[aliromman]

For every positive tensor width, every decoder of the actual partial-cache observer has worst-case Euclidean error at least 1/sqrt(19) on the physical input unit ball. Randomized decoders have worst-case expected squared error at least 1/19. This is a scoped quantitative extension of the exact observer kernel; kappa is unchanged.

Proof and scope, exact checks, source pins, registered replay. Base: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7. Contribution head: c732dfe12329f271b9d03278a40d600930091d16.

Pulling the 104-role raw witness through the actual provider entrance leaves 61 physical roles. The physical squared input norm is 76, while the hidden target squared norm is 4. The proof permits arbitrary nonlinear processing of every retained full-frame history and all eight endpoint caches. The constants are sharp on the exhibited whole-field kernel subspace; global minimax optimality is not claimed.

CPython 3.12.14 warning-strict registered normal and optimized checks pass. Five byte-exact source copies match their Git objects and SHA256 pins. Checks cover 244 full basis inverse products, 115200 literal monomial address actions, the target coefficient by two calculations, native unitarity, 117 rational square-loss identities and 14 rejection controls. Both modes print PASS_PARTIAL_CACHE_MINIMAX_FLOOR; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE.

python3.12 -B scripts/research_package.py --package research/partial-cache-minimax-floor

Python 3.12 and the standard library suffice. Committed-byte checks pass. The complete predecessor history-annihilation proof and all-width native realization are credited premises. Their complete scientific suites, human/proof-kernel review and selected cold replay were not run. The domain is continuous complex input with the stated Euclidean norms. No discrete rounding theorem, whole-machine blindness, paid-call floor, supplier admission or complete multiplication theorem follows. Hosted status is unconfirmed at publication.

Credit JoyboyBrian (@JoyboyBrian) for PR #5498, its pinned exact observer proof and kernel. Credit Dillon Amburgey (@dillona) for PR #5402 and its pinned physical entrance and provider proof. Their retained native, history and consumer lineage is preserved. Scoped claim.

Next: identify a versioned additional observation that separates this whole-field kernel, then price its physical availability and retention. The full-ball minimax constant and a discrete-input version remain open.

Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.