Technical Bridge
The one claim
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, Bridge Arguments, traces the across-volume bridge — how the trilogy's four equations compose into a single dependency chain from witnesses to mercy.
Erratum (2026-07). 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, i.e. how much convention two tools leave undisclosed at their seam. The one claim of this page — bilateral validity does not guarantee global coherence — is unaffected; only the empirical register's predictive reading is corrected below. See FALSIFICATIONS.md.
The four registers at a glance
| Register | What it says | Key artifact | Falsification |
|---|---|---|---|
| Philosophical | Across selected institutions, recurring witness functions make claims portable, and composing claims across contexts requires work whose cost and rent must be distinguished. | Vol I, especially the convergence thesis (Ch 1) and the scoped local-to-global mapping (bridge-arguments) | A durable coordination system that omits one of the five witness properties, or evidence that the categories erase the cases' material differences. |
| Formal | In specified sheaf and linear-algebraic models, bilateral compatibility need not produce a global section, and stated obstructions or rank identities are computable under their hypotheses. | The Proofs and the current formally verified technical reports, each within its declared model | 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 the disclosed corpus, the computed fee co-occurs with annotated convention mismatch. BABEL does not validate runtime-failure prediction. | The 703-composition calibration (bulla/calibration), read as annotation-derived co-occurrence | Met (2026-07): on execution-derived labels fee = 0 seams breach and the fee does not discriminate — reframed as disclosure. See FALSIFICATIONS.md. |
The registers carry related propositions with different warrants. The next section follows one term across them and marks the translation at each boundary.
Walking one row: the coherence fee
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 register
In Volume I, the coherence fee appears as a historical observation. There is an irreducible cost to composing local truths into global ones: the merchant's word, adequate among his creditors in Bruges, does not compose across the bill of exchange's journey to Florence without a notary's seal, an endorsement chain, and a standing relationship between the two parties' banking agents. The seal, the chain, and the relationship are not free. Someone pays. The Codex introduction names this cost directly:
There is a real cost to composing local truth into global coherence — call it the coherence fee — and someone must pay it.
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 properties describe what the selected cases repeatedly provide; they do not prove a universal price function.
Formal register
The Proofs contains two related formal constructions. The sheaf condition (A11) 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 coherence-cost framework indexed by scope and authority. It provides cost categories, tradeoffs, budgets, and an ordinal monotonicity result inside the model; it does not provide an absolute cost function or a calibrated comparison across institutions.
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 register
Bulla defines, under its stated assumptions on tool schemas, a rank difference:
fee(G) = rank(δ_full) − rank(δ_obs)
Where δ_full is the coboundary operator over all fields including internal_state, and δ_obs is the coboundary restricted to observable fields. The difference counts independent semantic dimensions hidden from at least one tool but required by an edge. The computation is polynomial time in the composition graph, exact under rational arithmetic, and requires no execution. It is a schema-level disclosure or omission invariant under those assumptions.
The move from the formal register to the computational is a modeling choice, not 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 register
BABEL and the calibration corpus measure how the computed fee relates to annotated convention-mismatch — schema-vs-convention labels, not execution outcomes. The registry calibration — 38 real MCP servers, 703 pairwise compositions — found Spearman ρ = 0.996 between boundary fee and the annotated mismatch rate, with zero annotated mismatches among the 678 compositions where boundary fee = 0; a separate pilot on 10 servers produced the annotation-derived zone table (fee = 0 vs fee ≥ 4).
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 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.
What the walk shows
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 an 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.
Gestured rows
Three other concepts traverse the four registers without requiring a full walk.
Witness structure
The five witness properties (binding, conditions, stakes, recourse, composition) recur across Vol I's selected institutions. 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 operationalizes a narrower choice through content-addressed WitnessReceipts whose three hashes bind composition, measurement, and judgment separately. BABEL's real-MCP tracks examine tool ecosystems. Bulla's 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) develops a formally verified finite-site core; its perimeter statements remain conditional.
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.
What cross-register coherence is evidence of
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.
Vulnerability
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.