[flyingviper]

At the fixed P112 marks {0,67}, {7,64}, {1,75}, every rational decoder whose queries lie in their canonical span has minimum invoice 18, regardless of dictionary, term count, dependent queries or asymmetric actual readers. Transposition gives the same result when the readers lie in that span. Thus any cheaper factorization at this placement must escape the canonical span on both sides.

The new argument uses all 441 source/reader support types, not a coefficient search. After restricting reader columns and merging parallel axial queries without increasing cost, there can be at most one cost-2 term. Every other term costs at least 3, and every term costing at most 3 has a single reader. Seven terms therefore cost at least 20. Six terms costing at most 17 would have to feed the six outputs individually, but the unique cost-2 term sends the wrong target to its output. Every hypothetical cheaper factorization consequently reduces to at most five terms.

The final five-term exclusion invokes the retained theorem from PR #3684 at fad2e401ff63c634fb04e47146b0323449af70dd. Its precise proof files and interface are pinned and audited. Its full historical classification/pricing suite was not rerun here; archival merging is not scientific approval. The new reduction to five terms is self-contained given the two foundational geometry inputs.

The package also proves budgets valid for any three ordinary marks in distinct paired roles: without axial queries, cost >= 12+n; after merging axial queries, cost >= 12+n-3a, where a<=3 and n are normalized active counts. A local cost<=17 witness must have at most 8, 11 or 14 terms for a=1, 2 or 3. These general bounds constrain other placements without claiming they save or are excluded.

All written arguments passed separate AI-agent audits. Two independent implementations rebuild the complete 441-type fixed-placement tariff table, using elimination versus binary span/annihilator sets. The general geometry checks cover 666 triad pairs, 153 paired-role unions, all 53 ordinary canonical tariffs and, independently, all 7,770 triad triples. Four registered replays, metadata and committed-byte/source-history checks pass at 2dd2a04d32be6958921b6b6228b4b386ec273219. The official Git-free export passes all four commands with empty PATH and read-only files; all 21 file hashes remain unchanged (0.89 seconds, Python 3.12.14 on Windows). Reproduce: python3 -B scripts/research_package.py --package research/p112-no-axial-three-mark.

The complete source-span floor fixes the displayed placement and exact endpoint. Other placements, exterior directions on both sides, larger interacting blocks and changed endpoints remain open. No decoder at 296, universal dense-factor floor, native schedule or improved multiplication bound is claimed. Canonical main was refreshed and remains 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7. A current PR delta supplements the earlier complete overlap snapshot; Discussions remain unavailable after HTTP 406, so no exhaustive novelty claim is made.

Prepared by flyingviper with substantial OpenAI Codex assistance and separate AI-agent mathematical/implementation checks. The P112 geometry and earlier five-term boundary retain their original attribution, including Nick Petalas' classification/transfer lineage. No independent human or formal-kernel verification is claimed. Apache-2.0.