Evidence for a Functional Proximity Law in Multilayer Networks

Version 5 — Corrections and presentation improvements

This version corrects documentation inconsistencies from v4 and improves readability of the manuscript. No changes were made to experimental procedures or underlying computed results.

Corrections:

• Added 3 missing rows to the supplementary evidence table: M_OSS1 (WordPress), M_OSS2 (Next.js), and M_TRANSFER_2 (Flask↔Express), which were present in the main text but omitted from the table
• Corrected external validation count: “All four” → “All five” (to reflect inclusion of M_TRANSFER_2)
• Corrected aggregate reporting: “16 confirmed” → “14 confirmed” (14 = 9 Tier A + 5 Tier B pre-registered; earlier value was a carry-over inconsistency)
• Fixed LaTeX rendering issues where equations appeared as raw text in v4

Presentation improvements:

• Abstract rewritten for structure and clarity (same result content, reduced length)
• Introduction reordered to present main finding earlier
• Pre-registration hash block on page 1 condensed into a single table format

Note: All experimental results and statistical computations remain unchanged from v4.