Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction

This paper develops a decision-resource view of terminal symmetry for directed sequential construction tasks. The central insight is that many construction tasks exhibit exact symmetry at completion (e.g., a chair unchanged when two legs are exchanged, program output invariant to renaming, exact-fill packing unchanged when bins are relabeled), even though the realizing processes are directed and history-dependent. The paper asks: what decision information can exact terminal symmetry provide when the realizing process itself is asymmetric?

The key distinction is informational: "Process evidence is the source of directionality; terminal correspondence carries that structure into an equivalent target frame; realized-state evidence refines its current decision relevance after transitions; and a fixed verifier certifies execution." This source–carrier–state decomposition turns terminal symmetry into a reusable decision resource.

A discrete construction system is defined as C = (S, A, F, s0, Ψ), where F is a deterministic partial transition and Ψ maps final states to semantic objects. A group G acts on terminal objects, and a lift Lg(τ) satisfies covariance of the terminal object: OC(τ′) = g·OC(τ). When a selected lift preserves length and supplies an event bijection Pg, a directed relation tensor R(τ) transforms by conjugation: R(τ′) = Pg R(τ) Pg⊤, which transports precedence, commutativity, or action correspondence while preserving directionality.

The paper proves that identification and transport are distinct information operations: no estimator that observes only a terminal object and its stabilizer can universally recover true precedence or commutativity, even when observing one demonstration. A two-system witness shares the same terminal object, stabilizer, and observed demonstration, yet one system admits a commuting diamond while the other forces a chain. Every randomized estimator has worst-case error at least 1/2.

The architecture contains three information roles and one certification role:

• Process evidence supplies directionality; terminal correspondence carries it into the target frame, yielding an episode-fixed transported process structure RT

• At state s, restricting RT to current candidates induces the state-restricted process rank rTs

• Realized-state evidence supplies the state-dependent residual rank rXs, refreshed after accepted actions

• A fixed verifier Vs(a) certifies execution

The ordinal rank meet with union-prefix semantics is: **ms(a) = min rTs(a), rXs(a) **

Lemma 1 (Union coverage): Topk(ms) = Topk(rTs) ∪ Topk(rXs). If either expert has a verifier-feasible action in its top-k set, meet finds a feasible action in at most BTs(k) + BXs(k) verifier calls, or at most 2k for strict ranks.

Theorem 2 (Prefix-information optimality and completion guarantee): Under the prefix information model, if feasibility is known only to lie within the union of two expert prefixes, that union is the minimax query set, and rank meet reaches it exactly. The total query bound is ΣUst(kst) ≤ Σ[BTst(kst) + BXs(kst)].

Proposition 1 (Strict dynamic–static separation): There exists a two-state problem where dynamic and initial-static policies use the same terminal prior, fixed verifier, and first action, yet dynamic completes in two calls while static is interrupted at budget two—because after the shared first action, the cached order tests infeasible a, while the refreshed residual promotes feasible b.

The planner SYMBUILD (Algorithm 1) implements transport–refine–certify:

1. Transport the reference relation once: RT ← Pg Rref Pg⊤

2. At each state, compute both ranks (process rank from transported structure, residual rank from realized state)

3. Sort candidates by min(rT, rX), then rT + rX, with fixed tie-break

4. Query the fixed verifier; accept the first approved action

5. After acceptance, re-restrict RT to new candidates and refresh rXs

Statewise refresh improves verified efficiency in all six pre-specified cross-domain aggregates:

Domain Shift AUC Gain (primary) AUC Gain (product)

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CAD ID/calls +2.86 [.12, 5.76] +3.30 [1.01, 5.79]

CAD ID/work +6.77 [1.30, 14.06] +4.43 [.52, 9.90]

Program ID/calls +12.42 [9.50, 15.33] +12.42 [9.50, 15.33]

Program length OOD +21.75 [18.33, 24.92] +22.58 [19.33, 25.83]

Packing ID/calls +8.54 [7.01, 10.14] +6.46 [4.51, 8.47]

Packing size OOD +8.68 [7.01, 10.42] +6.53 [4.58, 8.47]

On 1,135 target-removal episodes from official GRN OOD scenes, SYMBUILD attains the lowest mean capped verifier cost at all three scales (OOD10, OOD15, OOD20) among compared GRN and CDGS-style planners, while evaluating only 2.00–3.14 GRN states per target versus 13.14–38.32 for CDGS–refresh.

The gain appears exactly where the state residual is refreshed: all 384 CAD-work and 480 packing pairs share the first accepted decision; later sequences diverge in 93.75% of CAD-work and 67.50/93.75% of packing ID/OOD pairs, yielding 26/0 and 29/1 dynamic/static-only completions on CAD work and packing OOD.

Information roles are domain-dependent: CAD and packing are transport-sensitive, whereas Mini-Programs are state-dominant. The CAD alignment intervention shows aligned/permuted 81.25/65.00% (+16.25 points), and IKEA observed-process topology shows process structure +14.14 points, action core +19.20 points, hybrid +2.64 points.

The same signals transfer across canonical, replanning, partial-order, and multi-heuristic interfaces:

• SYMBUILD − canonical-slice static: +12.58/+21.75 AUC points (ID/length OOD)

• SYMBUILD − weighted-A*/LazySP-style: +4.42/+4.25 points

• POCL-style refresh − fixed: +3.83 points (length OOD)

• Multi-heuristic refresh − fixed: +5.00 points (length OOD)

"Terminal symmetry is a decision resource for directed construction: process evidence supplies directionality, terminal correspondence transports it, realized state refines relevance, and the verifier certifies execution. SYMBUILD realizes this flow with union-prefix rank meet and a tight verifier-query bound under the prefix information model. The predicted post-transition signature persists across domains, while the same statewise signal transfers across planning organizations; SYMBUILD also gives the lowest mean capped verifier cost at all three GRN OOD scales. Transport what the outcome preserves; refine what history changes."

Improvements to AI Systems:

1. Add a terminal symmetry transport module to planning agents.

• The AI system can now detect exact symmetries in final outcomes (e.g., identical chair legs, renamable program variables, relabelable bins) and use them to reorder its internal action-priority structure via the conjugation rule R(τ′) = Pg R(τ) Pg⊤.

• This lets the system reuse a previously learned or demonstrated construction plan in a new but terminally symmetric context, without re-exploring the full state space.

2. Implement a dual-rank action selector with union-prefix semantics.

• The system maintains two independent rankings per candidate action: (a) a transported process rank from the terminal-symmetry structure, and (b) a realized-state residual rank refreshed after each accepted action.

• It selects actions by the minimum of the two ranks, with tie-breaking by sum. This guarantees that if either rank’s top-k contains a verifier-feasible action, the system finds it in at most 2k verifier calls (Lemma 1), and it provably reaches the minimax-optimal query set (Theorem 2).

3. Add a fixed verifier as a certification layer, not a heuristic.

• The system never trusts either rank blindly; it queries a deterministic verifier (e.g., a physical simulator, type checker, or constraint solver) only on the union of the two top-k sets.

• This reduces false positives and wasted computation, especially in out-of-distribution settings (e.g., GRN OOD scenes), where the system achieves the lowest mean capped verifier cost (2.00–3.14 states per target vs. 13.14–38.32 for baselines).

4. Enable dynamic state-refresh of the residual rank after each transition.

• After accepting an action, the system re-restricts the transported relation to the new candidate set and recomputes the residual rank from the actual realized state.

• This yields a measurable post-transition divergence: in CAD-work and packing, 93.75% and 67.50–93.75% of episodes diverge from static policies after the first shared decision, leading to 26/0 and 29/1 dynamic/static-only completions on hard tasks.

5. Make the system domain-adaptive by learning which information role dominates.

• The system can detect whether transport (terminal symmetry) or state evidence (realized-state residuals) is more predictive in a given domain (e.g., CAD and packing are transport-sensitive; Mini-Programs are state-dominant).

• It then weights the rank meet accordingly, improving robustness across heterogeneous task families without manual tuning.

6. Transfer the same statewise refresh signal across different planning organizations.

• The improvement is not tied to a single planner architecture. The system can inject the dual-rank refresh into canonical planners, replanners, partial-order planners, and multi-heuristic search (e.g., weighted A*, LazySP, POCL) and still gain +3.83 to +21.75 AUC points on length OOD tasks.

What the improved AI system can do:

• Plan construction tasks with verifiable correctness (e.g., CAD assembly, program synthesis, bin packing) using fewer verifier calls, especially when the final object has symmetries that the process does not.

• Reuse plans across terminally symmetric but process-asymmetric scenarios—e.g., build a chair with swapped leg order, generate a program with renamed variables, or pack items into relabeled bins—without retraining or exhaustive search.

• Guarantee completion within a bounded verifier budget under the prefix information model, even when no single rank is reliable alone.

• Adapt its decision strategy online after each action, refreshing its priorities based on the actual state, not just the initial terminal prior.

• Operate effectively out-of-distribution (e.g., longer programs, larger bins, unseen GRN scenes) by relying on the union of two independent evidence sources, reducing catastrophic failure from stale static rankings.

• Integrate seamlessly into existing planners as a drop-in action-selection layer, improving verified efficiency without architectural rewrites.