Anchor Registry

The anchors constitute the book's mathematical spine. Each is a load-bearing definition that later chapters depend on. The dependency graph is acyclic by construction. For the full formal statement of each anchor, see the chapter where it is introduced.

Status key: formal — fully specified with standard mathematical content. preformal — specification only; formal machinery deferred. engineering — practical surrogate for a stronger theoretical claim.

Summary Table

ID	Name	Chapter	Status	Dependencies	Core Claim
A1	Commitment Set	Ch. 1	formal	—	Assertions extend a commitment set monotonically while preserving consistency
A2	Provenance Judgement	Ch. 2	formal	A1	Claims carry typed evidence: $(p, \pi)$ not bare $p$
A2b	Witnessed Assertion	Ch. 2	formal	A2	Assertions carry their own verification instructions
A2c	Witness Classes	Ch. 2	formal	A2b	Witnesses typed by verification regime: decidable, probabilistic, attested
A3	Schema as Signature	Ch. 3	formal	—	A schema is a signature $\Sigma = (T, P, I)$; adding a predicate is a language change
A3b	Model & Satisfaction	Ch. 3	formal	A3	Conservative extension = safe evolution; model-theoretic semantics for "safe"
A4	Epistemic Status	Ch. 4	formal	A3b	Truth relative to view and logic; CWA/OWA are inference rules within $L_U$
A5	Coherence Requirement	Ch. 5	preformal	A1–A4	Local extensions must agree on overlaps and preserve invariants
A6	Sense Boundary	Interlude I	preformal	A5	Context-indexed equivalence $\sim_U$; boundaries create equivalence classes
A7	Group Action & Invariant	Ch. 6	formal	A1, A3	Invariants—not coordinates—are the stable units of knowledge
A8	Isomorphism & Witness	Ch. 7	formal	A7	Identity is behavior under admissible probes, not a label
A9	Adjunction	Ch. 8	formal	A8	Translation between representations formalized as optimal compression/decompression
A10	Witnessed Sameness	Ch. 9	formal	A2b, A2c, A6, A8, A9	Three levels (equality, isomorphism, equivalence); all scoped, all witnessed
A11	Vocabulary Operator	Interlude II	formal	A3, A3b, A10	Vocabulary changes what distinctions you can state; the loop stabilized by invariants
A12	Cover & Restriction	Ch. 10	formal	A4, A5, A10	Context is a restriction structure; truth is indexed by view
A12b	Context Site Structure	Ch. 10	formal	A12	Contexts form a site $(\mathcal{C}, J)$ with Grothendieck topology; formalizes A5
A13	Sheaf Condition	Ch. 11	formal	A12, A12b	Coherence IS the sheaf condition: agreement on overlaps → unique global section
A14	Fibration	Ch. 12	formal	A12, A13	Dependent types as fibrations; validity varies with context
A15	Logic Selection	Ch. 13	formal	A4, A12	Different views may operate under different logics; merging requires reconciliation
A16	Transport Discipline	Ch. 14	engineering	A10, A13	Equivalence can license substitution with transport; engineering proxy for full univalence
A17	Predicate Invention	Ch. 15	formal	A3, A5, A11, A13, A16	Signature extension $\Sigma \to \Sigma'$ introducing predicate $q$ under sheaf obligations
A17b	Conservative Extension	Ch. 16	formal	A1, A3b, A15, A17	Conservative extension = safe evolution; breaking change requires migration
A18	Invariant Set	Ch. 16	formal	A1, A16, A17b	$I = I_{\text{hard}} \cup I_{\text{soft}}$; hard invariants reject, soft constraints penalize
A19	Proposal Operator	Ch. 17	formal	A17, A18	Statistical pattern matching produces hypotheses, not certified truths
A19b	Certification Contract	Ch. 17	formal	A2c, A19	Typed success or typed failure; the architect-grade interface
A20	Predicate Search Space	Ch. 18	formal	A11, A13, A17–A19	Inventing predicates is searching a constrained hypothesis space
A21	Coherence Cost Model	Ch. 19	formal	A13, A20	Global coherence is expensive; honest systems declare a coherence budget
A22	Context-Graph Substrate	Ch. 20	formal	A2c, A13, A17b, A21	Substrate stores locality, witnesses, constraints, and restriction maps
A23	Identity Maintenance	Ch. 21	formal	A10, A22	Identity emerges from witness networks, not brittle keys
A24	Predicate Package	Ch. 22	formal	A17, A17b, A21, A22	A predicate ships with signature, tests, invariants, provenance, scope
A25	Query Semantics	Ch. 23	formal	A19b, A24	Queries are contracts over locality, invariants, and acceptable witnesses
A26	Versioning Rules	Ch. 24	formal	A17b, A21, A24, A25	The Third Mode survives time via versioned predicates and explicit compatibility
A27	Refusal Obligation	Ch. 25	formal	A1, A18, A25	If constraints entail emptiness: produce unsat core, derivation chain, checkable witness
A28	Sense Gluing	Ch. 26	formal	A6, A13, A24	Disambiguation is gluing: local sense choices must agree on overlaps
A29	Predicate Acceptance	Ch. 27	formal	A17, A24, A26	Predicate invention safe only with tests, scope, invariants, versioning
A30	Scoped Equivalence	Ch. 28	formal	A10, A12, A23	Equivalence is context-indexed; transport valid only within scope
A31	N-ary Event Object	Ch. 29	formal	A22, A26	Higher-arity events are meaning atoms; binarizing destroys invariants
A32	The Third Mode	Ch. 30	formal	A10, A12, A12b, A13, A17, A17b, A21, A30	A system is Third Mode iff it supports predicate invention, witnessed equivalence, sheaf gluing, and coherence cost accounting

Critical Path

The longest dependency chain through the anchor graph:

A1 → A7 → A8 → A9 → A10 → A12 → A12b → A13 → A16 → A17 → A18 → A19 → A19b → A20 → A21 → A22 → A24 → A25 → A26 → A29

Every other anchor either feeds into this spine or branches from it. The path traces the argument: from commitment discipline through Erlangen invariants, witnessed sameness, the sheaf condition, predicate invention, and finally predicate acceptance. A32 (Third Mode Definition) is the argumentative culmination but reaches it by drawing on multiple branches of this spine rather than extending it linearly.

Normative Correspondence

The 32 anchors constitute the formal apparatus. The four governing equations of the Codex constitute the normative argument. The correspondence between them is not accidental: each equation draws on a specific region of the anchor graph for its formal grounding.

Equation	Normative Claim	Anchors	What They Formalize
Truth needs witnesses	Claims require verification; the cost of verification determines who gets to count	A1–A14	Commitment sets, provenance judgments, witness classes, schemas, epistemic status, the coherence requirement, sense boundaries, group invariants, isomorphism, adjunctions, witnessed sameness, cover structures, the sheaf condition, fibrations
Value needs work	You cannot create value without expenditure; expenditure leaves traces	A21	Coherence cost model, monotonicity theorem, coherence budget
Freedom needs receipts	Power that leaves no trace dominates in darkness; receipts make power legible	A17–A20, A22–A29	Predicate invention, conservative extension, invariant sets, proposal operators, certification contracts, predicate search, context-graph substrate, identity maintenance, predicate packages, query semantics, versioning, refusal obligation, sense gluing, predicate acceptance
Humanity needs mercy	Not formalized	—	The formal apparatus stops where mercy begins

The distribution is itself significant. Equation 1 (Truth) commands the largest formal territory because epistemology is the foundation: fourteen anchors specify what it means for local truth to compose into global coherence. Equation 3 (Freedom) commands the second largest because constitutional standing requires the most institutional machinery: thirteen anchors formalize the apparatus of predicate invention, certification, and accountable query that makes power inspectable. Equation 2 (Value) is compact because its formal content is concentrated: the coherence cost model in A21 captures the thermodynamic relationship between local assertion and global composition. Equation 4 (Mercy) has no anchors by design. The fourth equation marks the boundary where formal specification must yield to human judgment. Formalizing mercy would be a category error: mercy is precisely what you grant despite what the formal apparatus shows.

The absence of anchors under Equation 4 is not a gap in the formalization. It is the formalization's most important result. The first three equations can be mechanized; the fourth cannot. The boundary between what can be formalized and what must remain human is the mercy threshold, and the anchor registry's silence on Equation 4 is itself the formal expression of that boundary.

Witness Claim Index

Volume I states four witness claims explicitly (WC-08-01 through WC-08-04, all in Ch. 8 Similes of Symmetry). Each names a property of the witness frame and points at the formal anchor or anchors that bear its proof obligation. The table below resolves the references.

WC	Property	Anchor(s)	What it formalizes
WC-08-01	composition	A21, A24	Coherence cost model; predicate package
WC-08-02	recourse	A23, A10	Identity maintenance; witnessed sameness
WC-08-03	binding	A1, A2, A10	Commitment set; provenance judgement; witnessed sameness
WC-08-04	stakes	A21	Coherence cost model

WC-08-01 and WC-08-04 are intended to be read together: WC-08-01 predicts transitional concentration after a verification cost transition, WC-08-04 predicts eventual dispersal. The chapter prose makes this linkage explicit, and both claims rest on A21 — WC-08-01 invokes A21 alongside A24 (predicate package), WC-08-04 invokes it alone. The joint claim is that the rent component of the trust tax becomes measurable as verification cost approaches the coherence fee.

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Full formal statements, citations, and touchstone mappings are maintained in the project's anchor registry (references/anchors.ts).