Negative result: every scalar decoder on the PR2297 five-frame cover costs at least 554 (sharp; u=5 scalar row has no saving; no κ)

[thakrarsagar]

Statement

Negative result (exact sharp bound with proof; scoped proof plus exact finite certificate). No κ claim, no supplier, no change to the selected result.

On the exact PR2297 five-frame cover (22 orthonormal five-frames and one orthonormal pair, 112 ports, sum d_i = 554, no free (orthogonal) group pair, so K = c I_112), every finite scalar factorization c I_112 = A diag(c_t) B^T over Q, or over any field, has full-port cost

C = sum_t rank_F2(pairing on L_t x R_t) >= 554.

The direct one-term-per-port decoder attains 554, so the bound is sharp. The zero-saving threshold is floor(4432/9) = 492, so every scalar decoder has deficit (budget minus cost) at most -554/9. This holds for every number of terms, every support pattern (matched or unmatched spans), every sign and every denominator.

Proof (PROOF.md).

1. Lemma 1 (principal-rank count). For every group set S, K[S,S] = c I_{f(S)}, so at least f(S) = sum_{g in S} d_g terms are nonzero on S on both sides. This is a rank statement over the base field; no mod-2 reduction, so even denominators play no role.
2. Lemma 2 (LP duality over flats). If nonnegative weights y_S satisfy, for every ordered pair of flats (F, F'), sum_{S meets both} y_S <= rho(F, F'), then C >= sum_S y_S f(S). Arbitrary supports reduce to their closures (same spans and costs; hitting is monotone). The cover has 107 flats.
3. Certificate. CERTIFICATE.json: 88 group sets, integer weights over 10^6, all 11,449 ordered flat-pair constraints satisfied exactly (minimum slack 6/10^6), value 553996479/1000000 > 553. C is an integer, so C >= 554. (The certified LP value lies in [553.996479, 554]; that the LP optimum is exactly 554 is not certified.)

Declared milestone and decision informed

Milestone: local module (scoped proof), card coverage-law-change. Decision: whether the u=5 (k=5) coverage lever row can give any positive saving under the scalar-decoder contract on this cover. It cannot. This closes the general unmatched-span, even-denominator class left open by PR #2879 (Arham Tahir; equal spans only, plus a necessary condition of >= 16 unmatched terms with total asymmetry credit sum delta_t >= 62) and by PR #2532 / philippeVV's Theorem 1 (2-integral decoders only), and answers PR #2532's open question "even-denominator gain for k=5": the achievable saving is zero.

Scope of the consequence: scalar decoders only. A k-port saving on this row now needs a mask with free pairs, another tariff, another cover, or another endpoint contract (an operator/block-valued decoder such as PR2297's C_E^2 blocks or PR #3480 by Gonzalo Mier). Non-scalar endpoint contracts are not addressed.

Base and pins

• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90 (priority revision 6; canonical main unchanged at submission).
• Inputs: research/community-frontier1094-audit/selected-replay.tar.gz sha256 f1da8061…d529d6 (member DATA-PHASES.json sha256 c5916ff3…8251fa); research/kport-group-scalar-endpoint-audit/fixtures/PR2297-CANDIDATE.json sha256 5b69c7a8…a86c0. Full pins in SOURCES.json / VERIFY.json.
• Certificate: CERTIFICATE.json sha256 7c800588…adcc525.

Reproduce

python3 -B research/pr2297-principal-rank-floor/verify.py

Stdlib only, about 2 s and 35 MB. Final line PASS_PR2297_PRINCIPAL_RANK_FLOOR; JSON includes flats 107, flat_pairs 11449, certificate_sets 88, certificate_value 553996479/1000000, min_slack_numerator 6, floor 554, direct_decoder_cost 554, zero_saving_threshold 492, unpaid_budget 4432/9, exact_instance_cost 691. Generic hook: python3.12 scripts/research_package.py --package research/pr2297-principal-rank-floor (PASS_REGISTERED_RESEARCH_PACKAGE).

Second methods inside the verifier: flats enumerated twice (BFS closure; spans of all <= 5-group subsets with closedness check); pairing ranks twice (packed Gram elimination; dim F - dim(F ∩ F'^perp)); 2,000 seeded random non-closed supports checked with directly computed ranks; an exact rational decoder with denominators 2/4/8 (K = c0 I_112 checked entrywise, cost 691) satisfies the full inequality chain.

Negative controls (all rejected): one certificate weight inflated by 1; rho(ambient, ambient) set to 8; line-label tariff in place of the full-port tariff; doubled certificate; flat method B fed a group list missing one group (must disagree with method A); a mutated cover with a free (orthogonal) group pair (cover() must reject it, for that reason).

Scope, assumptions, unresolved obligations

• Exact PR2297 cover, scalar-decoder contract, full-port tariff (as in PR Audit (no κ): #2297/#2041 k-group port decoders are not scalar decoders — 6,608 parity-forced forbidden entries; k=5 needs a diagonal rank-112 map; 2-integral floor 554/334 exceeds every saving threshold #2532 / PR Prove rational cost floors for matched k-port spans #2879). Not a complete multiplication theorem.
• Inherited, not re-derived: the full-port tariff, K = c I_112 from the PR Audit (no κ): #2297/#2041 k-group port decoders are not scalar decoders — 6,608 parity-forced forbidden entries; k=5 needs a diagonal rank-112 map; 2-integral floor 554/334 exceeds every saving threshold #2532 mask rule, and the 8/9 portwise planning law of PR Prove rational cost floors for matched k-port spans #2879 §3 (budget 4432/9). The conclusion does not hinge on 8/9 alone: 554 also exceeds PR Audit (no κ): #2297/#2041 k-group port decoders are not scalar decoders — 6,608 parity-forced forbidden entries; k=5 needs a diagonal rank-112 map; 2-integral floor 554/334 exceeds every saving threshold #2532's incumbent-tie band 455-460 and PR Research: complex copy budgets for 3-6 unpaid coordinates with large local children (planning screen; no κ) #3344's (Matarener) u=5 rank-32 copy budget 488.
• Lemmas 1 and 2 are written out but not proof-assistant checked.
• Native frames, restoration, hosting, precision and the complete consumer are unpriced (moot for this row only because no decoder saves). Other covers, other u, masks with free pairs and non-scalar contracts are not covered.

Search record

Certificate search (find_certificate.py, provenance helper only, not a verifier requirement; numpy 2.5.4 / scipy 1.18.1 HiGHS): 3,252 candidate sets (all 23 singletons, all 253 pairs, the full set, 3,000 seeded random sets of size 3-20, seed 20261019) against all flat-pair rows; LP value 553.9999999999999; weights floored to 10^-6 and shrunk 2 ppm, then checked exactly. Rerun reproduced the shipped certificate byte for byte. A differently formulated LP (dominance-pruned rows) gave a different 93-set vertex with the same value. No decoder search was needed.

Overlap check (2026-10-19, ~00:55Z and refreshed ~01:20Z): gh PR searches (PR2297, k-port decoder, five-frame, unmatched span, even-denominator, principal rank, scalar decoder floor), open PRs since 2026-10-17, open PRs by thakrarsagar, and Discussions #1587, #2292, #2049 and #1166 (read-only). No competing result found. scripts/find_overlap.py was not refreshed. Concurrent related work on different objects: the lost-free plane-mask line (#3625 Matarener, #3674 ToikY, #3676 and #3756 jwr404) and the rank-57 triad line (#2757 and #3684 flyingviper; #3505, #3513, #3527, #3538 petalas).

Credit

• Arham Tahir (@Arham-Tahir64), PR Prove rational cost floors for matched k-port spans #2879 research/support-balanced-port-floor: matching lemma (its §1 already notes K[J,J] = cI for pairwise nonorthogonal J; Lemma 1 replaces the injection by a term count), equal-span floor, asymmetry budget.
• philippeVV, Discussion CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587 (comments 18927273, 18932906, 18942562): Theorem 1, top-layer and Smith lemmas, follow-up 4 barriers.
• Sagar Thakrar (@thakrarsagar), PR Audit (no κ): #2297/#2041 k-group port decoders are not scalar decoders — 6,608 parity-forced forbidden entries; k=5 needs a diagonal rank-112 map; 2-integral floor 554/334 exceeds every saving threshold #2532 research/kport-group-scalar-endpoint-audit: k-port masks, 2-integral floor, retained PR2297 fixture.
• Cinna Davis (@1337isnot1337), PR2297 cover. Adam Gall (@adamgall), PR1889 mask and tariff foundations. Tekalign (@tekuuu), PR2782 Gram denominator gate. ywz65, PR Bind mixed port coverage to actual endpoint dimensions #2528 weighted unpaid dimension.
• Method precedent on C5 masks: PR Prove the sharp Gram cost floor on all 23 physical C5s #3232 and Prove sharp arbitrary-term asymmetric cost on the named C5 core #3256 (thakrarsagar), PR Prove sharp arbitrary-term cost 20 on one residual physical C5 #3271 (Nick Petalas, @petalas), PR Certify four physical C5 cost floors by exact rational separation #3281 (Krzysztof Wójt, @krzysztofwojt). New here: the application to the PR2297 cover, the flat-indexed LP-dual certificate and the verifier.

Claim and review provenance

No Discussion claim was posted for this work (local-only claim record). Reviewed in two rounds of independent AI review passes in the same loop (reproducibility, mathematics, provenance, referee); two of those passes recomputed the certificate with separate scripts. All blocking/major findings were fixed; the final round's verdicts were "ship". There has been no human review.

Attribution

Prepared by Sagar Thakrar (@thakrarsagar) with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5) via Claude Code in an automated research loop. Apache-2.0.

🤖 Generated with Claude Code