Assessment 124: persistence proofs and charged substitution

7 October 2026. Dedicated primary-proof review, scheduled completion 30. This is a continuation of assessment123, not another catalog survey. The semantic Horn bridge and imported entry129 theorems are retained unchanged.

Primary argument comparison

Demaine, Langerman and Price, Confluently Persistent Tries for Efficient Version Control, Algorithmica 57 (2010), 462–483. Author PDF, sections1–2, Theorem1 and its complete proof, PDF pages2–9. With constantly many fingers, the compressed Steiner tree has constantly many distinguished vertices. Paths between them are deques; off-path subtries are retained separately. Moving a finger changes only constantly many path ends and adjacent child dictionaries. Copying the small top structure refreshes every distinguished finger. This proves O(log degree) time and space for movement and local modifications, including subtree copies. Fixed binary branching makes that bound constant. The argument requires located fingers, not arbitrary-name lookup in one operation; node payloads may include pointers. It does not refresh an unbounded collection of external cursors. The globally balanced alternative trades local constant updates for logarithmic-size bounds; it is not needed here. This proof supports the finite-active-cursor design, while leaving serialized routing and compiler locality to the Lab’s separate constructions.

Driscoll, Sarnak, Sleator and Tarjan, Making Data Structures Persistent, JCSS 38 (1989), 86–124. Author-hosted PDF, operation model pages87–90, complete node-copying construction and analysis section2.3/pages93–97, extension discussion pages122–123. Bounded indegree permits predecessor repair after a full modification box causes a node copy. Extra fields absorb updates; copying can cascade. For indegree bound p and e extra fields with e>=p, the potential is (e times live nodes minus unused extra fields)/(e-p+1). In the postprocessing calculation, the coefficient of the number of copied nodes cancels, leaving O(t) amortized cost for t initially pending copies. Access steps have constant overhead; update overhead is amortized. Version-indexed access pointers and stamps belong to its operation model. The stated write-once variant is relevant, so mutability alone is not a valid dismissal. It still does not establish bounded work on every input symbol or supply ordinary-name version selection. Unbounded indegree also invalidates the particular repair bound.

Consequence for our target

Both are established external results. Their proofs were read for mechanisms, not questioned for lack of formalization. The first makes a bounded set of active locations coherent; the second pays for repairing predecessor relationships over an operation sequence. A stored old pointer is not automatically a cursor into the current version. This comparison redirects the attack from a generic persistence compiler to an exact substitution/readout contract.

Three attempted transfers were examined: explicit materialization of renamed edges, aliases to later-changing columns, and lists of frozen singleton rows. Reader section13.4 gives their exact failure contracts and reopen gates. None excludes a different native representation. The materialization witness has input O(L+m log(m+2)) and output at least mL, even though its relation has an extremely compact shared representation. It is deliberately not used as an information lower bound.

The direct constructive response is section13: a checked finite terminal image allows arbitrary incoming fanout. Names receive finite slots during right-block input; intermediate rows and accumulated left rows carry masks, and later endpoint bits recover a slot. Nested dictionary paths use the already established capsule schedule. This also propagates through each fixed number of factors without fanout expansion. It is a supporting application of existing primitives and classical finite aggregation, not a claim of a new persistence technique or a general native characterization.

Evidence and continuation

The written invariant is independent of the finite checker tools/research/check_finite_image_composition.py, which uses the existing persistent capsule model against set products, including retained old versions, duplicate edges, missing/prefix names, overflow and multiple factors. Its counts and declared validation are recorded in the root checkpoint. The Python model does not implement the native finite controller or establish timing.

Seven surfaces: PDFs/extractions remain private scratch; no new root CLAIMS.yaml admission; the single reader proof owns the supporting statement; the checker is finite corroboration; failed routes have exact contracts and reopen gates; unbounded-image functional substitution remains open; the existing reader note and program entry convey the distinction without opening a new paper. All prior SAT, affine, replay, causal, unary-cubical and support-omission frontiers survive.

Accounting: scheduled29->30 once; dedicated review completed; next31 research and next dedicated35. Mathematical89/window3/last reassessment86 remain unchanged because this is a dedicated review and supporting construction, not a separately counted substantial research result. Next investigate the unbounded-image functional-right case with an explicit query-directed index and strict construction deadlines. No second count for publication or retries.