The Proofs
Companion to Similes of Symmetry
The trilogy's formal spine: thirty-two anchors that turn witnessed claims, scoped equivalence, gluing obligations, predicate invention, and coherence cost into one inspectable architecture.
32 anchors·6 parts·10 appendices·3 machine-checked·29 prose
Structure
The dependency graph of the thirty-two formal anchors. Each node is a definition or construction; each edge, a logical dependency. Hover to trace lineage and see how the line from commitment discipline to the Third Mode actually runs.
Volume I does not require notation. The Proofs exists for the reader who wants the machinery itself: where witness claims cash out formally, how the anchor dependencies fit together, and what it would mean for a system to reach the Third Mode rather than merely gesture toward it.
Reading Modes
Witness Claim Atlas
Volume I states four witness claims explicitly, all in Ch. 8 Similes of Symmetry. Each can be read as a bridge: historical argument on one side, formal burden on the other.
| Claim | Anchor(s) | Formal burden |
|---|---|---|
| WC-08-01 | A21, A24 | Composition becomes a priced formal obligation: the system must account for the cost of global coherence and specify the package that may travel across contexts. |
| WC-08-02 | A23, A10 | Recourse requires identities that survive transport and equivalences that remain witnessed and scoped rather than merely asserted. |
| WC-08-03 | A1, A2, A10 | Binding means more than attaching a name. Claims must extend a commitment set, carry typed provenance, and specify what kind of sameness they are actually licensed to assert. |
| WC-08-04 | A21 | Stakes become formal only when the system can expose the real cost of composition rather than bury it inside silent failure or political surcharge. |
Read horizontally for the Volume I bridge. Read vertically for the formal burdens that culminate in A32, where proposal, witnessed equivalence, gluing, and coherence accounting become one discipline.
Parts
Foundations
Commitment sets, provenance, witness typing, schemas, epistemic status, coherence
- A1Commitment SetsBinding claims to accountable identities
- A2Provenance and Witness TypingWitnessed assertions and witness classes
- A3Schemas and SatisfactionSignatures, models, and the conditions for validity
- A4Epistemic StatusWhat a system knows about what it knows
- A5The Coherence RequirementWhy local truth must compose into global consistency
- A6Sense BoundariesWhere one domain of meaning ends and another begins
Equipment
Group actions, isomorphisms, adjunctions, witnessed sameness, vocabulary operators
Coherence
Covers, sheaves, local theories, logic selection, univalence
Construction
Predicate invention, conservative extension, proposal vs. certification, search, cost
Specification
Context graphs, identity maintenance, predicate packages, query semantics, versioning
Demonstrations
Worked examples: refusal, sense gluing, predicate acceptance, scoped equivalence, n-ary events, third-mode capstone
From Anchors To Research
A13 + A21 + A24
Gluing, coherence cost, and predicate portability feed the coherence-fee line of work: Coherence Fee, Paper I, Paper II, and Paper III.
A17 + A17b + A20 + A29
Predicate invention, safe extension, constrained search, and acceptance gates become the research spine around SCPI.
A22 + A23 + A25 + A26 + A27
The specification anchors become protocol and benchmark work in Bridge, Seam, and BABEL.
A10 + A13 + A30 + A32
Witnessed sameness, gluing, scoped substitution, and the Third Mode define the boundary explored by Local Validity Does Not Compose, Interpretability Frontier, and Coherence Cliff.
Appendices
The appendices sit beside the anchor route rather than inside it. Use them for registries, terminology, interface notes, and background references.
After The Proofs
If you want the historical argument that motivates the anchors, return to Volume I. If you want the living technical program these anchors now support, continue to Research. If you want the capstone itself, end on A32: The Third Mode.