[Dugongue]

Prove an exact joint coframe optimization theorem for finite Gaussian programs, with the complete paid-minus-productive inventory. The inverse computational Lagrangian gives the right transition rank. Its powered distance is a metric, so binary moves to a common anchor produce submodular paid tables. If helper endpoint changes are unary, one minimum cut finds the best joint move, including every target debit. A feasible predecessor assignment remains available; no strict new instance improvement is asserted without a witness.

Extend the search in two ways. First, specialize the retained universal free-native-group word to identity and pay the resulting physical transitions, instead of requiring every native warp to stay ordinary. A common h-plane inverse-Lagrangian family has transition ranks at most h, including degenerate binary subspaces. Second, handle bivariate helper targets by signed-graph switching and finite branching; a spanning-forest change of coordinates concentrates fixed Clifford transport into explicit cycle edges. All phases, matchers, rank caps, independent inputs and owned scratch remain in the physical construction.

Prove a complete-inventory entropy-curvature lower bound and rational sufficient inequalities for the first-moment, higher-moment and completion guards. These provide a separate acceptance certificate for a candidate found with substantial computation. No optimizer output or numerical root is assumed correct.

Six informal mathematical documents include proofs and five concrete implementation/validation handoffs. Start with coframe-cut.md, then native-specialization.md and funding-certificate.md. coupled-labels.md handles the more general signed and transported families; handoff.md fixes outputs, acceptance criteria and stopping rules for each task.

This is an ideas-and-proof handoff, not an executed search, a new funded Gaussian caller or a claim of whole multiplication kappa >= 0.0035. The retained conditional BIT saving 0.003503 is unchanged. No mathematical computation, numerical evaluator, source replay, solver, Lean run, web search or remote refresh was performed. Source baseline: 5e0052c. Check current overlap before claiming implementation priority; no worldwide novelty is asserted for the underlying graph-cut, metric, switching, gauge or Taylor tools.

Credit the earlier restricted-C auxiliary-line construction, its same-price transpose B interface and uniform C/B decoder; the native-dual-suppliers restriction/quotient lineage; Cao Yifan PR4610/PR4470; Cinna Davis PR4694/PR4698 and SOURCE/frame/helper/controller work; DaysSky; Nick Petalas; Ryanson Jonathan; zxv; Jacob Alan Sussman; Dugongue's earlier transported-frame work; and all source Clifford, scalar-phase, ownership, matching, compiler, precision and analytic notices. Raw/coded-provider context retains Dillon Amburgey PR5402, Chris R / Clear1nsight PR5407/PR5413 and Ali Romman PR5383/PR5398. The unchanged BIT checkpoint retains ai-nanachi PR5512 and its credited deferred-input lineage. Local proof-tree precedents are identified as precedents, not unproved funding inputs.

New text is Apache-2.0 with substantial OpenAI Codex assistance. Only research/joint-gaussian-coframe-search is added.