Docket Map
Gateway
Two essays provide accessible entry points to the research program. Neither claims formal novelty; both derive consequences from first principles and ground them empirically.
The Coherence Fee — Pricing Composable Truth for Autonomous Coordination
Status: Demonstrated. Claims: C3 (Trust Tax), C4 (Compositional Failure). Chapters: 2, 3.
Derives the coherence fee as a topological invariant of tool compositions — the number of independent failure modes invisible to bilateral checking — and shows it is computable in polynomial time from schemas alone, independent of whether the nodes are agents, tools, services, or departments. Backed formally by the fourteen Lean-verified theorems in Papers I–III.
Bitcoin After Money — The Settlement Layer for Autonomous Coordination
Status: Demonstrated. Claims: C10 (Credible Exit). Chapters: 14.
Derives four settlement properties — censorship resistance, dilution resistance, low issuer dependency, permissionless verification — from first principles in autonomous coordination, then shows they converge with republican political theory. Constitutional requirement and technical constraint select the same four properties independently.
Spine
Seven papers constitute the formal core. Four carry machine-checked proofs in Lean 4.
SCPI — Predicate Invention Under Sheaf Constraints
Status: Verified (Lean 4). Claims: C4 (Compositional Failure). Chapters: 3.
Reduces predicate invention to a descent problem with three independent gates: topological, model-theoretic, and definability. The Extension Torsor Lemma shows that when global agreement is feasible, solutions form a principal homogeneous space over convention transformations. The separation enables targeted diagnostics: each gate can be tested independently.
Bridge — Empirical Witness for Compositional Failure
Status: Demonstrated. Claims: C3 (Trust Tax), C4 (Compositional Failure). Chapters: 2, 3.
Demonstrates that bilateral validity coexists with compositional failure in live LLM-database systems. Establishes the identification/specification gap: LLMs identify topologically prescribed bridge types with perfect accuracy but fail to specify them — topology determines what bridges are needed but not how to implement them.
Seam — Protocol Consequence of Compositional Failure
Status: Operational (seam-lint deployed on live MCP servers). Claims: C4 (Compositional Failure), C8 (Receipt Minimum), C9 (Predictable Failure). Chapters: 3, 11, 12.
Translates the compositional-failure diagnosis into an operational protocol: semantic manifests, fraud proofs, and a five-step verification procedure. When a pipeline fails at a seam, the protocol identifies which component produced the inconsistency and generates a fraud proof that is independently verifiable.
The Coherence Fee, Paper I: Hierarchical Decomposition — Measuring Hidden Convention Risk
Status: Verified (12 Lean theorems, 0 sorries; 703-corpus validation). Claims: C3 (Trust Tax), C4 (Compositional Failure). Chapters: 2, 3.
Proves fee = rank(δ_full) − rank(δ_obs): the coherence fee is the rank gap between what the composition requires and what bilateral observation can verify. A hierarchical decomposition theorem shows the fee splits into local sub-workflow fees plus a non-negative boundary fee, with an additive tower law across hierarchy levels. A minimum disclosure theorem prescribes exactly which fields to expose to eliminate all blind spots.
The Coherence Fee, Paper II: Column-Matroid Backbone — A Structural Identity
Status: Verified (12 Lean theorems, 0 sorries; 703 compositions). Claims: C4 (Compositional Failure). Chapters: 3.
The coherence fee equals the corank of the observable column set in the linear matroid on the full coboundary columns. This structural identity yields a closed-form pairwise formula — fee(A, B) = U(A) + U(B) + |shared_dims(A, B)| — verified on all 703 real-schema compositions. An empirical density law localizes the connecting homomorphism strictly in the interior of its attainable range.
The Coherence Fee, Paper III: The Witness Gram — Kron-Reduced Laplacian Characterization
Status: Verified (2 Lean theorems; 240/240 leverage predictions). Claims: C4 (Compositional Failure). Chapters: 3.
The witness Gram of a tool composition is always a Kron-reduced graph Laplacian. Effective resistance between hidden fields quantifies disclosure substitutability: the smaller the resistance, the more equivalent two fields are as repair choices. All 240 nontrivial compositions tested, including 37 natural regime breaks, match the predicted leverage values exactly.
Local Validity Does Not Compose — A Theorem on the Limits of Bounded AI Audit
Status: Demonstrated (proof complete; 500-composition spectral experiment). Claims: C1 (Tempo and Legitimacy), C4 (Compositional Failure). Chapters: 0, 3.
Bounded-radius local audits and low-order spectral summaries cannot, in general, certify global semantic coherence. The proof constructs semantic twins — compositions identical to every bounded local audit yet carrying arbitrarily many independent obstructions. The coherence fee is the exact computable alternative that bounded methods cannot replicate.
Empirical
Three papers test the formalism against real and synthetic compositions.
BABEL — Benchmark for Autonomous Bridge Evaluation and Localization
Status: Frozen (v0.1.0; 932 instances, 7 families, 3 provenance tiers). Claims: C4 (Compositional Failure), C9 (Predictable Failure). Chapters: 3, 12.
The first public benchmark for compositional semantic failure — locally correct systems composing into globally wrong outcomes. Five frontier LLMs cannot solve it. Topological awareness is the key ingredient; algebraic apparatus adds narrow dimension-specific value.
Coherence Cliff — Scaling Evidence for Sheaf Diagnostics
Status: Demonstrated (500 graphs, 7 scales, 5–50 nodes). Claims: C4 (Compositional Failure), C7 (Domination Without a Dominator). Chapters: 3, 10.
Documents the regime change where conventional testing collapses as composition size grows. Sheaf diagnostic maintains R² > 0.96 at all scales; the best of twelve baselines degrades from 0.83 (n=5) to 0.52 (n=50). Six convention dimensions grounded in real standards.
The Physical Coherence Fee — Constitutional Implications of Physical Compositions
Status: Conjectural. Claims: C6 (Bifurcation), C8 (Receipt Minimum). Chapters: 7, 11.
Extends the coherence fee to the physical domain, where convention mismatch between autonomous systems produces irreversible consequences externalized to non-participants. The augmented indifference curve — four cost terms that structurally push against voluntary adoption of compositional verification — shows why the coordination substrate becomes constitutional infrastructure at organizational boundaries.
Frontier
Two papers establish what alternative approaches cannot achieve.
SHEAF — Structured Heterogeneous Consensus Among Agents
Status: Conjectural (enriched Laplacian-Cohomology Bridge and topology auction unverified). Claims: C4 (Compositional Failure), C7 (Domination Without a Dominator), C8 (Receipt Minimum). Chapters: 3, 10, 11.
Extends diagnostics to heterogeneous agents with different observation spaces, vocabularies, and objectives. A topology auction assigns correction costs to the agents whose assessments contribute most to the obstruction. Impossibility certificates prevent indefinite negotiation when consensus is formally unachievable.
Interpretability Frontier — Can Mechanistic Interpretability Substitute for Structural Diagnosis?
Status: Demonstrated (240+ compositions; GPT-2 + Gemma 2 cross-model replication). Claims: C7 (Domination Without a Dominator), C11 (Quiet Foreclosure). Chapters: 10.
Edge-local interpretability — SAE, probing, circuit tracing — is provably incomplete for cyclic compositional failure. Structural diagnostic achieves ρ=1.0 in every condition; the best interpretability baseline never exceeds ρ=0.758. The gap is not an engineering shortfall but a structural impossibility: local tools cannot see global obstructions.
Claim Coverage
| Claim | Title | Ch | Formal Backing | Strongest Status |
|---|---|---|---|---|
| C1 | Tempo and Legitimacy | 0 | Local Validity | Demonstrated |
| C2 | Convergent Witnessing | 1 | — | Historical |
| C3 | Trust Tax | 2 | Coherence Fee, Bridge, Paper I | Verified |
| C4 | Compositional Failure | 3 | SCPI, Bridge, Seam, Papers I–III, BABEL, Coherence Cliff, Local Validity | Verified |
| C5 | Factor Prime | 6 | — | Empirical |
| C6 | Bifurcation | 7 | Physical Coherence Fee | Conjectural |
| C7 | Domination Without a Dominator | 10 | SHEAF, Coherence Cliff, Interpretability Frontier | Demonstrated |
| C8 | Receipt Minimum | 11 | Seam, SHEAF, Physical Coherence Fee | Operational |
| C9 | Predictable Failure | 12 | Seam, BABEL | Operational |
| C10 | Credible Exit | 14 | Bitcoin After Money | Demonstrated |
| C11 | Quiet Foreclosure | 10 | Interpretability Frontier | Demonstrated |
| C12 | Mercy Threshold | 17 | — | Architectural |
Three claims — C2, C5, and C12 — rest on historical, empirical, or architectural argument without formal substantiation. C2 is a convergence claim about civilizational evidence. C5 is an empirical observation about automation sequencing. C12 is an architectural requirement whose formal instantiation would require a theory of forgiveness that this program does not attempt.
Falsification Index
Each paper specifies the condition under which it is falsified. A reader who meets one of these conditions has refuted that portion of the framework.
| Paper | Falsified If |
|---|---|
| Coherence Fee | A composition with nonzero fee whose blind spots never produce observable failures across sufficiently diverse inputs |
| Bitcoin After Money | Regulatory substitution, stablecoin commitment, or delegated-access patterns proving sufficient for stateless agent settlement |
| SCPI | A predicate invention problem not decomposable into the three gates, or a composition failure not captured by H¹ |
| Bridge | LLM-generated bridge specifications achieving cycle closure rates comparable to hand-crafted typed specifications |
| Seam | A more efficient diagnostic mechanism that does not require the sheaf condition |
| Paper I | A composition with nonzero fee whose blind spots never produce observable failures across sufficiently diverse inputs |
| Paper II | The backbone identity fee = corank_M(O) fails, or the closed-form pairwise formula disagrees with coboundary rank |
| Paper III | Witness Gram rank disagrees with the coherence fee, or per-component leverage deviates from (card(H_C) − 1) / card(H_C) |
| Local Validity | A bounded-local method or low-order spectral statistic that certifies coherence on the semantic twins family |
| SHEAF | A non-topological diagnostic maintaining accuracy at scale, or a counterexample to the Laplacian-Cohomology Bridge |
| BABEL | A submission achieving positive R² on Track A without topological features |
| Coherence Cliff | A bounded-depth testing strategy that maintains accuracy at scale |
| Physical Coherence Fee | A market or regulatory mechanism inducing voluntary compositional verification at organizational boundaries |
| Interpretability Frontier | Probing classifiers achieving compositional signal, not just edge-local accuracy |