bibkey: scottsokal2003repulsive authors: “Alexander D. Scott; Alan D. Sokal” year: 2003 title: “The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma” doi: 10.48550/arXiv.cond-mat/0309352 url: https://arxiv.org/abs/cond-mat/0309352 claim: “For the hard-core independence polynomial, the positive component in the negative orthant characterizes simultaneous positivity on all induced vertex subsets.” strata_touched:

• D5/S3/Arith/Congruence/TwoOddPrimeUncoveredDensity license: citation-only triage: anchor

A positive ray certifies all induced subgraphs

The inspected version is v2, dated 16 September 2004; first version 2003. Checked 16 September 2026. Theorem 2.10(a) and (b′), with hard-core self-repulsion, identifies a positive path from zero to a negative weight vector with positivity of every induced-subset polynomial.

Consequently, if Z_V(t) is positive for all 0 <= t <= 1, then every Z_U(1) is positive. The Erdős #7 dossier also gives a finite deletion-recurrence argument for this specialization. Positivity only at the endpoint is insufficient. This criterion alone does not establish the required ray positivity for arbitrary congruence families.

Conditional avoidance under the same source

Theorem 4.1(a), equation (4.3), of the same v2 was checked against the original PDF on 20 September 2026. If the event-probability bounds satisfy the theorem’s conditional non-neighbor hypothesis (4.1) and lie in its strict region R(G), then, for any event-index sets Y and Z,

P(avoid Y | avoid Z) >= Z_G(-p 1_(Y union Z)) / Z_G(-p 1_Z) > 0.

Product-coordinate independence supplies (4.1) for events whose dependency graph joins overlapping coordinate supports. The strict-region condition must still be proved for the particular bounds; a positive value of the full polynomial alone is insufficient. This ratio concerns conditional avoidance in the same original probability space, not a resampling output law. The coupled first-root result uses it after checking the complete support-polynomial region for one specified block and its descendant-domain bounds.

Citation and source-boundary note only; no source text or code is vendored.

Literal laminar conflicts under actual conditioning

The same Theorem 4.1 permits any graph satisfying its lopsided hypothesis (4.1), as its Remark 1 explicitly states. For product laws on complete prime-power coordinates, a coordinatewise conditioning coupling preserves all compatible literal prefix events: compatible prefixes are nested. Thus the residue-conflict graph satisfies (4.1), even when the digits inside a coordinate are dependent. An independent rare query coordinate and (4.3) then give a query ratio for the actual law conditioned on avoiding the original events. The complete argument and its strict-region premise appear in the laminar-prefix application. This is an application of the cited theorem; it neither identifies the conditional law with a resampling terminal law nor supplies universal strict feasibility for AP families. The v2 primary theorem and Remark 1 were checked on 20 September 2026.

Ratio monotonicity and virtual upper response tables

Proposition 2.26(b), equation (2.74), and Corollary 2.27(c) of v2, printed pages30–31, were checked against the original PDF on26September2026. Write p(x)=Z_W(-x) for a hard-core independent-set polynomial. If 0<=x<=y and every induced polynomial at y is strictly positive, then Theorem2.10(b′) puts y in R(W), and Proposition2.26 yields

p(lambda x)/p(x) <= p(lambda y)/p(y),   0<=lambda<=1.

Corollary2.27(c) is an equivalent multiplicative route, but its function is the nonnegative extension defined in(2.75): it equals the raw polynomial only in the strict region. Full-polynomial positivity alone is insufficient, and zero boundary denominators do not permit this ratio argument.

For the graph L(K5), independent sets are matchings of at most two edges. If lambda_T retains only edges disjoint from the queried coordinate set T, and both response tables use the same positive coordinate masses Z_q, then

x_e=beta_e/(Z_q Z_s),
H_T=product_(q notin T) Z_q * p(lambda_T x).

Increasing all edge caps from beta to beta_plus within the strict region gives r=H_empty_plus/H_empty in (0,1], exact empty-support mass r H_empty=H_empty_plus, and simultaneous bounds r H_T<=H_T_plus. Multiplying one actual source by this same cellwise r preserves its domination and all queries. If r is independent of weak-marker indices, it also preserves both weak-marker priority inequalities. A virtual cap need not itself be a globally realizable phase layout. Positivity of its complete charged gate remains a separate obligation; this comparison does not supply it.

For the0/1 masks needed here, the ratio monotonicity also follows from a finite deletion induction. Let q_A be the induced polynomial and R_(A,v)=q_A/q_(A\{v}). The recurrence

q_A=q_(A\{v})-x_v q_(A\N_A[v])

and a successive deletion of the neighbors of v express R_(A,v)=1-x_v/product R_smaller. Induction on |A| proves R_(A,v)(x)>=R_(A,v)(y)>0 whenever x<=y and all induced polynomials at y are positive. This also proves positivity at x. Telescoping reciprocals over deleted vertices gives q_B(x)/q_A(x)<=q_B(y)/q_A(y) for B subset A. This is an elementary verification of the cited specialization, not a new originality claim or a proof for arbitrary fractional lambda.