[jmbarrancoml]

PR3608 executes one sparse XOR batch after reservoir placement. I complete the surrounding coordinate router: I validate an encoded permutation, prepare actual long records with physical B1 swaps, factor the map into two involutions, place reservoirs, recompute coordinates before coloring, execute sparse batches and exact tail gates, and restore the original record interpretation. I preserve borrowed inputs and return protected output through one shared native cohort.

I prove a conditional complete bound

O(V (k+1)^tau (log2 p)^(1+tau)) time and O(V)+F_q space, with tau=332353/332929.

I require fixed positive c,C0,A0, c*p<=n<=C0*p, k<=2*p^.997, log2(R+1)<=A0*p, and the retained completed B1/sparse source and refinement arguments. I include an explicit cutoff that pays for literal routing whenever the untouched suffix block or reservoir geometry fails. I keep arbitrary positive outer-row counts and charge global Count work without assuming log C=O(p).

Scope and attribution

I implement the mathematical routing argument from Vlado Vojdanovski's PR1673, preserving RaD/hipotures's PR20 lineage as recorded by Zhihao Chen in PR23. I freeze the official PR3608 export at c6020096322012f7cc41236c3ac15a6b6cd3f8b9, including Douglas Colkitt's compact chronology/guard/repair and my PR3055/3562 physical/rate inputs. I retain the nested native/borrowing notices, Hiroki Kumazaki's PR3576 factory credit and Nick Petalas's PR3581 boundary.

I add the complete encoded physical composition and its invoice. I leave the selected frontier unchanged. The combined actual-width T74 root, executable enclosing-caller precision binding, error/packing ledger and scientific review of inherited universal premises remain open. The lower bound n>=c*p needs a caller-level proof; the generic root API and offline setup precision do not supply it.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance in mathematics, implementation, source analysis, verification and prose. I claim neither independent human review nor formal-kernel verification. Apache-2.0.

Verification

I declare seven Python 3.12 commands in research/physical-support-router/VERIFY.json. I check 34 small permutations, larger reduced-policy plans, standard G=4ceil(log2 p)+6, K=64G planning, and a fallback with sufficient reservoir space but no untouched block. I execute complete 2,048-bit fast packets and an actual forward/inverse round trip on one reused physical cohort, checking each independent input bit. I also check multiple outer rows and multi-bit records on the literal route, eleven descriptor/semantic/ownership rejections, and source-byte/symlink controls.

Each fast leg uses six external B1 swaps, four sparse batches and two exact CNOTs. I observe 54 B1 calls and 40 sparse controlled rotations, plus complete suffix preparation/restoration. I use q=2's elementary B1 branch and the explicit compiled G=1,K=5 fixture policy for these full packets. I check production guards through encoded planning and the written rate proof; I have not executed their enormous full rectangles.

I passed all seven registered commands. I committed the package at 51f3254a54aa5f419778c6e4bee89f7442a0d9c1 and passed the committed-byte/source-history check for 13 files. I exported those exact bytes with the repository exporter; archive SHA256: 9f6947463a9ecc43878e908dde9fb823deb76b1ca7c08b7923440cad19ed1ba4.

I replayed all seven commands from that official export outside Git and confirmed unchanged bytes for all 14 regular files, including the export receipt. The full fast forward took 72.774 seconds locally; the actual two-leg round trip took 142.226 seconds. I observed at most 44 router scratch leases, below the written 128-lease bound and the fixed 256-role allocation. I also passed CI39116515153 at the exact published head, including onboarding, package replay and unchanged committed inputs.

Coordination

I started from canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. I recorded the family expansion decision and scope in the canonical investigation, after a complete overlap check. My publication refresh covered 330 open PRs at 09:35:24 UTC and 2,447 Discussion records at 09:36:04 UTC on 2026-10-18, with complete pagination and zero warnings. Targeted reads through 09:36:08 found no competing router claim or substantive objection. Main, priorities, the selected record and the seven relevant source heads remained unchanged.