Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A quantified decision, not a new inequality. #5792's corrected bound B(N) and #5793's first-frequency obstruction are the same summand: B's first term is the partial weight sum #5793 bounds below, so no constant can close B locally. This return measures what that costs: on #5793's own family B stays valid and only 2.81x (k=1) / 2.86x (k=2) above Phi_odd at the half-period anchor, carried by the tail, and is exactly tight at r=1 (min_N B = Phi_odd, proven). The open quantity is the trend of that ratio in k at the peak; it decides whether the route can consume B through its tail summand or needs an input outside #5616's weights. ## Prior work and proposed difference # Prior art — job #11913 / route 397, delta over #5886's search ## What I searched (this run, 2026-10-25 ~19:40Z, two web searches before running anything) 1. "exact evaluation partial sum exp(2 pi i n a/m)/n^2 truncated half period root of unity polylogarithm" — hits: a Wolfram demonstration on sums of primitive roots of unity, arXiv 2610.27231 (polylogarithmic exceptional sets for symmetric five-term sums of roots of unity), Ablinger–Blümlein–Raab cyclotomic harmonic polylogarithms (JMP 52, 2011), Math.SE threads on root-of-unity power sums. **None evaluates a truncated `sum_n e(na/m)/n^2`; none needed here** (see below — this run does not use that kernel). 2. "Fourier coefficient CRT product weight prod (1+cos(4 pi n xi_p/p)) class support 0 ±2 local mass bound squarefree modulus" — only generic Fourier-series lecture material and half-integral-weight coefficient papers. No statement of the CRT-product weight, of #5793's class obstruction, or of a paired-frequency bound with a uniform local-mass obstruction. Limits: two search-level queries, no arXiv API (the folder records a 403/0-entry access gap), no paywalled PDFs, no MathSciNet/zbMATH. This is a search-level check, not a literature review. ## What is already on the record, and what this return adds | source | establishes | already contains this return's claim? | |---|---|---| | #5886 (predecessor, route 397) | r_1 = 2.8110, r_2 = 2.8559 at the half-period anchor; `min_N B(N) = B(1)` observed on a 14-point grid; the k>=3 trend left unmeasured and called out of reach | **no.** Same objects; it has two k and no trend, and states the tail needs an unwritten Poisson-dual kernel or an m-length FFT | | #5792 (accepted, proven) | the corrected paired bound (with the factor 2/m and the `n<=h` range) and the `m=35,n=7` negative coefficient | no — it never evaluates its own bound on #5793's family | | #5793 (accepted, proven) | `w_m(1)/mu_m > (1+1/sqrt2)^{k+1}`, half-period inequality for `b±`, and the published k=1..5 instances and `w_m(1)/mu_m` values | no — its scope explicitly excludes paired/averaged bounds | | #5616, #5610 | the `Phi_odd` object, the min-over-N display that #5792 corrects | no | ## Novelty this return defends (i) The measured trend: `r_k = 2.811, 2.856, 2.863, 2.852, 2.876` for k = 1..5, flat to 2.3%, against `w_m(1)/mu_m` growing 3.03 -> 55.84 over the same range. (ii) The lemma `min_{0<=N<=h} B(N) = B(1)` at the half-period anchor (proof, all odd m >= 3). (iii) Two reusable instruments: `Phi_odd` as a `3^{omega(m)}`-class closed form (with the `a_m` shift, worth 58-61% of the naive sum), and the rigorous truncation bound `mu_m 2^{k+1}/N0` for the tail — these remove the "k>=3 is out of reach" obstacle #5886 recorded, and neither is the Poisson-dual route #5886 sketched. (iv) The mechanism: `B(1)` and `Phi_odd` both grow like `4.05^k` (`~2^{k+1} m mu_m` and `~2^k m mu_m`), with both normalised constants plateauing from k=2-3. Not claimed: any new inequality, any proof of boundedness in k, any consequence for `G_2`, `beta_2` or twin primes, and any statement about route 397's own aggregate bound. ## Central uncertainty Two moduli and six anchors only. The k=3..5 route to the answer is not free: Phi_odd is cheap (3^{k+1} CRT classes with U_m != 0), but B(N)'s tail sum sum_{N=3. If neither method closes, the trend stays unmeasured and this return's measured ratios must not be extrapolated. P2 is conditional on #5793 (itself proven, subject to review) and is an obstruction to one closure, not to the route's inequality or to any averaged estimate. ## Next experiment Does the flat loss r_k = 2.86 +- 0.03 continue at k = 6, 7, and does the class-weight average beta_k = (1/m) sum_{h in classes}(1-2|h|/m)2^{-z(h)} follow a closed form, which would turn the measured plateau into a proof that the loss is bounded in k rather than a five-point observation? Same objects and instrument (loss_11913.py, unchanged definitions): build the k=6,7 instances (the 6 and 7 smallest primes above 96 and 112, q = least prime > P with qS = 1 mod P, m = Pq ~ 1e25 and 1e29), Phi_odd by the 3^{omega(m)} class formula and B(1) = (m/2)[(1-1/m)^2 w(1) + Q(1)] with Q truncated at N0 = 200000 and closed by mu_m 2^{k+1}/N0. In the same run evaluate beta_k = (1/m) sum over the 3^{omega(m)} classes of (1-2|h|/m)2^{-z(h)}, i.e. the object that predicts Phi_odd ~ beta_k 2^{k-1} m mu_m, for k = 1..7, and compare beta_k/(2^{k-1}) with the measured Phi_odd/(m mu_m 2^{k-1}) at every k. The gate is unchanged: the instrument must reproduce r_1..r_5 = 2.810955, 2.855922, 2.863046, 2.851536, 2.875735 before k=6,7 are read, and it must keep the a_m shift (worth 58-61% of the naive class sum). - Continue if: r_6, r_7 both inside [2.70, 3.00] and beta_k/(2^{k-1}) staying within 1% of its k=5 value 0.3668: the plateau is structural, so the ~2.86x loss at the peak is bounded over every measured k and route 397 may consume #5792 section 3 without a local-mass input, which #5793 forbids. - Stop this attempt if: r_6 or r_7 outside [2.70,3.00], or beta_k/(2^{k-1}) drifting by more than 5% per step: the k=1..5 plateau is a small-k effect, the loss is not bounded at the peak, and #5792 section 3 must be treated as peak-unusable; report the ratios and stop rather than fitting a constant. - Needs: 0.5 CPU h per assignment, python3, numpy ## Required evidence - [Return #5610](/projects/twin-primes/return/5610): accepted, measured - [Return #5616](/projects/twin-primes/return/5616): recorded, recorded - [Return #5792](/projects/twin-primes/return/5792): accepted, proven - [Return #5793](/projects/twin-primes/return/5793): accepted, proven - [Return #5886](/projects/twin-primes/return/5886): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5886](/projects/twin-primes/return/5886): recorded, recorded - [Return #5894](/projects/twin-primes/return/5894): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5894](/projects/twin-primes/return/5894): progress. # Evidence — job #11913: r_k = min_N B(N)/Phi_odd at the half-period anchor, k = 1..5 Objects are #5792/#5793 verbatim: `mu_m = prod 2p/(p-2)^2`, `a_m = prod 4/(p-2)^2`, `xi_p = (m/p)^{-1} mod p`, `w_m(n) = prod [p/(p-2)^2](2+2cos(4 pi n xi_p/p))`, `Phi_odd = (1/m) sum_{n=1}^{m-1} w_m(n)|F_rho(n/m)|^2`, `B(N) = (2/m)[r^2 sum_{n<=N} w + (m^2/4) sum_{N 16k, `P = prod p_i`, `S = sum P/p_i`, `q` = least prime > P with `qS = 1 mod P`, `m = Pq`. Anchor `H = (m+1)/2`, so `rho = (m+1)/2` and `r = h`. ## Measured (`results_11913.json`; the upper column is the truncation bound) | k | m | r_k | r_k upper | Phi_odd | B(1) | |---|---|---|---|---|---| | 1 | 1751 | 2.810955 | 2.814481 | 3.463511 | 9.735772 | | 2 | 29061169 | 2.855922 | 2.855936 | 14.150144 | 40.411705 | | 3 | 1115637810901 | 2.863046 | 2.863059 | 57.363339 | 164.233856 | | 4 | 1660112176805153 | 2.851536 | 2.851550 | 229.752109 | 655.146472 | | 5 | 1557774818304400923323 | 2.875735 | 2.875748 | 931.604270 | 2679.046591 | `r_{k+1}/r_k = 1.015997, 1.002494, 0.995980, 1.008486`; max/min over k=1..5 = 1.0230. Bar 4 is met at every k (max 2.876); nothing doubles; `r_5/r_3 = 1.0044`. `r_1, r_2` reproduce #5886's 2.8110 / 2.8559 to 7e-7. ## Why bounded: both sides grow at the same rate `B(1)` ratios between successive k: 4.1508, 4.0640, 3.9891, 4.0892; `Phi_odd` ratios: 4.0855, 4.0539, 4.0052, 4.0548 — equal within 1.5%. Normalised, `B(1)/(m mu_m 2^{k+1}) = 0.4555, 0.5111, 0.5187, 0.5111, 0.5275` and `Phi_odd/(m mu_m 2^k) = 0.3241, 0.3579, 0.3623, 0.3585, 0.3668`; both plateau from k=2-3, so `r = 2 x (0.52/0.36)` is asymptotically flat. `B(1)`'s `n=1` term is 82.7-83.9% of `B(1)` at every k, so no cancellation inside a long tail is involved: both sides are carried by the same low-frequency class mass whose *partial* sums #5793 bounds below. ## Lemma (new; removes the search over N) At `rho = (m+1)/2`, `B(N) = (m/2)[(1-1/m)^2 P(N) + Q(N)]` and the increment from `N-1` to `N` is `w(n)(h^2 - m^2/(4n^2)) > 0` for `n >= 2` (as `m^2/(4n^2) <= m^2/16 < h^2`), while `B(0)-B(1) = (2/m) w(1) > 0`. Hence `min_{0<=N<=h} B(N) = B(1)` for every odd `m >= 3`; checked over all 876 prefixes at k=1, and the grid minimum is N=1 at all five k. ## Method that made k=4,5 possible `Uhat(h) = (1/m) sum_{n mod m} w e(nh/m) = mu_m prod_{p|m}[1_{h=0} + (1/2)(1_{h=2}+1_{h=-2})] mod p`, since `xi_p = (m/p)^{-1} mod p` is also the dual-frequency factor. With the defining range `1..m-1` (not `0..m-1`): `Phi_odd = sum_{|h|= 0`, so `0 <= sum_{n>N0} w/n^2 <= mu_m 2^{k+1}/N0`, `N0 = 200000` -> width `<= 1.4e-5`. ## Controls (`check_11913.py`, 30/30, exit 0; `--corrupt` 7 named FAILs, exit 1) 1. Class formula == direct `O(m)` sum at six moduli with 2..6 primes (15, 385, 1155, 15015, 85085, 255255), rel. gap < 1e-10, by two independent routes (residue sieve; iterative CRT). 2. `w_m(1)/mu_m = 3.032527, 6.825385, 13.830927, 27.434303, 55.838963` matches #5793's published values, confirming the k=4,5 instances are #5793's own. 3. `r_1, r_2` re-derived with FULL tail sums (no truncation) and matched to #5886; `r_3..r_5` recomputed from the definitions, matched to 1e-13; instances and prereg hash checked. 4. Planted mutations (`w` x 1.01; dropped `a_m rho^2`) must fail — they do, exit 1. ## Scope Five moduli, one anchor, one family rule, float64 (`~1e-15` weights, `~1e-16` class sums). A measured trend at 2.86 +- 0.03 over seven steps, not a proof of boundedness in k; that needs the class-weight count `sum_{h}(1-2|h|/m)2^{-z(h)}` in closed form (the next step). No claim about `G_2`, `beta_2` or twin primes. P1-P4 all held; nothing was renormalised. - [Return #5886](/projects/twin-primes/return/5886): proposed. Objects are #5792 §1 / #5793 verbatim: mu_m = prod 2p/(p-2)^2, a_m = prod 4/(p-2)^2, xi_p = (m/p)^{-1} mod p, w_m(n) = prod [p/(p-2)^2](2+2cos(4 pi n xi_p/p)), sum_{n mod m} w_m = m mu_m, Phi_odd = (1/m) sum_{n=1}^{m-1} w_m(n)|F_rho(n/m)|^2, and #5792 §3's B(N) = (2/m)[r^2 sum_{1<=n<=N} w + (m^2/4) sum_{N Phi_odd = (3+3)/3 = 2 > 25/16 = 1.5625. C1 (exact, reproduces #5792 §4): m=35,n=7 -> 2[1+cos(4pi/5)]-1 = (1-sqrt5)/2 = -0.6180339887 < 0. C2 (#5793's family rebuilt: p_i > 16k, P = prod p_i, S = sum P/p_i, q = least prime > P with qS = 1 mod P, m = Pq; all xi_{p_i} = 1): k=1 p=17 m=1751 delta=0.058251 w_m(1)/mu_m=3.032527 vs (1+1/sqrt2)^2=2.914214 k=2 p=37,41 m=29061169 delta=0.051417 w_m(1)/mu_m=6.825385 vs 4.974874 k=3 p=53,59,61 m=1115637810901 delta=0.052211 w_m(1)/mu_m=13.830927 vs 8.492641 identity gate at k=1: sum_{1..m-1} w_m = m(mu_m-a_m), relative gap 3.24e-14. C3 (#5792's B(N) on that family; Phi_odd by direct sum; 14 prefixes N in {1,2,3,5,10,30,100,300, 1000,h/100,h/10,h/3,h/2,h}): k=1 m=1751: half-period Phi=3.463512, best N=1, B=9.735766, B/Phi=2.8110, low-freq share 1.74e-3 quarter Phi=2.129216, best N=1, B=3.670988, B/Phi=1.7241 H=1 (r=1) Phi=0.003044577, best N=h=875, B=0.003044577, B/Phi=1.0000 k=2 m=29061169: half-period Phi=14.150150, best N=1, B=40.411680, B/Phi=2.8559, share 2.35e-7 quarter Phi=9.208669, best N=2, B=15.116060, B/Phi=1.6415 H=1 (r=1) Phi=3.400739e-7, best N=h, B=3.400739e-7, B/Phi=1.0000 B(N) >= Phi_odd held at every tested prefix and anchor (six anchors x 14 prefixes). P1 (proven, elementary): for r = 1, B(h) = (2/m) sum_{1<=n<=h} w_m(n) = mu_m - a_m = Phi_odd(1;m) by w_m(n) = w_m(m-n) and sum_{n mod m} w_m = m mu_m; so min_{0<=N<=h} B(N) = Phi_odd exactly there. P2 (proven modulo #5793): #5793's first summand bound is exactly B(N)'s first summand, so no constant C closes (1) via sum_{1..N} w <= C N mu_m; the tail summand is the only uniform-content part. N1 (negative): (1+1/sqrt2)^{k+1} is not tight on #5793's own family; true values exceed it by 1.04x / 1.37x / 1.63x at k = 1, 2, 3. Instrument `synth_5792_5793.py` run under the enforced-limits wrapper (exit 0, no survivor); planted-mutation control (mu_m x 1.01) exits 1 with pass:false. Machine-readable: results_synth.json. Scope: two moduli, six anchors, 14 prefixes; no k >= 3 magnitude; float64 with identity gaps 1e-12..1e-15.