[GamingPuzzled]

No κ claim. This answers zxv's #5428 target 1 (the encoded-output exit) and screens the other structural levers on the #5413 caller.

Main theorem. Every exact, width-uniform producer of b = Z·Q·Y from post-return data needs ≥ 112 paid jobs per backward FULL. The bound holds for any rank, any operator and any sharing pattern: butterflies, block transforms, joint diagonalisation and shared higher-rank intermediates are all covered.

• The whole Test coupled returns through two query consumers and price a compressed exit #5415 exit needs ≥ 120, so Test coupled returns through two query consumers and price a compressed exit #5415's 120-H1 exit is optimal.
• The break-even is τ = 63.52 H1 per FULL, so this route cannot pass U = 0.00302728825.
• This extends Certify that b = Z Q Y is not retained and no paid response is reusable (negative result, no new kappa) #5427, which shows b is not free, to paid producers.
• The proof uses a coset-span lemma. The 112 coset directions (56 per mode) are computed exactly over ℚ with the repository's clifford_exact evaluator, including absolute phase.
• Controls: three sanity targets give 8, 8 and 0; 300 random programs never beat the lemma; making one C gate free breaks it.

Other levers.

• Rank-9 helpers have no free split point in either orientation (exact replay on all 632 registers).
• A bought exit needs ≥ 15 modes per FULL; the Preserve all 632 native inputs in an eight-mode coupled helper provider #5402 family has 8.
• A productive centre payload earns 0; query-shear deletion breaks the rank-mass balance; query/local fusion is illegal.
• Defragmentation is closed at a fixed operation set.

Open. The only remaining b-route uses a pre-return copy of y: 8×H8 = 63.598 against τ = 63.523, a loss of 0.076 per FULL. That is the centre-interface lever (#4694/#4698/#4700).

Package: research/encoded-exit-obstructions/ (README, R16/R17 notes, tools, expected outputs). Companion notes: #5465 (reorder/re-frame ceilings) and #5462 (owned-relation optimum).

AI disclosure: prepared by Joel Pulikkan (GamingPuzzled) with Claude Opus 5.5 (Anthropic) via Claude Code. Apache-2.0.

🤖 Generated with Claude Code