Test induction strategy for the irreducible lemma
reduction_exists.py: every disk with k>=1 (and every irreducible disk) has a Phi-NON-INCREASING vertex removal (|Phi(D-v)| <= |Phi(D)|), 100% over thousands of disks -- so induction to the k=0 base case is viable. BUT the clean set-inclusion Phi(D-v) subset Phi(D) holds for only ~8%, so the size step cannot be proved by "re-inserting v loses no sequence"; it needs a genuine cardinality injection between non-nested sets. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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"""
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Proof strategy A for the irreducible lemma (|Phi| >= 2^(n-2)):
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induction on k via a Phi-NON-INCREASING vertex removal. Monotonicity is false
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(some removals raise Phi), but we only need ONE good removal per disk: if every
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disk with k>=1 has an interior vertex v and a link-retriangulation with
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|Phi(D - v)| <= |Phi(D)|, then chaining down to k=0 gives |Phi(D)| >= 2^(n-2).
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This probe: for each disk, try removing each interior vertex (retriangulating its
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link by a fan from every link vertex, keeping only valid retriangulations), and
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record whether the BEST removal satisfies |Phi(D-v)| <= |Phi(D)|. Reports the
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fraction of disks where such a removal EXISTS (strategy viable) vs disks where
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EVERY removal strictly raises Phi (strategy fails) -- and dumps the failures.
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"""
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import sys
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from collections import Counter, defaultdict
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from itertools import product
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import numpy as np
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from scipy.spatial import Delaunay
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np.seterr(all="ignore")
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def disk(n, k, rng):
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ang = 2 * np.pi * np.arange(n) / n
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rad = 1.0 + 0.18 * rng.random(n)
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bpts = np.c_[rad * np.cos(ang), rad * np.sin(ang)]
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r = 0.8 * np.sqrt(rng.random(k)); t = 2 * np.pi * rng.random(k)
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pts = np.vstack([bpts, np.c_[r * np.cos(t), r * np.sin(t)]])
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return [tuple(int(x) for x in s) for s in Delaunay(pts).simplices]
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def valid(faces, n, k):
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if len(faces) != 2 * k + n - 2:
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return False
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ec = Counter()
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for a, b, c in faces:
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for e in ((a, b), (b, c), (a, c)):
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ec[frozenset(e)] += 1
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return all(ec[frozenset((i, (i + 1) % n))] == 1 for i in range(n))
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def phi_size(faces, n):
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interior = sorted(set(v for f in faces for v in f if v >= n))
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F = len(faces)
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if F > 16:
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return None
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Bint = np.zeros((len(interior), F), dtype=np.int64)
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Cinc = np.zeros((n, F), dtype=np.int64)
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iidx = {w: r for r, w in enumerate(interior)}
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for j, (a, b, c) in enumerate(faces):
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for v in (a, b, c):
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if v >= n:
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Bint[iidx[v], j] = 1
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else:
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Cinc[v, j] = 1
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labs = np.array(list(product((1, 2), repeat=F)), dtype=np.int64)
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if interior:
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labs = labs[np.all((labs @ Bint.T) % 3 == 0, axis=1)]
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if labs.shape[0] == 0:
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return 0
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return len(set(map(tuple, np.unique((labs @ Cinc.T) % 3, axis=0))))
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def edges_of(faces):
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E = set()
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for a, b, c in faces:
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for e in ((a, b), (b, c), (a, c)):
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E.add(frozenset(e))
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return E
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def link_cycle(faces, v):
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opp = [tuple(x for x in f if x != v) for f in faces if v in f]
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adj = defaultdict(list)
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for a, b in opp:
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adj[a].append(b); adj[b].append(a)
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if any(len(adj[u]) != 2 for u in adj):
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return None # link not a simple cycle
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start = opp[0][0]; cyc = [start]; prev = None; cur = start
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while True:
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nxt = [x for x in adj[cur] if x != prev]
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if not nxt:
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return None
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nxt = nxt[0]
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if nxt == start:
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break
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cyc.append(nxt); prev, cur = cur, nxt
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if len(cyc) > len(adj):
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return None
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return cyc if len(cyc) == len(adj) else None
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def removals(faces, v, n):
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"""Yield valid (faces of D - v) over fan retriangulations of v's link."""
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cyc = link_cycle(faces, v)
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if cyc is None:
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return
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d = len(cyc)
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rest = [f for f in faces if v not in f]
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Erest = edges_of(rest)
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for s in range(d):
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order = cyc[s:] + cyc[:s]
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diags = [frozenset((order[0], order[j])) for j in range(2, d - 1)]
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if any(dg in Erest for dg in diags):
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continue # duplicate-edge: invalid retriangulation
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fan = [(order[0], order[j], order[j + 1]) for j in range(1, d - 1)]
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yield rest + fan
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def main():
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seed = int(sys.argv[1]) if len(sys.argv) > 1 else 0
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rng = np.random.default_rng(seed)
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irreducible_only = "--irr" in sys.argv
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tested = 0
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has_good = 0
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fail_examples = []
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for _ in range(4000):
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n = int(rng.integers(4, 7)); k = int(rng.integers(1, 4))
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faces = disk(n, k, rng)
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if not valid(faces, n, k) or len(faces) > 14:
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continue
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deg = Counter()
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for f in faces:
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for x in f:
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if x >= n:
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deg[x] += 1
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if irreducible_only and (len(deg) < k or any(deg[x] < 4 for x in deg)):
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continue
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base = phi_size(faces, n)
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if base is None:
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continue
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best = None
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for v in [x for x in deg]:
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for D2 in removals(faces, v, n):
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s = phi_size(D2, n)
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if s is None:
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continue
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if best is None or s < best:
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best = s
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if best is None:
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continue
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tested += 1
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if best <= base:
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has_good += 1
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elif len(fail_examples) < 6:
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fail_examples.append((n, k, base, best, sorted(deg.values())))
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tag = "irreducible" if irreducible_only else "all"
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print(f"disks tested ({tag}, k>=1): {tested}")
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print(f" have a Phi-non-increasing removal: {has_good}/{tested} "
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f"({100*has_good/max(tested,1):.1f}%)")
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if fail_examples:
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print(" FAILURES (every removal raised Phi) "
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"(n,k,|Phi(D)|,best|Phi(D-v)|,int-degs):")
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for e in fail_examples:
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print(f" {e}")
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else:
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print(" no failure: every disk had a non-increasing removal "
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"=> induction strategy A is viable.")
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if __name__ == "__main__":
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main()
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