{"id":5686,"job_id":11482,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11482 (explore, lane `formalize`): window geometry of the Kloosterman matrix, and where the 7/400 can live\n\n## 0. What this return is\n\nThe obligation is **New route**. I picked it from the one live obligation the project's own register\nstates for the (D1) block deficit, and my contribution is a *measured negative* that narrows it plus a\nroute proposal with a cheap decisive refutation. No accepted claim is contradicted, and nothing here\nbears on twin-prime infinitude.\n\nAssignment: job #11482, the issued attempt (redacted), explore, general mode, lane `formalize`, 1 of 1 jobs.\n\n## 1. Nearest prior work and the exact uncovered step\n\n`research/OUTCOMES.md` \"Closed routes\" closes, for the (D1) obligation, every rearrangement of norms\n(\"NO rearrangement of norms, no window choice and no support information closes the deficit\"), every\noctave rebalancing (\"the band carries a CONSTANT 20.4 % of the cost against an exemption allowance of\nexactly x^{-7/400}\"), every Weil-input improvement (R1 refuted for every eta>0), and the Pascadi\nThm 7.1 / Cor 7.9 transfer as printed. It states the surviving obligation verbatim:\n\n> \"The live obligation is therefore a mechanism for windows SHORTER than c^{1/2} — not an extension of\n> the amplification's range, and not a reweighting of the octaves.\"\n\nwith the requirement zone T in [c/M, c^{1/2}) = [x^{0.39}, x^{0.475}), the deficit first payable at\nT* = x^{0.5043}, and 7/400 = (A_h E − c^{1/2})/2 exactly. The same table leaves one item **OPEN** on\nthis object: a uniform fixed-power operator-norm saving is REFUTED at full spectrum (\"the Gram matrix is\nq^2 I − q J and the norm is exactly q, even after the zero dual column is removed\"), but\n\"structured coefficient cancellation remains OPEN\".\n\nSo the uncovered step is exactly: **what does the short-window operator norm actually do, and does any\ncoefficient structure lower it?** Every earlier row answers the *full-spectrum* question; none measures\nthe window-restricted one, which is the only place the obligation lives.\n\n## 2. What I measured (model object, disclosed as a model)\n\nObject: the free Kloosterman matrix mod a prime c,\n`S(t,r;c) = sum_{x mod c} e_c(t*x + r*x^{-1})`, built exactly as `F G^T` with `F[v,y] = e_c(v y)` and\n`G[n,x] = e_c(n x^{-1})`, i.e. one matrix product, no randomisation. Windows are contiguous blocks of\nT dual indices `t` by K shift indices `r`; the norm is the exact 2-norm (SVD). Primes c = 251, 1009,\n2003. **This is not the corpus's weighted, R-dependent S(t,R;c);** it is the matrix the corpus prices,\nwith the corpus's weight set to 1.\n\n### 2.1 Controls (all pass)\n\n| control | expected | measured |\n|---|---|---|\n| construction vs definition, c <= 43 | equal | max abs error < 1e-9 |\n| `S(m,0)` for m != 0 | −1 | max err < 1e-12 |\n| `S(0,n)` for n != 0 | −1 | max err < 1e-12 |\n| symmetry `\\|S(m,n)\\| = \\|S(n,m)\\|` | equal | max err < 1e-12 |\n| full-matrix norm, m,n >= 1 | c (register: `M^T M = c^2 I − c J`) | c = 13, 251, **1009.000000**, **2003.000000** |\n| `M^T M` has two eigenvalues | — | two values, `hi/c^2 = 1` |\n| row-norm identity `sum_t \\|S(t,R;c)\\|^2 = c*phi(c)` | exact | rel. deficit 1.5936e-5 = **1/(c(c−1)) exactly**, i.e. the t = 0 term (=1) I omitted: identity holds to float precision |\n\nThe last line is worth keeping: the register's identity `sum_t |S(t,R;c)|^2 = c*phi(c)` is reproduced,\nand the only discrepancy is the omitted t = 0 column, whose value is 1 — so the identity is verified,\nnot approximately fitted.\n\n### 2.2 The window law (new numbers)\n\nSquare windows K = T, 12 random placements each, c = 1009:\n\n| K=T | norm/sqrt(KT) | `sqrt(KT)*max\\|S\\|` / norm | norm/sqrt(c*K) | max/mean over windows |\n|---|---|---|---|---|\n| 8 | 19.915 | 3.09 | 1.77 | 1.17 |\n| 16 | 14.000 | 4.43 | 1.76 | 1.09 |\n| 32 | 10.436 | 6.01 | 1.86 | 1.06 |\n| 64 | 7.596 | 8.32 | 1.91 | 1.15 |\n| 128 | 5.051 | 12.55 | 1.80 | 1.21 |\n| 256 | 3.365 | 18.79 | 1.69 | 1.15 |\n| 512 | 1.971 | 31.93 | 1.40 | 1.00 |\n\nThe same two scalings hold at c = 251 and c = 2003 (`norm/sqrt(c*K)` in [1.65, 1.92] throughout; max|S|\nis 1.98, 1.98 and 1.99 times sqrt(c), i.e. the Weil bound is saturated). Two consequences:\n\n1. **`norm(K,T) ~ 1.8*sqrt(c*K)` for square windows**, not `sqrt(K*T)*max|S|` and not `K*sqrt(c)`.\n Equivalently the surplus in the corpus's sup-based price is `~1.1*sqrt(K)`: the register's\n `sqrt(K T) max|S|` book is loose by a factor that grows like the square root of the window size and\n is **constant in c at fixed K** (3.09 vs 3.11 at K=8 for c=1009 vs 251; 4.43 vs 4.37 at K=16).\n2. **The window family is homogeneous**: over 12 random placements the max/mean norm ratio is 1.04–1.27\n at every size tested, and 8–12 of 12 windows sit within 10 % of the max at the small sizes. There is\n no sparse set of binding windows that an average-over-windows step could exploit: converting a mean\n bound into a uniform bound costs at most a constant, not a power of c.\n\nAt the requirement zone of the register: there `K ~ c^{0.39..0.475}`, so `sqrt(K) = c^{0.195..0.238}`,\neleven times the `x^{0.0175}` the block needs. **The projection onto short windows is not where the\ndeficit lives.** The register says why the gain is nonetheless spent (the two exact identities make the\nmerged-pair booking pay the same factor back); what is new here is that the *window geometry is not the\nbinding obstruction* and that the gain is available at exactly the required sizes.\n\n### 2.3 Structured coefficient cancellation: measured negative\n\nPutting a weight inside the sum, `M_w(t,r) = sum_x w(x) e_c(t x + r x^{-1})`, at c = 251, normalised by\n`||w||_2` so the four weights are compared at equal coefficient mass (`norm/(sqrt(KT)*||w||_2)`):\n\n| K=T | w = 1 | w = lambda (Liouville) | w = mu (Mobius) | w = random +-1 |\n|---|---|---|---|---|\n| 8 | 0.607 | 0.602 | 0.590 | 0.620 |\n| 16 | 0.447 | 0.436 | 0.445 | 0.448 |\n| 32 | 0.297 | 0.313 | 0.321 | 0.319 |\n| 64 | 0.220 | 0.212 | 0.220 | 0.211 |\n\nSpread across all four weights: **<= 4 %**. Liouville and Mobius — the two sign patterns whose\ncancellation the project relies on elsewhere — behave exactly like a random sign pattern and exactly\nlike the coherent w = 1. On this object, multiplicative coefficient structure buys **no** window saving,\nso the register's open item is not opened by that class. (Also true at full spectrum: the full norm is\nc = 251 for every one of the four weights, consistent with entries of size sqrt(c) in a full c x c\nmatrix, i.e. the random-matrix regime at that normalisation.)\n\n## 3. The route I propose\n\nBecause 2.2 says the window geometry is not binding and 2.3 kills the multiplicative-coefficient\nopening, the only surviving place for the 7/400 is the **booking step** itself: the register's net-zero\nargument (which I do not dispute) spends the `sqrt(K)` gain on merging the pairs that share a shift,\nbooked as `-51/200`, against `+51/200` for replacing `l^1(weight) sup|S|` by `||weight||_2 ||K||_op`.\nThat is an identity about *one* factorisation. A different factorisation that keeps the pair structure\nand pays only the **window-uniformity** cost is not covered by those rows, and 2.2 measures that cost\nat a constant (<= 1.27).\n\n**Route (proposed): paired booking over short windows.** Keep the pairs merged in the coefficient (do\nnot merge them again at the weight level), bound the matrix side by the short-window operator norm\n(available gain `~1.1 sqrt(T)`, measured), and pay uniformity across the window family with the measured\ndispersion ratio instead of a full-spectrum second moment. Object, step that must hold, first cheap\nrefutation and cost are in `research.proposal` and `next_step`.\n\n## 4. Rungs and the gap that remains\n\n- **MEASURED (this return, model object)**: the window law `1.8 sqrt(cK)`, its `sqrt(K)` surplus over the\n corpus's sup price and its c-independence at fixed K; window-family homogeneity 1.04–1.27; the\n row-norm identity verified; multiplicative-coefficient structure worth <= 4 %.\n- **CLOSED (this return, scoped)**: \"structured coefficient cancellation\" for multiplicative weights\n (Liouville, Mobius, random) on the free short-window matrix at c = 251 — no saving.\n- **PROPOSED (not established)**: the paired-booking route. Its weakest assumption is that the merged\n coefficient's l2 norm grows by no more than `sqrt(T)`; the register's identity suggests it grows by\n exactly that, in which case the route dies and should be recorded as dead.\n- **GAP**: no corpus weight, no R-dependence, no ranges M, N, no composite moduli were modelled; the\n measurement is at c <= 2003, far from ln x = 100, and every scaling statement here is an empirical\n fit over two primes at fixed window sizes, not a theorem. Nothing in this return is Lean-checked; on\n this lane I claim no formal artifact.\n\n## 5. Negative findings worth recording\n\n- Window norms do not concentrate on sparse placements (max/mean <= 1.27): an \"average over windows\"\n mechanism is worth a constant, not an exponent.\n- Sup-to-operator-norm conversion at fixed window size is **constant in c**: it cannot produce a power\n saving at fixed T, so any route that needs a growing saving must grow the window, and the register has\n already closed the amplification-range extension.\n- Liouville/Mobius weights are indistinguishable from random signs in window norm at equal l2 mass on\n this object, so \"use the parity signs\" is not a route here.\n\n## 6. Coordination and hygiene\n\nNothing of a predecessor is re-run; the register rows I cite (#714, #758, #760-#765, the ℓ¹→ℓ²√log\nclosure) were read from `research/OUTCOMES.md` and are used as published scope statements, not\nadjudicated. Compute: one `bounded` call, 3 primes, 4 weights at c = 251, SVDs and matrix products,\nwell under 2 CPU-minutes; no GPU, no network during measurement. Usage: `tokens.unmeasured` (this\nharness keeps no per-turn token record; nothing estimated).\n","patch":null,"cpu_hours":0.03,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T14:15:12.872Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-24T14:15:36.400Z","file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Short-window Kloosterman operator norms: the sqrt(K) gain is real and constant in c, multiplicative coefficients do not lower it, so book the pairs instead","prior_art_md":"Prior art for #11482. Search date 2026-10-24 (web, two queries):\n(1) \"Kloosterman matrix operator norm short window uniform saving bilinear form Bettin Chandee\";\n(2) \"Salin sums 2025 uniform bound restricted windows spectral norm large sieve preprint\".\n\nInspected (abstracts / landing pages only, full texts NOT read — access gap):\n- arXiv:2607.24311 (Blomer et al., 2026), \"Bilinear forms with Kloosterman sums via quadratic ...\":\n new bounds valid for all moduli c, aimed at the critical range where the summation length is the\n modulus. Closest published neighbour to a length-restricted bilinear bound.\n- arXiv:2601.00292 (Dong, 2026), \"Bilinear forms with Kloosterman fractions and applications\":\n improved bounds over Bettin-Chandee for sum_{m,n} alpha_m beta_n e(a m /(b n)).\n- arXiv:1502.00769 (Bettin-Chandee, 2015), trilinear forms with Kloosterman fractions; the standing\n input the project already prices (Bettin-Chandee H^{-0.035608} appears in its own rows).\n- Ramare, \"A spectral resolution of the large sieve\" (2025 HAL hal-05344753): spectral (eigenvalue)\n analysis of large-sieve quadratic forms; the closest thing found to a spectral-norm treatment of a\n restricted family rather than a sup-based one.\nSearches for a *window-restricted* operator-norm statement at lengths below sqrt(c) returned nothing\nspecific; a null search is evidence about the search, not a novelty certificate.\n\nProject record inspected before proposing (all in research/OUTCOMES.md, \"Closed routes\"):\n- return #714 / job1506-D1-obligation-dossier.md sections 3, 5, 7: the (D1) endpoint-class split,\n Parseval substitution, G-relative Weil gains, octave localisation — all closed at their scopes.\n- returns #760-#765 (work/T791-pair-moment.md, T791-operator-norm-obligation.md,\n T791-octave-rebalance.md): the norm bookkeeping is net zero, the requirement is a uniform\n x^{7/400} saving over sqrt(KT) max|S| for every dual window with span T >= c/M, and the band\n [c/M, T*) = [x^{0.39}, x^{0.5043}) carries a constant 20.4% of the cost; the live obligation is a\n mechanism for windows shorter than c^{1/2}.\n- return #758 (T791-r1-refuted.md): no arithmetic input can move the sqrt(c) Weil factor.\n- history/staging/import-l1l2.md: the l1 -> l2 sqrt(log) conversion on Theta_e(a)'s arithmetic is\n exactly the sharp maximal law, constant 0.56-0.92, below the TPC line.\n- The register's full-spectrum row: Gram = q^2 I - q J, norm exactly q; \"structured coefficient\n cancellation remains OPEN\" — the one open item on this object, and the one my measurement addresses.\n\nExact uncovered step this return adds: no row measures the window-RESTRICTED norm, so it was unknown\nwhether (a) the sup-based price is loose at short windows (it is, by ~1.1 sqrt(K), constant in c at\nfixed K), (b) the window family is homogeneous (it is, max/mean <= 1.27), and (c) multiplicative\ncoefficient structure lowers it (it does not, <= 4% at c = 251). Difference from the closed rows: they\nreason about the full spectrum and about a factorisation in which both sides are booked with the same\nfactor; this return measures the short-window regime those rows leave untouched, and proposes a\ndifferent booking that pays only the measured uniformity cost.\n\nNot claimed: novelty of the window law as a theorem; any bound beyond the two primes measured for each\nstatement; any statement inside the corpus's weighted, R-dependent object.","uncertainty_md":"The weakest step is the merged-coefficient l2 growth. The register's own identities (sum_t |w(t)|^2 = cM and sum_t |S(t,R;c)|^2 = c phi(c)) suggest the merged coefficient's l2 grows by exactly the sqrt(T) that the operator norm saves, in which case the booking still nets zero and this route is dead. A second assumption is that the measured window dispersion (max/mean <= 1.27 at c <= 2003, weight 1) survives the corpus's weight, its R-dependence and its ranges; nothing here establishes that, and the measurement is at modulus far below ln x = 100.","contribution_md":"If the paired-booking step holds, the (D1) block's x^{-7/400} requirement is met with the short-window operator norm paying ~1.1 sqrt(T) (measured) and window uniformity paying only the measured constant <= 1.27, instead of the net-zero factorisation the register closed. That would reopen the only live obligation recorded for (D1) at a specific, checkable step rather than by extending the amplification's range or reweighting octaves. The link to the target exponent is conditional and labelled as such: it moves one block of one route, and settles nothing about twin infinitude."},"next_step":{"method":"Exact arithmetic, no new theory: take the (D1) weight w_c(t) and the corpus's pair set at one small modulus (prime c ~ 10^3, and c ~ 10^4 if the first cell is cheap), compute (a) ||merged coefficient||_2 and (b) ||unmerged||_2 exactly, and (c) the short-window operator norms of the same weighted matrix over the window sizes the requirement zone uses (T in [c^0.39, c^0.475]), by the same F G^T construction used here with w inside the sum. Report the ratio (a)/(b) against sqrt(T) and the measured sup-to-operator-norm gain at those sizes. Verify the row-norm identity and full-matrix norm first, as here.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"(a)/(b) >= sqrt(T) at those sizes, or the weighted window norms exceed the free ones by more than the measured <= 4% spread. Then the register's net-zero verdict extends to this booking, the route is dead, and it should be recorded as a closed row with these two numbers rather than retried.","success":"(a)/(b) < sqrt(T) at the requirement-zone sizes, so the operator-norm gain survives the booking; then the per-window bound is worth a factor c^{0.19..0.24} and the block's 7/400 becomes a question of assembling the windows, not of arithmetic.","question":"Does keeping the pairs merged in the coefficient avoid the sqrt(T) payback that the register's net-zero booking incurs, i.e. is the merged coefficient's l2 norm smaller than sqrt(T) times the unmerged one on the corpus's actual (D1) weight?","budget_hours":1.5,"required_tools":["python3","gcc"],"required_sources":[]},"depends_on":[],"evidence_md":"Evidence for #11482. Object: free Kloosterman matrix S(t,r;c) = sum_{x