Two-cluster moat kills and the thin-twin averaging row #
The moat family of two-cluster cut certificates: a low-degree tie-block S₁
whose closed neighbourhood forms a zeroed moat F, so that
algConn_le_two_of_two_clusters fires algConn G ≤ 2 with no diameter, census
or cell hypotheses. Each vertex of S₁ keeps an external degree budget of 2,
so the boundary hits the two-cluster tie ∂₁ ≤ 2|S₁| and the excess ledger
Σ_v (deg v − 3) = n − 8 (total_excess_eq) caps the bulk boundary.
Main results #
medge_moat_fires— anM-edge (a degree-3–degree-3 edge) fires atn ≥ 12.star_moat_fires(andz1_star_moat_fires) — a hubhhoardingdeg h − 2degree-3 twins fires at9·deg h ≤ n + 15.master_cycle_fires— a cycle with degree-sum tieΣ deg ≤ 4kfires atn ≥ 3·Σ(deg − 1) − 8.deco_edge_moat_fires(andz4c_fires) — an adjacent hub pair hoardingdeg u − 3,deg v − 3twins fires at9·(deg u + deg v) ≤ n + 42.- The thin-twin averaging row: the double-counting swap
∑_{t∈T} E₁(t) = ∑_v (deg v − 3)·|N(v) ∩ T|(twin_E1_sum_swap) with the excess ledger and a multiplicity capKyields∃ t ∈ T, |T|·E₁(t) ≤ K·(n − 8)(thin_twin_exists_of_multcap), with the unconditional instancesthin_twin_exists_deg5andthin_twin_exists_iso_of_multcap.
The thin-twin averaging row #
The averaging engine producing a thin twin (bounded 1-ball excess
E₁(t) = sphereExc G t 1). The double-counting swap twin_E1_sum_swap holds for
any root set T; combined with total_excess_eq and a multiplicity cap
|N(v) ∩ T| ≤ K it gives ∑_{t∈T} E₁(t) ≤ K·(n − 8) and the averaging existence
thin_twin_exists_of_multcap. The cap K stays explicit (a heavy vertex's
multiplicity is unbounded); the clean instances are thin_twin_exists_deg5
(K = 5 when Δ ≤ 5) and thin_twin_exists_iso_of_multcap (on isoTwins G).
The excess ledger. On the m = 2(n−2), δ ≥ 3 census (n ≥ 8) the total degree
excess is Σ_v (deg v − 3) = n − 8: the handshake Σ deg = 2·2(n−2) = 4n − 8 minus the
base 3n.
The M-moat kill #
Instantiates algConn_le_two_of_two_clusters with the tie-block S₁ = {u, p}
(an M-edge) against the bulk, moat F = (N(u) ∪ N(p)) ∖ {u,p}: ∂₁ = 4,
|F| ≤ 4, so medge_moat_fires gives algConn G ≤ 2 for every n ≥ 12.
QM1 — the M-moat certificate. A graph on Fin n (n ≥ 12) with 2(n−2) edges,
minimum degree ≥ 3, and one degree-3–degree-3 edge u–p has algConn G ≤ 2.
Instantiate the two-cluster law with the tie-block S₁ = {u, p} against the bulk
S₂ = ({u, p} ∪ F)ᶜ, where F = (N(u) ∪ N(p)) ∖ {u, p} is the moat. Boundary counts:
∂₁ ≤ 4, and (using that each moat vertex is adjacent to u or p, and the excess ledger
Σ_v (deg v − 3) = n − 8) ∂₂ ≤ 2·|S₂|; the Fiedler cut condition then holds since |F| ≤ 4
and n ≥ 12.
The star-moat kill #
The e(M) = 0 generalization: the star tie-block S₁ = insert h K (a hub h
with |K| = deg h − 2 degree-3 twins), each S₁-vertex keeping external budget
2, so crediting the hub's excess back gives star_moat_fires at
9·deg h ≤ n + 15 (z1_star_moat_fires: a degree-4 hub with two twins at
n ≥ 21). The twins need not be pairwise non-adjacent — an internal edge only
shrinks the boundary slices.
MZ1 — the star-moat certificate. A graph on Fin n with 2(n−2) edges, minimum
degree ≥ 3, a hub h and a twin set K ⊆ N(h) of degree-3 vertices with |K| = deg h − 2
has algConn G ≤ 2 whenever 9·deg h ≤ n + 15.
Instantiate the two-cluster law with the tie-block S₁ = insert h K against the bulk
S₂ = (S₁ ∪ F)ᶜ, F = (⋃_{x ∈ S₁} N(x)) ∖ S₁ the moat: each slice N(x) ∖ S₁ has ≤ 2
elements (hub loses K, twins lose the hub), so ∂₁ ≤ 2|S₁| and |F| ≤ 2|S₁|, and the
hub-credited excess ledger gives ∂₂ ≤ 2|S₂|; the Fiedler cut condition closes by ring.
The master-cycle kill #
A cycle c : ZMod k → Fin n (injective, cyclic adjacency) with degree-sum tie
Σ deg ≤ 4·k is a tie-block: each cycle vertex has two on-cycle neighbours so
external slice ≤ deg − 2, giving ∂₁ ≤ Σ(deg − 2) ≤ 2k; crediting the cycle
excess back, master_cycle_fires fires at n ≥ 3·Σ(deg − 1) − 8 (specializing
to C_k, the (3,4,5)-triangle at n ≥ 19, and alternating rows).
W1 — the master-cycle certificate. A graph on Fin n with 2(n−2) edges, minimum
degree ≥ 3, and an injective c : ZMod k → Fin n (k ≥ 3) forming a cycle
(c i ~ c (i+1) cyclically) whose degree sum satisfies the tie Σ deg (c i) ≤ 4·k has
algConn G ≤ 2 whenever 3·Σ (deg (c i) − 1) ≤ n + 8.
Instantiate the two-cluster law with the tie-block S₁ = image c (the cycle) against the bulk
S₂ = (S₁ ∪ F)ᶜ, F = (⋃_{x ∈ S₁} N(x)) ∖ S₁ the moat: each slice N(c i) ∖ S₁ has
≤ deg (c i) − 2 elements (the two cycle neighbours stay inside), so ∂₁ ≤ Σ(deg − 2) ≤ 2·k
and |F| ≤ Σ(deg − 2), and the full excess ledger gives ∂₂ ≤ 2·|S₂|; the Fiedler cut
condition closes by ring.
The decorated-edge moat kill #
The decorated-edge generalization of the star-moat certificate: the tie-block
S₁ = {u, v} ∪ Ku ∪ Kv (an adjacent hub pair with |Ku| = deg u − 3,
|Kv| = deg v − 3 twins), each S₁-vertex keeping external budget 2;
crediting both hubs' excess back, deco_edge_moat_fires fires at
9·(deg u + deg v) ≤ n + 42 (z4c_fires: adjacent degree-4 hubs with one twin
each at n ≥ 30).
W3 — the decorated-edge moat certificate. A graph on Fin n with 2(n−2) edges,
minimum degree ≥ 3, adjacent hubs u, v, and twin sets Ku ⊆ N(u), Kv ⊆ N(v) of degree-3
vertices with |Ku| = deg u − 3, |Kv| = deg v − 3 (disjoint, and avoiding the opposite hub)
has algConn G ≤ 2 whenever 9·(deg u + deg v) ≤ n + 42.
Instantiate the two-cluster law with the tie-block S₁ = {u, v} ∪ Ku ∪ Kv against the bulk
S₂ = (S₁ ∪ F)ᶜ, F = (⋃_{x ∈ S₁} N(x)) ∖ S₁ the moat: each slice N(x) ∖ S₁ has ≤ 2
elements (each hub loses the other hub and its twins; each twin loses its hub), so ∂₁ ≤ 2|S₁|
and |F| ≤ 2|S₁|, and the pair-credited excess ledger gives ∂₂ ≤ 2|S₂|; the Fiedler cut
condition closes by ring.
Z4c — the adjacent degree-4 decorated edge. Adjacent degree-4 hubs u, v, each with
a degree-3 neighbour (tu of u, tv of v, distinct and off the hubs), fire at every
n ≥ 30 (9·(4 + 4) = 72 ≤ n + 42 ↔ n ≥ 30). The twins may be adjacent to each other.