[Dugongue]

Specialize the concentrated spectral interpolation architecture in OpenAI's October 25, 2026 ETH preprint to the actual Gaussian tensor already specified by the project's GaussianTensorExecution source. The result is an explicit informal Gaussian precision and response compiler, not a new numerical kappa or a certification of the preprint's complete ETH claim.

For C=[[i,1],[1,i]], pi=1+i and A_n=(C/pi) tensor n, interpolate the actual four-phase spectrum i^j on a consecutive interval [a,a+R]. The explicit polynomial g(X)=i^a sum_(h=0)^R (i-1)^h binom(X-a,h) agrees exactly on those Walsh-weight spaces. Its ordinary numerator pi^R g(W) has a Gaussian-integer dynamic-program word, bounded coefficients, O(nR+R B_c+1) whole-array work and at most R+4 owned arrays. The exact residual has an emitted exceptional Walsh expansion, a binomial-tail rank bound, a norm bound and its dense correction bill. Computing the bulk modulo pi^p needs its numerator only modulo pi^(p+R), versus pi^(p+n) for the common numerator/correction: n-R fewer Gaussian precision units for the stated bulk. Small exceptional rank is not assumed to be a cheap native query, and generic dense-input runtime may be worse than existing fast transforms.

The complementary ramified Gaussian Hermite theorem reconstructs an exact origin from tapered nonzero-node jets, with explicit node precision P_t=P+16hD_t+16M(a+1). Under the stated inherited-precision hypothesis this is O(MD_t/u), versus Theta(MD_t) loss for the specified uniform-multiplicity formula. The package includes complete proofs, integral Cauchy local expansions, a hand-worked recovery, net mode weights after denominator debt, exact ordinary decoding, joint valuation/Taylor filtering and actual source-preserving query delivery. No global contraction runtime is assumed as an unpriced oracle.

A literal six-update P104 packet is affine in all six independent coefficient parameters: the full event-touch predicate proves that it has only a constant and six linear responses rather than 64 formal subset coefficients. This is an exact symbolic-construction saving. Its three pending-sum acquisitions are already PR5802 work and are not credited again. The full signed-invoice sufficient interface retains all target debits, ordinary passes, cache construction, dirty restoration, precision and selected guards.

Evidence is pure informal mathematics and literal source inspection, with independent cross-reviews among one coordinator and three workers. No solver, mathematical computation, operator/invoice replay, benchmark or Lean run was performed. Publication bytes were checked separately. There is no new paid/target histogram, supplier admission, selected-frontier change or rounded multiplier. All five inherited framework assumptions remain. Open next questions identify a literal exceptional-summary consumer, its net-weight table and the complete funded replacement; no favorable unknown gadget is assumed in the proved constructions.

Sources are pinned to OpenAI/math 7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a and canonical integer main de5ec2e, priority 8. Current guides, selected record and recent PRs were read. Final refresh found PR5811 at 883ab9c claiming conditional kappa=.00312418633064 and crediting the concurrent PR5806 baseline join. That is predecessor work, not this package's result. The selected .00310045 and admitted BIT .0034 remain distinct. PR5808's fixed immutable-channel exclusion and PR5809's accumulation audit are acknowledged within their actual scopes.

Eight mathematical/source notes are added only under research/eth-gaussian-spectral-transport. Start with gaussian-spectral-bridge.md B1-B4 and preprint-assessment.md G4-G6, then endpoint-lowering.md. The source map records existing Gaussian tensor/precision, linear invariant, TI/module, Schur and RAM precedents without claiming a fresh kernel audit. Classical component facts retain their attribution; no worldwide novelty or project-wide nonuse is claimed.

Credit OpenAI for the preprint architecture and its cited antecedents; Ali Romman, Joel Pulikkan/GamingPuzzled, Dugongue, Mason Hale, ToikY, flyingviper, Pierre-Loic Doulcet/hexapode, Cinna Davis, Cao Yifan, DaysSky, Nick Petalas, Ryanson Jonathan, Jacob Alan Sussman, zxv and all retained source/native/controller/compiler/ownership/BIT/precision/analytic contributors. Detailed predecessor notices are retained. New text is Apache-2.0 with substantial OpenAI Codex assistance.