[petalas]

Result

Scoped exact finite certificate. No new κ. This raises the error floor for the partial-cache observer of JoyboyBrian's #5498 from Ali Romman's 1/sqrt(19) ≈ 0.22942 (#5537) to sqrt(7/92) ≈ 0.27584. It also proves that 7/92 is the exact optimum over every scalar witness. #5537 used one fixed kernel vector.

• The scalar witnesses k ⊗ v (rational k on the 632 original roles, zero on the owned inputs) that annihilate every observed free-word row form a space of dimension 150. That is 4,858 distinct forms of rank 482.
• Over that space, max ||T J||^2 / ||J||^2 = 7/92 in Prove a physical 1/sqrt(19) error floor for the partial-cache observer #5537's physical norm. The optimum is isotropic: C^T G^-1 C = (7/92) I_8.
• Each mode has an integer witness that attains it. The mode-1 witness gives 784/10304: 56 X roles at ±13, 56 Y roles at ±1 and 44 helpers, with 113 physical roles.
• Prove a physical 1/sqrt(19) error floor for the partial-cache observer #5537's antipodal argument then gives uniform error at least sqrt(7/92) for every decoder, and randomized expected squared error at least 7/92. The Prove endpoint partial caches still fail to identify the actual consumer observer #5498 witness lies in the same kernel and gives exactly 1/19; this is checked as a control.

This answers the scalar-witness part of #5537's Open next question 2, the full norm of T on ker O. It is a lower bound on that norm, not the norm itself.

Package: research/partial-cache-scalar-witness-floor.

Evidence and reproduction

verify.py checks the SHA256 of every pinned input before importing the #5498 replay. It then:

1. builds every (observed row, free word) form,
2. computes the exact kernel,
3. solves G X = C with all 150 Gram pivots checked positive,
4. checks C^T X = (7/92) I,
5. constructs and checks eight attaining integer witnesses, and
6. runs the mutation controls.

RESULT.json was generated by verify.py --emit, and the verifier requires an exact match with it.

From the repository root (local interpreter ~/git/integer-mult-bounds/.venv/bin/python3.12, CPython 3.12):

python3.12 -I -W error -B research/partial-cache-scalar-witness-floor/verify.py
python3.12 -I -W error -B -O research/partial-cache-scalar-witness-floor/verify.py
python3.12 -B scripts/research_package.py --package research/partial-cache-scalar-witness-floor
python3.12 -B scripts/check_research_commit.py --package research/partial-cache-scalar-witness-floor --check-source-history

Local results:

• Both VERIFY.json commands pass, about 25 s each and 50 s total, under the 240 s per-command timeout. The output is PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed.
• --check-only passes.
• check_research_commit.py --check-source-history passes (PASS_RESEARCH_COMMIT_BYTES).

Controls, all checked:

• A perturbed witness leaves the kernel.
• Dropping the eight caches enlarges the kernel to dimension 158 and raises the diagonal optimum to 105/8.
• The raw norm gives 7/204 for the same witness instead of 7/92, so the helper-basis normalization matters.

Limitations and unrun obligations

• Only scalar witnesses k ⊗ v are covered. Operator-valued kernel elements may give a larger exact norm.
• Constraints use Prove endpoint partial caches still fail to identify the actual consumer observer #5498's reduced free-word grammar. Extra relations among the actual monomial operators could only enlarge the kernel and raise the floor.
• Cited premises, not re-proved here: Prove endpoint partial caches still fail to identify the actual consumer observer #5498's replay semantics and all-width coefficientwise lift; the PR5402 target formula, helper basis L, 2Km mode entrances and unitary wrappers, as used by Prove a physical 1/sqrt(19) error floor for the partial-cache observer #5537's physical norm; and their all-width physical realization.
• No supplier admission, paid-call bound, κ change or complete multiplication theorem.

Open next questions

1. The full norm of T on ker O: test at width one whether role-dependent unitary or monomial coefficients enlarge the 150-dimensional scalar kernel.
2. Why the optimum is isotropic, (7/92) I across all eight modes. A mode-boundary symmetry may give a closed-form witness.
3. How much one extra retained observation lowers 7/92. That is Prove a physical 1/sqrt(19) error floor for the partial-cache observer #5537's separation question, now measurable on this kernel. The retention cost is unpriced.

Related open work

No open PR studies the partial-cache observer, its kernel or the #5537 floor. I checked the open-PR list and searched for partial-cache, 5537 and 1/sqrt(19). This account's PRs opened since the CLAIM (#5997–#6005, #6007–#6011) study other objects: thin-query capacity, two-grid selectors, P104 sharing, Walsh-Hadamard agreement, Gaussian transitions, chirps, Pauli pairing, residue presentations, DUO connectors, mixed-point construction, group labels, char-2 jets, rank-5 deletions and shear targets. None computes this quantity.

Credit

Produced by Nick Petalas. Builds on JoyboyBrian (#5498: observation channel, free-word replay, first witness), Ali Romman (#5537: physical norm, minimax argument, open question) and Dillon Amburgey (#5402: provider, helper basis, target formula), with the lineage retained in those sources.

CI notes

Both hosted checks passed at head 947f4e425a6190dc38a3d3ea54f5f60564942843: plan and replay (research/partial-cache-scalar-witness-floor), in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40585106152.