diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/chained_seam_findings.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/chained_seam_findings.md index 8cd82fd..8e9c8f4 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/chained_seam_findings.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/chained_seam_findings.md @@ -221,12 +221,63 @@ still always holds. The load transfers to the genuine object — **per-interface selection respecting `R_T` coupling**, i.e. composing `R_T` along chains/trees with per-seam freedom rather than a global family. +## Finding 9 — `R_T` composition: the pigeonhole verified exhaustively for n ≤ 14 + +`kempe_rt_composition_probe.py` — the conjecture proper, with full per-interface +freedom. Two modeling facts were established first: + +- **The seam is exactly the singleton down apexes** of one inner face. A bite apex's + edge has parent tread faces on *both* sides, so it is internal to the parent's tile + and shares no medial vertex with the child. +- **Necklace states are exact, not approximate**: a child tile attaches with free + dihedral placement, so its realisable outer-sequence set is dihedral-closed and + "necklace match ⟺ some aligned sequence match". (This is the paper's own Def. of + medial boundary state.) + +Define `Ext(T)` = necklaces realisable on a subtree's outer seam by a compatible +Kempe-balanced selection: `Ext(T) = {o : ∃(o, i⃗) ∈ R_T^joint, i_j ∈ Ext(C_j)}`, +leaves = all-bite tiles (no singleton interfaces, degenerate inner boundaries). The +maps are monotone, so the **antichain of minimal reachable Ext sets per seam size** +decides whether ∅ is reachable. Relations are cached (`kempe_rt_relations_cache.json`). + +Result over all no-length-3-boundary tiles with `n ≤ 14` (7750 tiles, 1966 distinct +relations, 149 leaf, **27 branching**): + +- **∅ is NOT reachable.** Every tree assemblable from this universe — branching + included — admits a compatible Kempe-balanced selection. Since every real tire tree + whose treads have `n ≤ 14` and no separating triangles is such a tree, the + chain-pigeonhole conjecture is **verified exhaustively for that class**. +- **Composition saturates in 2 rounds.** The minimal antichains are already closed + under every tile map after one pass: restriction does *not* accumulate along chains + — it bottoms out immediately. This is exactly the structural behaviour the paper's + §6 programme hoped for ("restriction sets cannot remain mutually disjoint"). +- **Restriction is real but bounded.** Tightest subtrees per seam size: m=4 forces + 2 of 3 necklaces, **m=5 forces a single necklace `{00012}`**, m=7 forces 3 of 10, + m=8 forces 8 of 34, m=14 forces 133 of 7515. Yet no parent relation ever misses a + minimal set entirely. +- **The blocky states are always offered.** Every smallest minimal Ext set contains + the regular necklace of its size (`0^m` or `0^{m-2}12`-type). The regular family + failed (Finding 8) because a single tile sometimes cannot take blocky-in and + blocky-out *jointly* — but per-interface freedom routes around it, and the data + shows subtrees always keep the blocky option available downward. + +Caveats: (i) terminal facial triangles (innermost treads ending on a face of `G`) are +not yet modelled — our leaves are only the degenerate-inner-boundary all-bite tiles; +adding 3-faces as terminal children with `Ext = {012}` is a small extension. (ii) The +universe is abstract: it is a *superset* of real trees (good for the positive verdict; +an abstract obstruction, had one appeared, would still have needed a realisability +check). + ## Open threads -- **Per-interface `R_T` composition** along real chains/trees (per-seam freedom, - respecting coupling) — the paper's §6 conjecture proper, now the main line. -- **Structural lemma for the over-constraint.** Why does a monochromatic large rim - forbid the paired odd-inner necklace? A parity/structure argument on the annular - cycle for these specific failing tiles is small and concrete. -- (Set aside) whether *some* irregular uniform family still works at n=15 (full CSP) - — reframed away from; expensive (~hours) and not the conjecture. +- **Terminal-3-face leaves.** Extend the fixpoint with facial-triangle termination + (parent tiles with one 3-singleton face allowed as terminal); rerun. +- **Push `max_n`.** n=15 adds ~1 hr of one-time classification to the cache; the + 2-round saturation suggests verdicts stabilise quickly, but a deeper universe is + the only way an obstruction could still appear. +- **Why saturation?** The 2-round fixpoint convergence is the empirical shadow of a + provable statement: composing tire restriction relations stabilises after one + level. A proof of that, with the minimal antichains characterised, would *be* the + pigeonhole lemma for this class. +- **Structural lemma for the n=15 uniform failures** (why a monochromatic large rim + forbids the paired odd-inner necklace) — still open, now lower priority. diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_rt_composition_probe.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_rt_composition_probe.py new file mode 100644 index 0000000..3cb5a13 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_rt_composition_probe.py @@ -0,0 +1,277 @@ +"""R_T composition along chains/trees: the chain-pigeonhole fixpoint. + +The conjecture proper (paper section 6): in a nested chain/tree of tire treads, +boundary states must be selectable per interface so that every tile realises its +outer state and all its inner-face states jointly (with one Kempe-balanced +colouring). The uniform per-size family was refuted at n=15; this probe asks +the real question: composing the per-tile restriction relations with FULL +per-interface freedom, can the extendability set ever become empty? + +Model notes (worked out carefully): + + * The gluing seam between a parent tread and a child is EXACTLY the singleton + down apexes of one inner non-tooth face. A bite apex's edge has parent + tread faces on both sides, so it is internal to the parent's tile and + contributes no shared medial vertex. + + * Boundary states are NECKLACES (colour permutation + dihedral symmetry of + the boundary walk -- the paper's "medial boundary state"). This is exact, + not an approximation: a child tile attaches below a seam with free dihedral + placement, so its realisable outer-sequence set is dihedral-closed, and + "necklace match" is equivalent to "some aligned sequence match". + + * Ext(T) = set of necklaces o on T's outer seam such that the whole subtree + rooted at T admits a compatible Kempe-balanced selection: + Ext(T) = { o : exists (o, i_1..i_k) in R_T^joint with i_j in Ext(C_j) }. + Leaves are tiles with NO singleton-down interfaces (all downs in bites; + degenerate inner boundaries), whose Ext is their outer projection. + + * Kempe-balance is the locally-checkable necessary condition for a colouring + to be the restriction of a proper 3-colouring of M(G), so an Ext built from + valid colourings is the right restriction relation. + +Fixpoint: maintain, per seam size m, the ANTICHAIN of minimal reachable Ext +subsets of S_m (admissible necklaces). Ext maps are monotone, so minimal sets +suffice to decide whether the empty set is reachable. + + * empty set reachable => an explicit obstructed tree (witness printed): + the necklace-level local relations do not suffice -- the pigeonhole route + needs more invariants (or the tree is unrealisable as a triangulation). + * empty set unreachable => the chain-pigeonhole conjecture verified + exhaustively over every tree assemblable from this tile universe. + +Tiles: every no-length-3-boundary tile for n in [min_n, max_n] (internal seams +in the no-separating-triangle world have length >= 4). Relations are cached in +kempe_rt_relations_cache.json so reruns and extensions are cheap. + +Run: python3 kempe_rt_composition_probe.py --min-n 8 --max-n 13 +""" + +from __future__ import annotations + +import argparse +import itertools +import json +import os +import sys +import time +from collections import defaultdict + +from full_medial_tire_generator import generate, innermost_bite +from kempe_valid_colorings import classify_colorings +from kempe_transfer_relation_probe import necklace, admissible_necklaces +from kempe_universal_trend_probe import has_length3_boundary +from kempe_up_tooth_sequences import seq_str + +HERE = os.path.dirname(os.path.abspath(__file__)) +CACHE_PATH = os.path.join(HERE, "kempe_rt_relations_cache.json") + +Neck = tuple[int, ...] + + +def inner_faces(g): + faces = defaultdict(list) + for e in g.singleton_down_edges: + faces[innermost_bite(e, g.bites)].append(e) + return [sorted(es) for es in faces.values() if len(es) >= 3] + + +def tile_key(g) -> str: + b = ",".join(f"{i}-{j}" for i, j in sorted(g.bites)) or "-" + return f"{g.n}:{g.tooth_word}:{b}" + + +def compute_joint(g, faces) -> set[tuple[Neck, tuple[Neck, ...]]]: + """All (outer necklace, per-face inner necklaces) realised by one + Kempe-balanced colouring of the tile.""" + joint = set() + for c, v in classify_colorings(g, dedup_colors=True): + if not v.valid: + continue + o = necklace(tuple(c[f"u{e}"] for e in g.up_edges)) + ins = tuple(necklace(tuple(c[f"d{e}"] for e in f)) for f in faces) + joint.add((o, ins)) + return joint + + +# --------------------------------------------------------------------------- +# Tile collection with on-disk relation cache. +# --------------------------------------------------------------------------- + +def _parse(s: str) -> Neck: + return tuple(int(ch) for ch in s) + + +def collect(min_n: int, max_n: int): + cache: dict = {} + if os.path.exists(CACHE_PATH): + with open(CACHE_PATH) as fh: + cache = json.load(fh) + dirty = False + tiles = [] # (key, p, qsizes, joint) + for n in range(min_n, max_n + 1): + t0 = time.time() + cnt = 0 + for g in generate(n, min_up_teeth=3, dedup=True): + if has_length3_boundary(g): + continue + key = tile_key(g) + if key in cache: + rec = cache[key] + else: + faces = inner_faces(g) + joint = compute_joint(g, faces) + rec = { + "p": len(g.up_edges), + "q": [len(f) for f in faces], + "joint": [[seq_str(o), [seq_str(i) for i in ins]] + for o, ins in sorted(joint)], + } + cache[key] = rec + dirty = True + joint = frozenset( + (_parse(o), tuple(_parse(i) for i in ins)) + for o, ins in rec["joint"] + ) + tiles.append((key, rec["p"], tuple(rec["q"]), joint)) + cnt += 1 + print(f" n={n}: {cnt} tiles [{time.time() - t0:.0f}s]") + sys.stdout.flush() + if dirty: + with open(CACHE_PATH, "w") as fh: + json.dump(cache, fh) + print(f" (cache updated: {CACHE_PATH})") + return tiles + + +def dedup_by_relation(tiles): + """Merge tiles with identical (p, qsizes, joint) -- same constraint.""" + seen = {} + for key, p, qs, joint in tiles: + sig = (p, qs, joint) + if sig not in seen: + seen[sig] = key + return [(rep, p, qs, joint) for (p, qs, joint), rep in seen.items()] + + +# --------------------------------------------------------------------------- +# The fixpoint over minimal Ext antichains. +# --------------------------------------------------------------------------- + +def fixpoint(tiles): + fam: dict[int, set[frozenset]] = defaultdict(set) # size -> minimal Ext sets + prov: dict[tuple[int, frozenset], tuple] = {} + obstruction = None # (size, provenance) when empty set first reached + + def add(size, s, why) -> bool: + s = frozenset(s) + if (size, s) not in prov: + prov[(size, s)] = why + for e in fam[size]: + if e <= s: + return False # dominated by an existing (smaller) set + for e in [e for e in fam[size] if s < e]: + fam[size].discard(e) # s supersedes larger sets + fam[size].add(s) + return True + + def det(sets): + return sorted(sets, key=lambda s: (len(s), sorted(s))) + + rounds = 0 + changed = True + while changed and obstruction is None: + changed = False + rounds += 1 + for key, p, qs, joint in tiles: + if not qs: # leaf: no inner interfaces + proj = frozenset(o for o, _ in joint) + if add(p, proj, (key, [])): + changed = True + if not proj: + obstruction = (p, (key, [])) + break + continue + if any(not fam[q] for q in qs): + continue # some child seam size not yet seeded + for combo in itertools.product(*[det(fam[q]) for q in qs]): + O = frozenset( + o for o, ins in joint + if all(ins[j] in combo[j] for j in range(len(qs))) + ) + if add(p, O, (key, list(zip(qs, combo)))): + changed = True + if not O: + obstruction = (p, (key, list(zip(qs, combo)))) + break + if obstruction: + break + return fam, prov, rounds, obstruction + + +def print_witness(size, s, prov, depth=0): + key, children = prov[(size, frozenset(s))] + states = "{" + ",".join(seq_str(x) for x in sorted(s)) + "}" + print(" " + " " * depth + + f"seam size {size}: Ext={states or '{}'} via tile {key}") + for q, cs in children: + print_witness(q, cs, prov, depth + 1) + + +# --------------------------------------------------------------------------- +# Driver. +# --------------------------------------------------------------------------- + +def run(args): + print(f"R_T composition fixpoint over no-length-3-boundary tiles, " + f"n = {args.min_n}..{args.max_n}\n") + tiles = collect(args.min_n, args.max_n) + uniq = dedup_by_relation(tiles) + n_leaf = sum(1 for _, _, qs, _ in uniq if not qs) + n_branch = sum(1 for _, _, qs, _ in uniq if len(qs) >= 2) + print(f"\n{len(tiles)} tiles, {len(uniq)} distinct relations " + f"({n_leaf} leaf, {n_branch} branching)") + + t0 = time.time() + fam, prov, rounds, obstruction = fixpoint(uniq) + print(f"fixpoint: {rounds} rounds [{time.time() - t0:.0f}s]\n") + + print("minimal Ext antichains per seam size " + "(how restrictive a subtree can be):") + for m in sorted(fam): + adm = len(admissible_necklaces(m)) + sets = sorted(fam[m], key=lambda s: (len(s), sorted(s))) + sizes = [len(s) for s in sets] + smallest = "{" + ",".join(seq_str(x) for x in sorted(sets[0])) + "}" \ + if sets else "-" + print(f" m={m}: |adm|={adm} {len(sets)} minimal sets, " + f"sizes {sizes} smallest={smallest}") + unseeded = [m for m in {q for _, _, qs, _ in uniq for q in qs} + if not fam.get(m)] + if unseeded: + print(f" (seam sizes never seeded by any subtree: {sorted(unseeded)})") + + if obstruction: + size, why = obstruction + print("\n=> EMPTY EXT REACHED -- explicit boundary-state obstructed tree:") + prov[(size, frozenset())] = why + print_witness(size, frozenset(), prov) + print("\n (If this tree is realisable as a plane triangulation, the\n" + " necklace-level relations admit a genuine obstruction; if not,\n" + " realisability constraints are doing essential work.)") + else: + print("\n=> EMPTY EXT NOT REACHABLE: every tree assemblable from this " + "tile universe\n admits a compatible Kempe-balanced selection. " + "The chain-pigeonhole conjecture\n is verified exhaustively over " + f"all trees built from tiles with n <= {args.max_n}.") + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--min-n", type=int, default=8) + parser.add_argument("--max-n", type=int, default=13) + run(parser.parse_args()) + + +if __name__ == "__main__": + main() diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_rt_relations_cache.json b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_rt_relations_cache.json new file mode 100644 index 0000000..e4c312d --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_rt_relations_cache.json @@ -0,0 +1 @@ +{"8:UUUUUUUU:-": {"p": 8, "q": [], "joint": [["00000000", []], ["00000011", []], ["00001111", []], ["00001221", []], ["00011022", []], ["00012012", []], ["00110011", []], ["00120021", []]]}, "8:UUUUUDUD:5-7": {"p": 6, "q": [], 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