Conditional κ = 0.001673636244951726 via reordered T16 augmenting chain

research/t16-reordered-augmenting-chain/PROOF.md

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+# Scoped construction and admission argument
+
+The exact pending predecessor is PR3375 at `62fd8039cc40749cea6238e7807db9a63b4d6765`. Its retained source, source-bound owner delta and canonical reconstructed payload are authenticated before new construction. Predecessor attribution and conditional proof obligations carry forward.
+
+A call is identified by `(stage, replica)`. A permissible order contains each of the 144 calls exactly once, preserves stages 0 through 8 within each replica, and interleaves each first-bank call with its second-bank copy. The new order is obtained by swapping adjacent paired blocks at indices 22, 16 and 11. The associated neutral owner repairs are explicit data. Each repaired map independently satisfies capacity, contiguous slices, whole-helper ownership, strict event windows and per-call family uniqueness. Each neutral step has exactly the predecessor paid profile.
+
+The two-bank parent template is unchanged. Preserving each replica's stage order preserves its literal source-event projection, including bridges. The extended lift reconstructs `actual_windows` from the resulting event schedule and verifies every assigned helper against those windows. Provider timestamps are derived from call identities. The schedule hash and owner hashes are regenerated. Owner rows are serialized by invocation time and helper index at the documented steps so that emitter and independent auditor agree on the sigma stream. The selected call order in SPEC is bound to the same plan.
+
+The following ten-role augmenting chain concentrates two rank-3 residuals into rank 6. Two subsequent exchanges concentrate rank 6 and rank 7 into ranks 1 and 12, then ranks 12 and 7 into ranks 3 and 16. All roles retain their widths and call identities. The total idle-profile difference is `H1 - H3 - 2 H7 + H16`, with zero rank-mass difference. Exact pricing, rather than a floating objective or majorization screen, establishes the stated gain.
+
+The complete eleven-stage admission rebuilds the parent, binds suppliers, lifts the explicit order, emits the controller, independently audits its phases/owners/charts, derives the physical invoice and assembles both supplier bounds. It retains the 69 admission, 12 bit-supplier, 15 controller and six independent T16 rejection controls. Additional lift controls reject missing/duplicate calls, reversed replica chronology, broken bank pairing, incorrect role times and assignments outside windows.
+
+The actual chart count changes to 10,017. This does not change the charged finite stock. Comparison of complete typed invoices allows only three source-binding hashes and the permutation of the same complete call records; all other fields, including costs, remain equal. Source bindings are checked against generated files. The independent audit also checks prefix and row reserves directly.
+
+This establishes a finite source-bound packing improvement within the inherited conditional framework. It does not discharge all-size compilation or upstream analytic assumptions, certify global packing optimality, or change the repository's selected frontier. The unchanged word/coverage model retains its predecessor fixed-word ceiling.

research/t16-reordered-augmenting-chain/README.md

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+# Conditional κ through a reordered T16 augmenting chain
+
+This package composes three neutral paired-call reorderings, a ten-helper augmenting chain and two residual consolidations with the exact owner map in [PR3375](https://github.com/CrocSwap/integer-mult-bounds/pull/3375), commit `62fd8039cc40749cea6238e7807db9a63b4d6765`.
+
+The exact conditional value is **κ = 0.001673636244951726**, an increase of **0.000000007563044413** over PR3375. The conservative supplier gives **0.001673636242098946**. The complex supplier remains binding. These are packing improvements within the inherited conditional framework, not a complete multiplication theorem or a packing optimum.
+
+The new paired call order unlocks windows unavailable on the predecessor chronology. After three neutral owner repairs, a ten-role chain changes the idle profile by `-2 H3 + H6`; two further exchanges change it by `H1 - H6 - H7 + H12` and `H3 - H7 - H12 + H16`. The aggregate is `H1 - H3 - 2 H7 + H16`. The final profile is `{1:349710, 2:60192, 3:62326, 4:20448, 5:11808, 8:1152, 16:3, 48:4880}`. Rank mass remains 1,041,408, deficit 5,120, with 69,120 whole-helper roles in 144 complete invocations.
+
+`REPAIRS.json` records the explicit steps and row-order conventions. The verifier reconstructs the predecessor from retained, authenticated source evidence and reconstructs every new map. No optimizer runs during verification. The private search tools and solver dependencies are not part of this package.
+
+The admission wrapper extends the attributed T16 lift to accept an explicit complete, causal, paired call order. It checks every replica's stage order and derives windows, provider times and the literal schedule digest. The downstream controller, physical invoice, both supplier checks, 47 outer margins and independent T16 audit are retained. The final map uses 10,017 actual charts; the finite invoice still charges the same stock, 512 prefix bits and physical row coefficient 16,469. Invoice comparison permits the actual call permutation and source-binding hashes only; all complete call records and cost fields must agree.
+
+Run `python3 scripts/research_package.py --package research/t16-reordered-augmenting-chain`. See `VALIDATION.json` for the exact executed checks and export replay. `PROOF.md` explains the schedule extension and scope.
+
+Credit: the direct-sum T16 controller and independent audit originate with Hao Yan (@FanPUYen), PR858; the retained strict admission follows Philippe Van Velzen's PR1087 and the PR3308 lineage. The predecessor incorporates the contributions credited in retained PR3375/3366/3359 sources, including Nick Petalas's PR3338 and the preceding explicit owner cycles. Source authorship and Apache-2.0 attribution are preserved. Pierre (hexapode) prepared this contribution with substantial OpenAI Codex assistance; no independent human review is claimed.
+
+## Open next questions
+
+Can the three empty rank-16 bins support a further legal concentration? Begin with the exact final map and price both suppliers after an explicitly audited whole-helper exchange. Window geometry and per-call family uniqueness remain constraints. Additional causal schedule changes are another candidate domain; their owner map, event projection and complete invoice must be rechecked.