First correction to the critical mean degree in extreme long-range percolation

Summary

Place a point at every integer along a line, and independently connect pairs with a probability that decreases with distance. How many neighbours, on average, are needed before an infinite connected cluster can appear?

When the distance penalty becomes very weak, the critical mean degree approaches one. This candidate identifies the first correction above one: the probability that two independent steps drawn from the connection kernel return to their starting point. Immediate returns do not discover new vertices. The proof makes that familiar obstruction quantitative in a fixed one-dimensional, small-exponent limit.

Reading the banner: the highlighted out-and-back route returns to an already discovered vertex; the quieter arcs represent other possible connections, not a sampled cluster. The formulas show the leading threshold correction and how an added nearest-neighbour weight changes its coefficient. No curve or critical threshold has been fitted to simulations.

Summary for specialists

For independent undirected bonds on $\mathbb Z$ with exact opening probability $C|x-y|^{-1-\sigma}$, put $Z_\sigma=\zeta(1+\sigma)$ and $D_\sigma(x)=|x|^{-1-\sigma}/(2Z_\sigma)$ for $x\ne0$, with $D_\sigma(0)=0$. The normalized parameter $p=2Z_\sigma C$ is expected degree. The written proof gives constants $K,\sigma_0>0$ such that, for every real $0<\sigma<\sigma_0$,

Consequently $(p_c-1)/\sigma^2\to\pi^2/12$. In the original bond-amplitude parameter, $C_c-(2Z_\sigma)^{-1}=\pi^2\sigma^3/24+O(\sigma^4)$.

Section 7 proves a sufficient criterion for even kernels with maximum weight $O(\epsilon)$, $J_{2,3}=O(\epsilon^2)$ and $J_{3,2}=O(\epsilon^3)$, together with full stated structural and admissibility assumptions. It applies to arbitrary nonnegative summable additions of uniformly bounded total mass and to probability mixtures of exponent ratios in a fixed compact positive interval.

Technical account

The lower bound uses a nonbacktracking exploration: it removes the immediate return along the edge just traversed. The upper bound uses the positive remainder after the first odd truncation of the percolation lace expansion. Uniform small-exponent Fourier estimates bound the relevant closed diagrams more sharply than their supremum norms.

The deleted-bond restriction matters. Retaining it produces the subtractive two-step term; a generic triangle bound alone would lose the coefficient. The proof controls the first coefficient one-sidedly and uses positivity of the finite-order remainder. It does not require a matching expansion for every lace coefficient.

For a concrete extension, add weight $\lambda\ge0$ at distance one before normalizing the pure-power kernel. The coefficient of $s^2$ becomes

Thus keeping the same far-tail exponent does not keep the correction unchanged. For mixtures $D_s=\int D^0_{cs}\,\nu(\mathrm dc)$ with $0<c_-\le c\le c_+<\infty$, the coefficient instead is $\frac{\pi^2}{12}(\int c\,\nu(\mathrm dc))^2$—the square of the mean exponent ratio, not its second moment.

Evidence, assurance and limitations

This is an unrefereed analytic written-proof candidate. The supplied review has an itemised response, and the source applicability, diagram estimates and additional kernel families have been checked internally. Finite numerical or algebraic diagnostics check their declared identities and controls; they do not prove the infinite-volume threshold theorem.

There is no end-to-end formal verification, external specialist endorsement or unaffiliated reproduction established. Neither a DOI nor media supplies that assurance. The constants $K$ and $\sigma_0$ are not numerically evaluated, so the expansion is not a certified approximation at a specified positive exponent. The result does not determine the cubic coefficient, cover oriented percolation, or replace the linear bond law by an exponential one.

Relationship to earlier work

Amit's motivating paper gives explicit bounds through monotone paths. Earlier work of van der Hofstad and Sakai already identifies the two-step return correction in a spread-out regime. The present contribution is not discovery of that mechanism or a new lace expansion: it supplies uniform control in the small-exponent regime at fixed dimension one, plus the stated transferable criterion and examples. The infrared theorem and event decomposition are inherited from the cited primary sources with their hypotheses retained.

Who should care, and why

Why the problem matters

The branching approximation discards repeated discoveries. Identifying its first error explains which part of the geometry survives when connections become extremely long range. The kernel extensions make that comparison more informative: local alterations can change the coefficient even when the broad tail exponent is unchanged. No empirical network application or measured downstream benefit is claimed.

How to inspect or reproduce the recorded checks

Begin with README.md and AI_INDEX.md in the linked versioned repository. Read the model normalization and main theorem, then the nonbacktracking bound, imported infrared assumptions, deleted-bond argument and positive-remainder threshold step. Section 7 checks the additional kernels, including existence of an admissible percolating parameter.

Follow the package's verification instructions for the finite diagnostics and negative controls. Inspect their scope in ASSURANCE.md; do not substitute their successful execution for reading the analytic proof. REVIEW_RESPONSE.md identifies the supplied review's requests and their dispositions; the full supplied review is not redistributed.

The most valuable next projects

The immediate assurance priority is an unaffiliated specialist audit of the source-to-model bridge and the uniform diagram estimates. Further mathematical work could seek an explicit useful error constant and exponent range, or a third-order coefficient. Other bond laws and oriented models need their own analysis rather than a change of notation.

What is in the evidence package

The package includes the manuscript and LaTeX source, source and citation audits, the response to the supplied review, scoped verification material, an environment record and an agent-readable index. Original prose and diagrams are CC0; original code is MIT. The banner and synthetic audio explain the result but are not additional scientific evidence.

@misc{smallexponentpercolation2026,
  title        = {First correction to the critical mean degree in extreme long-range percolation},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23294525},
  url          = {https://doi.org/10.5281/zenodo.23294525},
  version      = {1.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/small-exponent-percolation/}
}