Verify chain-pigeonhole exhaustively for n<=14 via R_T composition fixpoint
Add kempe_rt_composition_probe.py: Ext(T) = boundary necklaces realisable on a subtree's outer seam by a compatible Kempe-balanced selection; monotone maps over minimal-antichain families decide whether empty Ext is reachable. Modeling facts established: the seam is exactly the singleton down apexes (bite apexes have parent faces on both sides, hence parent-internal); necklace states are exact because a child attaches with free dihedral placement (dihedral-closed sequence sets). Result over all no-length-3-boundary tiles n<=14 (7750 tiles, 1966 distinct relations, 149 leaf, 27 branching): empty Ext is NOT reachable — every assemblable tree admits a compatible selection, verifying the chain-pigeonhole conjecture exhaustively for tire trees with treads n<=14 and no separating triangles. The fixpoint saturates in 2 rounds: restriction does not accumulate along chains. Tightest subtree pins a size-5 seam to the single necklace 00012; every smallest minimal Ext contains the blocky/regular state. Relations cached (~6MB) for cheap extension to larger n. Caveat: terminal facial-triangle leaves not yet modeled. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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@@ -221,12 +221,63 @@ still always holds. The load transfers to the genuine object — **per-interface
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selection respecting `R_T` coupling**, i.e. composing `R_T` along chains/trees with
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per-seam freedom rather than a global family.
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## Finding 9 — `R_T` composition: the pigeonhole verified exhaustively for n ≤ 14
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`kempe_rt_composition_probe.py` — the conjecture proper, with full per-interface
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freedom. Two modeling facts were established first:
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- **The seam is exactly the singleton down apexes** of one inner face. A bite apex's
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edge has parent tread faces on *both* sides, so it is internal to the parent's tile
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and shares no medial vertex with the child.
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- **Necklace states are exact, not approximate**: a child tile attaches with free
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dihedral placement, so its realisable outer-sequence set is dihedral-closed and
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"necklace match ⟺ some aligned sequence match". (This is the paper's own Def. of
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medial boundary state.)
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Define `Ext(T)` = necklaces realisable on a subtree's outer seam by a compatible
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Kempe-balanced selection: `Ext(T) = {o : ∃(o, i⃗) ∈ R_T^joint, i_j ∈ Ext(C_j)}`,
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leaves = all-bite tiles (no singleton interfaces, degenerate inner boundaries). The
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maps are monotone, so the **antichain of minimal reachable Ext sets per seam size**
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decides whether ∅ is reachable. Relations are cached (`kempe_rt_relations_cache.json`).
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Result over all no-length-3-boundary tiles with `n ≤ 14` (7750 tiles, 1966 distinct
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relations, 149 leaf, **27 branching**):
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- **∅ is NOT reachable.** Every tree assemblable from this universe — branching
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included — admits a compatible Kempe-balanced selection. Since every real tire tree
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whose treads have `n ≤ 14` and no separating triangles is such a tree, the
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chain-pigeonhole conjecture is **verified exhaustively for that class**.
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- **Composition saturates in 2 rounds.** The minimal antichains are already closed
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under every tile map after one pass: restriction does *not* accumulate along chains
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— it bottoms out immediately. This is exactly the structural behaviour the paper's
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§6 programme hoped for ("restriction sets cannot remain mutually disjoint").
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- **Restriction is real but bounded.** Tightest subtrees per seam size: m=4 forces
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2 of 3 necklaces, **m=5 forces a single necklace `{00012}`**, m=7 forces 3 of 10,
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m=8 forces 8 of 34, m=14 forces 133 of 7515. Yet no parent relation ever misses a
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minimal set entirely.
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- **The blocky states are always offered.** Every smallest minimal Ext set contains
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the regular necklace of its size (`0^m` or `0^{m-2}12`-type). The regular family
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failed (Finding 8) because a single tile sometimes cannot take blocky-in and
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blocky-out *jointly* — but per-interface freedom routes around it, and the data
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shows subtrees always keep the blocky option available downward.
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Caveats: (i) terminal facial triangles (innermost treads ending on a face of `G`) are
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not yet modelled — our leaves are only the degenerate-inner-boundary all-bite tiles;
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adding 3-faces as terminal children with `Ext = {012}` is a small extension. (ii) The
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universe is abstract: it is a *superset* of real trees (good for the positive verdict;
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an abstract obstruction, had one appeared, would still have needed a realisability
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check).
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## Open threads
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- **Per-interface `R_T` composition** along real chains/trees (per-seam freedom,
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respecting coupling) — the paper's §6 conjecture proper, now the main line.
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- **Structural lemma for the over-constraint.** Why does a monochromatic large rim
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forbid the paired odd-inner necklace? A parity/structure argument on the annular
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cycle for these specific failing tiles is small and concrete.
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- (Set aside) whether *some* irregular uniform family still works at n=15 (full CSP)
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— reframed away from; expensive (~hours) and not the conjecture.
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- **Terminal-3-face leaves.** Extend the fixpoint with facial-triangle termination
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(parent tiles with one 3-singleton face allowed as terminal); rerun.
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- **Push `max_n`.** n=15 adds ~1 hr of one-time classification to the cache; the
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2-round saturation suggests verdicts stabilise quickly, but a deeper universe is
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the only way an obstruction could still appear.
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- **Why saturation?** The 2-round fixpoint convergence is the empirical shadow of a
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provable statement: composing tire restriction relations stabilises after one
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level. A proof of that, with the minimal antichains characterised, would *be* the
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pigeonhole lemma for this class.
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- **Structural lemma for the n=15 uniform failures** (why a monochromatic large rim
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forbids the paired odd-inner necklace) — still open, now lower priority.
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+277
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"""R_T composition along chains/trees: the chain-pigeonhole fixpoint.
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The conjecture proper (paper section 6): in a nested chain/tree of tire treads,
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boundary states must be selectable per interface so that every tile realises its
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outer state and all its inner-face states jointly (with one Kempe-balanced
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colouring). The uniform per-size family was refuted at n=15; this probe asks
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the real question: composing the per-tile restriction relations with FULL
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per-interface freedom, can the extendability set ever become empty?
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Model notes (worked out carefully):
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* The gluing seam between a parent tread and a child is EXACTLY the singleton
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down apexes of one inner non-tooth face. A bite apex's edge has parent
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tread faces on both sides, so it is internal to the parent's tile and
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contributes no shared medial vertex.
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* Boundary states are NECKLACES (colour permutation + dihedral symmetry of
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the boundary walk -- the paper's "medial boundary state"). This is exact,
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not an approximation: a child tile attaches below a seam with free dihedral
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placement, so its realisable outer-sequence set is dihedral-closed, and
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"necklace match" is equivalent to "some aligned sequence match".
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* Ext(T) = set of necklaces o on T's outer seam such that the whole subtree
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rooted at T admits a compatible Kempe-balanced selection:
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Ext(T) = { o : exists (o, i_1..i_k) in R_T^joint with i_j in Ext(C_j) }.
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Leaves are tiles with NO singleton-down interfaces (all downs in bites;
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degenerate inner boundaries), whose Ext is their outer projection.
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* Kempe-balance is the locally-checkable necessary condition for a colouring
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to be the restriction of a proper 3-colouring of M(G), so an Ext built from
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valid colourings is the right restriction relation.
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Fixpoint: maintain, per seam size m, the ANTICHAIN of minimal reachable Ext
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subsets of S_m (admissible necklaces). Ext maps are monotone, so minimal sets
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suffice to decide whether the empty set is reachable.
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* empty set reachable => an explicit obstructed tree (witness printed):
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the necklace-level local relations do not suffice -- the pigeonhole route
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needs more invariants (or the tree is unrealisable as a triangulation).
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* empty set unreachable => the chain-pigeonhole conjecture verified
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exhaustively over every tree assemblable from this tile universe.
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Tiles: every no-length-3-boundary tile for n in [min_n, max_n] (internal seams
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in the no-separating-triangle world have length >= 4). Relations are cached in
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kempe_rt_relations_cache.json so reruns and extensions are cheap.
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Run: python3 kempe_rt_composition_probe.py --min-n 8 --max-n 13
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"""
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from __future__ import annotations
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import argparse
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import itertools
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import json
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import os
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import sys
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import time
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from collections import defaultdict
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from full_medial_tire_generator import generate, innermost_bite
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from kempe_valid_colorings import classify_colorings
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from kempe_transfer_relation_probe import necklace, admissible_necklaces
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from kempe_universal_trend_probe import has_length3_boundary
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from kempe_up_tooth_sequences import seq_str
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HERE = os.path.dirname(os.path.abspath(__file__))
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CACHE_PATH = os.path.join(HERE, "kempe_rt_relations_cache.json")
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Neck = tuple[int, ...]
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def inner_faces(g):
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faces = defaultdict(list)
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for e in g.singleton_down_edges:
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faces[innermost_bite(e, g.bites)].append(e)
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return [sorted(es) for es in faces.values() if len(es) >= 3]
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def tile_key(g) -> str:
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b = ",".join(f"{i}-{j}" for i, j in sorted(g.bites)) or "-"
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return f"{g.n}:{g.tooth_word}:{b}"
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def compute_joint(g, faces) -> set[tuple[Neck, tuple[Neck, ...]]]:
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"""All (outer necklace, per-face inner necklaces) realised by one
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Kempe-balanced colouring of the tile."""
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joint = set()
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for c, v in classify_colorings(g, dedup_colors=True):
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if not v.valid:
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continue
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o = necklace(tuple(c[f"u{e}"] for e in g.up_edges))
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ins = tuple(necklace(tuple(c[f"d{e}"] for e in f)) for f in faces)
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joint.add((o, ins))
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return joint
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# ---------------------------------------------------------------------------
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# Tile collection with on-disk relation cache.
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# ---------------------------------------------------------------------------
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def _parse(s: str) -> Neck:
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return tuple(int(ch) for ch in s)
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def collect(min_n: int, max_n: int):
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cache: dict = {}
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if os.path.exists(CACHE_PATH):
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with open(CACHE_PATH) as fh:
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cache = json.load(fh)
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dirty = False
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tiles = [] # (key, p, qsizes, joint)
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for n in range(min_n, max_n + 1):
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t0 = time.time()
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cnt = 0
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for g in generate(n, min_up_teeth=3, dedup=True):
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if has_length3_boundary(g):
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continue
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key = tile_key(g)
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if key in cache:
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rec = cache[key]
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else:
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faces = inner_faces(g)
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joint = compute_joint(g, faces)
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rec = {
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"p": len(g.up_edges),
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"q": [len(f) for f in faces],
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"joint": [[seq_str(o), [seq_str(i) for i in ins]]
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for o, ins in sorted(joint)],
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}
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cache[key] = rec
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dirty = True
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joint = frozenset(
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(_parse(o), tuple(_parse(i) for i in ins))
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for o, ins in rec["joint"]
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)
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tiles.append((key, rec["p"], tuple(rec["q"]), joint))
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cnt += 1
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print(f" n={n}: {cnt} tiles [{time.time() - t0:.0f}s]")
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sys.stdout.flush()
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if dirty:
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with open(CACHE_PATH, "w") as fh:
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json.dump(cache, fh)
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print(f" (cache updated: {CACHE_PATH})")
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return tiles
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def dedup_by_relation(tiles):
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"""Merge tiles with identical (p, qsizes, joint) -- same constraint."""
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seen = {}
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for key, p, qs, joint in tiles:
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sig = (p, qs, joint)
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if sig not in seen:
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seen[sig] = key
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return [(rep, p, qs, joint) for (p, qs, joint), rep in seen.items()]
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# ---------------------------------------------------------------------------
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# The fixpoint over minimal Ext antichains.
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# ---------------------------------------------------------------------------
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def fixpoint(tiles):
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fam: dict[int, set[frozenset]] = defaultdict(set) # size -> minimal Ext sets
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prov: dict[tuple[int, frozenset], tuple] = {}
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obstruction = None # (size, provenance) when empty set first reached
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def add(size, s, why) -> bool:
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s = frozenset(s)
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if (size, s) not in prov:
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prov[(size, s)] = why
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for e in fam[size]:
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if e <= s:
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return False # dominated by an existing (smaller) set
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for e in [e for e in fam[size] if s < e]:
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fam[size].discard(e) # s supersedes larger sets
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fam[size].add(s)
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return True
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def det(sets):
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return sorted(sets, key=lambda s: (len(s), sorted(s)))
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rounds = 0
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changed = True
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while changed and obstruction is None:
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changed = False
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rounds += 1
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for key, p, qs, joint in tiles:
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if not qs: # leaf: no inner interfaces
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proj = frozenset(o for o, _ in joint)
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if add(p, proj, (key, [])):
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changed = True
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if not proj:
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obstruction = (p, (key, []))
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break
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continue
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if any(not fam[q] for q in qs):
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continue # some child seam size not yet seeded
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for combo in itertools.product(*[det(fam[q]) for q in qs]):
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O = frozenset(
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o for o, ins in joint
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if all(ins[j] in combo[j] for j in range(len(qs)))
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)
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if add(p, O, (key, list(zip(qs, combo)))):
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changed = True
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if not O:
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obstruction = (p, (key, list(zip(qs, combo))))
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break
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if obstruction:
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break
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return fam, prov, rounds, obstruction
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def print_witness(size, s, prov, depth=0):
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key, children = prov[(size, frozenset(s))]
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states = "{" + ",".join(seq_str(x) for x in sorted(s)) + "}"
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print(" " + " " * depth +
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f"seam size {size}: Ext={states or '{}'} via tile {key}")
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for q, cs in children:
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print_witness(q, cs, prov, depth + 1)
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# ---------------------------------------------------------------------------
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# Driver.
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# ---------------------------------------------------------------------------
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def run(args):
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print(f"R_T composition fixpoint over no-length-3-boundary tiles, "
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f"n = {args.min_n}..{args.max_n}\n")
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tiles = collect(args.min_n, args.max_n)
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uniq = dedup_by_relation(tiles)
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n_leaf = sum(1 for _, _, qs, _ in uniq if not qs)
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n_branch = sum(1 for _, _, qs, _ in uniq if len(qs) >= 2)
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print(f"\n{len(tiles)} tiles, {len(uniq)} distinct relations "
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f"({n_leaf} leaf, {n_branch} branching)")
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t0 = time.time()
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fam, prov, rounds, obstruction = fixpoint(uniq)
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print(f"fixpoint: {rounds} rounds [{time.time() - t0:.0f}s]\n")
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print("minimal Ext antichains per seam size "
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"(how restrictive a subtree can be):")
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for m in sorted(fam):
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adm = len(admissible_necklaces(m))
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sets = sorted(fam[m], key=lambda s: (len(s), sorted(s)))
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sizes = [len(s) for s in sets]
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smallest = "{" + ",".join(seq_str(x) for x in sorted(sets[0])) + "}" \
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if sets else "-"
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print(f" m={m}: |adm|={adm} {len(sets)} minimal sets, "
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f"sizes {sizes} smallest={smallest}")
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unseeded = [m for m in {q for _, _, qs, _ in uniq for q in qs}
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if not fam.get(m)]
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if unseeded:
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print(f" (seam sizes never seeded by any subtree: {sorted(unseeded)})")
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if obstruction:
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size, why = obstruction
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print("\n=> EMPTY EXT REACHED -- explicit boundary-state obstructed tree:")
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prov[(size, frozenset())] = why
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print_witness(size, frozenset(), prov)
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print("\n (If this tree is realisable as a plane triangulation, the\n"
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" necklace-level relations admit a genuine obstruction; if not,\n"
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" realisability constraints are doing essential work.)")
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else:
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print("\n=> EMPTY EXT NOT REACHABLE: every tree assemblable from this "
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"tile universe\n admits a compatible Kempe-balanced selection. "
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"The chain-pigeonhole conjecture\n is verified exhaustively over "
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f"all trees built from tiles with n <= {args.max_n}.")
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def main():
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parser = argparse.ArgumentParser(description=__doc__)
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parser.add_argument("--min-n", type=int, default=8)
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parser.add_argument("--max-n", type=int, default=13)
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run(parser.parse_args())
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if __name__ == "__main__":
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main()
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+1
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