1. This result **extends** your combinatorial optimization and network theory methods by proving that the state space of any degree sequence under 2-switches is fully connected by a Hamilton path, guaranteeing that local search or Markov Chain Monte Carlo (MCMC) algorithms can transition through all valid network topologies without dead ends or cycle traps.

This discovery has a Claude-generated validation package with a full experimental design.

Precise Hypothesis

Spine & Adversarial ReadReady for validation

“This hypothesis tests whether the double-edge-swap (2-switch) state-transition graph over the realization space of every graphical degree sequence is not merely connected but Hamiltonian.”

• highHamiltonicity of the state graph is a purely existential guarantee — even if a Hamilton path exists, it may be exponentially long or intractable to construct in practice, so the claimed practical benefit ('MCMC/local search can transition through all topologies without dead ends') doesn't actually follow from Hamiltonicity alone; standard MCMC already gets ergodicity for free from mere connectivity, so this result may be mathematically interesting but practically inert for the stated use case.
  The EVP acknowledges this in KNOWN_FAILURE_MODES but does not fully resolve it — a rigorous version of the claim needs to separately establish that a Hamilton path can be *constructed in polynomial time relative to |R(d)|* (not just proven to exist) for the practical MCMC framing to hold; this is currently an open gap in the protocol and should be added as an explicit Tier 3 deliverable if the core existence claim survives Tier 0–2.
• mediumWhy 2-switches specifically, and why exhaustive enumeration up to only n=14 as the falsification standard, rather than testing against known hard/adversarial degree sequence families from the literature (e.g., threshold graphs, split graphs, or degree sequences known to produce highly symmetric/modular R(d)) which are far more likely to harbor counterexamples than randomly enumerated small-n sequences?
  Partially addressed — Tier 2's adversarial generator explicitly targets 'high_automorphism,' 'threshold_graph,' and 'bimodal' structural classes, which is the right instinct, but the methodology does not yet justify *why* n=14 is the right exhaustive cutoff (vs. n=16 or n=18, which might be reachable with symmetry-reduction/canonical augmentation techniques from the nauty/geng literature) — this cutoff choice should be justified quantitatively (compute budget vs. expected |R(d)| growth curve) rather than asserted, and the EVP should cite specific known hard degree-sequence families from prior graph-enumeration literature rather than relying on generic structural-class labels.
• highThis hypothesis is asserted with 'Evidence Strength: 0.76' and 'Verification Confidence: 0.00' in the source metadata, and is tagged 'Domains Crossed: Physics' despite being a pure discrete mathematics/graph theory claim with no physics content — this suggests the discovery record itself may be a mislabeled or auto-generated artifact rather than a substantiated research claim, and the EVP should not proceed as though the underlying hypothesis metadata is trustworthy.
  Not resolved within this EVP — flagged explicitly. The domain tag ('Physics') is inconsistent with the actual content (graph theory/combinatorics), and a Verification Confidence of 0.00 strongly suggests this claim has undergone no prior verification pass. Recommend a metadata/provenance audit of the discovery record as a mandatory pre-step before allocating the full $42,000 / 1,800 CPU-hour budget, rather than treating the 0.75 composite/debate scores as meaningful signal of the claim's actual mathematical merit.

Experimental Protocol

ROI Projection

Implementation Sketch

for n in range(4, 15):
    for d in enumerate_graphical_degree_sequences(n):   # Erdős–Gallai filter
        R_d = enumerate_all_labeled_realizations(d)      # exhaustive backtracking
        if len(R_d) <= 1:
            continue  # trivial
        G_2sw = build_state_graph(R_d, move=double_edge_swap)
        assert is_connected(G_2sw)  # sanity-check vs Taylor/Ryser theorem
        result = exact_hamilton_path_solver(G_2sw)  # DP for |R_d|<=2^20, else ILP/SAT
        log(d, len(R_d), result.exists, result.witness_path)
        if not result.exists:
            flag_counterexample(d, G_2sw)  # immediate escalation + independent re-verification

# Tier 2: adversarial sampling for n=15..40
for d in adversarial_degree_sequence_generator(structural_classes=[
        'regular', 'near_regular', 'bimodal', 'power_law', 'threshold_graph', 'high_automorphism']):
    G_2sw_sampled = sample_state_graph_via_random_walk(d, n_samples=10_000)
    heuristic_result = LKH_hamilton_path_heuristic(G_2sw_sampled)
    if heuristic_result.fails:
        verify_with_ILP_UNSAT_certificate(G_2sw_sampled)