Enumerate colour/tread phase over the residue graphs

residue_phase_sweep.py exhaustively enumerates the two colouring control knobs -- the per-annulus tread phase {0,1}^A and the root-DFS colour order perms(0,1,2) -- on top of every insertion-site combo, for the graphs the random-phase site sweep still fails. canonical_coloring_explicit makes this deterministic. Result (residue_phase_sweep_results.txt): the two hub graphs are RESCUED once phase is enumerated rather than sampled (so the random-phase fail count overstates difficulty); the genuine obstructions that survive sites x phases x colour-orders are exactly the face-leaf graphs (terminal-triangle leaf gadget). Smallest is seed2 #26 [3,6,3] face (1 combo, 24 settings, all fail at gadget-removal) -- a minimal obstruction target. Caveat: try_establish is a bounded local Kempe search, so STILL FAILS means unreachable by the bounded search from canonical-even over all knob settings, not that no Kempe path exists. Findings note updated. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
2026-06-13 00:03:49 -04:00
parent 9d296eb9c8
commit faf9e01139
3 changed files with 209 additions and 0 deletions
@@ -85,6 +85,52 @@ knobs. Building this joint solver — and finding the smallest configuration, if
 forcing a self-loop or odd cycle — is the next step and the exact thing a proof
 would need to rule out.

+## Exhausting the control knobs over the residue
+
+The site sweep above counts a graph `ok` if some placement works over only **4
+random** colour phases per combo. `residue_phase_sweep.py` takes the graphs that
+sweep still fails (the residue) and **exhaustively enumerates the colour/tread
+phase and the root-DFS colour order** on top of every insertion site:
+
+```
+phases   in {0,1}^A          (A = # non-root annuli; the tread phases)
+colorder in perms(0,1,2)     (root-region DFS colour priority)
+```
+
+over all site combos (cap 512). Result on the seed-1/seed-2 residue
+(`residue_phase_sweep_results.txt`):
+
+```
+seed1 #16 [3,8,3,5]    hub   RESCUED      (720 settings, 18 ok)
+seed1 #51 [3,7,3,3,7]  hub   RESCUED    (42336 settings, 196 ok)
+seed1 #3  [3,7,4,6,3]  face  STILL FAILS  (672 settings,  0 ok)
+seed1 #4  [3,4,5,5,3]  face  STILL FAILS (2400 settings,  0 ok)
+seed2 #26 [3,6,3]      face  STILL FAILS   (24 settings,  0 ok)
+seed2 #30 [3,3,6,7,3]  face  STILL FAILS (2016 settings,  0 ok)
+seed2 #54 [3,3,5,3]    face  STILL FAILS  (720 settings,  0 ok)
+```
+
+Two things fall out:
+
+1. **Phase reachability explains part of the residue.** The two `hub` graphs are
+   *rescued* once the phase/colour-order is enumerated rather than sampled — they
+   were never genuine obstructions, just unlucky random phases. So the
+   random-phase `fail` count overstates the true difficulty.
+2. **The genuine obstructions are exactly the `face`-leaf graphs.** Every graph
+   that survives exhausting sites × phases × colour-orders has `leaf='face'` —
+   i.e. an inner terminal triangle carrying a leaf gadget. The smallest is
+   `seed2 #26 [3,6,3]` (one site combo, 24 settings, all fail at
+   `gadget-removal`): a minimal target for the joint solver / an obstruction
+   hunt. (#26 fails at the gadget step; #3/#4/#30/#54 at `diamond-switch`.)
+
+**Caveat on "STILL FAILS".** `try_establish` is a *bounded* local Kempe search
+(≤3 components anchored at the quad support). So `STILL FAILS` means *no (site,
+phase, colour-order) lets the bounded search from the canonical-even colouring
+reach a descendable one* — not that no Kempe path exists. A descendable colouring
+provably exists (M(G) is 3-colourable); whether it is reachable under a principled
+(joint, unbounded) switch is the open question, now sharply localised to the
+`face`-leaf family.
+
 ## Caveats / domain

 - Real plantri triangulations mostly `skip:chord-level-edge` under BFS-from-outer
@@ -0,0 +1,154 @@
+"""Exhaustively enumerate the colour/tread-phase control knobs over the residue
+graphs -- the synthetic ring triangulations the SITE sweep still fails on while
+only *sampling* random phases (kempe_even_program_harness.run_graph draws 4
+random phase vectors + colour orders per site combo).
+
+For each residue graph we sweep, per insertion-site combo:
+    phases   in {0,1}^A   (A = number of non-root annuli; the tread phases)
+    colorder in perms(0,1,2)  (the root-region DFS colour priority)
+and ask whether SOME (combo, phases, colorder) yields a full reduction. This
+isolates how much of the residue is a phase-reachability artifact vs a genuine
+obstruction.
+
+Usage:  python3 residue_phase_sweep.py            # the seed1/seed2 residue
+        python3 residue_phase_sweep.py SEED IDX    # one graph
+"""
+import sys
+import random
+import itertools as it
+import kempe_even_program_harness as H
+
+# residue from the random-phase site sweep (seed, idx) -> reported failure
+RESIDUE = [(1, 3), (1, 4), (1, 16), (1, 51), (2, 26), (2, 30), (2, 54)]
+MAX_COMBOS = 512
+
+
+def reconstruct(seed, idx):
+    """Replay the synthetic loop to get graph #idx (rng state is sequential)."""
+    rng = random.Random(seed)
+    g = outer = sizes = leaf = None
+    for _ in range(idx + 1):
+        sizes, leaf = H.random_profile(rng)
+        g, outer = H.ring_triangulation(sizes, leaf, rng)
+    return g, outer, sizes, leaf
+
+
+def descend_explicit(template, outer, orig, gadgets, combo, phases, colorder):
+    gg = template.copy()
+    dia = [gg.insert_diamond(a, b) for (a, b) in combo]
+    an = H.Analysis(gg, outer)
+    if an.degenerate:
+        return f"skip:post-diamond-{an.degenerate}"
+    if any(len(c) % 2 for _, c in an.seams):
+        return "fail:seam-evening"
+    col, reason = H.canonical_coloring_explicit(gg, an.level, outer, phases, colorder)
+    if col is None:
+        return f"fail:canonical-{reason}"
+    for (w, u, v, x, t) in dia:
+        adj = H.medial_adj(gg)
+        quad = H.quad_of(gg, w, u, v)
+        if quad is None:
+            return "fail:diamond-quad"
+        support = [H.e(quad[i], quad[(i + 1) % 4]) for i in range(4)]
+        if not H.try_establish(col, adj, support,
+                               lambda c: H.diamond_condition(c, quad) is not None):
+            return "fail:diamond-switch"
+        H.collapse_degree4(gg, col, w, u, v)
+        if not H.verify_proper(gg, col):
+            return "fail:improper-after-diamond"
+    for (y, z, u, v, x, t) in gadgets:
+        done = False
+        for first, second, tri in ((z, y, (x, u, v)), (y, z, (u, v, t))):
+            gt, ct = gg.copy(), dict(col)
+            adj = H.medial_adj(gt)
+            quad = H.quad_of(gt, first, u, v)
+            if quad is None:
+                continue
+            support = [H.e(quad[i], quad[(i + 1) % 4]) for i in range(4)]
+            if not H.try_establish(ct, adj, support,
+                                   lambda c: H.diamond_condition(c, quad) is not None):
+                continue
+            H.collapse_degree4(gt, ct, first, u, v)
+            if not H.verify_proper(gt, ct):
+                continue
+            adj2 = H.medial_adj(gt)
+            a_, b_, c_ = tri
+            support2 = [H.e(a_, b_), H.e(b_, c_), H.e(c_, a_)]
+            if not H.try_establish(ct, adj2, support2,
+                                   lambda c: H.rainbow_condition(c, tri)):
+                continue
+            H.collapse_degree3(gt, ct, second)
+            if not H.verify_proper(gt, ct):
+                continue
+            gg, col = gt, ct
+            done = True
+            break
+        if not done:
+            return "fail:gadget-removal"
+    if set(gg.rot) != orig:
+        return "fail:vertex-mismatch"
+    if not H.verify_proper(gg, col):
+        return "fail:final-improper"
+    return "ok"
+
+
+def phase_arity(template, outer, combo):
+    gg = template.copy()
+    for (a, b) in combo:
+        gg.insert_diamond(a, b)
+    an = H.Analysis(gg, outer)
+    sk, _ = H.coloring_skeleton(gg, an.level, outer)
+    return len(sk['nonroot']) if sk else None
+
+
+def sweep(seed, idx):
+    g, outer, sizes, leaf = reconstruct(seed, idx)
+    orig = set(g.rot)
+    prep = H._prep_gadgets(g.copy(), outer)
+    if isinstance(prep, str):
+        print(f"seed{seed} #{idx} {sizes} {leaf}: prep -> {prep}")
+        return
+    template, an_g, gadgets = prep
+    sites = H._candidate_sites(an_g)
+    if sites is None:
+        print(f"seed{seed} #{idx} {sizes} {leaf}: no diamond site")
+        return
+    combos = list(it.product(*sites))
+    n_full = len(combos)
+    truncated = n_full > MAX_COMBOS
+    combos = combos[:MAX_COMBOS]
+    tried = wins = 0
+    first = None
+    for combo in combos:
+        nph = phase_arity(template, outer, combo)
+        if nph is None:
+            continue
+        for phases in it.product((0, 1), repeat=nph):
+            for colorder in it.permutations((0, 1, 2)):
+                tried += 1
+                res = descend_explicit(template, outer, orig, gadgets,
+                                       combo, phases, list(colorder))
+                if res == "ok":
+                    wins += 1
+                    if first is None:
+                        first = (combo, phases, colorder)
+    verdict = "RESCUED" if wins else "STILL FAILS"
+    msg = (f"seed{seed} #{idx} {sizes} {leaf}: {verdict}  "
+           f"(combos={n_full}{'[capped]' if truncated else ''}, "
+           f"settings tried={tried}, ok={wins})")
+    if first:
+        msg += f"\n    first ok: site={first[0]} phases={first[1]} colorder={first[2]}"
+    print(msg)
+
+
+def main():
+    if len(sys.argv) == 3:
+        sweep(int(sys.argv[1]), int(sys.argv[2]))
+    else:
+        for seed, idx in RESIDUE:
+            sweep(seed, idx)
+            sys.stdout.flush()
+
+
+if __name__ == "__main__":
+    main()
@@ -0,0 +1,9 @@
+seed1 #3 [3, 7, 4, 6, 3] face: STILL FAILS  (combos=7, settings tried=672, ok=0)
+seed1 #4 [3, 4, 5, 5, 3] face: STILL FAILS  (combos=25, settings tried=2400, ok=0)
+seed1 #16 [3, 8, 3, 5] hub: RESCUED  (combos=15, settings tried=720, ok=18)
+    first ok: site=((11, 13), (14, 15)) phases=(0, 1, 0) colorder=(0, 1, 2)
+seed1 #51 [3, 7, 3, 3, 7] hub: RESCUED  (combos=441, settings tried=42336, ok=196)
+    first ok: site=((6, 7), (10, 11), (14, 15), (21, 22)) phases=(0, 0, 1, 0) colorder=(0, 1, 2)
+seed2 #26 [3, 6, 3] face: STILL FAILS  (combos=1, settings tried=24, ok=0)
+seed2 #30 [3, 3, 6, 7, 3] face: STILL FAILS  (combos=21, settings tried=2016, ok=0)
+seed2 #54 [3, 3, 5, 3] face: STILL FAILS  (combos=15, settings tried=720, ok=0)