# Constituents, order, and path identity

What must an encoding retain to distinguish paths that reach the same apparent state?

Separate visited symbols, the order of operations, path-dependent state, and reduced projections; explore whether repeated operational patterns can be recognized without losing reversibility.

Status: open problem. Formulation and comparisons: reconstruction. Development: historical branch.

## Alternative representations

### A product encoding

Affords: Offers a compact composition.

Limits or losses: Commutative products erase order; general matrix products need not be injective.

### Words over explicitly defined generators

Affords: Retains sequence before choosing a reduction.

Limits or losses: Finite representations, cancellation, and bounded precision require separate checks.

### A reduced state or projection

Affords: Supports operational comparison.

Limits or losses: The same projection may correspond to different complete paths.

## Tension

Order sensitivity, unique decoding, reversibility, and behavioral equivalence are separate demands.

## Challenge

Which distinct paths collide, and does a projected recurrence imply recurrence of the complete relevant state?

This challenge is an editorial prompt, not a claim that the evidence already resolved it.

## Proposed next move

Run the supplied counterexample script, then define a restricted generator system with a stated decoding guarantee.

Status: new model-proposed experiment or extension; not a reported result.

## Related problems

- [A conclusion’s evidential path ends at that conclusion](causal-reading-paths.md) — distinguishes arrival from history (retrospective_connection). Two paths reaching the same visible place can differ in their relevant history. Algebraic path encoding and evidential exposure are analogues, not identical mechanisms.
- [Composing operations with obligations inherited from the route](path-dependent-composition.md) — makes route identity operational (retrospective_connection). Inherited compatibility can depend on the route; the right equivalence relation must preserve relevant obligations.
- [The next description becomes another object](objectifying-the-meta-level.md) — constrains claims about computational limits (retrospective_connection). A projected loop or a new encoding alone does not establish a general decision procedure.
- [Reading that changes what is stored](observation-that-changes-state.md) — challenges reversible inspection (corpus_evidence). The read-disturbance example explicitly complicates an attempted inspection of the encoded route.

## Evidence anchors

- basis-order-sensitive-path-encoding: An ordered encoding of traversed symbols is proposed using complex primes and matrix multiplication. After acknowledging that these products may not be unique, the proposal switches to freely generating elements. It then distinguishes returning to the same symbol from recurring operational patterns visible in a projection. [Editorial paraphrase; presence of a proposal does not verify its truth.]

Root evidence and private recovery mappings are not part of this public release.

## Recorded revisions

- [From assumed unique products to a requirement for free generation](../REVISIONS.md#revision-path-encoding)

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