[thu-nmrc]

The odd-E8 family in #5439 leaves a physical side producer and its caller unresolved. This PR supplies constructive native components: the whole K core, both source-consuming low-rank E adapters, and a retained directed cross-cube primitive. The E construction supplies an exact typed three-stage caller with one decoded Gaussian half. Its negative coded half remains unresolved, so there is no new kappa or complete Gaussian supplier claim.

Source-consuming low-rank E adapters

The new research/odd-e8-lowrank-e-native package combines the actual whole-K word with spectral-mode queries. On the central line Q, fresh rank24 Gaussian D, and R=DQ, let C=F3(C_base) in the physical real-normalizer similarity and E=K(I-C)^-1. The actual C spectrum is 0, 98, -1078, with exactly 20,480 nonzero modes. K is a real symmetric involution, KC=-C, and E commutes R; E need not commute Q.

For H(T)=diag(T,Q^-1 T Q), the exact local contracts are

W_F = H(E) diag(D,D) L(Q^-1)
W_B = H(E^-1) diag(D,D) U(-Q)
paid per FULL: {8:v,9:v,15:v,16:v,25:20480}, v=112^3

Each invocation has two banks of arbitrary independent original physical ground inputs. The backward correction gathers actual original Y before precode/overwrite, pays one R25 Gaussian per nonzero mode, and scatters -R C Y before postX(E^-1). An explicit rational 112-owner basis and its in-place ADD/SCALE/SWAP words define the global gather/scatter. These are structured loops, rather than an expansion of hundreds of millions of repeated scalar events.

The forward construction does not assume access to a thin query on Q^-1 X. It independently constructs the backward inverse: save each source at its actual common-frame contact, finish the source inverse to recover original Y, undo the correction, and execute reverse target work against those saved contacts. The owned program is then linearly transposed, with explicit COPY/CLEAR dual rules and ordinary D-squared/J boundaries. It never inverts a destructive erasure.

The exact root512 checker authenticates 512 contacts and 38,912 reads, emits literal inverse/inverse-transpose unary realizations and checks both full main-operator composition orders. Complete small Q(i) fixtures check original-input acquisition, absolute phases, all owned lifetimes and transpose boundaries. A separate fixture binds two actual E8 cubes, 32 independent fields, 192 operator blocks and 264 phase checks. The separate AI peer controls import no author code; they independently check the mode basis, physical normalizers, native contact structure and exact small operator identities. The reviewer authored the two-cube fixture, so this is scoped peer separation, not independent human or full-chain reconstruction.

Ordinary work is explicit: the global mode gather/scatter alone has 129,905,664 ADDs,1,204,224 SCALEs and1,053,696 SWAPs, plus normalizers and mode operations. The local unified upper bill also counts two E recodings, native boundaries and source contacts. Its conservative owned peak is v+20480+512, in addition to original main fields. These are field-level upper bills. The finite norm products bound whole-field-l2 error amplification; converting coordinatewise fixedpoint errors to that ABI and completing outer TM, routing and storage costs remain open.

The actual F1,B2,F3 word has no IDLE or thin query. After its ordinary final signed swap the main endpoints are

Q^-1 B_E D123             and             B_E D123 Q
B_E = E3 E2^-1 E1

An ordinary B_E^-1 decoder makes the positive half independent H73 jobs. The negative half is still coupled/coded. A conditional screen granting it H73 has positive moment near .04382074131724228 per port per local call, certified by rational log/exp bounds. That grant is UNKNOWN and is not productive target credit or an exponent claim.

• All-width construction and exact limits
• Explicit modes, costs and conditional screen
• Separate scoped peer review
• Precision ABI qualification

Whole K and retained cross-cube components

research/odd-e8-whole-k-native constructs the entire block-diagonal K core on arbitrary independent ground inputs:

F_K=diag(D,D)L(K Q^-1), B_K=diag(D,D)U(-Q K)

Each port pays H8+H9+H15+H16, with 76 ADDs and no helper, owned field, COPY or CLEAR. The structured constructor covers 2,744 cubes and 1,404,928 ports; the root cube and a second distinct cube were fully emitted/replayed. All-family geometry and generic absolute-phase proofs are distinguished from executed instances. Construction, invoice, backward word, review.

Its terminal support appendix proves a scoped upper support bound for Q^-1 B D with distinct-fresh fitting K stages: each path has Gaussian rank n-2. Fixed finite ordinary monomial/owner-scalar adapters on that single cohort cannot produce higher-rank Gaussian jobs at all widths. The proof has primary and separate portable controls, including all 38,912 committed edges. It is not a full-caller lower bound; nonfitting E codes and changed protocols are outside its hypothesis. Existing signed-normalizer and terminal-support predecessors are credited.

research/odd-e8-polarized-native is byte-identical to c9875dc. It retains the directed H_ab transfer with paid ranks 7/8/16/17 per port. Its incoming line services remain caller obligations. It is a separate component; the new E route uses spectral queries. Retained interface.

Reproduction and scope

Python3.12 and the standard library suffice:

python3.12 -B scripts/research_package.py --package research/odd-e8-lowrank-e-native
python3.12 -B scripts/check_research_commit.py --package research/odd-e8-lowrank-e-native
python3.12 -B scripts/research_package.py --package research/odd-e8-whole-k-native
python3.12 -B scripts/research_package.py --package research/odd-e8-polarized-native

The E package's four registered hooks and exact 41-file committed-byte check pass. Official committed export/check and separate Git-free, read-only, empty-PATH replay pass. Its scientific code, inputs, programs and receipts are byte-identical to the reviewed local author seal; only declared publication metadata and portable reviewer path mapping differ. The unchanged K package's eight grouped hooks/twelve scientific subcommands and the retained cross-cube package's earlier cold replay also pass on their exact bytes. Canonical decompressed scientific payloads are checked independently of gzip container metadata. A full source-history ancestry check is not claimed for the shallow local checkout. Hosted CI is separate execution evidence.

The remaining decisive gate is a profitable all-width negative E Gaussian decoder or a genuinely changed caller/target inventory, followed by complete ordinary/precision/storage admission. Neither a conditional positive screen, CI nor this component PR is scientific promotion. At canonical main 32890bc, selected conditional kappa remains 0.003027288235 under its retained framework assumptions.

Bounded primary overlap read the coded-native GAUGE/PHYSICAL sources, #5439's separated-lift result and #5467's native/decoder/precision components. No identical low-rank R correction plus source-contact inverse/transpose construction was found there. Generic gauge identities, Clifford transposition, root modes and support-counting methods are inherited. A final exact-head check also read #5469 at3dc2edc4044bad430581c1fafa6d472cab480b78. Its b=ZQY theorem assumes only post-return data and explicitly leaves pre-return copies outside that model. This construction retains original ground-Y modes before overwrite and computes a different rank20480 R25 correction, so it does not contradict that theorem or construct its prescribed exit. The pre-return lever is an acknowledged existing escape, not a new general idea. No exhaustive literature or unpublished priority claim is made.

Credit Ali Romman; Dugongue and Dillon Amburgey; Cao Yifan; Cinna Davis, zxv, Jacob Alan Sussman and the retained native/Clifford/compiler/analytic lineage. Prepared for THU New Media Research Center (@thu-nmrc) with substantial OpenAI Codex assistance, including separate AI-worker reviews. Apache-2.0; retained input notices remain in place. No independent human or proof-kernel acceptance is claimed.