# Verification report for Sharp Graham-Sequence Reconstruction for Spiders

Version 1.0, 17 October 2026. Author: Alper Ferudun.

## Accepted mathematical scope

For every spider with n vertices and r >= 3 legs, the prefix through index
floor((n-r-1)/2)+2 determines it among spiders. The bound is sharp for each
n-r-1 >= 2. Three-leaf trees are recognizable among all trees and have sharp
uniform cutoff floor(n/2), with sharpness for n >= 6.

The proof is the marked-core decomposition, inversion identity, active-port
symmetry induction and excess-partition argument in the manuscript. The
explicit extremal pairs prove sharpness. This is a written proof accepted
after the originating researcher's self-audit, not a proof-assistant result
or an independent review. No known mathematical gap is recorded in this
scoped argument. Graham's conjecture for arbitrary trees remains unresolved.

## Supplementary exact checks

- C++17 leaf-extension enumeration and centre-rooted canonical encodings:
  all 1,346,024 unlabelled trees on one through twenty vertices.
- Independent Python integer/NetworkX 3.5 WROM enumeration: every tree code
  and every retained sequence prefix are checked; prefixes are prefix-free.
- C++ degree-moment formula and Python unaggregated wedge formula agree.
  Four literal line-graph iterations are also checked on all 25 trees of
  order at most seven.
- Largest retained term: 50,858,095,643. C++ uses signed 128-bit integer
  arithmetic, not floating point; Python uses arbitrary-precision integers.
- Blind reconstruction: 351 spiders, 995 inversion steps, 42 sharp pairs.
  Surviving ports are chosen only from the decoded count, without consulting
  the unknown leg lengths.
- These finite computations support the implementation and establish only
  the stated finite separation result. They do not prove the unrestricted
  conjecture and do not replace the all-parameter spider proof.

The source supplement includes the original programs, all certificates,
historical replay receipts, dependency specification and a portable replay
wrapper. Run without Python optimization, because the original checker uses
assertions. The wrapper rejects optimized mode. Independent implementations
are not independent mathematical reviewers.

## Presentation and prior art

The seven-page PDF was compiled with the built-in compiler and Tectonic.
All seven rendered pages were directly inspected; no clipping, broken math
or overfull boxes were found. The package builder separately recompiles the
extracted source archive and compares PDF text and page content streams.

The 2020 Balch source was read completely. Cooper--Kay--Swifton and
Weatherspoon--Zeilberger are credited for relevant earlier work. The full
2024 King--Lane-Harvard--Milligan chapter was unavailable, so overlap with its
detailed results remains unexcluded. This limitation is explicit in the
paper and metadata. Publication is not a novelty certification, peer review
or a claim to have closed the full source problem. No third-party full text
is included in the public supplement.