Recorded variable-use references

Recorded variable-use references

Reference model and candidate reader integration for the next geometric rule. The deployed mixed interfaces are a separate result. This package uses existing public compiled records and owned calibration controls; it contains no original private mathematical source or new native Lean run.

Representation problem

A context extension does not establish a typing judgment for a new source variable occurrence. A shared variable node can also refer to different original binders and have different recorded roles. A useful reference link therefore needs an exact source occurrence, its lexical target and its own native owner.

index.cjs extracts existing direct and construction-premise observations of bound variables. A source use keeps its packet/graph hashes, original route, node, context length and binder route/ordinal. Repeated observations at that same occurrence retain separate evidence hashes, original recording owners, fields, type components and roles. They are not counted as independent uses. Agreement is checked against the accepted records, not inferred from a node ID or binder name. The catalogue does not infer missing observations.

The selected lookup entry and a construction’s native owner can have different evidence hashes after supplementary let records are attached. The index takes a premise observation’s owner from the construction entry, not from the enclosing lookup entry. The public paged check opens all 304 observations and rejects the three cases where substituting the lookup entry’s evidence hash would misidentify that native owner. Matching type syntax alone would not detect this error.

Geometric rule

A recorded-use reference points from one original source use to its exact original binder. Within a mixed context frame it belongs to exactly one of the disjoint header/body regions. There are three cases:

  1. The target is a retained binder slot in the current frame.
  2. The target precedes the frame and remains an original-context boundary.
  3. Both target and use are inside a hidden region. This is an internal reference, not a requirement on the region’s enclosing context.

The reference contributes zero source occurrences. It is a link/badge for an occurrence already counted inside its region, not a second retained source vertex. Its native observations type that particular use in its own original context. They do not type the region or create a fresh variable judgment. If a later view exposes that use as a retained vertex, its ordinary source cut must be refined as well; the same occurrence cannot stay hidden and be added again. Reference links express lexical use, not mathematical necessity.

references(frame, result) preserves the frame’s original source partition. Selecting a reference can recover its source route and every original native observation. Choosing between multiple observations remains explicit.

Bounded delivery

Source pages contain full use records once. Binder-order pages contain only references back to those rows. Both orders and their shared source identity are bound by an expected manifest hash. Each page has an expected hash and decoded byte count. This authenticates the producer’s catalogue; it does not make the producer, codec or renderer formally verified.

The consumer retains at most one source and one binder page. A page has at most 8 rows and 64 KiB; a manifest has at most 64 KiB. A query returns at most 8 uses and 128 KiB, scans at most 32 rows and reads at most 8 pages. Continuation is tied to the same issued consumer, immutable catalogue and query. Changed queries, foreign cursors, wrong source identities, damaged pages and secondary references to another occurrence are refused. Oversized records/manifests fail explicitly. Disposal releases both page caches and prevents further use.

The finite audit covers 51 profiles: 26 project/AI extraction profiles and 25 owned calibration declarations. Its 340 observations describe 170 distinct uses; 36 uses target original binders outside the current typed-interface inventory. The audit checks 131 binder queries, 575 source-region queries, 909 mixed frames and 12,642 reference placements. Its storage-depth control is synthetic and does not claim a native judgment at that deep position. Input identities and reproducible catalogue hashes are in verification.json.

These are delivery-component bounds. The producer and reference tests load the accepted source/native tables. Decoded JSON sizes and page-cache counts do not measure whole-process or browser memory. The candidate consumer and its browser checks are described in reader.md.

Syntax counts are a separate measure

support.cjs counts syntactic occurrences of one original binder in its body. For a selected binder, start its body at target index zero. At a nested binder, the target index increases only in its body; domains and let values keep their prior context. For each (node, targetIndex) pair, a matching bound variable contributes one and other constructors sum their original child contributions. Shared nodes may contribute through several paths, so these are expanded occurrence counts, not unique DAG-node counts.

The dynamic traversal retains at most 65,536 states. Validated neededScope metadata and conservative 64-residue free-index masks prune impossible matches. The masks never drop residue zero when it might also stand for index 64 or larger. They add one fixed-width value per accepted source node. A budget refusal returns unavailable, with no partial or zero count. This reference counter uses a complete source table; it is not yet a paged browser counter.

The finite check completes 296 of 303 binder counts; seven counts in the large working-snapshot polynomial proof remain unavailable. Ninety-five small counts match an independent original-path/binder enumeration; 274 match an evaluation without either pruning rule. Scope-only pruning reproduces 278 completed counts; residue pruning completes 18 more at the same state limit. Shadowing, let-value scope, index-64 wraparound and forced-budget controls are separate checks. support-verification.json retains each packet, original binder route and result.

In 229 completed cases, the syntax count exceeds the recorded typed-use count. Consequently a future display must distinguish recorded typed uses from all syntactic occurrences, and keep an unavailable syntax count separate from both. A syntactically unused binder is not a proof that an assumption is mathematically unnecessary. None of these comparisons adds native judgments or establishes semantic abstraction, general preservation or human comprehension.

Reproduction

From the repository root, using the existing compiled fixtures:

node --max-old-space-size=4096 research/proof-atlas/recorded-variable-uses/check.cjs
node --max-old-space-size=4096 research/proof-atlas/recorded-variable-uses/check-support.cjs

No compiler build, Mathlib download or new private source publication is required. The general research program remains active. No blog result is claimed here.