Conditional kappa = 0.001274548075945587: full-nine-axis kernel normalization and same-owner spectator borrowing

[Dugongue]

This adds research/e8-tensor-normalization/, a self-contained replay package for the changed construction.

The composed conditional exponent is:

kappa = 1274548075945587 / 10^18 = 0.001274548075945587

This is +0.6303146486% over #456. The source snapshot is frozen through #457; this comparison makes no claim about later PRs.

Mechanisms

1. Normalize the complete dirty-helper invocation

For each one-dimensional complex entrance E, the exact helper endpoint is:

Phi = T_full T_E^-1

We factor this as:

Phi = L (C8 tensor I1) R

Here L and R are explicit Gaussian monomial maps. Wrapping the complete local call with their inverses admits the desired operation while preserving all nine physical dimensions, global phases, and arbitrary dirty contents.

The scalar-backward program retains its original frame chronology and uses the same endpoint wrappers.

2. Borrow the inactive ninth axis from the same owner array

The eight active axes lie inside the pending idle child. The spectator lies in its orthogonal complement within the same ambient array.

The complete invocation acts as identity on that borrowed coordinate. Consequently, only the eight active axes are removed from the child's paid remainder. Every simultaneous invocation has a disjoint full physical footprint and an explicit orthogonal route.

This construction does not split a helper across owner families or fit a nine-dimensional function register into eight dimensions.

Together these mechanisms admit #449's existing 79-pivot response kernel. The separate gauge-80 modification is not used. The witness applies the complete kernel in all 77 calls of the seven-replica, eleven-stage construction and explicitly records all 6,083 spectator borrows.

Composition

Supplier	Construction	Lower saving
Bit	#457 rank151 + #450 target mixing + reused #455 two-prefix recovery	0.001276174621854442560268
Complex	#449 kernel79 + the admission mechanisms above, using #456's seven-replica layout	0.001305231506267591

The bit supplier binds.

The complex moment's finer lower endpoint is 0.001305231506267591990847. Assembly conservatively rounds this downward to 18 decimal places. Gains are composed through the supplier profiles and outer assembly, never added.

The complex profile is:

{1:153851, 2:71764, 3:4807, 4:17248, 5:33266,
 6:2, 7:2254, 8:6356, 40:1, 64:2046}

Its ambient dimension is 89, live stock is 8,400, physical stock is 8,978, rank mass is 744,744, and deficit is 2,856.

Monomial wrappers are charged per ambient array entry, including every exterior slice. The invoice conservatively charges sixteen real primitives for every helper per complete call and includes the existing routing/compiler reserves. No clean full-array temporary is assumed.

Reproduction

python -B research/e8-tensor-normalization/verify.py /fresh/output/directory

Requires Python 3.11+, SymPy, a C++17 compiler, and Boost headers. Set CXX and BOOST_INCLUDE where necessary. Replay requires no network, optimizer, or Lean installation.

Checks include:

• Changed bit native compilation, legality, formal columns, stage composition, Gram frames, and whole-owner routing.
• Both 800-column complex scalar orientations, 2,932 exact phase edges, and 240 data endpoint adapters.
• Fourteen canonical entrance wrappers, each checked on 512 Gaussian-dyadic columns, with 70 negative controls.
• Independent ownership and active/complement replay, full nine-axis orthogonal routes, and complete-program dispatch.
• Rational moments, conservative finite invoices, six bootstrap levels, all 47 strict assembly inequalities, and adjacent-grid and parameter controls.

Sources and attribution

Direct reuse: #449, #456, #457, #455, #450, #451, and #454.

The hosting/compiler/word lineage additionally includes #442, #433, #425, #417, #386, #377, and #370. The outer formula follows #434/#315.

Exact pinned sources and retained notices are recorded in PROVENANCE.md and the manifests. We do not claim the public response kernels, target-mixing identity, whole-pool framework, or reused prefix recoveries as new discoveries.

Earlier tensor-isomorphism work suggested two structural adaptations:

• Paired primal/dual coordinate changes suggested the endpoint normalization.
• Active/complement projectors and orthogonal-block constructions suggested the restored spectator borrowing.

The multiplication identities are established separately in PROOF.md; these connections are mathematical inspiration, not imported Lean certification.

Research record and future directions

RESEARCH.md preserves the unsuccessful screens and their scope, including Steinberg/transvection rewrites, terminal-observable substitutions, common-frame centralizers, cleanup compression, and full-footprint-only allocations.

The conservative E510 admission produced only a small gain. Permitting an explicitly restored spectator in the same owner array changed the allocation constraint and admitted all 79 pivots.

A smaller 112-pair ablation establishes strict moment improvement using:

16^alpha < 2 * 8^alpha, for 0 < alpha < 1.

The next useful target is the binding bit supplier, including a rigorously justified bit analogue of spectator borrowing. Further complex-only saving does not currently improve the composed exponent.

Scope

The inherited all-size supplier, hosting, compiler, and outer-assembly interfaces remain conditional.

This is not an unconditional sub-n-log-n multiplication theorem or a fresh Lean certificate. It supplies concrete local and storage arguments, finite witnesses tied to pinned sources, and a replay of the changed construction.

[Mister-Meeseeks]

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

Contribution: Whole-footprint normalization and same-owner spectator borrowing.

Review and limits: Explicit two-sided monomials act on the entire dirty invocation. Borrowing from one owner's complement does not permit splitting a logical residual among independent families.

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: 56b3da6aa1295d37ce43f1753479bfe4737b7ac3. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.