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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0528, E0529.

Walks: E0528 and E0529

Let c_n count n-step nearest-neighbor self-avoiding walks from zero in fixed Z^k. A walk of length m+n determines its first m steps and its final n steps translated to zero. This map is injective into the product of the two unrestricted self-avoiding-walk sets; cross intersections exclude some pairs. Therefore

$$c_{m+n}\leq c_m c_n,\qquad C_k=\inf_{n\geq1}c_n^{1/n}$$

by subadditivity of log c_n. Exact enumeration at a finite n certifies an upper bound on C_k. It supplies no lower bound merely by concatenating its walks, since the concatenation may self-intersect. There are k^n walks using only positive coordinate directions, so C_k>=k. The first step has 2k choices and later steps at most 2k-1, giving C_k<=2k-1 for k>=2. In dimension one there are exactly two walks at each positive length, giving C_1=1. The Clisby source card's enumeration scope was read, but no tables were replayed. The next test is an injective bridge or finite-state concatenation with a proved comparison to unrestricted walks.

For E0529, Hara--Slade Theorem 2.1(b), inspected in the primary announcement, states for each k>=5 E||omega_n|| squared=D n(1+O(n^(-epsilon))). Cauchy--Schwarz gives

$$d_k(n)\leq\sqrt{D n}\,(1+o(1)).$$

The source explicitly defers its proofs to later works, which were not inspected here. The dimension restriction excludes k=3,4.

Duminil-Copin--Hammond Theorem 1.1, inspected at statement level, gives for every fixed v>0 an epsilon(v)>0 with an exponentially small probability of maximum displacement at least vn. Since endpoint distance is at most n,

$$d_k(n)\leq vn+n\exp(-\epsilon(v)n).$$

First send n to infinity and then v to zero to obtain d_k(n)=o(n). There is no supplied uniform rate as v shrinks with n, so substituting v=n^(-1/2) is invalid. This implication neither gives the desired square-root upper bound in dimensions three and four nor a planar super-square-root lower bound. Different lattices, weakly self-avoiding models and subsequence results need separate transfer arguments.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/_index|clisby 2007 self avoiding walk enumeration via lace]]: Equation (1), enumeration scope and Section 7.2 account (account reading).
• [[../library/discrete_geometry/duminilcopin_2013_self_avoiding_walk_is_sub_ballistic/_index|duminilcopin 2013 self avoiding walk is sub ballistic]]: Section 1.1, Theorem 1.1 and Corollary 1.2 (statement reading).
• [[../library/discrete_geometry/hara_1991_critical_behaviour_self_avoiding_walk_five_more_dimensions/_index|hara 1991 critical behaviour self avoiding walk five more dimensions]]: Section 2, Theorem 2.1(b); Section 3 announces later proofs (statement reading).