Isolate the irreducible case: reduces the floor proof to one lemma

irreducible_floor.py: over 10k+ irreducible disks (k>=1, min interior
degree >=4), |Phi| never violates 2^(n-2) and never sits on it -- min is
5*2^(n-4) = (5/4)2^(n-2), the wheel being the minimizer. Universal toggles
are dead (99.9% have zero boundary-only faces). Since un-stacking degree-3
vertices preserves Phi and terminates at a k=0 or irreducible residue, the
whole lower bound reduces to: every irreducible disk has |Phi| >= 2^(n-2).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/heawood_restrictions_on_nested_tire_graph_duals/experiments/irreducible_floor.py

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+"""
+The open case for the 2^(n-2) lower bound is the IRREDUCIBLE disk: k>=1 interior
+vertices, all of degree >=4 (un-stacking already settles everything reducible to
+a degree-3 vertex). This probe isolates irreducible disks and asks:
+
+  (a) does the floor |Phi| >= 2^(n-2) hold there (it must, but it's the open case);
+  (b) is it STRICT (|Phi| > 2^(n-2)) -- if irreducible disks never sit ON the floor,
+      the proof only needs ">= floor" via "any nontrivial slack";
+  (c) how many UNIVERSAL toggles each has -- a "boundary-only" face (all 3 vertices
+      on C) can be flipped feasibly regardless of the labelling, giving a guaranteed
+      doubling. n-2 independent ones would prove the bound outright. We count them
+      to see whether universal toggles alone can carry the irreducible case (a wheel
+      has zero, so we expect NOT).
+"""
+
+import sys
+from collections import Counter
+from itertools import product
+
+import numpy as np
+from scipy.spatial import Delaunay
+
+np.seterr(all="ignore")
+
+
+def disk(n, k, rng):
+    ang = 2 * np.pi * np.arange(n) / n
+    rad = 1.0 + 0.18 * rng.random(n)
+    bpts = np.c_[rad * np.cos(ang), rad * np.sin(ang)]
+    r = 0.78 * np.sqrt(rng.random(k)); t = 2 * np.pi * rng.random(k)
+    pts = np.vstack([bpts, np.c_[r * np.cos(t), r * np.sin(t)]])
+    return [tuple(int(x) for x in s) for s in Delaunay(pts).simplices]
+
+
+def valid(faces, n, k):
+    if len(faces) != 2 * k + n - 2:
+        return False
+    ec = Counter()
+    for a, b, c in faces:
+        for e in ((a, b), (b, c), (a, c)):
+            ec[frozenset(e)] += 1
+    return all(ec[frozenset((i, (i + 1) % n))] == 1 for i in range(n))
+
+
+def interior_degrees(faces, n):
+    deg = Counter()
+    for f in faces:
+        for v in f:
+            if v >= n:
+                deg[v] += 1
+    return deg
+
+
+def phi(faces, n):
+    interior = sorted(v for f in faces for v in f if v >= n)
+    interior = sorted(set(interior))
+    F = len(faces)
+    if F > 18:
+        return None
+    Bint = np.zeros((len(interior), F), dtype=np.int64)
+    Cinc = np.zeros((n, F), dtype=np.int64)
+    iidx = {w: r for r, w in enumerate(interior)}
+    for j, (a, b, c) in enumerate(faces):
+        for v in (a, b, c):
+            if v >= n:
+                Bint[iidx[v], j] = 1
+            else:
+                Cinc[v, j] = 1
+    labs = np.array(list(product((1, 2), repeat=F)), dtype=np.int64)
+    if interior:
+        labs = labs[np.all((labs @ Bint.T) % 3 == 0, axis=1)]
+    if labs.shape[0] == 0:
+        return set()
+    return set(map(tuple, np.unique((labs @ Cinc.T) % 3, axis=0)))
+
+
+def boundary_only_faces(faces, n):
+    return sum(1 for (a, b, c) in faces if a < n and b < n and c < n)
+
+
+def main():
+    seed = int(sys.argv[1]) if len(sys.argv) > 1 else 0
+    rng = np.random.default_rng(seed)
+    by_n = {}
+    n_irred = 0
+    floor_fail = 0
+    on_floor = 0           # irreducible AND exactly at 2^(n-2)
+    bof_ge = 0             # irreducible with >= n-2 boundary-only faces
+    bof_zero = 0
+    min_bof = 99
+
+    for _ in range(6000):
+        n = int(rng.integers(4, 7)); k = int(rng.integers(1, 4))
+        faces = disk(n, k, rng)
+        if not valid(faces, n, k) or len(faces) > 16:
+            continue
+        deg = interior_degrees(faces, n)
+        if any(deg[v] < 4 for v in deg) or len(deg) < k:
+            continue                            # reducible (has a degree-3 vertex)
+        P = phi(faces, n)
+        if P is None or not P:
+            continue
+        n_irred += 1
+        floor = 2 ** (n - 2)
+        if len(P) < floor:
+            floor_fail += 1
+        if len(P) == floor:
+            on_floor += 1
+        bof = boundary_only_faces(faces, n)
+        min_bof = min(min_bof, bof)
+        bof_ge += (bof >= n - 2)
+        bof_zero += (bof == 0)
+        d = by_n.setdefault(n, {"cnt": 0, "min": 10**9, "floor": floor})
+        d["cnt"] += 1
+        d["min"] = min(d["min"], len(P))
+
+    print(f"irreducible disks (k>=1, all interior degree >=4): {n_irred}\n")
+    for n in sorted(by_n):
+        d = by_n[n]
+        print(f"  n={n}: {d['cnt']:5d} disks   min|Phi|={d['min']}   "
+              f"2^(n-2)={d['floor']}   floor-held={d['min']>=d['floor']}")
+    print(f"\n  floor violations (|Phi| < 2^(n-2)) : {floor_fail}")
+    print(f"  irreducible disks sitting ON the floor: {on_floor}  "
+          f"(if 0, irreducible => strictly above floor)")
+    print(f"  universal toggles (boundary-only faces):")
+    print(f"     >= n-2 such faces : {bof_ge}/{n_irred}")
+    print(f"     exactly 0         : {bof_zero}/{n_irred}   (min over all = {min_bof})")
+    print("  => universal toggles alone "
+          + ("CAN" if bof_zero == 0 else "CANNOT")
+          + " carry the irreducible case.")
+
+
+if __name__ == "__main__":
+    main()