Introduction to The Proofs

#Work That Need Not Be Repeated

Much of what an institution knows was established by somebody else. The report arrives without the laboratory, the calculation without the people who chose its assumptions, the identification without the inquiry that distinguished one person or object from another. This separation makes cooperation possible. A recipient who had to begin every investigation again would often be unable to use its result at all.

The difficulty is to inherit the result without silently enlarging it. A measurement can answer a new question when the question requires what was measured. A translation can preserve a property without preserving every other property of the thing translated. The receiver needs to know which of these achievements it has obtained, and what further work its intended use requires. A familiar name for the result—proof, certificate, verified record—cannot settle the matter in advance.

The Proofs develops a connected specification for this passage from an earlier result to a receiving operation. Its principal question is what the operation may claim on the strength of work already done. The specification follows that question through proposals, checks, substitution, changing definitions and subsequent correction. It also supplies mathematical models in which some of the preservation requirements can be stated and proved. Their hypotheses matter because the practical work is often to establish whether those hypotheses describe the proposed use.

A proof, a statistical estimate, an attestation and an authorization do not become interchangeable when placed in the same record. A valid calculation can proceed from false empirical premises. An authenticated report can preserve exactly what its author asserted without establishing that assertion. Permission can authorize an act without predicting its success. The receiving contract must preserve the differences that its own conclusion depends upon.

This is a formal companion to Similes of Symmetry, Volume I of Res Agentica. The historical argument follows institutions that make evidence usable beyond the circumstances of its production. Here the reader can examine selected models and proposed operations motivated by that problem. The models do not prove the history, establish a unique institutional design or confer political standing on the people described in a database. Where the trilogy asks who may decide and what an institution must relinquish, a computational certificate cannot answer in the institution's place.

#What a Receiver Gains

Consider the advantage of an established substitution. The next operation can use it without repeating the identification, provided the property it needs falls within the mapping's scope and the supporting grounds remain adequate. A record that binds those grounds to that use can therefore save real work. It can also prevent a later operator from mistaking the saved investigation for an investigation nobody needed to perform.

The same record becomes useful when something changes. A correction to one premise may remove the only support for one conclusion while leaving another untouched. An independent test may supply what the withdrawn report no longer does. Following those differences requires more than retaining the old document beside its correction. It requires knowing what each recipient consumed and what it has now established instead. Where that knowledge is missing, the limit on completed correction must remain visible.

The proposed contribution lies in making these established disciplines govern the same use. Provenance, incomplete outcomes, scoped equivalence, versioning and correction already have substantial technical and institutional histories. What We Claim states the comparison the companion must meet. If a simpler combination of existing tools preserves the necessary distinctions and satisfies the receiving contracts, that counts in favor of the obligations. It supplies no reason to demand this book's ontology as the price of compliance.

Several mathematical constructions give parts of the inquiry an exact form. A specified sheaf guarantees unique amalgamation of exactly matching local sections. A property-preserving map can justify a substitution. A declared cost framework can identify checking expenditures and show which duties a budget leaves unfinished. These are different achievements, with different assumptions. Agreement within a tolerance can be useful statistical evidence without inheriting the exact-gluing theorem. A smaller bill for checking can reflect a useful saving, or the removal of a comparison the new representation no longer asks anyone to make.

The latter distinction is developed through A21's counterexample: refinement does not impose a universal ordering on verification cost. Its failure matters to the surviving argument because it changes how a saving must be understood. Likewise, the exact/approximate boundary changes what an interface may certify. Other withdrawn constructions have their present status and missing requirements identified where they appear. The durable correction record preserves their history; understanding the current specification does not require replaying it.

#Three Ways Through the Work

The publication order develops definitions, mathematical equipment, constructions, specifications and demonstrations. It is one route through the inquiry. The dependency graph helps locate a hypothesis; it is not the only order in which a reader can discover why the hypothesis matters.

For an operational reader, begin with the Part V entrance. Its hypothetical component dossier follows the saving and the obligations created by reuse. Continue to Appendix I, where the ten operations distinguish successful checks, established failures and unfinished work. Use query semantics, A25 and correction and versioning, A26 to examine the receiving contract, then the demonstrations of refusal and scoped substitution. Appendix G's formats are proposed records, not a report of shipped behavior.

For a mathematical reader, begin with commitment sets, witness typing and the context construction, consulting the notation and vocabulary appendices as needed. Follow the hypotheses into A13's exact gluing, transport and predicate admission. Appendix K distinguishes the written results from the declarations actually checked by the companion projects. Appendix L separates standard consequences and finite boundary calculations from constructions still missing their objects or laws. An unfinished construction is not used as a theorem merely because it has retained a place in the book.

For a trilogy reader, begin with the institutional question that prompted the visit. Volume I's Witness Claims and the Anchor Registry identify related models where a pointer is supplied. Read the Part V dossier or the relevant worked passage before following that pointer into its definitions. The relation is an invitation to test the model against the problem. No formal pointer is required where the institutional claim has no appropriate mathematical counterpart.

These routes meet at the receiving operation. A reader interested in implementation needs to know what its result promises. A reader interested in proof needs to know what the premises establish. The trilogy reader needs to see which further judgment the formal result leaves to an institution. None can borrow the others' answer merely because the records have been joined.

#The Standing of the Work

The registry retains 32 primary anchors and six subanchors. They identify definitions, results and proposed contracts; their presence in a dependency graph does not give them one common verification status. The statement, its hypotheses and its evidence determine what follows. Appendix K supplies the declaration-level boundary for the machine-checked material. A written proof remains assessable as a proof; absence from a checked project is neither refutation nor verification.

A fixed vocabulary can support much of this work. Its users can preserve the scope of an earlier result, obtain additional grounds and reassess the conclusions that consumed a corrected premise without inventing a predicate. A32's Third Mode describes a fuller design that includes governed vocabulary extension; Appendix I specifies its ten-operation contract. Adequate reuse does not require every implementation to adopt that whole profile.

The operational contracts also remain proposals. Bulla supplies some related record, verification and bounded enforcement capabilities. The correspondence in Appendix I distinguishes what the inspected implementation records, checks and enforces from what it does not supply or decide. A receipt can bind a report to an identified artifact without establishing that the reported execution occurred. A policy checker can decide whether a submitted record meets its rule without deciding whether the institution should have adopted that rule. Complete Third-Mode compliance is not claimed.

The notation keeps equivalences scoped and reasoning rules explicit. Witness classes name regimes of assurance, not a universal numerical ladder: combining statistical claims requires joint assumptions, while accepting an attestation requires the relevant trust and authority. Stronger standing can be earned through additional grounds. What is prohibited is promoting a claim beyond those grounds merely by copying it into a more authoritative-looking record.

The demand becomes hardest after the result has been used. A receiving system must stop advertising support withdrawn under its applicable correction process. It must disclose the reassessment it has completed and the dependencies it has not reached. If its earlier answer helped authorize an action, changing the answer may begin a further obligation rather than discharge one. The record can make that obligation harder to evade. It cannot fulfill it on behalf of whoever has the power to act.