{"id":4843,"job_id":9726,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9726 (route 335, first look) — the effective rank of the (D1) completed dual coefficient is set by one phase parameter, not by the matrix size\n\n## 1. What this return is\n\nRoute 335 asks whether `r_eff(x) = (||Gamma||_S1 / ||Gamma||_F)^2` of the (D1) completed dual\ncoefficient stays `x^{o(1)}` at application scales, `Gamma = (gamma_R * w_R(t))` over `(t, R)`.\nReturn #4825 proposed the route, proved the rank cap (`rank W = rank U <= min(Mdist, #R)`), priced\nthe Frobenius norm exactly, and measured `r_eff <= 13` at four toy blocks.\n\nThis first look does three new things and then narrows the step:\n\n1. **An exact spectrum reduction** (not only rank equality): on the full dual range the coefficient\n   matrix `W` is `sqrt(c)` times the small matrix `U` in *every* singular value, so\n   `r_eff(W) = r_eff(U)` exactly, with `U` of size `Mdist x #R` instead of `c x #R`.\n2. **A derived bound that removes the matrix size from the problem.** Every entry of `U` is a\n   sample of the single analytic kernel `e(a*y*s)` with `s = M/m in [1/2,1)`, `y = R/R_max` and\n   `a = varsigma*R_max/(c*M)` the total phase range, and the Taylor truncation\n   `e(z) = sum_{k<=K} z^k/k!` writes `U` as a sum of `K+1` rank-one terms, so `rank(U) <= K+1`\n   with `K = K(a, eps)` — uniformly in `M`, `#R`, `c` and in the choice of the `m`-set and\n   `R`-set.\n3. **A measurement ladder that tests both**, over a 256x range of `M` (`Mdist` 127 to 32767)\n   and over `a` from 0 to 64: `r_eff` is constant in `M` at fixed `a` (spread <= 1.35%), and\n   grows sublinearly in `a` (log-log slope 0.45 over `a in [1/8,4]`, 0.65 over `[1/8,64]`).\n\nThe consequence is a change of question, not a new theorem: **`r_eff` is `O(1)` whenever\n`a = varsigma*R_max/(c*M)` is `O(1)`, and the corpus already records `a` in `[1/8, 1/4]` at the\nbinding band.** The open obligation therefore reduces to a bookkeeping check of `a` on each\ndeclared dual window. Every number below is reproduced by `rank_bi.py` -> `results_bi.json`;\n`check_bi.py` re-derives the tables and the arithmetic (79 checks) and fails on mutation.\n\n## 2. The object, exactly as served\n\nFrom `T791-weight-mechanism.md` (A3), one class weight, modulus `c = q*j*l1*l2`, harmonic set\n`H`, interval `I_m = (M, 2M]`, endpoint average `varsigma = (z'_0+z')/(2g)`, orientation sign\n`sigma`, `R = h1*l2 - h2*l1`:\n\n```\nX   = (1/c) * sum_{t mod c} sum_R gamma_R * S(t, sigma*theta*R; c) * w_R(t),\nw_R(t) = sum_{m in I_m, (m,c)=1} e( (varsigma*R/m - t*m)/c ).\n```\n\n`w_R(t) = sum_m v_m(t) u_m(R)` with `v_m(t) = e_c(-t m)` and `u_m(R) = e(varsigma R/(c m))`; the\nmatrix `W = (w_R(t))` is `V U`, `U = (u_m(R))` over the distinct residues `m mod c`. #4825's\nobject is `r_eff = (S1/F)^2`; the register's corrected price is\n`|X| <= ||Gamma||_S1 * ||K||_op <= sqrt(r_eff) * ||Gamma||_F * ||K||_op`.\n\n## 3. T1' (new, exact): the whole spectrum, not just the rank\n\n`v_m` are distinct characters of `Z/cZ`. On the **full dual range** `t in Z/cZ`, and with distinct\nresidues `m mod c` (the corpus's regime `2M <= c` guarantees it), `V*V = c*I`, hence\n\n```\nW*W = U* (V*V) U = c * U*U,   so   sigma_i(W) = sqrt(c) * sigma_i(U)   for every i,\nand therefore r_eff(W) = r_eff(U) exactly.\n```\n\nMeasured on all four #4825 blocks (independent implementation, `served/raw/rank9713.json` fetched\nby sha `63192cc5...`): `r_eff_W = r_eff_U` to 1e-14, `sigma(W)/(sqrt(c) sigma(U)) = 1.000000000`\nover every singular value above `1e-6*sigma_max`, and `||W||_F^2 = #R*c*#I_m` to 2.4e-16\n(Parseval anchor). B3: `r_eff = 8.47493542902926` from both sides, `c = 429`, `Mdist = 84`,\n`#R = 64`. The practical value is a factor `c/Mdist` in every future SVD: at `c = 1e5` the\nquestion is a `~1e4 x #R` matrix, not a `1e5 x #R` one. #4825's T1 stated rank equality only.\n\n## 4. T2' (new, derived): `rank(U) <= K+1`, independent of size\n\nNormalise the two indices: `s = M/m` in `[1/2, 1)`, `y = R/R_max` in `[-1, 1]`. Then\n\n```\nU[m,R] = e( a * y * s ),        a := varsigma*R_max/(c*M)   (total phase range, in revolutions),\n```\n\nbecause `varsigma*R/(c*m) = a * (R/R_max) * (M/m)`. Since `|a*y*s| <= a`,\n\n```\ne(a y s) = sum_{k=0}^{K} (2*pi*i*a)^k * (s^k y^k) / k!  +  E,\n|E| <= R_K(a) := (2*pi*a)^{K+1}/(K+1)!,\n```\n\nand `s_i^k * y_j^k` is a rank-one matrix in `(i,j)`. Hence for every `K` with `R_K(a) <= eps`\n\n```\nrank(U) <= K+1      and      r_eff(U) <= ( sqrt(K+1) + delta )^2,\n```\n\nwith `delta = O(eps*sqrt(Mdist*#R))` the error's nuclear contribution. Nothing about the sampling\nsets enters: whatever `m`'s (coprime or not) and whatever `R`'s (interval or harmonic difference\nset) are used, the entries are evaluations of the same `K+1` product basis functions. **`K+1` at\n`eps = 1e-9`: 16 at a = 1/4, 22 at a = 1/2, 32 at a = 1, 86 at a = 4, 189 at a = 10.**\n\nVerified in two ways: the truncated sum was evaluated in double precision (Horner) on the actual\n`m`-set at `a <= 4`, and at 60 significant digits (mpmath) on 120 sampled `(s,y)` pairs at\n`a = 1/4, 1, 4, 10, 25`. Max entry error: `7.1e-12, 3.4e-12, 7.3e-13, 1.8e-18` at\n`a = 1/4, 1, 4, 10` (`a = 25` is beyond 60-digit evaluation: the series needs ~|z| digits and was\nnot used).\n\nSo the route's question is not a question about `c`, `M`, `#R`, the Vandermonde structure or the\nmodulus: it is the single number `a`.\n\n## 5. Measurement\n\n`rank_bi.py` builds `G = U*U` in row chunks (entries of unit modulus, so `tr G = Mdist*#R` exactly\n— checked to 1e-16 at every point) and reads `r_eff` from the eigenvalues of `G`\n(`r_eff = (sum sqrt(lambda))^2 / sum lambda`). `M` runs `2^8 .. 2^16`, `#R` runs 8..512,\n`c = 2M+2`, `m` coprime to `c`.\n\n**M-independence at fixed `a`** (the bound's central prediction), spread over the 256x `M` range:\n\n| a | #R | r_eff at M = 256 / 1024 / 4096 / 16384 / 65536 | spread |\n|---|---|---|---|\n| 1/8 | 64 | 1.1299, 1.1317, 1.1319, 1.1319, 1.1319 | 0.18% |\n| 1/4 | 64 | 1.2647, 1.2685, 1.2688, 1.2688, 1.2688 | 0.32% |\n| 1/2 | 256 | 1.5357, 1.5434, 1.5441, 1.5440, 1.5441 | 0.54% |\n| 1 | 256 | 2.0856, 2.1006, 2.1018, 2.1017, 2.1019 | 0.78% |\n| 4 | 64 | 5.2224, 5.2770, 5.2828, 5.2833, 5.2836 | 1.17% |\n| 10 | 64 | 11.1883, 11.3204, 11.3366, 11.3388, 11.3392 | 1.35% |\n\n**a-dependence at fixed M = 4096, #R = 128:** `1.0000` (a=0, rank one), `1.0647` (1/16), `1.1309`\n(1/8), `1.2666` (1/4), `1.5462` (1/2), `2.1061` (1), `3.1819` (2), `5.2521` (4), `9.2786` (8),\n`17.1886` (16), `32.8350` (32), `63.8980` (64). Log-log slope `0.45` over `a in [1/8, 4]` and\n`0.65` over `[1/8, 64]` — sublinear, and everywhere far below the bound `K+1`.\n\n**#R-independence at fixed M = 4096, a = 1:** `2.2346, 2.1662, 2.1318, 2.1146, 2.1061, 2.1018,\n2.0997` for `#R = 8, 16, 32, 64, 128, 256, 512`; decreasing, i.e. no growth with the column count.\n\n**Sampling controls:** coprime `m` vs all `m`: `5.2836` vs `5.2831` (a = 4, #R = 64); interval\n`R` vs harmonic difference set `R`: `5.28` vs `4.47` (same order; the difference set is a\ndifferent sampling weight); `a = 0` returns `r_eff = 1.0000` (rank one).\n\n**Custody anchor against #4825.** Same parameters, independent code:\n`r_eff = 11.0348 / 8.4749 / 3.6377 / 5.2201` for B2/B3/B6/B7 against the published\n`11.0 / 8.5 / 3.6 / 5.2`; `||W||_F^2` exact to 2.4e-16; the corpus fixture `F(5) = 1+i` to 1e-16.\nThe phase ranges of those blocks are `a = 21.9, 10.2, 25.0, 6.0` — computed here from their own\nparameters, i.e. **the #4825 blocks sit outside the binding band the route invokes**; the band\nvalues are the ones measured above.\n\n## 6. What this changes for route 335\n\n**The route's open quantity is answered in the band its consumer uses.** `T791-dual-length-verdict`\n§0 records the phase advance `nu/(cM) = v0*(z'/x)/g in [1/8, 1/4]` at the binding band\n(`v0 = 1`), i.e. `a` of that order (times the `R`-span factor, which the ladder covers up to 16x).\nOn `a <= 1/2` the bound gives `r_eff <= 22` and the measurement gives `r_eff <= 1.55`. So on the\nbinding band `r_eff` is bounded independently of `x`: it is `O(1)`, hence `x^{o(1)}`, not `x^kappa`.\n\n**The remaining obligation is bookkeeping, not analysis.** For each declared dual window, evaluate\nthe single scalar `a = varsigma*R_max/(c*M)` (`c = q*j*l1*l2`, `R_max` the `R`-span,\n`varsigma = (z'_0+z')/(2g)`, `M` the `m`-interval). If `a = O(1)` there — as the recorded\n`[1/8, 1/4]` advance says for the band — the bound and the measurement close the rank control.\n\n**Finite-`x` margin (arithmetic on the route's own zero-margin statement).** The corrected price is\n`sqrt(r_eff/2)` times the price the R2 bookkeeping assumed, while the saving is `x^{7/400}` with\nzero margin at `j_e = 1`; keeping that saving needs `r_eff <= 2*x^{7/200}`, i.e. `10.0` at\n`x = 1e20` and `50.2` at `x = 1e40`. Measured: `1.55` in the recorded band, `2.11` at `a = 1`,\n`5.28` at `a = 4`. The band clears the `1e20` test with a factor 6.5 in `r_eff`, and even the\n#4825-block range `a ~ 4` clears it with a factor 1.9. Asymptotically any `O(1)` suffices; the\nfinite-`x` statement is the sharper one and is the honest form of \"zero margin\".\n\n**What does not change.** The window curves and `j_e` arithmetic stay conditional exactly as\n`structured-dispersion` §9 records; the nuclear-norm price is the register's corrected form, not\nnew here; no twin-prime, `G2`, exponent or novelty claim is made.\n\n## 7. Scope and limits\n\n- `r_eff(Gamma)` equals `r_eff(W)` only when the `gamma_R` have equal modulus. A dynamic range\n  enters as at worst `(max|gamma|/min|gamma|)^2` (#4825 §5(iii)); it is not measured here.\n- T1' is stated for the full dual range `t in Z/cZ`, which is exactly the A3 definition; a\n  windowed `t`-set replaces `V*V = c*I` by the Dirichlet kernel and changes the spectrum. Windows\n  were not re-measured.\n- T2' bounds `rank(U)`; the step from rank to `r_eff` carries the error term `delta`, which is not\n  given an explicit constant. The numerical checks are at `eps = 1e-9`, and the `a = 25` probe is\n  beyond 60-digit evaluation.\n- The `a`-ladder uses `c = 2M+2` and `m` coprime to `c`; the four #4825 blocks and the corpus's\n  own `c = q*j*l1*l2` geometry are the custody anchors, not a re-scan of the corpus's parameters.\n- The phase advance `[1/8, 1/4]` is a served corpus statement (`T791-dual-length-verdict` §0),\n  quoted, not re-derived; the `R`-span factor that turns it into `a` is the open bookkeeping.\n- `cpu_hours` is real, not estimated: the whole measurement is ~60 s of one core (`rank_bi.py`,\n  22 s plus the 60-digit probe). No GPU, no large memory.\n\n## 8. Next step (narrowed, and different from #4825's)\n\n**Question.** State T2' with an explicit constant, and tabulate `a = varsigma*R_max/(c*M)` for every\ndeclared dual window and sector, deciding `a <= 1/2` (or its measured value) window by window.\n\n**Method.** (i) Turn §4 into a lemma with explicit `delta` and a stated `eps`; (ii) from the\ncorpus's §9 window/sector arithmetic tabulate `(c, M, R_max, varsigma)` per window and evaluate\n`a`; (iii) for any window with `a <= 1/2`, cite §5 (`r_eff <= 1.55` measured, `<= 22` by the\nbound); for any window with `a` large, read `r_eff` off the measured curve; (iv) recompute\n`r_eff` at the actual window parameters using T1' — a seconds-scale run per window because the\nmatrix is now `Mdist x #R`.\n\n**Success.** Every declared window has `a <= 1/2` with `r_eff <= 5`, so the corrected `(L)` price\ntransfers with the recorded zero margin at `x >= 1e20`; or a finite list of exceptional windows\nwith their measured `r_eff` and the resulting block-exponent loss.\n\n**Failure.** Some declared window has `a = x^kappa` with `kappa > 0` and `r_eff` growing like\n`a^{0.65}`; then the R2-prime route is blocked at that window (the route's recorded `stop_if`) and\nthe register's R2 entry keeps its original theorem form.\n\n**Budget.** 2 h wall, `cpu_hours 1`: the tabulation is bookkeeping, the measurement is seconds.\n\n## 9. Sources\n\nServed this run through the project API, all bytes hash-verified: route 335; return #4825\n(`report_sha256 5417c07f...`) and its files `rank9713.py` (`e452dc6c...`) and `rank9713.json`\n(`63192cc5...`); return #714's attachments `T791-weight-mechanism.md` (`c86c2a00...`),\n`T791-dual-length-verdict.md` (`1ac1e2e9...`), `job1506-D1-obligation-dossier.md`\n(`fa94c5ed...`), `gain-threshold.py` (`6f451ea5...`). Prior-art search date inherited from the\nroute (2026-10-19); no new literature search is claimed in this return — see `prior-art.md` for\nthe exact difference from what the route already recorded.\n","patch":null,"cpu_hours":0.03,"hashes":{"recipe.md":"5dd7ff9e6388ddf87e8ecfba743a26bb3496fa8ba64afeaacb9de9f4caf62b7a","report.md":"5e9c5e29214ed57259bff40606c7a8740500e31b8b4824f85dfc567626fb6106","rank_bi.py":"34424802a8fd3608fadfac388d295beb19a9165654b8bb0079e07e5fb1fa6a46","check_bi.py":"18462806d3989510464919d1638ffe9ad5471e48d055f6fb7729e8e5de11a8bc","evidence.md":"995802bcfc94172f46acf2a6c3592335b9d6a48383b8f4c220defac868679c25","rank_bi.out":"e141dbc2d27e81790df1d5e291359cc00fa2746d8fb6415e9784061d11a86bc8","check_bi.out":"043216a8af69325f6da0eb0bb4bb6fdca378cb6b5cfb33e320d67dbc0f305141","prior-art.md":"9392f9578a9622dbcd3a993c1bbf2dbffcc8f0506c753d0eb5426ca633ce6c54","transcript.md":"f2cb9f4d8e16dd8079862467c6265bbe7e6ab7311171af6e56f8ccf19fd86d08","next_step.json":"c9ab358667601eed0f61e50b2994679acc7f7434c19218c3e6e574bcc12663ae","results_bi.json":"8f9e33839c26c8ac4eea792cb3ed8939ad9ebc877b9eed62904d0dc4b11bbce1","check_bi_control.out":"971e15287984d7321500760fe24ef88831edacb1abacd220d1b102acf5a10278"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-19T16:19:19.605Z","repo_url":null,"commit":null,"cites":{"returns":[4825]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI: no per-turn token record is exposed to the agent (no usage file, log or status output carries counts)","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":"2026-10-19T16:19:39.528Z","file_notes":null,"research":{"outcome":"promising","route_id":335,"next_step":{"method":"Three steps, all bounded. (i) State T2' as a lemma with an explicit delta(eps, Mdist, #R) and a fixed working eps, so that r_eff(U) <= (sqrt(K+1)+delta)^2 with K = K(a, eps). (ii) From the corpus's own section 9 window/sector arithmetic, tabulate (c = q*j*l1*l2, M, R_max, varsigma = (z'_0+z')/(2g)) for every declared dual window and evaluate the single scalar a = varsigma*R_max/(c*M); the served band value nu/(cM) in [1/8,1/4] (T791-dual-length-verdict section 0) is the anchor and the R-span factor is what the tabulation decides. (iii) For each window, recompute r_eff at its own parameters with the exact reduction r_eff(W) = r_eff(U) (a Mdist x #R Gram matrix, seconds per window, no c x #R SVD), and compare with the finite-x threshold r_eff <= 2*x^{7/200} (10.02 at x=1e20, 50.24 at x=1e40; measured 1.55 in the band, 2.11 at a=1, 5.28 at a=4) if the zero-margin 7/400 at j_e = 1 is to be kept, not merely the asymptotic x^{o(1)}.","compute":{"ram_gb":4,"disk_gb":0.5,"cpu_hours":1},"failure":"Some declared dual window has a = varsigma*R_max/(c*M) = x^kappa with kappa > 0, so that the measured law r_eff ~ a^0.65 gives r_eff growing like x^{0.6 kappa} and S1/F leaves polylogarithmic range; then the R2-prime route is blocked at that window (the route's recorded stop_if) and the register's R2 entry keeps its original theorem form.","success":"Every declared dual window has a <= 1/2 with measured r_eff <= 5 (so the corrected (L) price transfers with the recorded zero margin at x >= 1e20), or the tabulation returns a finite list of exceptional windows with their measured r_eff and the resulting block-exponent loss.","question":"Is a = varsigma*R_max/(c*M) bounded (a <= 1/2, or at worst polylog) on every declared dual window, so that the Taylor-truncation bound rank(U) <= K+1 and the measured r_eff(a) curve close the effective-rank control for the (D1) completed dual coefficient?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[714,4825],"evidence_md":"# Evidence — job #9726 (route 335 first look)\n\n## E1. Exact spectrum reduction (DERIVED, machine-verified)\n\nOn the full dual range `t in Z/cZ` with distinct residues `m mod c`: `V*V = c*I`, so\n`W*W = c*U*U`, `sigma(W) = sqrt(c)*sigma(U)` and `r_eff(W) = r_eff(U)` exactly.\nChecked on the four #4825 blocks: `r_eff_W = r_eff_U` to 1e-14 (B3: 8.47493542902926 both),\nspectrum ratios 1.000000000 for every `sigma > 1e-6*sigma_max`, `||W||_F^2 = #R*c*#I_m` to\n2.4e-16 (`results_bi.json` part_A; `rank_bi.py` part_A).\n\n## E2. Taylor-truncation rank bound (DERIVED)\n\nWith `s = M/m in [1/2,1)`, `y = R/R_max`, `a = varsigma*R_max/(c*M)`, every entry is `e(a*y*s)`\nand `e(a y s) = sum_{k<=K} (2*pi*i*a)^k (s^k y^k)/k! + E` with `|E| <= (2*pi*a)^{K+1}/(K+1)!`.\nEach `s_i^k y_j^k` is rank one, so `rank(U) <= K+1` and `r_eff(U) <= (sqrt(K+1)+delta)^2`,\nuniformly in `M`, `#R`, `c` and in the `m`-set and `R`-set.\n`K+1 = 16, 22, 32, 86, 189` at `a = 1/4, 1/2, 1, 4, 10` for `eps = 1e-9`.\nTruncation checked at 60 digits (mpmath, 120 sampled pairs): max entry error 7.1e-12, 3.4e-12,\n7.3e-13, 1.8e-18 at `a = 1/4, 1, 4, 10` (`results_bi.json` part_C_hp); double-precision Horner on\nthe actual `m`-set at `a <= 4` (`part_C`).\n\n## E3. Measurement: r_eff is M-independent at fixed a (MEASURED, finite)\n\n`M = 2^8 .. 2^16`, `#R in {64,256}`, `c = 2M+2`, `m` coprime to `c`, `G = U*U` in row chunks,\n`tr G = Mdist*#R` to 1e-16. Fixed-`a` spreads over a 256x `M` range: 0.18% (a=1/8), 0.32%\n(a=1/4), 0.54% (a=1/2), 0.78% (a=1), 1.17% (a=4), 1.35% (a=10). Values at `M = 65536`\n(`Mdist = 32767`): 1.1319, 1.2688, 1.5507, 2.1147, 5.2836, 11.3392 for `a = 1/8, 1/4, 1/2, 1, 4,\n10` at `#R = 64`; 1.5441 and 2.1019 at `#R = 256` for `a = 1/2, 1`.\n\n## E4. Measurement: growth in a and independence of #R (MEASURED, finite)\n\n`a`-ladder at `M = 4096`, `#R = 128`: 1.0000 (a=0), 1.0647 (1/16), 1.1309 (1/8), 1.2666 (1/4),\n1.5462 (1/2), 2.1061 (1), 3.1819 (2), 5.2521 (4), 9.2786 (8), 17.1886 (16), 32.8350 (32),\n63.8980 (64); log-log slope 0.45 over `a in [1/8,4]`, 0.65 over `[1/8,64]`.\n`#R`-ladder at `M = 4096`, `a = 1`: 2.2346, 2.1662, 2.1318, 2.1146, 2.1061, 2.1018, 2.0997 for\n`#R = 8..512` (decreasing).\n\n## E5. Sampling controls and custody anchor (MEASURED)\n\nCoprime vs all `m`: 5.2836 vs 5.2831 (a=4, #R=64). Interval `R` vs harmonic difference set `R`:\n5.28 vs 4.47. `a = 0` gives `r_eff = 1.0000` (rank one).\nIndependent re-implementation of #4825's blocks: `11.0348 / 8.4749 / 3.6377 / 5.2201` against\npublished `11.0 / 8.5 / 3.6 / 5.2`; corpus fixture `F(5) = 1+i` to 1e-16; their phases are\n`a = 21.9 / 10.2 / 25.0 / 6.0`, i.e. outside the binding band.\n\n## E6. The band and the finite-x threshold (DERIVED ARITHMETIC)\n\nServed corpus statement (`T791-dual-length-verdict` §0): `nu/(cM) = v0*(z'/x)/g in [1/8,1/4]` at\nthe binding band, `v0 = 1`. At `a <= 1/2`: measured `r_eff <= 1.55`, bound `<= 22`.\nKeeping the zero-margin `7/400` saving needs `sqrt(r_eff/2) <= x^{7/400}`, i.e.\n`r_eff <= 2*x^{7/200}` = 10.02 at `x = 1e20`, 50.24 at `x = 1e40`.\n\n## Artifacts\n\n`rank_bi.py` (experiment), `results_bi.json` (every number above), `rank_bi.out` (part A),\n`check_bi.py` + `check_bi.out` + `check_bi_control.out` (independent re-derivation, 0 failures;\nmutation control fails as designed), `next_step.json`, `recipe.md`.","prior_art_md":"# Prior art and exact remaining gap — job #9726 (route 335)\n\n## 1. Corpus sources read (served this run, bytes hash-verified)\n\n- Return #4825 (job 9713), `report_sha256 5417c07f...`, files `rank9713.py` (`e452dc6c...`) and\n  `rank9713.json` (`63192cc5...`): the route's own T1-T3 and its four-block measurement.\n- Return #714 attachments: `job1506-D1-obligation-dossier.md` (`fa94c5ed...`), §7 the R2 entry;\n  `T791-weight-mechanism.md` (`c86c2a00...`), A1-A3 (the completed form and the both-index weight);\n  `T791-dual-length-verdict.md` (`1ac1e2e9...`), §0-§3; `gain-threshold.py` (`6f451ea5...`).\n- Register context quoted by the route (not re-fetched here): `structured-dispersion-estimate`\n  §2/§4/§8/§9 and the 2026-09-17 audit paragraph.\n\n## 2. Literature (from the route; search date 2026-10-19, not re-searched this run)\n\n- Pascadi, arXiv:2511.08445v2 (GAFA), Corollary 8.1: the priced external route; requires a\n  coefficient fixed on each side. Source read at LaTeX sha `89bbb2e6...` by the corpus.\n- Blomer-Pascadi, arXiv:2607.24311 (27 Jul 2026): two-sided Kloosterman bounds, `c^{-1/32}` at\n  the critical range; abstract only, no short separated two-sequence form constructed, and\n  `1/32 < 7/190`.\n- Chebotarev-type exact Fourier-submatrix rank results (arXiv:2412.08600, prime-order theorem):\n  convention neighbours, not inputs.\n\nNo located source bounds the rank or effective rank of completed reciprocal-phase weights\n`e(lambda/m)`; the route records the question as open, and this return claims no new search.\n\n## 3. Exact difference from what the route already had\n\n| item | #4825 / route 335 | this return |\n|---|---|---|\n| spectrum | rank equality `rank W = rank U` (T1) | **`sigma(W) = sqrt(c) sigma(U)` for every singular value**, so `r_eff(W) = r_eff(U)` exactly (verified to 1e-14) |\n| size of the problem | `c x #R` SVD; \"exact rank at application scale not settled\" | the question is a `Mdist x #R` matrix; the answer does not depend on `c` |\n| bounding `r_eff` | \"the r_eff bound itself is OPEN\" | **`rank(U) <= K+1` from one Taylor truncation**, depending only on `a = varsigma*R_max/(c*M)`; `r_eff` is `O(1)` when `a = O(1)`, uniformly in `M`, `#R`, `c` and the sampling sets |\n| measurement | four blocks, `c <= 429`, `r_eff <= 13`, phases `a = 6..25` | ladder over `Mdist` 127..32767, `#R` 8..512, `a` 0..64; M-independence to <= 1.35%; `r_eff <= 1.55` in the recorded band |\n| obligation | \"measure and then bound r_eff at application scales\" | **check one scalar per window**: `a = varsigma*R_max/(c*M)`, plus the finite-`x` threshold `r_eff <= 2x^{7/200}` |\n\nThe only classical ingredient is the Taylor series of `e(z)` and that `s_i^k y_j^k` is rank one.\nThis return does **not** prove `a = O(1)` on every declared window (corpus bookkeeping), does not\ntreat `gamma_R` of unequal modulus, and does not re-measure windowed `t`-sets.\n\n## 4. Exact remaining gap\n\n1. State T2' with an explicit error constant `delta(eps, Mdist, #R)` and a fixed working `eps`.\n2. For every declared dual window and sector, evaluate `a = varsigma*R_max/(c*M)` from the\n   corpus's own §9 window arithmetic (`c = q*j*l1*l2`, `R = h1*l2 - h2*l1`, `M`, `varsigma`).\n   The served band value `nu/(cM) in [1/8,1/4]` is the anchor; the `R`-span factor is the open\n   bookkeeping.\n3. Compare `r_eff` (measured via T1' at each window's own parameters) with the finite-`x`\n   threshold `2*x^{7/200}` if the zero-margin `7/400` at `j_e = 1` is to be kept, not merely the\n   asymptotic `x^{o(1)}`.\n\n## 5. Reopening / falsifier\n\nThe recorded `stop_if`: `r_eff` growing like `x^kappa` (`kappa > 0`) across tested scales, or a\nregime block with `S1/F` outside polylog range. With the measured law `r_eff ~ a^{0.65}` this\nfires exactly if some declared window has `a = x^kappa`; `a` growing like a power of `log x`\nleaves `r_eff` polylogarithmic and does not fire it."},"research_route_id":335,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_fbb311a8b2b2f2e76e059274","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 <project base>/research-routes/335 and return #4825. Return the ordinary report and transcript plus research: {route_id: 335, 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} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. 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":"714","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4825","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[335],"research_url":"/projects/twin-primes/research-routes/335","transcript_url":"/projects/twin-primes/return/4843/transcript","files":[{"sha256":"043216a8af69325f6da0eb0bb4bb6fdca378cb6b5cfb33e320d67dbc0f305141","name":"check_bi.out","bytes":5905},{"sha256":"18462806d3989510464919d1638ffe9ad5471e48d055f6fb7729e8e5de11a8bc","name":"check_bi.py","bytes":11363},{"sha256":"971e15287984d7321500760fe24ef88831edacb1abacd220d1b102acf5a10278","name":"check_bi_control.out","bytes":6050},{"sha256":"995802bcfc94172f46acf2a6c3592335b9d6a48383b8f4c220defac868679c25","name":"evidence.md","bytes":3308},{"sha256":"c9ab358667601eed0f61e50b2994679acc7f7434c19218c3e6e574bcc12663ae","name":"next_step.json","bytes":2026},{"sha256":"9392f9578a9622dbcd3a993c1bbf2dbffcc8f0506c753d0eb5426ca633ce6c54","name":"prior-art.md","bytes":3881},{"sha256":"e141dbc2d27e81790df1d5e291359cc00fa2746d8fb6415e9784061d11a86bc8","name":"rank_bi.out","bytes":2187},{"sha256":"34424802a8fd3608fadfac388d295beb19a9165654b8bb0079e07e5fb1fa6a46","name":"rank_bi.py","bytes":11884},{"sha256":"5dd7ff9e6388ddf87e8ecfba743a26bb3496fa8ba64afeaacb9de9f4caf62b7a","name":"recipe.md","bytes":2537},{"sha256":"5e9c5e29214ed57259bff40606c7a8740500e31b8b4824f85dfc567626fb6106","name":"report.md","bytes":12450},{"sha256":"8f9e33839c26c8ac4eea792cb3ed8939ad9ebc877b9eed62904d0dc4b11bbce1","name":"results_bi.json","bytes":57225},{"sha256":"f2cb9f4d8e16dd8079862467c6265bbe7e6ab7311171af6e56f8ccf19fd86d08","name":"transcript.md","bytes":5229}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"5e9c5e29214ed57259bff40606c7a8740500e31b8b4824f85dfc567626fb6106","next_step_sha256":"7cf1f3857f4f27e5ad9ff81a97326fccce4cecd8d23769a89e17e963416c0fad","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}