research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3

[Dugongue]

Result and frozen comparison

This adds research/e8-response-subspaces, a self-contained finite conditional construction with

kappa = 101331088745971 / 100000000000000000 = 0.001013310887459710.

The comparison is #399 at b974e481efd2f771cda2dcdd71c91f133793558f, claiming 0.001002661106042177. The relative improvement is 1.0621516436%. The source sweep is frozen through #399; this does not claim priority over later submissions. No separately reported gains are added.

What changes

The starting word is #398's R781 signed-prefix construction. Instead of restricting entrance gauges to existing frames or odd rank-one vectors, we synthesize 26 new response subspaces of dimensions 1, 2, 4 and 5. The fixed witness has 119 pivot helpers, 143 donors and 240 literal entrance shears, with unit numerators and denominators 1 or 2.

For each pivot, the exact target-response columns satisfy K_p + sum c_pd K_d = 0. Disjoint group role sets make the entrance transformation explicitly invertible. The constructor checks the complete identity M_new = M_old diag(I,I,S) in all 1,021 raw columns and checks every nested frame transition; clean-input agreement is not the acceptance criterion.

The additional mechanism is two-sided residual normalization for even or degenerate entrance spaces. For the actual phase-sensitive chart T_E, exact Clifford decomposition gives F T_E^-1 = L C_(9-dim E) R, with explicit rank-zero monomial unitaries. Applying R^-1 before the complete old-response phase and L^-1 after complete cleanup leaves the data unchanged and exposes the standard coordinate residual. This uses a separately checked block-diagonal dirty endpoint, not just a zero response column, and does not commute adapters through internal gates. Both orientations and all Gaussian phases are checked. All 17,136 boundary adapters are emitted and charged.

The mixed residual inventory per stage is {4:120, 5:208, 7:64, 8:560, 9:5296}. Explicit coordinate routes place it into 748 width-73 banks with padding 482 x rank1 + 1 x rank10. The literal inverse owner matrices, actual owner injectivity, inherited interleaved data/idle schedule, cover corrections and private namespaces are independently decoded.

Exact accounting

Quantity	Value
Local replicas / stages	8 / 9
Complete local invocations	72
Banks per stage	748
Live / physical stock	14,412 / 15,203
Paid child count	316,107
Rank mass	1,049,580
Deficit	2,496
Maximum child rank	48
Complex coarse lower bound	507169363968149971159 / 500000000000000000000000
Unchanged bit coarse lower bound	255384852701747 / 250000000000000000

The finite reserve bridge is recomputed from the changed emitted source and independent decoder. The required row coefficient is 15,445, below the retained 20,161 allowance. Two exact rational moment engines and six fixed ordinary bootstrap levels yield the stated rational kappa through all 47 strict outer inequalities. The complex supplier still binds.

Reproduction and evidence

From the added directory:

python -B verify.py /absolute/path/to/a-new-output-directory

The default replay uses Python 3.11+ and GCC/Clang C++17 (CXX may be set). It is offline, source-hash checked, and does not rerun public PR pipelines or discovery searches. It freshly checks the changed scalar programs, both entrance orientations, 532 new/reflected local Clifford edges, all 13,312 residual address columns, emitted banks and invoice, eight adverse structural controls, the new exact reserve bridge, moments and outer assembly. The unchanged bit component and old local phase edges are reused only through pinned prior receipts and exact source/witness bindings. --replay-bit optionally reruns the bundled bit component with its documented dependencies.

VALIDATION.json records executed source hashes and the final result; MANIFEST.json covers the upload bytes. No compiled binaries or discovery outputs are required. Research notes retain sufficient identities, bounded search results and unpriced routes for future workers.

Direct sources and related mechanisms

• Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 supplies the exact signed-prefix R781 input and prior phase evidence.
• Conditional kappa 0.001001098432210898: signed-prefix donor gauges #396 directly inspires the entrance response-kernel route. This package extends it to synthesized larger/even/degenerate spaces using explicit two-sided normalization; it does not relabel entrance kernels as an unrelated invention.
• Conditional κ=0.0011208649522582786: split centres with complete signed admission #376, 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, and their Reduce E8 recursive cost with four coupled frame changes #365/Conditional κ = 9.54395132518163e-4: coupled complex retiming after #375 (+0.01588%) #374 retiming lineage supply the helper substitutions and frame-repaired baseline; the reviewed new Conditional κ=0.0011208649522582786: split centres with complete signed admission #376 head is distinguished from the exact older baseline in NOTICE.md.
• κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377 supplies the nine-stage complex geometry, phase-sensitive representatives, owner routes and invoice framework.
• 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 and Conditional κ=0.0011208649522582786: split centres with complete signed admission #376 supply the unchanged bit composition and lifetime reuse.
• E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390 supplies the positive-prefix mechanism extended to signed prefixes in Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398.
• Conditional kappa 0.000998634658090835: finite bootstrap and exact coarse precision #387, Conditional kappa 0.001000220224856110: six fixed levels on #390 #392, Finite leaf bootstrap above PR182: conditional kappa = 0.000660627334074291 #185 supply the ordinary bootstrap/slack construction; Conditional κ = 9.98634656132003·10⁻⁴: #377's nine-stage E8 complex component with our #378 re-timed E8 unit, paired with our #382 bit word (+0.039 % over #382) #384 carries the exact outer engine, with 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/Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144/Semantic precision and bulk resampling for a conditional 1099/10^8 > 2^-17 bound #23 ancestry retained.
• Conditional κ = 1.000385913854264·10⁻³: bank padding fused into three new free sink helpers on #390's E8 complex program (+0.0166 % over #390) #394, Conditional kappa 0.001001256225581264: signed prefixes and padding fusion #395, Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398) #399 were reviewed for sink/prewhitening/padding fusion. That different cleanup bracket is not composed here. Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398) #399 is the frozen benchmark, not an extra gain added to the price.

Further compiler/supplier ancestry includes #352, #363, #373, #370, #375, #266 and #234. All bundled upstream headers and notices are retained.

Scope and future work

This is a finite conditional construction, not a Lean certificate or an unconditional integer-multiplication theorem. The inherited all-size compressed-window lift, arbitrary-correlated dirty-bank semantics, recursive compiler, ordinary-call/fixed-tape/restored-row, precision and analytic interfaces remain assumptions. Finite endpoint/phase checks do not discharge those contracts.

The new route exposes a broader entrance-space design space. Research notes explain the failed simultaneous-target basis/readout/overlapping-bank screens and the sufficient auxiliary prewhitening identity, without claiming they yield extra kappa. Potential next steps are compatibility with the distinct #399 cleanup bracket, further actual mixed-rank inventories, and controlled donor-group overlap. None is assumed here, and bounded searches establish neither global optimality nor literature-wide novelty.

[Mister-Meeseeks]

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

Contribution: Response-subspace gauges and two-sided normalization.

Review and limits: Exact complete dirty operator, phase pairs, banks and outer assembly. Default verifier explicitly reuses the unchanged source-bound bit receipt; optional bit replay is separate.

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: 7844f72c6051a4408284c3092490687061675c1a. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.