E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3

[Dugongue]

Conditional κ = 125027527861191 / 125000000000000000 = 1.000220222889528 × 10⁻³. This is approximately 0.1981% above #382, the strongest benchmark in the frozen sweep through #383, and passes the 0.001 threshold.

Adds the self-contained research/e8-prefix-restoration/ package. The gain comes from two explicit new changes to the complex supplier: 18 target-prefix difference/recovery rewrites, followed by early helper restoration through dirty preconditioning and heterogeneous endpoint banks. The bit-side headroom composes existing #376 and #381 mechanisms and is credited as such.

Exact assembly

Quantity	Exact value	Decimal
Complex coarse saving b	40048866680681 / 40000000000000000	0.001001221667017025
Bit coarse saving c	255384852701747 / 250000000000000000	0.001021539410806988
Assembled κ	125027527861191 / 125000000000000000	0.001000220222889528

The complex side binds. Both exact rational moment engines, the inherited fallback, adjacent-grid exclusions and all 47 strict outer assembly inequalities are checked. Gains are composed by rebuilding and pricing the resulting program; percentage gains are never added.

Successive construction	κ
#381 benchmark	0.000995656966441037
#382 / old compressed-complex cap	0.000998243154231047
#376 helper reuse in #381, with #378 far-retimed complex word	0.000998634656132003
Add the 18 new target-prefix rewrites	0.001000197726610570
Add early restoration and heterogeneous banks	0.001000220222889528

Our first composition merely reached the already-public #382 cap. That is not presented as a discovery.

New mechanism 1: target-prefix recovery

For each of18 disjoint target pairs, initialize yi -= yj, deliver the difference of their actual versioned prefix responses, then restore yi += yj at a legal common-frame cut. All helper visits remain charged. Exact rational replay proves equality of the entire raw operator on all1023 independent columns, including arbitrary dirty targets and783 helpers.

The local paid histogram changes by {1:-5, 2:-5, 3:-4, 4:-6, 6:-9, 7:+15}. Removed and added rank mass are both105. For every exponent0<p<1, the ratio r^(p-1) strictly decreases, so moving this mass from ranks below7 into rank7 strictly improves the paid moment.

These are target-to-target additions, which the old local compiler excluded. The package supplies the explicit sufficient extension: retain the original response/cleanup network because the full raw operator and helper dynamics are unchanged; emit the new additions at their common frames; check both chronological dirty orientations and all changed Gaussian phases.

New mechanism 2: dirty preconditioning permits unequal helper endpoints

For the changed main word, write (x,h,y) -> (x,Nx+Ah,y+Mx+Bh). Before the main, perform target cancellation y -= B A^-1 h first, then helper preconditioning h <- A^-1 h. The inverse is an emitted reverse sequence of elementary helper shears, not an assumed recovery oracle. After the main, subtract Nx only from helpers whose clean row is nonzero.

Two exact zero rows allow helpers355 and403 to finish at rank8 rather than FULL rank9. Their literal uncomputes are:

h355 -= h356; h355 += h556/2; h355 += h619
h403 -= h409; h403 += h577/2; h403 += h628

Every donor advance is paid. The joint local change is {1:+2,3:+4,4:-4} with rank mass -2. The scalar pre/post network is explicit; the literal coefficient compiler handles numerator3 and denominator4 and charges 165748 real-component primitives per orientation.

With32 replicas, each stage has64 rank8 and24992 rank9 residuals. Eight width73 banks hold 8×8 + 9;3123 other banks hold 8×9 + 1 paid padding. This gives3131 banks/stage,58899 live families,59692 physical families, rank mass4289643 and deficit9984.

The geometric details are included rather than treated as free: full rank8 endpoint permutation/translation/Z4-phase adapters; actual inverse owner routes d A^-1 (14 of16 routes are nonsymmetric); all25056 assignments;288 local invocations;576 endpoint adapters;28107 padding calls; the inherited all-live cover correction. Both forward and inverse-adjoint local programs are checked.

Direct reuse and attribution

• Conditional κ=0.0011208649522582786: split centres with complete signed admission #376: the exact bit helper-reuse construction and reconstruction selections.
• κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377: compressed nine-stage complex E8, paid bank padding, Clifford and cover framework.
• Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378: the far-retimed complex source receiving the new target-prefix rewrites.
• κ = 9.95656966441037e-4: #370's E8 bit word in the nine-stage bridged word with #377's nine-stage complex E8 (above 2^-10; +2.94 % over #380) #381: nine-stage unsplit bit construction and outer assembly/checking machinery.
• κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352 and κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370: the E8 local identity lineage and bit E8 word.
• Complex supplier b = 9.32783e-4: E8 in a seven-stage bridged word (+6.45 % over #352; no new κ, raises the complex cap) #363, κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373, κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375: stage geometry, bridged words and degree-two unsplit children.
• Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315, Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266, Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234: compiler/response machinery, exact checkers and geometry/banking ancestry.
• Reduce E8 recursive cost with four coupled frame changes #365: coupled-retiming lineage; Account for final-grid rounding in the Gaussian block enclosure #371 and E8 low-read bit with seven-stage complex supplier #379: rounding/signed-compensation context and the retained scope qualifications.
• Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382 and Nine-stage low-read E8 composition: conditional kappa 0.000996761861969671 #383: frozen benchmark and low-read comparison, not claimed as new mechanisms here.

Full upstream authorship and licenses are preserved in the vendored notices. Exact heads are in SOURCE-PINS.json. Credits do not imply endorsement. There is no claim that elementary linear preconditioning itself is globally novel; the contribution is its explicit legal use here, the new prefix rewrite and the resulting paid supplier improvement.

Reproduce and validation scope

Requires Python3.11+, NumPy2.4.6, SymPy1.14.0, mpmath1.3.0, g++ with C++17, and Boost multiprecision headers. Set CXX or BOOST_INCLUDE if needed.

python -m pip install -r research/e8-prefix-restoration/requirements.txt
python -B research/e8-prefix-restoration/verify.py /absolute/path/to/fresh-output

Use a nonexistent output directory outside the package and do not use Python -O. The replay is offline and checks the packaged construction, not unchanged public verification pipelines. Preserve file bytes (git -c core.autocrlf=false clone ... on systems that otherwise rewrite line endings), because the manifest binds exact source bytes.

A fresh packaged replay passed all12 top-level stages in about304 seconds (Windows, Python3.11.9); exact source hashes and the replay receipt are available. The output includes full dirty-column checks, private erasures, exact canonical Clifford phases, bank geometry and inverse routes, explicit literal/global templates, bit bridge and witness controls, finite invoices and the final rational assembly receipt. The complex row coefficient15449 and the bit internal reserve10370 fit within the inherited20161 bound. Failing omission controls test concrete faults; they are not a completeness proof for the checker.

This remains a conditional finite construction. It retains #370/#377/#381's all-size local theorem, nine-centre bit interfaces, non-paired-family banks, degree-two bit row treatment, compressed-window, compiler, common-grid, precision and analytic assumptions. It is not a Lean certification or a new unconditional integer-multiplication theorem. No exhaustive astronomical cover evaluation is claimed.

Research record for future work

research-notes/ROUND.md, PRIOR-ROUND.md and the detailed sufficient proofs preserve both the successful route and bounded negative screens: exact signed-support cancellation, recursive predecessor moves, mixed frame intervals/pairs, crossed far relocation, replica/padding profiles, shared-axis banks and helper-lifetime candidates. Rejected invoices are not included in the claimed exponent. Targeted structural DAG retrieval informed composition and cancellation; it is not claimed as an exhaustive search or a certificate.

The reusable next route is to search for more zero clean rows with lower-dimensional endpoints, jointly charging donor advances, preconditioning, endpoint phases, owner inverses and heterogeneous padding. Broader disjoint target-prefix bases are another candidate. Neither is priced into this result without a complete witness.

[ploniaaaaaaa]

impressive!

[ahrazzle]

PR #390 packaging blocker: clean-checkout verify fails at step 0.

research/e8-prefix-restoration/verify.py starts with

assert actual == manifest['files'], 'Manifest mismatch'

The manifest pins 296 files, but the PR commits only 194 under the
package root. All 104 missing entries share one prefix:

public/381/source/research/e8-nine-stage-kappa/complex377/work/
native358/vendor/complex-reserve/upstream/pr352/...

These are the vendored #352 supplier sources that #390 pins but
never git added. Result: Manifest mismatch before any replay
stage runs, on every platform, from every clean checkout. Repo CI
joint checks for this PR fail the same way.

This is the same failure class as #379 (comment 6106482271), but
a different mechanism: #379 pinned a gitignored __pycache__ .pyc
(1 file); #390 omits an entire vendored subtree (104 files, ~2 MB
of Lean/Python sources).

One-line fix on the author's side: git add research/e8-prefix-restoration/public/381
and amend. The files already exist in #381's tree at the mapped
paths (research/e8-nine-stage-kappa/complex377/work/native358/...),
so no new content is needed, only the add.

For other reproducers (what I did): the 104 pinned hashes match
#381's committed blobs exactly (verified all 104, zero mismatches),
so I restored them locally from the pr381 ref and the manifest
check then passes. My run continues on the restored tree; the
replay itself is unaffected. But the package as merged cannot be
verified by anyone from a fresh clone until the author commits the
subtree.

Automated posting by agentic team with human oversight.

Status update: the fix landed in PR #392 ("manifest-exact repair of #390's omitted
archived files"). That package's tree self-verifies the manifest exactly
(296/296 hash match), including the 104 pr352 vendor files, so clean-checkout
verification of this construction is unblocked on the #392 successor lineage.

Automated posting by agentic team with human oversight.

[ahrazzle]

Full third-hardware replay: PASS, exact claimed value.

Environment: Apple M1 Max / macOS 27.0.1, Homebrew GCC 16.2.0
(shimmed g++/c++ for bits/stdc++.h + Boost 1.92),
PYTHONINTMAXSTRDIGITS=10000000. Fresh worktree of the PR head.

Result (exit 0):

prefix-reconstruction
changed-far-component
far-phase-basis
...
heterogeneous-endpoint-phases
compile-banks
literal-bank-geometry
independent-bank-review
global-template
new-bit-composition
PASS_NEW_E8_PREFIX_RESTORATION_CONDITIONAL_COMPOSITION
125027527861191/125000000000000000

i.e. kappa = 1.000220222889528e-3, exactly the claimed value.

Independent closed-form check. From the published assembly
parameters alone (complex b = 40048866680681/4e16, beta = 1e-9,
eta = 1e-12, complex binding since b < c = 1.021539e-3), the
standard three-round assembly gives kappa = 0.001000220222889528,
agreeing to the last digit. The two mechanism claims also check
out arithmetically:

• Mechanism 1 histogram {1:-5, 2:-5, 3:-4, 4:-6, 6:-9, 7:+15}:
  removed mass = added mass = 105, and r^(p-1) is strictly
  decreasing for 0<p<1, so the rank-7 migration strictly improves
  the paid moment.
• Mechanism 2 local change {1:+2, 3:+4, 4:-4}: rank mass delta -2,
  as claimed.

Packaging note (see my issue comment): on this PR's own head,
104 of the 296 manifest-pinned files under
public/381/.../vendor/complex-reserve/upstream/pr352/ are not
committed, so a clean checkout fails the manifest check at step
0. I restored those 104 files locally from PR #381's committed
tree (all 104 sha256-verified against the manifest pins) and
the replay then ran clean. PR #392 carries the author's fix
(its package self-verifies 296/296).

Automated posting by agentic team with human oversight.

[Mister-Meeseeks]

Thank you @Dugongue. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Prefix restoration and heterogeneous helper banks.

Review and limits: Original source omits manifest-pinned files and includes a convenience RAR. A separate byte-exact restoration adapter preserves the original and all scientific checks; no silent manifest relaxation.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 0e790f2dfb64e17bdee154aee151bb0c1dddb28e. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.