The starved-world census and the owner-choke rows #
The pure-counting census of the starved world — the e(M) = 0 stratum: a
never-firing graph on Fin n with 2(n−2) edges, minimum degree ≥ 3, and no
degree-3–degree-3 edge. Its degree-3 vertices are twins (all M-isolated), the
degree-4 vertices the sea, the degree-≥ 4 vertices the hubs. The
contrapositive of the star-moat law caps twin-hoarding (starved_cap), and
feeding the caps into the twin-incidence total against the degree-excess ledger
Σ_h(deg − 3) = n − 8 produces the census rows.
Main results #
starved_cap,hoarding_law,slots_law— the cap and the two aggregate rows; the hoarding row gives the giant censusn_g ≤ 1atn ≤ 49.starved_owner_choke_35_49— with the owner-choke7·t₄ + 3X ≤ 4n − 32, the35 ≤ n ≤ 49band dies by pure counting.slots_law_sharp— the per-psharpened slots row24 + 2X ≤ t₄ + p + h₆₊ + 4n_gsplit along the honest per-heavy caps (p= saturated degree-5 hubs).choke_count,p_choke_count,p_choke_row— the owner-choke and its per-psharpening7·t₄ + 8·p + 3X + 32 ≤ 4n, with the independence rowsowner_independence,owner_sat_independence.deco_edge_shared_twin_fires,sat_sat_independence,p_choke_row_unconditional— the shared-twin decorated-edge moat (fires at9·(deg u + deg v) ≤ n + 56) discharging the last moat-provenance hypothesis.
The starved cap (contrapositive of star_moat_fires). In a never-firing census world
(m = 2(n−2), δ ≥ 3) a hub h under the cap horizon 9·deg h ≤ n + 15 owns at most
deg h − 3 M-isolated twins: if it owned deg h − 2, that twin star would fire.
The cap ledger. Over any hub set S, the total twin ownership is bounded by the
degree excess plus a 3-per-giant credit: a capped hub (9·deg ≤ n + 15) owns
≤ deg − 3 twins (starved_cap), while a giant contributes the trivial ≤ deg = (deg − 3) + 3.
The twin total. Under hs0 (e(M) = 0) the degree-3 set is the M-isolated twin
set, so the hub-to-twin incidence sum is 3·(8 + X) = 24 + 3X.
The hub excess ledger. The degree excess is carried entirely by the hubs:
Σ_{h ∈ Hub}(deg h − 3) = n − 8, since every non-hub is a degree-3 twin (total_excess_eq).
The hoarding law (SW §S2). Twin-slot supply meets capacity: the 24 + 3X twin
incidences are hosted by the hubs, capped at deg − 3 apart from the 3-per-giant credit, so
3X + 32 ≤ n + 3·n_g where n_g is the number of giants (hubs above the cap horizon
9·deg ≤ n + 15).
The slots row (SW §S2 / §4 slots). Splitting the twin total by degree class: the
deg-≥ 5 hubs absorb at most X + h slots (h = number of heavies), plus the 3-per-giant
credit, so the deg-4 owner-slot total t₄ is forced up to 24 + 2X ≤ t₄ + h + 3·n_g.
The giant census bound (SW §S2). A giant h (n + 15 < 9·deg h) has
9·(deg h − 4) ≥ n − 20, and every giant is a heavy, so summing gives
n_g·(n − 20) ≤ 9·X. With hoarding this pins n_g ≤ 2 at n ≤ 49.
The per-p sharpened slots row (D1) #
Sharpens slots_law by splitting the deg-≥ 5 twin demand along the per-heavy
caps starved_cap (no independence law). The twin total 24 + 3X splits into the
deg-4 owner slots t₄ and a deg-≥ 5 remainder ≤ X + p + h₆₊ + 4n_g, giving the
sharpened row 24 + 2X ≤ t₄ + p + h₆₊ + 4n_g (p = saturated deg-5 hubs,
h₆₊ = non-giant deg-≥ 6 hubs, n_g = giants).
D1 — the per-p sharpened slots row. In a never-firing starved census (m = 2(n−2),
δ ≥ 3, hs0) the twin total 24 + 3X splits along the honest per-heavy caps into
24 + 2X ≤ t₄ + p + h₆₊ + 4·n_g, where t₄ = ∑_{deg-4 hubs} |N(h) ∩ Iso|, p is the count of
saturated deg-5 hubs (deg 5, owning exactly 2 M-isolated twins), h₆₊ the count of
non-giant deg-≥ 6 hubs (6 ≤ deg, 9·deg ≤ n + 15) and n_g the number of giants
(n + 15 < 9·deg). Each deg-≥ 5 hub contributes at most (deg − 4) plus one of the class
credits, so the deg-≥ 5 remainder is ≤ X + p + h₆₊ + 4·n_g; the star-moat cap starved_cap
supplies the non-giant per-hub bounds.
Independence and triangle helpers #
Triangle test.
Owner independence. In a never-firing starved census (m = 2(n−2), δ ≥ 3) at
n ≥ 30, any two distinct degree-4 vertices o₁, o₂, each carrying a degree-3 neighbour
(t₁ resp. t₂), are non-adjacent. If they were adjacent: distinct twins (t₁ ≠ t₂) fire the
(4,4) decorated edge z4c_fires; a shared twin (t₁ = t₂) makes {o₁, o₂, t₁} a
degree-(4,4,3) triangle which fires the master cycle tri_deg445_fires — either way
contradicting ¬ algConn G ≤ 2.
The owner-choke count. In a starved census (m = 2(n−2), δ ≥ 3, hs0), under the
starved degree-4 twin cap (hcap: each degree-4 hub owns ≤ 1 twin) and owner independence
(hindep: distinct twin-carrying degree-4 hubs are non-adjacent), the degree-4 owner-slot
total t₄ = ∑_{deg 4 hubs} |N(h) ∩ Iso| satisfies the choke 7·t₄ + 3X + 32 ≤ 4n.
Each owner o (degree-4 hub with its unique twin) spends 1 edge on that twin and, by
independence, 0 on other owners, so its remaining 3 edges land on non-owner hubs
(|N(o) ∩ NH| = 3). The bipartite double count cross_count transposes ∑_{o} |N(o) ∩ NH| = 3·t₄ into ∑_{w ∈ NH} |N(w) ∩ O| ≤ ∑_{w ∈ NH} deg w, and the hub degree total
∑_{Hub} deg + 3X + 32 = 4n splits as ∑_{NH} deg + 4·t₄; combining gives the choke.
The per-p sharpened owner-choke #
Sharpens the owner-choke choke_count (7·t₄ + 3X + 32 ≤ 4n) by 8·p, where p
counts saturated degree-5 hubs (owning exactly two twins). A saturated degree-5
hub spends 2 edges on its twins and, by the decorated-edge law, 0 on owners and 0
on other saturated hubs, so its remaining 3 edges land on non-owner
non-saturated hubs; the transposed slice count gives p_choke_count /
p_choke_row (7·t₄ + 8·p + 3X + 32 ≤ 4n). The (4,5) independence is
owner_sat_independence; the (5,5) independence is threaded as hindep55.
N5 — the per-p sharpened owner-choke (counting core). In a starved census
(m = 2(n−2), δ ≥ 3, hs0) under the degree-4 twin cap (hcap4) and the three independence
rows — degree-4 owners pairwise non-adjacent (hindep44), owner–saturated non-adjacent
(hindep45), saturated–saturated non-adjacent (hindep55) — the degree-4 owner-slot total
t₄ = ∑_{deg 4 hubs} |N(h) ∩ Iso| and the saturated degree-5 count p satisfy the sharpened
choke 7·t₄ + 8·p + 3·X + 32 ≤ 4n.
Owners O (degree-4, one twin) each spend 3 edges on non-owner non-saturated hubs NH;
saturated degree-5s P (two twins) each spend 3 edges on NH (the 2 twin edges and the
0 owner/saturated edges being excluded by independence). The bipartite double count
cross_count transposes ∑_{O ∪ P} |N(·) ∩ NH| = 3·t₄ + 3·p into
∑_{NH} |N(w) ∩ (O ∪ P)| ≤ ∑_{NH} deg w, and the hub degree total
∑_{Hub} deg + 3X + 32 = 4n splits as ∑_{NH} deg + 4·t₄ + 5·p; combining gives the choke.
The (4,5) decorated-edge independence. In a never-firing starved census
(m = 2(n−2), δ ≥ 3) at n ≥ 39, a degree-4 owner o (carrying a twin) and a saturated
degree-5 hub s (owning exactly 2 twins) are non-adjacent. If they were adjacent: a shared
twin makes {o, s, t} a degree-(4,5,3) triangle (Σdeg = 12 = 4·3) firing master_cycle_fires
(n ≥ 19); distinct twins fire the (4,5) decorated edge deco_edge_moat_fires
(9·(4 + 5) = 81 ≤ n + 42, i.e. n ≥ 39) — either way contradicting ¬ algConn G ≤ 2.
N5 — the per-p sharpened owner-choke (assembled row). A never-firing starved census
(m = 2(n−2), δ ≥ 3, hs0) on n ≥ 48 satisfies the sharpened choke
7·t₄ + 8·p + 3·X + 32 ≤ 4n. The degree-4 twin cap is the contrapositive of the star moat
(starved_cap at deg = 4), the (4,4) independence is owner_independence, and the (4,5)
independence is owner_sat_independence; the (5,5) independence hindep55 is threaded as the
single moat-provenance hypothesis (its shared-single-twin subcase requires a shared-twin
decorated-edge moat, a separate node — scratchpad/ahl_import.md §1.3/§4.4).
The shared-twin decorated-edge moat and the unconditional row #
Reworks the decorated-edge moat for the overlapping-twin case and discharges the
last hypothesis of the per-p owner-choke. deco_edge_shared_twin_fires: adjacent
hubs sharing exactly one twin fire at 9·(deg u + deg v) ≤ n + 56 (the shared
twin has external budget 1, so the threshold is easier than the disjoint
n + 42). sat_sat_independence: two adjacent saturated degree-5 hubs are
non-adjacent at n ≥ 48. p_choke_row_unconditional: the census-only per-p
choke row, discharging hindep55 via sat_sat_independence.
The (5,5) saturated-pair independence. In a never-firing starved census
(m = 2(n−2), δ ≥ 3) at n ≥ 48, two saturated degree-5 hubs s₁, s₂ — each owning exactly
2 M-isolated twins — are non-adjacent. If they were adjacent: disjoint twin sets fire the
disjoint decorated edge (deco_edge_moat_fires, 90 ≤ n + 42); one shared twin fires the
shared-twin decorated edge (deco_edge_shared_twin_fires, 90 ≤ n + 56); two shared twins form a
(5,3,5,3) 4-cycle s₁, t₁, s₂, t₂ firing master_cycle_fires (Σdeg = 16 = 4·4,
3·12 = 36 ≤ n + 8) — either way contradicting ¬ algConn G ≤ 2.
The fully census-only per-p owner-choke row. A never-firing starved census
(m = 2(n−2), δ ≥ 3, hs0) on n ≥ 48 satisfies the sharpened choke
7·t₄ + 8·p + 3·X + 32 ≤ 4n, with the (5,5) saturated-pair independence discharged internally
by sat_sat_independence (no moat-provenance hypothesis remains).