Conditional kappa 0.001001256225581264: signed prefixes and padding fusion

[rohanarun]

Conditional κ = 0.001001256225581264 (62578514098829/62500000000000000), 0.016580% above this PR's signed-prefix-only result and 0.086998% above #394.

This combines our 25 signed target-prefix pairs with chafreaky's #394 three free sink helpers and padding fusion, retaining Dugongue's #390 dirty preconditioning, early restorations and heterogeneous banks, plus the #392 exact finite bootstrap. Both constituent packages remain separately replayable in this branch.

The 25-pair construction generalizes #390's prefix identity to lambda in {-1,1}. On arbitrary initial values, yi -= lambda*yj; yi += U-lambda*V; yi += lambda*yj restores the original increment U. Tokens may cancel only when physical donor, exact write version and address frame agree. Its extra local histogram change over #390 is {1:-3,2:-3,3:-3,4:-2,6:-5,7:+8}.

#394's nine free helper shears make three additional clean rows zero. After exact prewhitening and final repair, these helpers need no subsequent scalar or climb. Each final rank-two climb is tensored with one bank-padding C, forming a rank-three child. We add three exact ten-coordinate Clifford compiler certificates (nine local coordinates and one exterior coordinate, with 63 identity spectators), including the complete Gaussian global phase.

The fresh global template maps all 864 fused occurrences to literal bank owners. Their routes fix coordinate 72, their stage/bank pairs are distinct, and they replace exactly 864 of the original 28107 standalone paddings, leaving 27243. Five global mutation controls reject missing fusion, double padding, wrong route, wrong rank and omitted cover correction. Eight local controls cover timing, cleanup and scalar signs. The profile delta from the signed-prefix baseline is {1:-864,2:-864,3:+864}; live stock 58899, physical stock 59692, rank mass 4289643 and deficit 9984 are unchanged.

The finite invoice is recomputed, including literal coefficient expansion: 165812 real-component primitives per local orientation, plus an extra conservative 55296 compiler groups for fusion. The per-vertex bound is 2626110500158867 and the row coefficient remains 15449 < 20161. Both rational moment engines keep the full positive 10^-16 fallback, and the six-level bootstrap checks all 47 strict outer constraints.

Full combined validation passes on Python 3.11 and 3.14: GitHub Actions run 38123176066, commit c60df7a806d1525c461b4b674598c37685c5ff3a. Both saved artifacts report PASS_FULL_SIGNED_PREFIX_PADDING_FUSION_AND_BIT_COMPOSITION. Each run freshly executes the entire signed-prefix baseline (including all twelve #390 stages), rebuilds the fused local programs and global placement template, checks the exact mixed-child phases, rejects all 13 fusion controls, recomputes the finite invoice and two-engine coarse enclosure, reruns the bit composition at the changed complex bound, and checks all 47 outer constraints. Expected records are compared only after execution. The complete artifacts retain the intermediate receipts and logs.

The broader repository matrix currently has 58 checks passed and four still running; this PR remains a draft pending that matrix.

Current comparison: later #398 reports a larger conditional value, 0.001002494659629864, using an R781 supplier. This PR's fully checked construction is below that later result; it retains the independently verified fusion refinement documented above.

python -m pip install -r research/e8-prefix-restoration/requirements.txt
python -B research/e8-signed-padding/verify.py /fresh/output

C++17 and Boost headers are required. Original supplier code and licenses remain unchanged, including #392's manifest-exact repair of #390's omitted archived files.

Conditional boundary: finite construction, not a new Lean theorem or unconditional multiplication bound. All inherited all-size local identity, compiler, arbitrary-dirty banking/cover, owner-route, fixed-tape, precision/recovery, restored-row, nine-centre bit and analytic interfaces remain assumptions. PR394's end-of-stage padding timing is explicitly replaced by its operator-level commutation argument under the arbitrary-dirty invocation property. The exterior child is explicitly compiled above; unchanged intermediate exterior canonical labels are not asserted. Floating matching weights only nominated the signed selection; no global optimality claim is made.

Credits: Chafik Boukhalfa / chafreaky (#394 fusion, free sinks and bracket compiler), Dugongue (#390 construction), the #378/#377/#381/#376 supplier lineage and all inherited notices; Rohan Arun (signed prefix extension, matching discovery, combination, exact mixed-child certificates, global template and recomputed invoice), with OpenAI Codex assistance. Apache-2.0; unchanged #394 sources and original notices are in research/e8-signed-padding/vendor394/.