Prescribed-zero prefixes in equal-arm spiders

Prescribed zero on a contiguous prefix

Let S(k^n,1^m) have a designated center, n arms of k edges and m additional center leaves. Depth means distance in edges from that center. A graceful labeling uses every vertex label 0,…,N and every absolute edge difference 1,…,N exactly once, with N=nk+m.

For every integer k≥19 define

D8(k)=11                         for 19≤k≤34;
D8(k)=27                         for 35≤k≤52;
D8(k)=25+16 floor((k−35)/18)      for 53≤k≤124.

For k≥125 put p=floor((k−3)/2)=61+11a+r, with a≥0 and 0≤r≤10, and set D8(k)=107+20a+2r.

For all integers n≥2,m≥0, any selected long arm and every depth 2≤d≤D8(k), there exists a graceful labeling with zero at that specified vertex. Different requests may use different labelings. The statement includes both arm-length parities and n=2,m=0.

The bound 0≤10k−11D8(k)≤261 gives D8(k)/k→10/11 through all integers. This is a sufficient prefix. It does not establish full zero-rotatability or an optimal threshold. The earlier D7 thresholds K7 remain sufficient, without an assertion that they are the exact inverse of D8.

Construction

Compatible cores of sizes 16 and 17 have reflected extreme pairs starting at depth 26. The q9,q10,q11 insertions preserve the required side, sum, endpoint and window partitions. Applied before the final reverse-complement, they increase reflected depths by 16,18,20, respectively.

Seven recipes fill the missing D7-to-D8 intervals: five give pairs 108/109,112/113,116/117,120/121,124/125 at core sizes 62,64,66,68,70. At size 71, one recipe supplies 124/125 and a second 126/127; both are needed to fill the four-depth gap. Additional q11 insertions increase size by 11 and depths by 20, translating these certificates to every later block. Odd and even shells, residual grafting and whole-graph complementation give the stated actual-spider quantifiers.

The first increase over D7 is at k=127,128, from 107 to 109. At k=145,146 the endpoint rises from 123 to 127. The limiting proportion remains 10/11.

Proof and scope

The package contains the written construction, executable graph checks, a Lean proof of the actual-spider D8 theorem and a separate build from copied sources. Formal ratio verification uses an integer precision criterion. These are internal project checks, without external peer review; historical priority remains unresolved. Classical path, reflection and alpha-amalgamation methods and their prior authors are credited in the source comparison.

Proof and source files

Construction and certificates · D8 proof · Separate internal check · Source comparison · New proof archive.

The earlier growing-depth proof archive remains available unchanged: SHA-256 3be4ef1da7645c2e8024ebf8b126434c1554e50a67dfbaa5051f7a830b86d450, 360302 bytes.