slug: zhang-zhao-2026-domirank-entropy-monotonicity
bibkey: zhangzhao2026domirank
doi: 10.48550/arXiv.2610.00107
url: https://arxiv.org/abs/2610.00107v1
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/Graph/DomiRankEntropyRefutation.result

DomiRank entropy need not decrease with competition

Problem

Yingying Zhang and Chengye Zhao, Theoretical Analysis of DomiRank Centrality: Automorphism, Entropy, and Graph Transformations, arXiv:2610.00107v1, Section 4.1, Remark 4.6, identify the unconditional assertion

the DomiRank entropy of a connected non-regular graph decreases monotonically with σ

as an open problem. The claim quantifies over every finite connected nonregular unweighted undirected simple graph and every pair $0<\sigma <\tau <-1/{\lambda }_{\min }(A)$. It asserts $H_G(\tau )\le H_G(\sigma )$, with the actual inverse equilibrium $\Gamma (s)=s(I+sA{)}^{-1}A1$, positive clipping, normalization, and Shannon entropy in bits. There is no raw-positivity or extra nonzero-normalizer premise. Conjecture 4.7 is excluded.

Motivation

Preregistration #13267 classifies this explicit single-paper question as Tier 1. The conditional Theorem 4.5 and four numerical networks motivate the question of removing sufficient conditions. The sole public theorem negates the full source-faithful universal assertion.

Gap

The qualification in #13267 found no settlement in the inspected primary v1, target-bounded arXiv material, original DomiRank v2 paper, companion v1/v3, OpenAlex exact record and cited-by query, or fourteen target-specific GitHub issue/PR searches. This is bounded reported evidence. Scholar was blocked, Semantic Scholar returned 429, MathDB was unavailable, and only the formal-conjectures path index was searched. Exhaustive novelty and priority are ASSUMED-UNVERIFIED. The identifier/display-date mismatch is retained without inferring chronology.

Route

The connected graph on $\{0,1,2,3,4,5\}$ has edges $\{0,4\},\{0,5\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\}$ and degree vector $(2,3,3,2,3,1)$. Its spectral minimum lies in $[-3,-1]$. The inverse equilibrium gives

$$\Gamma (1/20)=\frac{(397,577,577,378,576,198)}{4357},\Gamma (1/6)=\frac{(33,45,45,28,44,16)}{129}.$$

All coordinates are strictly between zero and one. Rational logarithm bounds with integral remainders prove $H_G(1/6)-H_G(1/20)>1/(500\log 2)$.

Let $d=A1$, $D={\sum }_i{d}_i$, $w_i=d_i/D$ and $q_i=(Ad{)}_i/d_i$. For every finite nonempty graph with positive degrees, the normalized inverse extension at zero has derivative

$$H^{\prime }(0)=\frac{{\sum }_i{w}_i{q}_i\log d_i-({\sum }_i{w}_i{q}_i)({\sum }_i{w}_i\log d_i)}{\log 2}.$$

For the six-vertex graph this is $\log (27/2)/(49\log 2)>0$. Explicit derivative estimates prove strict increase for every $0<a<b<\delta$, where $\delta =1/5000000$; all such points are spectrally admissible with scores strictly between zero and one. Zero is used only in this normalized inverse extension and is not a source-domain point or a counterexample parameter.

For every natural $m>0$, form the actual connected nonregular independent-clone graph $G_m$ on $\mathrm{Fin}(6)\times \mathrm{Fin}(m)$. It has $6m$ vertices and spectral minimum in $[-3m,-m]$. The proof establishes the inverse equilibrium replication and identities

$${\Gamma }_{G_m}(s{)}_{(i,j)}={\Gamma }_G(ms{)}_i,P_{G_m}(s{)}_{(i,j)}=P_G(ms{)}_i/m,H_{G_m}(s)=H_G(ms)+\log (m)/\log 2$$

for $0<s<1/(3m)$. Consequently every $m>0$ has the exact pair $1/(20m),1/(6m)$ with the same strict entropy gap, and the whole increasing interval $0<a<b<\delta /m$, with admissible parameters and scores strictly between zero and one.

Falsifier

The certificate must concern the source adjacency and inverse, the full stable spectral domain, and clipped normalized entropy. A scalar proxy, a floating-point comparison, a witness at zero, or a statement with an added positivity premise would fail this target. The cubic reduced coordinate denominator in the proof is not identified with the adjacency determinant.

Evidence

DomiRankEntropyRefutation.lean exports the source-facing definitions Nonregular, clipped, normalized, entropyBits, spectralMinimum, graphAdj, graphDegrees, graphGamma, graphEntropyBits, claim, and the sole closed theorem result : ¬ claim. The covariance, exact pair, whole intervals and every-$m$ family are proof-local certificates. The theorem consumes the family at $m=1$, transfers both exact-pair and interval increments through its entropy identity, and contradicts the claimed nonincrease using their sum. Its axiom closure is propext, Classical.choice, and Quot.sound.

Triage

Tier 1; resolution Refuted. Admission basis is open-problem-resolution, preregistration #13267. The judgement form is proof_shape: result: bind-only, escape_witness: none: the proof composes existing finite-graph, inverse, spectral, calculus and logarithm results with explicit calculations and estimates. The external open-problem resolution supplies the admission basis. Utility is certified-instance; basis=refutes, with the closed claim and result. Information-escape registration and enrollment remain paused and unfinished.

What the settlement shows

The failure mechanism is a positive degree-weighted covariance of neighbor-degree ratio with log degree. Competition initially moves normalized mass toward a more even distribution in this connected nonregular graph. This produces a whole positive interval of entropy increase, and independent cloning preserves that failure at every order $6m$ after rescaling the parameter. These are one problem settlement, not separate open-problem resolutions.

The source’s conditional Theorem 4.5 is not contradicted by negating its proposed unconditional extension; its sufficient conditions are not proved here. The four sampled numerical networks do not establish a universal law. The separate star-minimum Conjecture 4.7 remains outside this result. A classification of graphs or additional conditions ensuring entropy nonincrease throughout the stable domain is unresolved here.

ASSUMED-UNVERIFIED

The bounded literature qualification is source-reported; exhaustive novelty and priority are not established. The identifier/display-date discrepancy remains unresolved. The source’s conditional Theorem 4.5 and separate Conjecture 4.7 are not verified by this module.