> **2026-09-21 update:** The active GCTS and GCTS+RL lanes now use the > [shared pair-corona marking learner](3d-gcts-marking-learning.md). > Conservative markings and default GCTS lookahead described below are historical > v2.0/v2.1 behavior; retain those semantics when interpreting recorded results. # 3D Lattice Tiler v2 V2 replaces the main workbench at both existing entry points: `/3d-lattice-tiler/` and `/apps/3d-lattice-tiler/`. The complete previously published explorer remains at `/apps/3d-lattice-tiler/legacy.html`; its engine, builders, catalog and controls are unchanged. The new workbench reuses the published exact tile adapters, Three.js renderer dependencies and separate periodic verifier. ## V2.1: 2D tiles in one slab The named hat, turtle and A₂ hexagon prism presets now use a single slab. This replaces their v2.0 half-weight 3D point-window model. It does not modify the original explorer, ordinary 3D tiles, or historical results. For the thin prism, the old cap point had weight `2*a / 48`, where `a / 12` is its planar angle value. Both end-cap copies now have `4*a / 48`. Thus cap-interior points have `48 / 48 = 1`; one placed tile saturates them immediately and the scheduler omits them from the frontier. Rim points carry the corresponding planar-angle values. There are no intermediate sampled layers or vertical-edge interior samples in this thickness-one construction. Hat and turtle use exactly the demo's index-3 predicate, relative to the normalized first vertex: `x ≡ y ≡ z (mod 3)` on `x+y+z=0`. Sampling is applied to every orientation's t-support before lifting it to the caps; it does not multiply values by three. Hat retains ten boundary points and one interior point per cap; turtle retains eleven boundary points and two interior points. The hexagon control uses the full A₂ point domain. The caps are at coordinate sums 0 and 3. Allowed translations have sum zero, and for hat/turtle belong to the index-3 lattice. This is an explicit point placement domain, not a hidden polygon-intersection test. Candidates that would move either cap outside the slab are excluded during complete domain enumeration; the independent verifier enforces the same condition separately. Both caps' finite targets are hexagonal windows in the declared lattice basis, with 14, 38 or 74 required points. Exterior points remain constrained by capacity, but only lateral extension is allowed. Every required point begins at generation zero, and all four search lanes share this model. The orientation group is the planar A₂ group, with six rotations and optional reflected planar tiles (twelve orientations for hat/turtle when enabled). This is explicitly different from v2.0's proper cubic prism rotations. Prism geometry is just the display of these planar orientations, extruded by `(1,1,1)`. The existing three-dimensional quotient probes cannot certify this new rank-two slab problem: their button is disabled for slab presets, and programmatic requests return an explanatory unsupported-model error. They remain available for ordinary 3D tiles and in the original explorer. `test-3d-lattice-slab.mjs` compares the support/weights with the demo engine, checks all aligned slab candidates against an independent enumeration, checks that full-cap points have no remaining legal incidence or frontier obligation, tests exact rollback, and replays all four lanes' solutions. It also rejects shifted-slab and wrong-coset certificates. Historical v2.0 benchmarks below and on the page retain their original weights and targets; none is presented as evidence for the new slab model. ## V2.0 experiment and scope (historical) The unit of comparison is a **finite exact point window**, not a tile-count growth milestone. Every point in a centered integer cube is a required root obligation of generation zero. All integer translations of the supplied orientations that touch this window are admissible. Selected placements are distinct; integer t-values must sum to the common capacity at every required point and must never exceed capacity anywhere, including the exterior. The exterior is open: no other point is required to be filled. The target is finite; a solution is not an infinite tiling certificate or a geometric solid-cover certificate. Disconnected external components are not introduced. The original catalog's orientation group is retained and exported explicitly. In particular, the A₂ prism group has six layer-preserving orientations, not all cubic rotations. Reflections are optional and shared by all lanes. Numerical and symbolic weights are rejected rather than rounded. The SCD geometry therefore remains available in the original explorer, but does not enter v2's exact comparison without a suitable exact adapter. Its geometric aperiodicity theorem would not prove aperiodicity of a different finite point model or a different orientation group. ## Shared engine `v2/search.js` enumerates the complete candidate universe by every support alignment at every required point. One candidate node is shared by all its incident points. Forward incidence and reverse t/m dependencies include marking-only interactions outside the target. Applying a placement invalidates every affected candidate at every incident point. Unaffected domains are preserved. The same trail restores validity, degrees, selected placements, totals and marking reference counts. The scheduler checks all dead points first, then all forced points, then minimum generation (minimum degree only breaks generation ties). Every target point is an explicit generation-zero root, so this finite experiment has no later-generation target points. Placement generation is one plus the minimum incident root generation. No chronological birth order is used. Future unbounded growth would need a separate fair activation rule and additional generation trails; v2 does not claim to implement it. ## GCTS and cluster proposals GCTS synthesizes orientation-equivariant vector markings by equating slots for **every capacity-legal pair**. No bounded-window dead pair is generalized into a global forbidden pair. These fields preserve all finite-window solutions. The 256-slot synthesis budget uses a constant field fallback; constant fields are elided at runtime because they cannot disagree. Rank, fallback status, version and scope are exported. These conservative fields may have no effect at all. V2 does not claim useful learned markings where the fields are constant. The additional GCTS pruning consists of proved residual-capacity tests and context-local failed-move lookahead. At each unfilled point, available contributions must have enough total mass, and their gcd must divide the residual. At a genuine branch, up to eight candidates are tried with up to three forced continuations. A resulting dead point or residual contradiction proves only that candidate impossible in its current parent context. Eliminations use the ordinary graph trail and are undone on leaving that context. No sampling or probe limit truncates the base domains. These costs and eliminations are separately recorded; they must not be described as marking-derived eliminations. The RL lane is an experimental **tabular cluster-return bandit** with bounded rollout proposals. After the reference scheduler selects a branch, up to four candidates seed irregular rollouts of up to three base placements. Each continuation uses the same global scheduler, complete domain and point legality. Every rollout is rolled back exactly. Expansions can mix supplied species and orientations; they are not limited to periodic, isohedral or known motifs. No catalog identity enters the policy. Proposals are ranked using completed obligations plus their online mean continuation return. The selected first placement orders the complete base list; the remaining base alternatives are always retained. Execution returns to the global scheduler after every placement. Proposal expansions, visits, return values, training seed and provenance are exported. The table is capped at 512 retained clusters; excess proposals may still guide the current choice without being retained. This is finite irregular cluster learning, not a hierarchical grammar learner or a trained cross-tile policy. Rollout lookahead and learned returns are distinct components. The reference experiment includes an ablation with learned values ignored during ordering (`learnReturns: false`). A win by the cluster lane alone does not establish that reinforcement learning caused the win. No held-out generalization claim is made; every lane begins with empty learning state. ## Protocol and reproducibility The browser runs the four lanes sequentially in fresh workers. The recorded reference rotates lane order across seeds. Total time includes model preparation, marking synthesis, complete graph construction, online learning, proposal validation and final independent verification. Module loading and message/renderer transport are excluded. The same time limit and 10,000 base placement-attempt limit apply to every lane. Probe work is charged to wall time and reported separately; it does not masquerade as a base attempt. Each engine also enforces a 180,000-candidate resource limit and a depth limit of 1,024. Hitting either bound cannot prove failure. A browser watchdog can stop a worker at the selected time limit plus 15 seconds, covering synchronous preprocessing and graph construction. Time is checked at safe points, so a run may exceed its nominal limit slightly. Cancellation is unknown. No interrupted result is relabeled untileable. The graph chart samples actual current completion, including decreases on backtracking, rather than reporting only maxima. A budget result can display the best retained partial state; it is explicitly unverified as a complete window. Memory is a rough graph/trail footprint estimate, not JS heap or process RSS; it excludes engine code, workers, renderer and runtime overhead. Result labels: - `finite_exact`: independent replay verifies the entire required window and every capacity/marking condition, including exterior interactions. - `exhausted_finite`: the entire finite tree is exhausted under this model. - `unknown`: time, attempt, depth, memory, watchdog or cancellation limit. - Unsupported exact input: the comparison is unavailable, not a negative. Periodic and isohedral probes use the existing exact quotient engine as separate structural controls (up to eight motif copies). Their certificates and scope are exported separately. Failure within the bounded family is unknown, not evidence of aperiodicity. Research cases that miss those probes are separated from cases where a small certificate was found. Reproduce: ```sh node scripts/test-3d-lattice-v2.mjs node scripts/benchmark-3d-lattice-v2.mjs /tmp/lattice-v2-benchmark node scripts/verify-3d-lattice-v2-evidence.mjs exported-evidence.json ``` The reference evidence and full summary are linked from the live workbench. No timing is guaranteed across devices. Three seeds and a 27-point window are a short diagnostic experiment, not a broad performance study. ## Recorded reference results Five cases, seeds 1–3, a 27-point target, proper rotations only, and two seconds / 10,000 attempts per lane produced 6 / 15 verified windows for free-range, 6 / 15 for GCTS, 6 / 15 for RL clusters and 7 / 15 for the combination. The combined lane solves hat-prism seeds 1 and 2 and turtle seed 1 while the baseline remains unknown. The baseline wins other seeds; there is no overall speedup theorem or statistically established superiority. Ignoring learned returns during RL ordering still solves the same 6 / 15 instances. That ablation retains table update cost and isolates the ordering term. The present gains are evidence for rollout-assisted cluster ordering, not an isolated reinforcement-learning benefit. Useful learned markings also remain an open research task; constant fields cannot explain a gain. The eight-copy periodic and isohedral probes both remain unknown within two seconds for hat, turtle and tuning fork. They both find point certificates for buckled ring and twisted H, which the UI therefore labels periodic stress controls. The 16-vertex tile and cube remain explicit known-periodic controls. None of these bounded misses is a proof of nonperiodicity. [All measurements](../../apps/3d-lattice-tiler/v2/reference/summary.json) and [full replay evidence, compressed JSON](../../apps/3d-lattice-tiler/v2/reference/evidence.json.gz) include every reported seed, successes and failures, all base placement data, marking metadata, cluster expansions and structural probe certificates. Timings are local observations on this machine, with ordinary host variability. ## Conformance evidence and remaining gaps `scripts/test-3d-lattice-v2.mjs` checks independently exhaustive tiny-model agreement for all four lanes, fractional residuals, shared candidates, untouched required points, full rollback, explicit-zero and extended marking-only interactions, dead/forced/generation precedence, budget labels, and cluster/probe transaction correctness. `verify()` independently rebuilds the point sums and markings directly from exported base placements; it does not use graph state or search caches. Browser checks cover running, stopping, lane selection, geometry controls, reflection changes, periodic probes, custom JSON, export, unsupported input and mobile layout. Remaining limits: no geometric-solid verifier for general returned patches; no unbounded frontier activation; no learned marking synthesis from higher order obstruction certificates; no hierarchical cluster composition; no proof of nonperiodicity for the showcased point problems; no statistical claim that the new lanes outperform free-range across the tile population. The original engine and historical benchmark semantics are preserved behind the original-explorer link and are not retroactively declared conforming.