coloring_nested_tire_graphs: empirical findings on the König-lift conjecture (negative)
Tested the candidate induced γ-partition from
worst_case_proof_sketch.tex (Conj t2-induces-partition).
Findings:
1. AT k = k_2 = 6 (antipodal chord, faces 3+3): Candidate
partition (next-D or prev-D) gives Latin ⊆ π_U. ✓
But this is partly coincidental: |π_U| = 90 is so large that
ALL 10 triple-partitions of {0,..,5} have Latin ⊆ π_U.
2. AT k = k_2 = 9 (chords (0,3)(3,6), faces 3+3+3): Candidate
partition FAILS. Only 8 of all 280 triple-partitions of
{0,..,8} have Latin ⊆ π_U, and the candidate is not one of
them. The 8 surviving partitions have no obvious geometric
interpretation.
3. ASYMMETRIC k ≠ k_2 (e.g. k=6, k_2=3): Candidate doesn't
produce a triple-partition at all, and no triple-partition
has Latin ⊆ π_U. Conjecture as stated doesn't cover the
case where the empirical worst-case overlap lives.
Implication: The candidate construction is wrong past k = 6.
Step 3 (prove inclusion) is not the right next move -- we'd
be proving a false statement.
Reassessment of Approach 2: the König-overlap proposition (when
both tires give direct γ-face partitions) is still cleanly proven,
but applies to fewer cases than hoped. The asymmetric pairs that
witness the empirical worst case are not covered.
Both approaches now have known structural obstacles:
- Approach 1 (2-SAT): single open Conjecture 1.5, empirically true.
- Approach 2 (König): natural construction empirically wrong past
k=6, plus asymmetric pairs out of scope.
Honest status: chain pigeonhole has no full proof yet.
Files:
experiments/induced_partition.py
notes/induced_partition_findings.tex (3 pages)
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
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"""Verify the candidate induced γ-partition conjecture for T_2 (the inner
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tire whose chord is on B_in^(2), not on γ).
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Setup:
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T_2 is an SP tire with B_out^(2) = γ (length k) and B_in^(2) = C_{k_2}
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(length k_2 divisible by 3) plus chord(s) creating O^(2)-faces of
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size 3 each. Balanced annular triangulation interleaves D-triangles
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(one per B_in^(2)-edge) and U-triangles (one per γ-edge) on the dual
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cycle T'_ann of length k + k_2.
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The conjecture is that π_U(C(T_2)) ⊇ L(γ, ~F_2), the Latin subset for
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some induced γ-face partition ~F_2. Candidate construction (from
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worst_case_proof_sketch.tex): for each O^(2)-face F^(2) (a triple of
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B_in^(2)-edges = a triple of D-triangles), the associated γ-edges are
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the U-triangles that lie in the cyclic span of F^(2)'s D-positions on
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T'_ann, with boundary U-triangles assigned by some rule (e.g. "next
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F^(2) cyclically").
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This script:
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- Constructs the candidate induced partition.
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- Computes π_U(C(T_2)) by brute enumeration.
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- Checks whether L(γ, induced partition) ⊆ π_U.
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- Reports findings.
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"""
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from itertools import permutations, product
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from collections import defaultdict
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from tire_fiber_chords import (
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fiber_distribution,
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u_positions_for,
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projection_support,
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)
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from tire_fiber_chunked import projection_support_streaming
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def induced_gamma_partition_candidate_1(k, k_2, chords, d_positions, u_positions):
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"""Candidate 1: assign each U-triangle to the O^(2)-face whose
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D-triangle interval on T'_ann contains it. Boundary U-triangles
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are assigned to the cyclically-next F^(2)."""
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n = k + k_2
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# First find face structure of O^(2)
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from tire_fiber_chords import compute_faces_from_chords
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o_faces = compute_faces_from_chords(k_2, chords) # list of lists of B_in^(2)-edge indices
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# For each B_in^(2)-edge a, find which face it's in
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edge_to_face = {}
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for face_idx, edges in enumerate(o_faces):
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for a in edges:
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edge_to_face[a] = face_idx
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# d_positions[i] = T'_ann position of B_in^(2)-edge i.
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# Walk T'_ann cyclically. At each U-position, find the next D-position
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# cyclically, and assign U to that D's face.
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d_pos_set = set(d_positions)
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d_pos_to_edge = {p: i for i, p in enumerate(d_positions)}
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u_to_face = {}
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for u_pos in u_positions:
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# Find the next D-position cyclically after u_pos
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for offset in range(1, n):
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check_pos = (u_pos + offset) % n
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if check_pos in d_pos_set:
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next_b_in_edge = d_pos_to_edge[check_pos]
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u_to_face[u_pos] = edge_to_face[next_b_in_edge]
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break
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# Each U-position corresponds to a γ-edge. γ-edges are indexed by
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# u_positions in cyclic order.
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# u_positions[i] ↔ γ-edge i in the cyclic order on T'_ann
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u_pos_to_gamma_edge = {p: i for i, p in enumerate(u_positions)}
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# Build the partition: face_idx → list of γ-edge indices
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partition = defaultdict(list)
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for u_pos, face_idx in u_to_face.items():
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partition[face_idx].append(u_pos_to_gamma_edge[u_pos])
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return [sorted(v) for v in partition.values()]
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def induced_gamma_partition_candidate_2(k, k_2, chords, d_positions, u_positions):
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"""Candidate 2: assign each U-triangle to the O^(2)-face whose
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D-triangle interval on T'_ann contains it (using PREVIOUS D for
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boundary assignment instead of next)."""
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n = k + k_2
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from tire_fiber_chords import compute_faces_from_chords
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o_faces = compute_faces_from_chords(k_2, chords)
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edge_to_face = {}
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for face_idx, edges in enumerate(o_faces):
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for a in edges:
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edge_to_face[a] = face_idx
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d_pos_set = set(d_positions)
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d_pos_to_edge = {p: i for i, p in enumerate(d_positions)}
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u_to_face = {}
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for u_pos in u_positions:
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for offset in range(1, n):
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check_pos = (u_pos - offset) % n
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if check_pos in d_pos_set:
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prev_b_in_edge = d_pos_to_edge[check_pos]
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u_to_face[u_pos] = edge_to_face[prev_b_in_edge]
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break
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u_pos_to_gamma_edge = {p: i for i, p in enumerate(u_positions)}
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partition = defaultdict(list)
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for u_pos, face_idx in u_to_face.items():
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partition[face_idx].append(u_pos_to_gamma_edge[u_pos])
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return [sorted(v) for v in partition.values()]
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def latin_set(partition, k):
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"""Set of σ ∈ {1,2,3}^k where σ restricted to each partition block
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is a permutation of {1,2,3} (= each block has exactly 3 elements
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using all 3 colors)."""
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L = set()
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if any(len(b) != 3 for b in partition):
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return None # not all triples
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blocks = [sorted(b) for b in partition]
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for assignment in product(permutations((1, 2, 3)), repeat=len(blocks)):
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sigma = [0] * k
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for block, perm in zip(blocks, assignment):
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for pos, color in zip(block, perm):
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sigma[pos] = color
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L.add(tuple(sigma))
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return L
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def all_partitions_into_triples(elements):
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"""Yield all ways to partition a set of 3n elements into triples."""
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elements = list(elements)
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if not elements:
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yield []
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return
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if len(elements) % 3 != 0:
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return
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if len(elements) == 3:
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yield [tuple(elements)]
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return
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first = elements[0]
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rest = elements[1:]
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from itertools import combinations
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for combo in combinations(rest, 2):
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triple = (first,) + combo
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remaining = [x for x in rest if x not in combo]
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for sub_partition in all_partitions_into_triples(remaining):
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yield [triple] + sub_partition
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def check_all_triple_partitions(pi_U, k):
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"""Return all triple-partitions of {0,...,k-1} whose Latin set ⊆ π_U."""
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if k % 3 != 0:
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return []
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good = []
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for partition in all_partitions_into_triples(range(k)):
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L = latin_set([list(b) for b in partition], k)
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if L is None:
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continue
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if L <= pi_U:
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good.append([sorted(b) for b in partition])
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return good
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def main():
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cases = [
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(6, 6, [(0, 3)], "k=k_2=6, antipodal chord (faces 3+3)"),
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(6, 6, [(0, 2), (3, 5)], "k=k_2=6, two chords (faces 2+2+2)"),
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(6, 3, [], "k=6, k_2=3 (T_2 inner = C_3, no chord, SP)"),
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(9, 9, [(0, 3), (3, 6)], "k=k_2=9, two chords (faces 3+3+3)"),
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]
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for k, k_2, chords, desc in cases:
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print(f'\n=== {desc} ===')
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print(f' k = γ length = {k}, k_2 = {k_2}, chords = {chords}')
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# Get T_2's projections. T_2's outer boundary = γ, so we use π_U.
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from tire_fiber_chords import d_positions_for
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d_pos = d_positions_for(k, k_2)
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u_pos = u_positions_for(k, k_2)
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try:
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if k + k_2 <= 14:
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fibers, _, _ = fiber_distribution(k, k_2, chords)
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pi_U = projection_support(fibers, u_pos)
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else:
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# chunked streaming for larger cases
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pi_U = projection_support_streaming(k, k_2, chords, u_pos)
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except Exception as e:
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print(f' computation failed: {e}')
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continue
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print(f' |π_U| = {len(pi_U)}, 3^k = {3**k}')
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# Try candidate induced partitions
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for cand_name, cand_fn in [
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('Candidate 1 (next-D)', induced_gamma_partition_candidate_1),
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('Candidate 2 (prev-D)', induced_gamma_partition_candidate_2),
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]:
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partition = cand_fn(k, k_2, chords, d_pos, u_pos)
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partition_str = ", ".join(str(sorted(b)) for b in partition)
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print(f' {cand_name}: partition = {partition_str}')
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L = latin_set(partition, k)
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if L is None:
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print(f' partition not all-3-triples; skipping Latin test')
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continue
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print(f' |L| = {len(L)}, L ⊆ π_U? {L <= pi_U}')
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# Enumerate all triple-partitions and find which give Latin ⊆ π_U
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if k % 3 == 0:
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good_partitions = check_all_triple_partitions(pi_U, k)
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print(f' # triple-partitions with Latin ⊆ π_U: {len(good_partitions)}')
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for p in good_partitions[:5]:
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print(f' Latin ⊆ π_U: {p}')
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if __name__ == '__main__':
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main()
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\title{Empirical findings on the K\"onig-lift conjecture\\
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(Conj.\ \emph{t2-induces-partition} from
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\texttt{worst\_case\_proof\_sketch.tex})}
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\author{}
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\maketitle
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\section*{What was tested}
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The K\"onig-lift approach to chain pigeonhole
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(\texttt{worst\_case\_proof\_sketch.tex}) conjectures that for an SP
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tire $T_2$ with $B_{\mathrm{in}}^{(2)}$-chord structure such that
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every $O^{(2)}$-face has exactly $3$ $B_{\mathrm{in}}^{(2)}$-edges,
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there is an \emph{induced} face partition $\widetilde{\mathcal{F}_2}$
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of $\gamma$ into triples with
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\[
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\pi_U(\mathcal{C}(T_2)) \supseteq
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\mathcal{L}(\gamma, \widetilde{\mathcal{F}_2}).
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\]
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attempt'') groups $\gamma$-edges by the $O^{(2)}$-face whose
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Script: \texttt{experiments/induced\_partition.py}.
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\section*{Findings}
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\begin{center}
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\small
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\begin{tabular}{lll|cc|cc}
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\toprule
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$k = |\gamma|$ & $k_2 = |B_{\mathrm{in}}^{(2)}|$ & chords on $B_{\mathrm{in}}^{(2)}$
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& $|\pi_U|$ & $3^k$
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& candidate 1 OK? & candidate 2 OK? \\
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\midrule
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$6$ & $6$ & $(0,3)$ (faces $3{+}3$)
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& $90$ & $729$ & \textbf{Yes} & \textbf{Yes}\\
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$6$ & $6$ & $(0,2)(3,5)$ (faces $2{+}2{+}2$)
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& $456$ & $729$ & N/A (not triples) & N/A\\
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$6$ & $3$ & none (face $3$, single)
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& $84$ & $729$ & N/A (not triples) & N/A\\
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$9$ & $9$ & $(0,3)(3,6)$ (faces $3{+}3{+}3$)
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& $978$ & $19683$ & \textbf{No} & \textbf{No}\\
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\bottomrule
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\end{tabular}
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\end{center}
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\subsection*{(a) The candidate works at $k = 6$}
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the same kind of $\gamma$-partition (a two-block partition of size
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$3$ each), and the Latin subset $\mathcal{L}(\gamma, \widetilde{
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\mathcal{F}_2})$ of size $36$ is verified $\subseteq \pi_U$.
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$\{0, \ldots, 5\}$ (there are $10$ such) has Latin subset
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$\subseteq \pi_U$ here, because $|\pi_U| = 90$ is much larger than
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$36$ and absorbs all of them.
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\subsection*{(b) The candidate \emph{fails} at $k = 9$}
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\begin{itemize}
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|
\item Candidate $1$ (next-$D$):
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|
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|
$|\mathcal{L}| = 216$, but $\mathcal{L} \not\subseteq \pi_U$
|
||||||
|
(some Latin elements are missing).
|
||||||
|
\item Candidate $2$ (prev-$D$):
|
||||||
|
$\widetilde{\mathcal{F}_2} = \{\{0,1,2\},\;\{3,4,5\},\;\{6,7,8\}\}$.
|
||||||
|
$|\mathcal{L}| = 216$, but $\mathcal{L} \not\subseteq \pi_U$.
|
||||||
|
\item Of all $280$ possible triple-partitions of $\{0, \ldots, 8\}$,
|
||||||
|
only $8$ have $\mathcal{L} \subseteq \pi_U$.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
The eight surviving partitions are not contiguous blocks. Examples:
|
||||||
|
$\{\{0,2,3\},\{1,6,8\},\{4,5,7\}\}$,
|
||||||
|
$\{\{0,2,4\},\{1,6,8\},\{3,5,7\}\}$, etc. They do not have an
|
||||||
|
obvious geometric interpretation in terms of $T_2$'s annular
|
||||||
|
triangulation.
|
||||||
|
|
||||||
|
\subsection*{(c) The asymmetric case ($k \ne k_2$) is outside scope}
|
||||||
|
|
||||||
|
For $k = 6, k_2 = 3$ (the configuration with $T_2 = (3, -, \mathrm{SP})$),
|
||||||
|
the candidate construction collapses to a single block of $6$
|
||||||
|
$\gamma$-edges (since there is only one $O^{(2)}$-face), so it is not a
|
||||||
|
triple-partition. Moreover, \emph{no} triple-partition of
|
||||||
|
$\{0, \ldots, 5\}$ has Latin subset $\subseteq \pi_U$ here.
|
||||||
|
|
||||||
|
So Conj.\ \emph{t2-induces-partition} as currently stated does not
|
||||||
|
cover $k \ne k_2$, and the empirical data shows there is no
|
||||||
|
``rescue'' partition of any kind.
|
||||||
|
|
||||||
|
\section*{Implications}
|
||||||
|
|
||||||
|
\subsection*{The K\"onig lift's natural construction breaks past $k = 6$}
|
||||||
|
|
||||||
|
The candidate $\widetilde{\mathcal{F}_2}$ from the worst-case note is
|
||||||
|
the geometrically natural one (group $\gamma$-edges by their nearest
|
||||||
|
$O^{(2)}$-face $D$-triangle), and it succeeds at $k = 6$ partly by
|
||||||
|
coincidence: $|\pi_U|$ is so large that every triple-partition fits.
|
||||||
|
At $k = 9$ the gap between $|\pi_U|$ and $3^k$ widens, and the
|
||||||
|
candidate's specific partition is no longer in the small set of
|
||||||
|
``correct'' partitions.
|
||||||
|
|
||||||
|
The fact that only $8 / 280$ partitions work at $k = 9$ suggests
|
||||||
|
that whatever the right $\widetilde{\mathcal{F}_2}$ is, it is
|
||||||
|
\emph{not} just a function of $T_2$'s outerplanar face structure ---
|
||||||
|
it must encode finer information about the annular triangulation.
|
||||||
|
|
||||||
|
\subsection*{Asymmetric pairs not covered at all}
|
||||||
|
|
||||||
|
The empirical worst-case overlap $|S_1 \cap S_2| = 6$ in step-$2$
|
||||||
|
data comes from \emph{asymmetric} pairs (e.g.\
|
||||||
|
$T_1 = (6, (0,3), \mathrm{SP})$ vs $T_2 = (3, -, \mathrm{SR})$)
|
||||||
|
where $k \ne k_2$. Even if the K\"onig lift were proved for the
|
||||||
|
symmetric case, it would not handle the asymmetric pairs that
|
||||||
|
witness the worst case.
|
||||||
|
|
||||||
|
\subsection*{Step 3 (proof) is not the right next move}
|
||||||
|
|
||||||
|
Plan-step 3 from \texttt{two\_approaches\_comparison.tex} was
|
||||||
|
``prove inclusion via transfer matrix / fibre lifting,'' assuming
|
||||||
|
the candidate partition was empirically correct. The candidate is
|
||||||
|
\emph{not} empirically correct beyond $k = 6$, so trying to prove
|
||||||
|
the wrong statement is futile. Instead the right next move is:
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Find the right induced $\widetilde{\mathcal{F}_2}$ at
|
||||||
|
$k = 9$: study the $8$ surviving triple-partitions, see if
|
||||||
|
they have a common structural description (e.g.\ via the
|
||||||
|
$T_2$ annular triangulation).
|
||||||
|
\item Or abandon the ``$\widetilde{\mathcal{F}_2}$ is a partition''
|
||||||
|
framing entirely and look for a different structure on
|
||||||
|
$\gamma$ that $T_2$ induces and that suffices for chain
|
||||||
|
pigeonhole.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
\section*{Reassessment of Approach 2}
|
||||||
|
|
||||||
|
Approach 2 (K\"onig lift) was preferred in
|
||||||
|
\texttt{two\_approaches\_comparison.tex} on the grounds that ``the
|
||||||
|
hard step is already proven, only the induced-partition piece is
|
||||||
|
conjectural.'' These findings show:
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item The induced-partition piece is \emph{not} just conjectural ---
|
||||||
|
the specific construction in the worst-case note is
|
||||||
|
\emph{wrong} for $k > 6$.
|
||||||
|
\item The K\"onig-overlap proposition (when both tires give direct
|
||||||
|
$\gamma$-face partitions) is still cleanly proved; it just
|
||||||
|
applies to fewer cases than was hoped.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsection*{Updated ranking}
|
||||||
|
|
||||||
|
Both approaches now have known structural obstacles:
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item \textbf{Approach 1 (2-SAT, \texttt{rainbow\_proof.tex})}:
|
||||||
|
single open conjecture (2-SAT solvability), empirically true
|
||||||
|
for all tested $\sigma \in \mathcal{P}_m$ at $m \in \{4, 6\}$.
|
||||||
|
Limited to $m \in \{4, 6\}$ (SP feasibility) but at least
|
||||||
|
empirically holds throughout that range.
|
||||||
|
\item \textbf{Approach 2 (K\"onig lift,
|
||||||
|
\texttt{worst\_case\_proof\_sketch.tex})}: K\"onig-overlap
|
||||||
|
prop proved, but the natural induced-partition construction
|
||||||
|
is empirically wrong at $k = 9$. Asymmetric pairs (where
|
||||||
|
the worst case actually lives) are not covered at all.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
Both approaches give partial structural results. Neither closes
|
||||||
|
the chain-pigeonhole step in its full generality. The honest
|
||||||
|
status: chain pigeonhole has no full proof yet, and both attempted
|
||||||
|
attacks have specific empirical limits.
|
||||||
|
|
||||||
|
\end{document}
|
||||||
Reference in New Issue
Block a user