is eligible cannot replace the required determination.
Unfinished instructions are part of this decision. Suppose a payment command still names Alice when Bob takes the associated right. Changing the register’s payee does not change the provider’s accepted instruction. The transfer must preserve a valid forwarding or discharge arrangement, close the old command conclusively, or use an effective transfer of execution capacity. Otherwise, the same obligation can acquire two incompatible payment paths.
The “Transfer preservation” proposition in § 7.3 states what an admitted transfer retains. Its “Enabled transfer” proposition adds a conditional completion result. Completion requires a reservation, a stable guard admitting the transaction, and weak fairness, meaning continuously enabled work eventually receives service. A permission that disappears before use is not a stable guard. Neither result supplies a missing consent, provider acknowledgment, or legal effect.
Payment and title can become final together only under a stronger contract. Allocated Title, § 6, requires reserved metal, funded settlement entitlement, and a common durable decision with the stated legal effect. Prepared rights must survive interruption and authorized succession. Even then, settlement entitlement is a claim on the settlement issuer. Later withdrawal remains a separate obligation. Without these premises, the record preserves completed legs and the remaining duties instead of claiming a completed exchange.
4 When forty kilograms earn a payment
A buyer contracts for 100 kilograms at a total price of 1,000 currency units. The governing acceptance rule pays ten units per accepted kilogram. Forty conforming kilograms are delivered and accepted. They earn 400 units. Sixty kilograms remain due, and only 400 units have become earned consideration under that rule.
The acceptance cannot rest on a weight report alone. It must bind the relevant quantity, measurement, assay where required, title and delivery conditions, receiving decision, and current authority. Any conversion between reported physical units and contractual quantity units must be exact, and the payment rule must be explicit. The same accepted kilograms cannot earn the full price again through a second report.
Suppose instead that an offered forty-kilogram batch fails the required assay. Its offered quantity earns nothing under this acceptance rule. If ten kilograms are later accepted, they earn 100 units. The other thirty kilograms of that offer remain disputed or unaccepted. The original hundred-kilogram duty retains the remainder that the governing disposition requires.
A distinct handling obligation can use a different basis. A handling fee of 50 units for the full hundred kilograms could earn 20 for forty handled kilograms, if its own instrument permits that allocation. Delivery acceptance and handling are separate duties. One observation cannot silently supply both legal predicates.
These calculations illustrate “Conserved earned consideration” in Allocated Title, § 8.2. The result assumes one serialized allocation authority, stable occurrence identities, exact conversion, and the complete acceptance conditions. Serializing allocation means competing uses are decided in one order against the same current state. A later acceptance cannot reuse quantity already consumed by an earlier one.
Correction presents a different problem from initial acceptance. Suppose 400 units were paid before an authoritative correction set that delivery’s target consideration to 100. The original 400-unit entitlement and payment remain recorded. The correction creates the applicable 300-unit repayment or recovery obligation. It does not erase the payment, manufacture a refund, or make that cash available again. Sections 8.3 and 8.4 carry these distinctions through correction and durable handover.
5 An award, a stay, and actual receipts
Recourse, § 2, follows a seller who invoices 100,000 units after delivery [2]. The buyer disputes part of the delivery. Under the example’s assumed valid agreement and arbitral process, the tribunal awards 80,000. The buyer voluntarily pays 20,000, and a properly authorized custody release supplies another 60,000. Both recipient credits reach the required legal finality. The applicable reversal windows close, and a receipt fee is 100:
The award, gross receipt, and net recovery are different quantities. Their relationship depends here on actual payment, funded escrow, proper release authority, finality, and fees. An award alone establishes none of those payment facts. The “Award is not recovery” corollary in § 9.7 proves that an accepted award gives no positive recovery lower bound in any unit. A valid record can simply stop after the award.
Suppose a court later stays a proposed execution. The stay changes present permission to act. It does not undo the voluntary payment or the earlier custody release merely by entering the record. Conversely, historical permission does not authorize a new execution after the stay.
The paper keeps four components: accepted historical acts, current permissions, economic effects, and explicit obstructions. Its fold is the deterministic calculation that derives these components from observed evidence. It distinguishes when an event happened, when evidence became available, and the time at which the record is being assessed. Reading or replaying the record never dispatches a payment.
This design does not declare every historical conclusion immutable. A review-only extension with fresh event identities preserves accepted history and economic effects while potentially changing current permissions. New legal attestations can revise historical legal conclusions. Evidence of key compromise has its own validity consequences. The theorem’s preservation statement is deliberately narrower than “nothing can ever change.”
Recovery also requires evidence beyond a provider’s acceptance of an instruction. Recipient credit, the governing legal-finality evidence, and a complete account of applicable reversal windows must support collection at the stated cutoff. Missing window metadata does not mean that no windows apply. The authoritative certificate must establish completeness even when its window set is empty.
“Bitemporal authority and effect integrity,” in § 9.7, assumes authentication, correct identity binding, canonical ordering, authoritative economic evidence, and single-successor attribution journals. Bitemporal here refers to the separate treatment of event and observation times. Consent, arbitral process, recognition, coercive actionability, custody release, and authorized claim adjustments remain external legal predicates. The theorem checks how their determinations are used. It proves none of those predicates true.
6 Who may release property and who pays the decision maker
Escrow is property held under a mandate specifying when and how it may be released. In the Recourse escrow profile, the trustee’s participation is mandatory. Release requires the trustee together with either the bonded party or the award executor, subject to the qualifying basis and mandate. The bonded party owns the bonded interest. This is not an unrestricted two-of-three signature rule. The bonded party and executor cannot bypass the trustee.
The signers must authorize the same instruction under current authority. The guard includes funded property, the committed custody predecessor, replay protection, operative holds, and the required release basis. A search finding no stay is not an affirmative determination that release is permitted. The intake record must meet its declared completeness requirements. Those requirements do not prove that no unobserved external stay exists.
Disputed property requires independent review under the stated profile. Holding property in suspense does not pay the beneficiary. A legal deadline cannot be silently extended, and its arrival does not itself authorize distribution. Separate operating reserves support continued case service without treating contested collateral as operating cash.
The institution’s remuneration rule addresses a different incentive. A protected term fixes a person’s total remuneration, payment dates, full funding, and reserved capacity before assignment. The total includes related benefits and cannot vary through case, hour, appeal, or outcome additions. Replacement capacity needs its own funded provision.
“Protected remuneration under a funded term,” in Recourse, § 10.4, establishes invariance to case assignment within this contract. Equal compensation receipts additionally require the stated provider execution and finality premises. The proposition is not an empirical finding of impartiality. Ownership, conflicts, appointment, and removal controls still matter. A fixed amount cannot by itself establish independent judgment.
7 A successor inherits the unfinished command
Consider the continuity example in Recourse, § 10.7. An obligation is 125 units, of which 40 have been paid. An old command can still pay the remaining 85, and 85 units remain funded. A successor inherits an 85-unit obligation, but has no capacity for an additional independent 85-unit command. The existing command already occupies it.
A late receipt then establishes another payment of 60. Twenty-five units remain due, and the old command may still perform those twenty-five. Only authoritative closure of that remainder, or an effective transfer of its capacity, permits a replacement under the corresponding rule. A local timeout establishes neither. An instruction can remain in flight after local authority to issue further instructions has ended.
Admissible Obligation Transitions, § 5.2, gives the more general “Exact joint replacement bound” [3]. It tests all old and proposed commands sharing the affected cash or performance capacity. Performance capacity starts with the unpaid duty and subtracts observed performance awaiting discharge and reservations outside the tested scope. Cash capacity is the usable funding remaining after outside commitments. Neither baseline subtracts the reservations for commands whose sufficiency the joint test is checking. The two capacities need not share a unit or numerical value.
The test covers every permitted intermediate outcome, including effects already made but not yet reflected at the accounting cutoff. In each coordinate, the least upper bound on possible consumption must fit the available capacity. The bound need not be attained by any one outcome. With an exact description of possible outcomes, the condition is necessary and sufficient. A sound overestimate gives a sufficient test, but can reject a safe replacement.
For example, suppose only 25 units of both cash and performance capacity remain. An old command can use 25, and a proposed replacement can independently use another 25. Their possible total is 50, so replacement fails. Now suppose an admitted provider contract limits remaining debit across both commands to 25 and separately limits remaining performance to 25. If both limits cover in-flight effects, the capacity test passes. The replacement still requires current action authority. An estimate that the commands are unlikely to execute together is not such a contract.
Every intermediate outcome matters. A later refund cannot excuse an earlier funding shortfall. A cash credit offsets exposure only when available for use. Returned performance restores capacity only under the governing obligation rule. Admission must atomically replace reservations for the complete tested scope and record the new command. Atomically means that these bookkeeping changes commit together or none commits. It must not count one joint reservation several times or deduct an already reflected debit again.
These conditions concern real provider behavior as well as local records. They cannot be obtained by renaming a command or moving it to a successor’s queue.
8 Changing the institution without resetting the obligation
Institutional change can alter the operator, interpreter, custody route, or technical representation. The continuing duty needs an enduring identity across these changes. Admissible Obligation Transitions, § 7, binds that identity to its constituting act and original occurrence. A new representation does not constitute a fresh obligation.
The successor needs more than a document archive and current balance. The transferred state includes the duty journal, current beneficiaries and units, exact performed portions, pending commands, reservations, and the evidence governing current authority. A continuation is the unfinished computation together with the state needed to resume it. An unfinished program also needs its instructions, working environment, pending control state, and remaining work count. A saved result cannot reconstruct unfinished execution.
The complete continuation cut in Recourse, § 10.7, gathers this state across the affected namespaces and histories. One authorized successor must have exclusive execution control. Old external commands remain possible until the required evidence limits or closes them. Durable local transactions and effective external controls are distinct premises. A database handover does not itself supply a provider-side fence against further execution.
Preservation also differs from amendment. Carrying an existing duty into another representation preserves its currently effective terms and payment history. Changing the beneficiary, amount, priority, or discharge rule requires the authority for that substantive change. Equal balances are insufficient. Different allocations of a past payment can leave different people entitled to different remaining portions.
Useful completion needs additional service assumptions. Repeatedly transferring a duty must not reset its original deadline or erase work already counted. The progress results in Admissible Obligation Transitions, § 7.4, track the original duty episode and require the specified productive service. Fair scheduling alone cannot make unavailable funding, unlawful instructions, or an unresponsive provider deliver performance.
The constructions therefore preserve obligations through change without promising that change will resolve them. A complete handover can retain a shortage, a stayed remedy, a disputed receipt, or an unavailable route. These are specific remaining duties or obstructions. Removing them from the successor’s record would conceal the work that still needs an authorized actor.
9 Technical reading map
Allocated Title supplies the property and physical-performance construction. Read § 4 for assumptions A1–A8, then §§ 5.1–5.6 for quantities, cash attribution, and conservation. Section 6 states “Coupled final trade entitlements.” Sections 7.1–7.3 define the four transfer modes, their admission contract, “Transfer preservation,” and “Enabled transfer.” Sections 8.1–8.4 cover acceptance, “Conserved earned consideration,” corrections, and “Durable physical performance.” Section 8.6 extends the construction through “Continuation with current obligations.”
Recourse separates adjudication from execution and receipt. Section 2 contains the invoice example. Section 8 specifies escrow and its mandatory-trustee rule. Sections 9.1–9.5 define evidence, authority, economic effects, and the fold. Sections 9.6–9.7 state assumptions F1–F5, external predicates L1–L6, “Bitemporal authority and effect integrity,” and “Award is not recovery.” Section 10.4 treats funded protected remuneration. Section 10.7 states complete continuation cuts, durable continuation, and resource conservation.
Admissible Obligation Transitions supplies the general obligation model. Section 5.2 defines the “Joint remaining-effect contract” and proves the “Exact joint replacement bound.” Section 7.2 states “Preservation across carriage.” Sections 7.4–7.7 distinguish progress assumptions, concrete productive service, and durable execution. Sections 7.8–7.9 treat exact interval representation and structural continuation with one economic ledger. These are specified constructions, not equivalence results for arbitrary replacement programs.
The cited results have written mathematical arguments with explicit hypotheses. Their scope must be read statement by statement. In particular, Recourse, § 13, states that none of its results is machine-checked. No result cited here proves legal effect, physical custody, universal provider reliability, or a positive minimum cash recovery from an award.
References
[1] Raeez Lorgat. Allocated Title. September 2026.
[2] Raeez Lorgat. Recourse. September 2026.
[3] Raeez Lorgat. Admissible Obligation Transitions. September 2026.