Warrant Does Not Travel
Aa
Bilateral Validity, Global Failure
Bilateral validity does not guarantee global coherence. The research program studies related forms of that proposition in institutional, formal, computational, and empirical settings, then asks what follows when the mapping between them holds.
The proposition appears in four registers across the corpus, but the objects are not identical. Volume I develops a historical and institutional argument about the work required to make claims portable. The Proofs supplies defined local-to-global structures and scoped obstruction results. Bulla computes a rank measure over tool schemas under stated assumptions. BABEL reports how that measure co-occurs with annotated convention mismatch on a disclosed corpus. The last is a disclosure or omission result, not a prediction of runtime failure; the erratum below records the distinction.
The page that follows maps what survives as the proposition moves from one register to another and what changes with the object being studied. The mappings organize the research program; they do not allow evidence in one register to certify a claim in another. Historical recurrence does not prove a theorem, a theorem does not validate an institutional identification, and an annotation result does not establish runtime performance.
This is the across-register bridge. Its companion, No Equation Governs Alone, traces the across-volume bridge — how the trilogy's four equations constrain one another without pretending to form a single deduction from witnesses to mercy.
Erratum (2026-07; scope note corrected 2026-08). The Empirical register below originally read the coherence fee as predicting real compositional failure (ρ = 0.996, "fee ≥ 4 unsafe at ~100%"). That predictive framing is retired. Those correlations were against annotation-derived labels; on execution-derived labels the fee does not predict runtime failure. It is a disclosure/omission measure: how much convention two tools leave undisclosed at their seam. A reproduction audit separately established that the archived 703-composition index is the complete pairwise closure of the 38 servers in a 57-manifest registry that meet the documented
MIN_SCHEMA_FIELDS = 3threshold:C(38,2) = 703. Every one of the 893 excluded pairs contains at least one of the 19 below-threshold servers. Statistics from that index describe the declared subset and do not establish behavior on its excluded complement. The one claim of this page — bilateral validity does not guarantee global coherence — survives both corrections. The repository records arebulla/FALSIFICATIONS.mdandscripts/REPRODUCIBILITY.md.
Four Registers of Warrant
| Register | What it says | Key artifact | Falsification |
|---|---|---|---|
| Philosophical | Across selected institutions, the witness vocabulary distinguishes support for a claim, responsibility for its use, and the work required at a change of context. | Vol I, especially Chapter 7 and No Equation Governs Alone | Evidence that the categories erase material differences, or that the comparison adds no useful distinction beyond existing accounts. Omission of a function is not by itself a refutation of the narrower claim. |
| Formal | In specified sheaf and linear-algebraic models, bilateral validity need not produce a global section, and stated obstructions or rank identities are computable under their hypotheses. | The Proofs and machine-checked technical artifacts, each limited to its exact theorem statements | A counterexample to a stated result under its exact hypotheses, or failure of an institutional application to satisfy those hypotheses. |
| Computational | Under Bulla's assumptions, a rank difference measures semantic dimensions left undisclosed across a tool-schema seam. | Bulla, especially coboundary.py and witness_geometry.py | A counterexample to the rank identity under its assumptions, or a material class of schemas for which the assumptions do not hold. |
| Empirical | On an archived, scope-constrained corpus, the computed fee co-occurs with annotated convention mismatch. BABEL does not validate runtime-failure prediction. | The archived 703-composition index (bulla/calibration), read as annotation-derived co-occurrence on the complete documented ≥3-field subset | Met (2026-07): on execution-derived labels fee = 0 seams breach and the fee does not discriminate — reframed as disclosure. The 703-index scope excludes the 19 below-threshold servers. |
The registers carry related propositions with different warrants. The next section follows one term across them and marks the translation at each boundary.
Coherence Fee Across Four Registers
The coherence fee is the clearest through-line and the easiest place to mistake analogy for identity. The name refers to related but non-identical quantities across the registers. The walk below states what is preserved and what is not.
Philosophical Warrant
In Volume I, the coherence fee appears as a historical observation. There is an irreducible cost to composing local truths into global ones. A merchant's claim could travel across distance through correspondent relationships, documentary form, presentation, and recourse. When a bill was dishonoured, notarial protest could make refusal travel as evidence; in later regimes, endorsement allowed transfer among holders. These mechanisms did not arise together or operate uniformly. None was free. 1 The Codex introduction now gives the claim its necessary scope:
Verification has always had an institutional price. A notary authenticates an act, a bank maintains accounts, and a platform vouches for a stranger. Each performs work at a boundary between what someone needs to know and what that person can check alone. That boundary work changes with the domain and has no universal physical or monetary unit. Its irreducible remainder is the coherence fee.
This is not a theorem. It is an interpretation of institutional history: notarial infrastructure, guild chartering, double-entry bookkeeping, credit bureaus, and platform governance all perform work at boundaries where claims must travel. Diffusion, legal borrowing, local adaptation, and independent response can all contribute to the recurrence. The five witness functions organize comparisons among the selected cases; Chapter 7 rejects treating them as a universally necessary package. Neither their recurrence nor that comparison proves a universal price function.
Formal Warrant
The Proofs contains two related formal constructions. The sheaf condition (A13) states when local data glue into a global section within its defined structure, and specified obstructions describe some failures of gluing. A21 is a separate proposed coherence-cost framework indexed by scope and authority. It supplies cost categories, tradeoffs, and budgets, but its former refinement-monotonicity theorem was false under the stated hypotheses and has been withdrawn. It supplies no ordinal guarantee. Chapter 19 and Appendix K.4 retain the counterexample and the surviving cost framework.
Correction (2026-09-11): A21. A cover can refine another while reducing the number of nonempty overlap checks. The previous cost-ordering proof incorrectly assumed that every old comparison survived as a distinct new comparison. An earlier review preserved the formal statements while correcting their references, and accepted this bridge's rejection without propagating it into Chapter 19, Appendix K.4 and their consumers. The present correction supersedes that part of the cross-edition clearance; the earlier record remains visible. Cost categories and declared budgets survive. Refinement supplies no universal cost ordering, and the separate rank and edge-addition results retain their own hypotheses.
The formal register therefore makes a chosen local-to-global structure exact. It does not turn the notary's fee into a cohomology class, prove that every historical witness system instantiates the model, or establish a real-world cost law. Those identifications remain arguments that must be defended in the domain where they are made.
Computational Warrant
Bulla defines a rank difference under its stated assumptions on tool schemas:
fee(G) = rank(δ_full) − rank(δ_obs)
Here δ_full is the coboundary operator over each tool's declared internal_state, while δ_obs is its column restriction to declared observable fields. Their rank difference is the codimension introduced by that restriction under the model's loaded convention vocabulary. Under the declared representation, the computation is polynomial in the composition graph, exact under rational arithmetic, and requires no execution. It is a schema-level disclosure or omission invariant, not a census of every convention the schemas failed to name.
Moving from the formal register to the computational is a modeling choice rather than an identity of definitions. The hierarchical fee theorem — fee(G) = Σ fee(Gᵢ) + boundary_fee(P) for a stated partition construction — decomposes the rank measure under its hypotheses. The checked results support that scoped calculation. They do not show that the number prices institutional work, measures A21's organizational cost, or predicts whether a live composition will fail.
Empirical Warrant
BABEL's archived calibration measures how the computed fee relates to annotated convention mismatch — schema-vs-convention labels, not execution outcomes. Within its 703-composition index, the original analysis reported Spearman ρ = 0.996 between boundary fee and annotated mismatch count per composition, with zero annotated mismatches among the 678 compositions where boundary fee = 0. A separate 10-server pilot supplied the annotation-derived zone table (fee = 0 vs fee ≥ 4). These are descriptions of the 38-server subset selected by the documented three-field threshold. The index is complete within that domain; it says nothing about pairings involving the 19 excluded servers.
On execution-derived labels the picture is different, and it is the one that governs. When the label is a real round-trip rather than an annotation, the fee does not predict failure: fee = 0 seams breach at real execution, and the fee does not discriminate a breaking seam from a safe one (see bulla/FALSIFICATIONS.md). The rank quantity remains a defined disclosure/omission invariant, and the reported annotation-derived association remains an observation on that corpus. Neither makes the fee a runtime-failure oracle.
Resemblance Is Not Validation
The walk preserves a family resemblance while keeping four distinct claims in view. Institutional history motivates the boundary problem. The formal work makes selected local-to-global structures exact. Bulla computes a scoped omission measure. BABEL reports a scope-constrained, annotation-derived co-occurrence and, through its failed execution prediction, establishes a limit. The later registers were designed in conversation with the earlier ones, so their agreement organizes inquiry rather than independently validating the historical thesis.
Witnesses, Fees, New Predicates
Three other concepts traverse the four registers without requiring a full walk.
Witness Structure
Volume I uses binding, conditions, stakes, recourse, and composition as a comparative vocabulary, with dependencies defended for particular uses rather than a general hierarchy. The Proofs defines commitment, evidence, schema, epistemic status, coherence requirements, and sense boundaries for its own formal program; these objects model parts of the witness problem without reproducing the historical categories one for one. Bulla makes a narrower implementation choice through content-addressed WitnessReceipts: a composition_hash binds what was proposed, a diagnostic_hash binds what was measured, and a receipt_hash covers those references together with policy profile, disposition, and the rest of the witnessed record. BABEL's real-MCP tracks examine tool ecosystems. This receipt format is one implementation over tool compositions, not a universal realization of the five properties.
Boundary Fee
Vol I identifies chokepoints at which the trust tax may accrue: the notary, the bureau, and the platform sit at boundaries across which claims must travel. The Proofs studies agreement across overlaps within its formal structures. Bulla's hierarchical fee theorem decomposes its rank measure into intra-component and boundary terms under the stated partition construction. A cross-server example can show what the schema-level omission measure reports; it does not demonstrate that the omission causes runtime failure or that it is identical to an institutional trust tax.
Predicate Invention
Vol I's convergence thesis asks how new witness predicates — novel ways of naming what a witness attests — emerge under institutional need. The Proofs gives that question a particular formal treatment in A17 (predicate invention) and A17b (conservative extension): within the model, admissibility requires local grounding, overlap agreement, and conservative extension. Predicate Invention Under Sheaf Constraints (SCPI) has machine-checked finite constructions, including a concrete overlap-incompatibility witness and scoped Z/2 torsor results. Its present conservativity-descent and generalized counterexample files do not formalize the propositions their comments announce, so those claims remain unverified.
The empirical register is currently empty. No BABEL-equivalent benchmark measures predicate invention in the wild. This is the gap in the cross-register matrix, and it should be named rather than hedged: the philosophical, formal, and formal-adjacent registers converge on a claim that the empirical register has not yet tested. A future benchmark family — compositions in which a new predicate is introduced mid-pipeline and bilateral validity must be re-established — would close the row. Until then, predicate invention travels three registers, not four.
Correspondence Without Cross-Validation
Agreement across registers can reveal a productive correspondence: it shows where an institutional question admits a formal model, where the model admits a computation, and where the computation can be confronted with observations. It is not independent cross-validation when later artifacts inherit definitions and examples from earlier ones. Each proposition must retain the warrant proper to its register.
This is the methodological reason to keep the artifacts distinct. A failed empirical prediction can narrow Bulla's interpretation without erasing a correct rank identity. A correct theorem can survive even when an institutional application violates its hypotheses. A persuasive historical recurrence can motivate formal work without being proved by it. The bridge is useful when it exposes those boundaries, not when it makes them disappear.
Failure at the Identifications
The cross-register argument has specific failure conditions.
If the selected historical cases do not support the five-property recurrence, the institutional thesis weakens without deciding the formal results. If an application of the sheaf model does not satisfy its hypotheses, the identification fails even if the theorem remains true. If Bulla's assumptions exclude a material class of real schemas, its computational scope narrows. BABEL has already met the runtime-prediction falsifier: the rank measure does not discriminate breaking seams from safe ones on execution-derived labels and must remain a disclosure or omission measure unless new evidence earns a different claim.
The bridge fails whenever one register borrows another's warrant without an explicit identification. Its surviving claim is modest and exact: bilateral validity can coexist with global failure in several defined settings, and every move between those settings remains open to audit.