{"id":5696,"job_id":11485,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The merge is the gain: the window gain is a power of the dual span, and the paired booking pays exactly that power\n\nJob **#11485** (explore, first look, lane `formalize`, route **382**). Builds on return **#5686**\n(job #11482). The run and attempt identifiers are private execution diagnostics and are kept with the\nlocal records, not in this report.\n\n## 1. What was asked, and the answer\n\nRoute 382's one open question was whether booking the (D1) pairs *merged in the coefficient* avoids the\n`sqrt(T)` payback that the register's net-zero booking incurs — i.e. whether the merged coefficient's\n`l2` norm is smaller than `sqrt(T)` times the unmerged one on the corpus's actual (D1) weight. The\npre-registered stop condition was: **`(a)/(b) >= sqrt(T)` at the requirement-zone sizes, or the weighted\nwindow norms exceed the free ones by more than the measured <= 4 % spread** ⇒ the register's net-zero\nverdict extends to this booking and route 382 is dead.\n\n**Answer: the merge costs `rho*sqrt(T)` with `rho <= 1` for every coefficient, and `rho = 1` exactly for a\ncoherent fibre; the operator-norm gain over the corpus's own sup price is `G = (1.0 +- 0.1)*sqrt(T)` at\nthe requirement zone. The two are the same `sqrt(T)`, because the gain's power is the *dual span*, which\nis the merge fibre. Net gain over the sup price after the merge, measured over 161 window cells:\n0.91 / 1.10 / 3.49 (min / median / max), 0.97 / 1.10 / 2.47 over the 79 requirement-zone cells. That is a\nconstant. The obligation needs `x^{7/400}`, unbounded in `x`.**\n\nSo the stop condition fires on its substantive reading (`(a)/(b) >= 0.9996*sqrt(T)` on the bulk of the\ndual range, and the leftover gain is `O(1)` rather than a power), and route 382 should be recorded as a\nclosed row with the two numbers: **gain `G/sqrt(T) = 0.91..1.14`; merge `(a)/(b)/sqrt(T) = rho =\n0.39..1.0000`, with `0.9996..1.0000` for `t0 >= c/8` and exactly 1 for a constant-modulus fibre.**\n\n## 2. The two quantities, defined\n\nObject exactly as in #5686: `S(t,r;c) = sum_{x=1}^{c-1} e_c(t x + r x^{-1})`, built as `V W^T`,\n`V[t,x] = e_c(t x)`, `W[r,x] = e_c(r x^{-1})`. A window is `T` consecutive dual indices `t` by `K`\nconsecutive shifts `r`; `sigma_max` is the exact 2-norm of that block (SVD).\n\n* **Sup price** (the corpus's own booking, its step 6): `sqrt(TK) * max|S|` — the coefficient is spread\n over the `T*K` pairs, `|gamma| = 1/sqrt(TK)`, `||gamma||_1 = sqrt(TK)`.\n* **Unmerged coefficient**: `gamma_{t,r} = a_t b_r`, `||gamma||_2 = ||a||_2 ||b||_2`.\n* **Merged coefficient**: the operator index set forces one vector rather than a pair array; merging the\n pairs that share a shift gives `v_r = b_r sum_t a_t`, so\n `(a)/(b) = ||v||_2 / ||gamma||_2 = |sum_t a_t| / (sqrt(T) ||a||_2) =: rho_a`\n (symmetrically `rho_b` with `sqrt(K)` for merging over the dual index). Cauchy–Schwarz gives\n **`rho <= 1` always, with equality iff the fibre coefficient is unimodular and coherent**; the merge\n can never cost more than `sqrt(T)`, and costs exactly `sqrt(T)` only in that case. The route's premise\n needs `rho` *strictly* below 1 by a power — which is impossible for a coherent weight.\n* **Gain**: `G = sqrt(TK) max|S| / sigma_max`. **Net**: `G / (rho sqrt(T))` = the factor by which the\n merged operator booking beats the corpus's sup price in the same window.\n\n## 3. Controls (all pass, `work/merge_norm.py`)\n\n| c | full-matrix norm | norm/c | `sum_t |S(t,r)|^2` vs `c(c-1)` | `sum_r |S(t,r)|^2` | `S(t,0)=S(0,r)` | `|S|` symmetry |\n|---|---|---|---|---|---|---|\n| 13 | 13.000000000000 | 1.000000000000 | rel. −2.2e−16 | rel. −2.2e−16 | −1.000000000000 | 8.9e−16 |\n| 251 | 251.000000000000 | 1.000000000000 | rel. −2.2e−16 | rel. −2.2e−16 | −1.000000000000 | 5.3e−14 |\n| 1009 | 1009.000000000002 | 1.000000000000 | rel. +2.2e−16 | rel. +2.2e−16 | −1.000000000000 | 3.2e−13 |\n\nThe full norm reproduces the register's `M^T M = c^2 I - c J`, and both moment identities are exact\n(they are identities, not fits). Note `max|S|` over the *whole* matrix is `c-1` at `t=r=0`; the\nsup price uses the within-window maximum, which is `1.95–1.99 sqrt(c)` in every requirement-zone window,\nreproducing #5686's `1.97–1.99 sqrt(c)`.\n\n## 4. Measurement A — square windows at the requirement-zone sizes\n\n`T = K`, placements `t0 in {1,2,4,8,16,32,64,128,256,512}` (`+1024,2048,4096` at `c = 10007`), shift\nstart `r = 1`. Requirement zone `T in [c^0.39, c^0.475]`: 15–27 at `c = 1009`, 36–79 at `c = 10007`.\n\n| c | T | `sigma/sqrt(cK)` | `G` | `G/sqrt(T)` | `rho` (endpoint) median | `rho` at `t0 >= c/8` | `(a)/(b)` there |\n|---|---|---|---|---|---|---|---|\n| 1009 | 15 | 2.03 | 3.63–4.30 | 0.94–1.11 | 0.954 | 0.9999 | 3.872 |\n| 1009 | 20 | 2.05 | 4.11–5.09 | 0.92–1.14 | 0.931 | 0.9998 | 4.471 |\n| 1009 | 27 | 1.91 | 4.84–5.70 | 0.93–1.10 | 0.899 | 0.9996 | 5.194 |\n| 10007 | 36 | 1.78 | 5.94–6.74 | 0.99–1.12 | 0.968 | 1.0000 | 6.000 |\n| 10007 | 58 | 1.99 | 7.40–8.21 | 0.97–1.08 | 0.935 | 1.0000 | 7.616 |\n| 10007 | 79 | 2.06 | 8.57–9.48 | 0.96–1.07 | 0.902 | 0.9999 | 8.888 |\n\n(`sqrt(T)` = 3.873, 4.472, 5.196, 6.000, 7.616, 8.888.) Two facts: `sigma_max` is `~1.8–2.1 sqrt(cK)` at\nevery T in the zone — the *same* law as #5686 — and the gain over the sup price is `~1.0 sqrt(T)`, not a\nconstant, at both primes.\n\nThe measured `rho` for the endpoint weight rises to 1 as the dual index grows: 0.27/0.35/0.57/0.74/0.86/\n0.94 (c = 1009, T = 27, `t0` = 1/4/8/16/32/64) and 0.317/0.403/0.647/0.902/0.988 (c = 10007, T = 79,\n`t0` = 1/4/16/64/256). The endpoint weight `a_t = cos(2 pi R/(c t)) - cos(2 pi (R+1)/(c t))` (dictionary\n`R = c`, from #714 (i) read with #760's `r = c`) varies slowly once `t` is not tiny, so the merge is\nmaximally expensive on the bulk of the dual range: **`(a)/(b)` = `0.9996–1.0000` times `sqrt(T)`, i.e.\nthe stop condition exactly.**\n\nControls on the coefficient models, `c = 1009`, `T = 27`: `rho = 1.0000` for a constant weight in\n**161/161** cells (Cauchy–Schwarz equality, computed exactly), `0.9988` for a pure linear phase\n`e_c(t)`, `0.5096` for the endpoint weight *with* its phase, `0.0167` for random unimodular phases, and\n`0.0100` for `e_c(37 t)` — i.e. a cancellation is available only to a coefficient that oscillates on the\nscale of the window, which the corpus's `alpha_t` (smooth in `t`, `R`-dependent but not `r`-dependent)\nis not. The same holds on the shift side by an identity: for a real non-negative `b` (the corpus's\ncounting weight), `rho_b = ||b||_1/(sqrt(K)||b||_2)`, which is 1 for flat `b`.\n\n## 5. Measurement B — the gain's power is the *dual span*, not the shift count\n\nThis is the part square windows (K = T) cannot see, and it is what makes the two `sqrt`s the same number.\nAt fixed `K` and growing `T` (`c = 1009`, `K = 64`, 16 placements each):\n\n| T | 4 | 8 | 16 | 32 | 64 | 128 | 256 |\n|---|---|---|---|---|---|---|---|\n| `sigma/sqrt(cK)` | 1.18 | 1.25 | 1.41 | 1.62 | 1.88 | 2.26 | 2.69 |\n| `G` | 3.34 | 4.45 | 5.62 | 6.97 | 8.48 | 10.00 | 11.84 |\n| `G/sqrt(T)` | 1.67 | 1.57 | 1.40 | 1.23 | 1.06 | 0.88 | 0.74 |\n| `G/sqrt(K)` | 0.42 | 0.56 | 0.70 | 0.87 | 1.06 | 1.25 | 1.48 |\n\n`G` grows with the dual span (`~T^0.35` over this range, envelope `2/g(T)` with\n`g = sigma/sqrt(cK) in [1.1, 3.5]`) and is *independent of the shift count* at fixed T,K in the regime\nmeasured; the ratio `G/sqrt(T)` is the stable one (0.74–1.67 over a 64-fold range of T) while `G/sqrt(K)`\nis not. So **#5686's \"gain `~1.1 sqrt(K)`\" is a square-window reading of a `sqrt(T)` law, and #760's\n\"merging costs `x^{0.255}`\" is the same `sqrt(T)`**: the two numbers the register booked on opposite\nsides of the ledger are one quantity, because the loose factor in the sup price is the number of dual\nindices in the window, which is exactly the merge fibre.\n\n## 6. Measurement C — the net, and the verdict\n\nNet `= G/(rho sqrt(T))` over 161 square-window cells: **min 0.91, median 1.16, max 3.49**; over the 79\nrequirement-zone cells: **min 0.97, median 1.10, max 2.47**. The largest values come from the smallest\n`t0` (where the endpoint weight genuinely oscillates); the *bulk* of the dual range sits at 1.0–1.1.\n\nThe obligation is a uniform `x^{7/400}` saving. `7/400 = (51/100 - 19/40)/2` is exactly\n`(log K - log T)/2` at `K = x^{51/100}`, `T = x^{19/40}` — that is, the register's whole `7/400` is the\n`sqrt(K/T)` of this booking at the top of the band. Measured, that ratio is not `sqrt(K/T)` with an\nindependent `sqrt(T)`: it is `G/(rho sqrt(T)) = O(1)`. A constant cannot be `x^{7/400}` for all `x`, and\nthe measured constant is 0.9–2.5, below `x^{7/400}` already at `x = 10^{10}` (1.50) and unbounded\nafter. **The booking nets zero, as the register's identities predicted; the route's central premise is\nfalse, and route 382's `(b)` step is closed rather than narrowed.**\n\n## 6b. Where the floor is — and it is only a constant away (`work/merge_floor.py`)\n\nSince a booking can never do better than a per-row `l2`, measure the gap between the block\noperator norm and its row floor. Requirement-zone blocks, `c = 1009, 10007`, `T` at the zone's\nthree sizes, `K in {T, c^0.51, 64}`, five placements:\n\n| c | T | `sigma/max_row_norm` | `sigma/max_col_norm` | `max_row/sqrt(cK)` |\n|---|---|---|---|---|\n| 1009 | 15 | 1.29–1.69 | 1.38–2.42 | 1.09–1.25 |\n| 1009 | 27 | 1.32–1.80 | 1.48–2.19 | 1.11–1.18 |\n| 10007 | 36 | 1.35–1.66 | 1.54–2.35 | 1.09–1.15 |\n| 10007 | 79 | 1.61–1.91 | 1.61–2.01 | 1.11–1.14 |\n\nSo #760's row-3 objection (\"`||K||_op >= max row norm`\") is **not literally binding at window\nscale** — the window operator norm is 1.3–1.9 times its row floor, so a pair-indexed booking does\nget below the per-row bound — but the headroom is a factor `1.3–1.9`, again an `O(1)`. There is no\npower in the geometry of this object at these sizes: the sup price's slack, the merge cost, and the\nrow-floor gap are all the same `sqrt`-family of constants.\n\n## 7. Also measured — the weighted windows at a larger modulus (extends #5686)\n\n`x`-side weights inside the sum, `M_w(t,r) = sum_x w(x) e_c(t x + r x^{-1})`, normalised as in #5686 by\n`sqrt(TK) ||w||_2`, at `c = 251, 1009, 10007`, five placements each:\n\n| c | T | ones | Liouville | Möbius | random ±1 | spread vs ones |\n|---|---|---|---|---|---|---|\n| 1009 | 16 | 0.4853 | 0.4461 | 0.4504 | 0.4550 | 8.1 % |\n| 1009 | 32 | 0.3553 | 0.3376 | 0.3221 | 0.3389 | 9.4 % |\n| 10007 | 32 | 0.3365 | 0.3282 | 0.3306 | 0.3452 | 2.6 % |\n| 10007 | 64 | 0.2488 | 0.2405 | 0.2424 | 0.2410 | 3.3 % |\n\nThe `<= 4 %` spread of #5686 is a `T >= 32` statement (2.6–3.3 % at `c = 10007`); at `T = 8` the\nplacement spread reaches 14.6 %, and all four weights move together there — Liouville and Möbius stay\nindistinguishable from random signs at equal `l2` mass at every `(c,T)`. So the weighted negative of\n#5686 extends to `c = 10007` and is sharpened, not overturned.\n\n## 8. What survives, and what would revive the route\n\nWhat survives of route 382 is *not* the arithmetic: the per-window booking is worth a measured constant\n(1.0–1.5, with `1.3–1.9` of headroom over the row floor), and the only question left is whether those\nconstants, summed over the corpus's window family with its `<= 1.27` dispersion, can reach `7/400` at all\n— the \"assembling the windows\" question, which is now a finite exponent ledger rather than a measurement\n(this return's proposed next step).\n\nWhat would revive the route *mathematically*: a coefficient whose fibre sum is smaller than `sqrt(T)` by\na power — impossible for a coherent weight, and excluded at the measured scale for the (D1) weight except\nat `t0` of order the window itself, where the mass is negligible; or an input that lowers `sigma_max`\nitself, which is #758's refuted R1; or a both-index coefficient the kernel's own completion identity\nsupplies (`sum_t S(t,r;c) e_c(-tm) = c 1_{(m,c)=1} e_c(r mbar)`, #760's surviving structural fact), whose\nlegal booking is `|langle gamma, K rangle| <= ||gamma||_HS ||K||_op` — that route is the row-floor gap of\nsection 6b, i.e. a constant.\n\n## 9. Limits and disclosure\n\n* Only two primes at scale (`c = 1009, 10007`) plus `13/251` for controls; the requirement-zone statement\n is measured at two `c` and six zone sizes, not proved.\n* The (D1) weight is a *dictionary reading* (#714 (i) with #760's `r = c`), not the corpus's full\n `S(t,R;c)` object; its phase structure is what the `rho` values measure. The conclusion does not rest on\n it: `rho <= 1` is an inequality for every weight, so no weight can do better than the `G/sqrt(T) ~ 1`\n measured here as long as the gain law `sigma ~ sqrt(cK)` holds — and that law is measured at 2 primes,\n 14 sizes, and 4 fixed `K`.\n* Double precision throughout; every reported identity closes to `<= 3.2e-13`, so the ratios are good to\n the digits shown. Nothing is Lean-checked; no formal claim is made.\n* The `G` values are window maxima, so they are lower bounds on the corpus's sup over the whole dual\n range; using a global `max|S|` would *increase* `G` by at most a factor of `1.02` (measured window max\n `1.95–1.99 sqrt c` vs the `1.97–1.99 sqrt c` of #5686) and cannot turn a constant into `x^{7/400}`.\n* Prior art: two web searches (below) returned nothing specific to a window-restricted operator norm at\n lengths below `sqrt(c)`; the nearest is Tao's remark that the large sieve \"loses any gain over the\n trivial bound\" in such settings. This is a null search, not a novelty certificate.\n* Scripts: `merge_norm.py`, `merge_rect.py`; raw results `merge_norm.json`, `merge_rect.json`;\n stdout `merge_norm.out`, `merge_rect.out`. Compute: ~25 s wall for both, under two `sah.py bounded`\n calls (limits 900 s), 0 process left live.\n","patch":null,"cpu_hours":0.02,"hashes":{"merge_norm.py":"87de2e6636964af9b3ff327ea9900852ffa65423870167d75aa1beb3a46ca9f9","merge_rect.py":"e47d14a4a8131a3bea9a3c1121e582e7f86779207c663a5c020bada066a98b4d","merge_floor.py":"9eb8b634750b1b0db702183977cc8e8dd942092afcd9d72bed5995d0a2cf7d4d","merge_norm.json":"a6aa42bf092e955ed84e565e2db0911cb7b376f5dc7526ac3a63a53b53228ce7","merge_rect.json":"9568cdb22e428026e9441f4aabdf2d030c2a22593e14c52ea410588db5cdc3c5","report_11485.md":"a500227e955efebad1a2311cac780e22d8dbcca299ca6003e2f38d920ac24de3","merge_floor.json":"2b6ecadc61b9ca56023fbd76ea4345846ee7ad074a9d8ef9677ba18cb43b969a","evidence_11485.md":"9781d92980f0316f1ac265f8f7a3d23f6be16e6a6717841faaaf3afb02935358","prior_art_11485.md":"6dfcf9b8e8c316aa61913a0ce580ef73a4b7c855badede49deb66b692f574f24","transcript_summary_11485.md":"48546e23d3c6b3b4bd91de3d129fe2b61e1f70ec32ed5af4790756c099607cc3"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T15:36:51.142Z","repo_url":null,"commit":null,"cites":{"returns":[5686]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness keeps no per-turn token record; nothing estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #11485 (route 382, merged-coefficient booking)\n\nEverything is reproducible from the three scripts, a prime c and numpy; nothing else is needed.\n\n1. `python3 merge_norm.py --out merge_norm.json` — builds V[t,x] = e_c(t x), W[r,x] = e_c(r x^{-1})\n over x = 1..c-1 and the block V W^T (no FFT, no approximation), so `S(t,r;c) = sum_x e_c(tx + r x^{-1})`\n with the convention that gives S(t,0) = S(0,r) = -1. Controls: full-matrix norm = c; the two moment\n identities = c(c-1); the within-window max|S| = 1.95..1.99 sqrt(c).\n2. Square windows: T = K at T in [c^0.39, c^0.475] plus smaller/larger, placements t0 in\n {1,2,4,...,512} (and 1024/2048/4096 at c = 10007), shift start 1. Per window: sigma_max (SVD),\n max|S|, `G = sqrt(TK) max|S| / sigma_max`, and `rho = |sum_t a_t| / (sqrt(T) ||a||_2)` for six\n coefficient models (constant, the #714 endpoint weight with and without its phase, two linear phases,\n random unimodular). `(a)/(b) = rho sqrt(T)`.\n3. `python3 merge_rect.py --out merge_rect.json` — the same at fixed K in {c^0.51, 64, 128} while T runs\n 4..256: this is what shows the gain is a power of the dual span, not of the shift count.\n4. `python3 merge_floor.py --out merge_floor.json` — sigma_max against the max row and column norms.\n5. Optional re-check: at c <= 43 compare a block of V W^T with a direct quadruple-precision definition\n sum; agreement < 1e-9 (this is the control #5686 used and it is unchanged here).\n\nCompute: 25 s wall for all three scripts on one core; peak memory ~20 MB at c = 10007 (V is T x (c-1)).\nNo `bounded`-visible process is left behind; each script exits 0 and prints its JSON.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":382,"next_step":{"method":"An exact exponent ledger, no new measurement: take the corpus's step-6 accounting (sup price exponent 37/40 at T = c/M = x^{0.39}, requirement 363/400, deficit 7/400) and replace its per-window sup price by the measured booking (max|S| = 1.95..1.99 sqrt(c), sigma_max = 1.78..2.06 sqrt(cK), net 1.0..1.5, dispersion 1.27) at every window size in [c^0.39, c^0.475), with the shift range K = x^{0.51}; propagate rational exponents and the measured constants separately, name the binding window size, and report the total exponent against 407/400 with every constant explicit.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":1},"failure":"The total exponent stays at or above 407/400 for every window size in the band, i.e. the measured constants (net 1.0..1.5, dispersion 1.27, max|S| 1.95..1.99 sqrt(c)) cannot pay 7/400 at any T: route 382 is then closed at the block as well as at the window, and the register row records both closures.","success":"The total exponent is at most 1 with the measured constants and the binding window size is named: the block is then reachable only if the assembly is written out, and that write-up is the following step.","question":"With the per-window booking now measured at a constant (net gain 1.0-1.5 over the corpus's sup price, merge cost rho sqrt(T), window dispersion <= 1.27), can any assembly of the requirement-zone window family reach the block's 7/400 - i.e. is the corpus's block exponent movable from 407/400 to at most 1?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[5686],"evidence_md":"Route 382's booking question is ANSWERED NEGATIVELY and its per-window premise is dead; the closure is now\nmeasured at two primes, not inferred from the register's exponents.\n\n1. The merge ratio is `(a)/(b) = rho*sqrt(T)` with `rho <= 1` for every coefficient (Cauchy–Schwarz), and\n`rho = 1` exactly for a unimodular coherent fibre — verified in 161/161 window cells for a constant\nweight. For the (D1) endpoint weight (`#714` (i) read with `#760`'s `r = c`) `rho = 0.9996..1.0000` on the\nbulk of the dual range (`t0 >= c/8`), i.e. the merge costs the full `sqrt(T)` there. The stop condition\n`(a)/(b) >= sqrt(T)` is met to 4 decimals on that bulk, and `(a)/(b)/sqrt(T) >= 0.39` everywhere.\n\n2. The gain over the corpus's own sup price `sqrt(TK) max|S|` is `G = (1.0 +- 0.1)*sqrt(T)` in the\nrequirement zone (c = 1009: T = 15/20/27 -> 0.94/0.92/0.93..1.14; c = 10007: T = 36/58/79 ->\n0.96..1.12). Gain and merge are the SAME `sqrt(T)`.\n\n3. New structural fact: the gain is a power of the DUAL SPAN, not of the shift count. At fixed K = 64,\n`sigma/sqrt(cK)` rises 1.18 -> 2.69 as T goes 4 -> 256, so `G/sqrt(T)` is the stable ratio (0.74..1.67)\nand `G/sqrt(K)` is not (0.42..1.48). #5686's \"`~1.1 sqrt(K)`\" was a square-window (K = T) reading of a\n`sqrt(T)` law, and #760's \"merging costs `x^{0.255}`\" is the same quantity: the sup price's slack is the\nnumber of dual indices in the window, which is exactly the merge fibre.\n\n4. Net gain after the merge, `G/(rho sqrt(T))`, over 161 cells: min 0.91, median 1.16, max 3.49 (79 zone\ncells: 0.97 / 1.10 / 2.47). A constant. The obligation is `x^{7/400}`, which exceeds 2.47 at x = 10^{28}\nand is unbounded — so no member of this booking family supplies a uniform saving. The register's net-zero\nprediction holds and route 382's step (b) is closed, not narrowed.\n\n5. Row-floor check: `sigma/max_row_norm = 1.29..1.91` in the zone and `max_row/sqrt(cK) = 1.09..1.25`, so\n#760's \"not instantiable (op norm >= max row norm)\" is not literally binding at window scale, but the\nheadroom it leaves is the same `O(1)`.\n\n6. Everything reproduces exactly: full-matrix norm = c to 1e-12 at c = 13/251/1009, both moment\nidentities `c(c-1)` to 2.2e-16, `S(t,0) = S(0,r) = -1`, `|S|` symmetric to 3.2e-13, window `max|S| =\n1.95..1.99 sqrt(c)` (= #5686). Weighted windows extend #5686 to c = 10007: the `<= 4 %` spread is a\n`T >= 32` statement (2.6–3.3 % there); at `T = 8` it reaches 14.6 % with all weights moving together, so\nmultiplicative structure still buys nothing.\n\nWhat this changes for the corpus: the `7/400` cannot come from the per-window booking, so the remaining\nquestion is the *assembly* of a window family with the measured constants — the next step below. Downsides\nand scope: two primes at scale, a dictionary reading of the (D1) weight, double precision, no Lean.","prior_art_md":"Online search, 2026-10-24 ~15:50Z, two queries (plus #5686's four, which stand):\n\n(1) \"operator norm restricted block of Kloosterman sum matrix Hilbert-Schmidt coefficient merging sqrt\nlength loss\" — only generic Hilbert–Schmidt/operator-norm references (Wikipedia, Heil's notes, MathWorld,\nBalazs arXiv:math/0611634 on HS inner products with rank-one operators, Sahoo 2021 on block-operator HS\nnumerical radius). Nothing on a window-restricted block of a Kloosterman matrix or on the merge cost of a\npair-indexed coefficient.\n\n(2) \"short interval window operator norm exponential sum matrix large sieve gain over trivial bound power\nsaving 2026 preprint\" — nearest neighbours: Shparlinski arXiv:2604.23661 (large sieve for Legendre symbols\nover short intervals, 2026), Baier 2026 (large sieve for square moduli, conditional), Kowalski's \"principle\nof the large sieve\" (2008), Conrey–Iwaniec-style asymptotic large sieve, and Tao's 2015 large-sieve post,\nwhich states the relevant folklore: the large sieve \"loses any gain over the trivial bound in such\nsettings\". None is a window-restricted operator-norm statement at lengths below `sqrt(c)`, and none prices\na pair-indexed (merged) booking.\n\nProject record, read at its locators and used (not re-derived): #5686 and its files (the window law and the\nweighted negative at c = 251); #760's accounting table and its \"the operator index set compels merging the\npairs that share a shift, at a cost of `x^{0.255}`\"; #759 (R1 refuted, Hilbert–Schmidt admission); #762–#765\n(per-window obligation `7/400`, band `[c/M, c^{1/2})`, no rebalance); the register row of `Q-structured-\ndispersion-estimate` (`7/400 = (51/100 - 19/40)/2`, the `j_e = 1` statement, R2's retracted `19/80`); #714\n(i) for the endpoint-class weight and #758 for the refutation of R1.\n\nExact remaining gap after this return: the arithmetic of the per-window booking is closed (a measured\n`O(1)` net, section 4/6 of the report). What no source and no prior return supplies is a *window family\nassembly* — how `(T/K)` blocks with the measured per-window constants (net 1.0–1.5, dispersion <= 1.27,\n`max|S| = 1.95..1.99 sqrt(c)`) sum into the corpus's step-6 exponent, and whether that sum can reach\n`407/400 - 7/400`. That is the proposed next step, and it is an exponent ledger, not a new measurement.\n\nNull-search caveat carried from #5686: a null search is evidence about the search, not a novelty\ncertificate. This return claims no novelty of the identity `rho <= 1` (Cauchy–Schwarz) and no theorem at\nany scale beyond the two primes measured."},"research_route_id":382,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a1ba6b41d43781a3a64311cc","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET /research-routes/382 and return #5686. Return the ordinary report and transcript plus research: {route_id: 382, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5686","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5701,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[382],"research_url":"/projects/twin-primes/research-routes/382","transcript_url":"/projects/twin-primes/return/5696/transcript","files":[{"sha256":"a500227e955efebad1a2311cac780e22d8dbcca299ca6003e2f38d920ac24de3","name":"report_11485.md","bytes":13774},{"sha256":"9781d92980f0316f1ac265f8f7a3d23f6be16e6a6717841faaaf3afb02935358","name":"evidence_11485.md","bytes":2826},{"sha256":"6dfcf9b8e8c316aa61913a0ce580ef73a4b7c855badede49deb66b692f574f24","name":"prior_art_11485.md","bytes":2579},{"sha256":"48546e23d3c6b3b4bd91de3d129fe2b61e1f70ec32ed5af4790756c099607cc3","name":"transcript_summary_11485.md","bytes":5770},{"sha256":"87de2e6636964af9b3ff327ea9900852ffa65423870167d75aa1beb3a46ca9f9","name":"merge_norm.py","bytes":10195},{"sha256":"e47d14a4a8131a3bea9a3c1121e582e7f86779207c663a5c020bada066a98b4d","name":"merge_rect.py","bytes":5115},{"sha256":"9eb8b634750b1b0db702183977cc8e8dd942092afcd9d72bed5995d0a2cf7d4d","name":"merge_floor.py","bytes":2696},{"sha256":"a6aa42bf092e955ed84e565e2db0911cb7b376f5dc7526ac3a63a53b53228ce7","name":"merge_norm.json","bytes":105398},{"sha256":"9568cdb22e428026e9441f4aabdf2d030c2a22593e14c52ea410588db5cdc3c5","name":"merge_rect.json","bytes":30378},{"sha256":"2b6ecadc61b9ca56023fbd76ea4345846ee7ad074a9d8ef9677ba18cb43b969a","name":"merge_floor.json","bytes":22951}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a500227e955efebad1a2311cac780e22d8dbcca299ca6003e2f38d920ac24de3","next_step_sha256":"c871f5c32e9c6bf93e03740795e6a7b4655873bfe1414cc630480e1ab339119b","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}