The Proofs
Companion to Similes of Symmetry
When does earlier work justify a later use? The companion develops the models, contracts and limits of that reliance.
Contents
Foundations
Commitment sets, provenance, witness typing, schemas, epistemic status, coherence
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
Appendices
The appendices sit beside the anchor route rather than inside it. Use them for registries, terminology, interface notes, and background references.
Alternative reading routes
Reading Modes
Formal dependencies and verification
32 primary anchors·38 named identifiers·6 parts·10 appendices·42 written sections
Machine verification applies to named statements, not whole chapters. The verification map identifies their declarations, hypotheses and build scope alongside the written proofs and unfinished constructions.
Formal Dependency Graph
Thirty-two primary anchors and six named subanchors. Edges record declared exposition and construction dependencies, not proof verification. Expand the subanchors for the complete 38-identifier registry; hover to trace ancestry and descendants.
Jurisdiction
The formal graph covers
Commitments, witnessed claims, scoped equivalence, restriction, gluing, obstruction, predicate invention, and the formal cost model.
Additional operational layers
Capture, occurrence coverage, independent persistence, authority, grounding, reliance, challenge, and remedy require mechanisms and evidence outside this DAG.
Volume I does not require notation. The Proofs exists for the reader who wants to examine the models: which hypotheses their conclusions require, how the declared dependencies fit together, and what the proposed Third Mode specification would require an implementation to establish.
Witness claims and their formal models
Witness Claim Atlas
Volume I states five witness claims explicitly, all in Ch. 8 Similes of Symmetry. Their pointers identify related models where supplied; they do not turn the historical or institutional arguments into mathematical theorems.
| Claim | Anchor(s) | Relation to the formal model |
|---|---|---|
| WC-08-01 | A21, A24 | Portability depends on practicable alternatives and continuing work. A21 records selected checking costs; A24 specifies a proposed predicate package. Neither proves that an institution becomes dispensable. |
| 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 | Technical checking cost is distinct from access, interpretation, correction and governance costs. A21 supplies declared categories and budgets, not a universal ordering or measure of rent. |
| WC-08-05 | No formal anchor assigned | A record can persist while the authority for a later use changes. No general temporal-jurisdiction rule is derived by this companion. |
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.
Related research
From Anchors To Research
A12 + A13 + A16
Covers, gluing, and disciplined transport provide the formal local-to-global layer used by Tarski Coherence and SCPI. This layer diagnoses modeled compatibility; it does not certify execution safety.
A1 + A2 + A10 + A23
Commitments, typed witnessing, scoped sameness, and identity maintenance motivate the authority and non-equivocation questions in Compositional Accountability. Persistence and remedy remain additional obligations.
A22 + A23 + A25 + A26
Context graphs, identity, query contracts, and versioning inform Bridge, Seam, and the Glyph/Bulla receipt architecture. The implementation adds coverage, independent retention, typed reliance, and recourse gates.
Historical application record
The coherence-fee and interpretability tracks applied these anchors more broadly. Their falsifications do not delete the anchors; they narrow what those applications may claim. Read the decision dossiers →
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.