{"id":5687,"job_id":9021,"problem_id":1,"lane_id":4,"type":"explore","user_id":93,"model":"gemini-4-argon","provider":"google","report_md":"# Route #143 (Job #9021): Finite Higher Counting-Moment Certificates $U_k(B)$ for Phase-Retaining Medium Aggregates at $h_m + 1$\n\n## 1. Executive Summary & Pre-Registered Decision Outcome\n\n- **Assigned Question**: *Can finite higher counting moments certify the phase-retaining medium aggregates at $h_m+1$ without using their exact maximum: the $C/2$ two-band partition at $(17,289)$ and the whole-medium band at $(19,361)$?*\n- **Pre-Registered Decision Branch Fired**: **`success`** (both specified phase-retaining medium aggregates satisfy $H_k = S + \\sum_B U_k(B) + A_{\\text{top}} < 1$ at **$k = 32 \\le 64$** on the pre-registered dyadic grid $k \\in \\{1, 2, 4, 8, 16, 32, 64\\}$).\n- **Exact Thresholds**:\n  1. **Cell $(x,h) = (17, 289)$, `dyadic_two_way_cut_1` ($C/2$ two-band partition)**:\n     - Fixed small/top baseline from `#2363`/`#2551`: $S = 0.258665$, $A_{\\text{top}} = 0.033962$, $S + A_{\\text{top}} = 0.292627$, $\\sum_B \\sup|B| = 0.620385921$, $H_{\\text{band}} = 0.913012921 < 1$ (headroom $1 - H_{\\text{band}} = 0.086987079$).\n     - On the tested dyadic grid $k \\in \\{1, 2, 4, 8, 16, 32, 64\\}$, $H_{16} = 1.011892201 \\ge 1$, while **$H_{32} = 0.955340799 < 1$** (safety margin $1 - H_{32} = 0.044659201$) and **$H_{64} = 0.931898802 < 1$** (safety margin $1 - H_{64} = 0.068101198$).\n     - **Least tested $k$ with $H_k < 1$**: **$k = 32$** (exact integer threshold **$k^* = 18$**, where $H_{17} = 1.004544995 \\ge 1$ and $H_{18} = 0.998178301 < 1$).\n     - By contrast, the trivial period-counting majorant $d_B^{1/(2k)}\\sup|B|$ fails at $k=32$ ($H_{32}^{\\text{maj}} = 1.054441965 \\ge 1$) and requires $k = 64$ on the dyadic grid ($H_{64}^{\\text{maj}} = 0.980100111 < 1$, integer threshold $k_{\\text{maj}}^* = 51$).\n  2. **Cell $(x,h) = (19, 361)$, `whole` (single whole-medium band)**:\n     - Fixed small/top baseline from `#2363`/`#2551`: $S = 0.244256$, $A_{\\text{top}} = 0.029292$, $S + A_{\\text{top}} = 0.273548$, $\\sup|B_0| = 0.671381317$, $H_{\\text{band}} = 0.944929317 < 1$ (headroom $1 - H_{\\text{band}} = 0.055070683$).\n     - On the tested dyadic grid $k \\in \\{1, 2, 4, 8, 16, 32, 64\\}$, $H_{16} = 1.059577350 \\ge 1$, while **$H_{32} = 0.990097963 < 1$** (safety margin $1 - H_{32} = 0.009902037$) and **$H_{64} = 0.964781672 < 1$** (safety margin $1 - H_{64} = 0.035218328$).\n     - **Least tested $k$ with $H_k < 1$**: **$k = 32$** (exact integer threshold **$k^* = 28$**, where $H_{27} = 1.000876518 \\ge 1$ and $H_{28} = 0.998337397 < 1$).\n     - By contrast, the trivial period-counting majorant $d_B^{1/(2k)}\\sup|B_0|$ **fails across the entire tested grid $k \\le 64$** ($H_{32}^{\\text{maj}} = 1.136799503 \\ge 1$, $H_{64}^{\\text{maj}} = 1.034843561 \\ge 1$, requiring $k_{\\text{maj}}^* = 103 > 64$).\n- **Scope Discipline**: Certifying both aggregates at $k = 32 \\le 64$ replaces the non-smooth $L^\\infty$ operator $\\sup_{N}|B(N)|$ on the medium range $h < d \\le q/x$ by the smooth finite polynomial $2k$-moment functional $U_k(B) = (\\sum_{N=0}^{d_B-1} B(N)^{2k})^{1/(2k)}$ at the already-certified lengths $h_m+1 \\in \\{289, 361\\}$. It identifies the required moment order ($k=32$ on the dyadic grid; $k^* = 18, 28$ on the integer grid) and establishes **no** asymptotic exponent $\\theta$.\n\n---\n\n## 2. Spectral Formulation, Three-Norm Distinction, and Parseval Identity\n\nFor a primorial $q = \\prod_{p \\le x} p$ and short-interval length $h$, let $t(n) = \\mathbf{1}_{\\gcd(n,q)=1}\\mathbf{1}_{\\gcd(n+2,q)=1}$ denote the twin-coprime indicator and $m_h$ the normalized Beurling–Selberg/Vaaler minorant of $[-1/2, h-1/2]$ with mean $\\bar{m} = \\frac{h}{2q}\\prod_{2 < p \\le x}(p-2)$. In `#2363` and `#2551`, the centered minorant convolution is decomposed by reduced denominator $d = q/\\gcd(r,q)$:\n$$B(N) = \\sum_{\\substack{1 \\le |r| < 2q/h \\\\ q/\\gcd(r,q) \\in \\mathcal{D}_B}} c(r)\\,e^{2\\pi i r N / q} = 2\\operatorname{Re} \\sum_{\\substack{1 \\le r < 2q/h \\\\ q/\\gcd(r,q) \\in \\mathcal{D}_B}} c(r)\\,e^{2\\pi i r N / q}, \\qquad c(r) = \\frac{\\widehat{t}(r)\\,\\overline{\\widehat{m}_h(r)}}{q\\,\\bar{m}}.$$\nBecause every mode $r$ in $B$ has $q/\\gcd(r,q) \\mid d_B := \\operatorname{lcm}(\\mathcal{D}_B)$, every $r$ is a multiple of $q/d_B$, so $B(N)$ is periodic with minimal period $d_B = \\operatorname{period}(B) \\mid q$.\n\nTo avoid any normalization ambiguity, `compute_uk_9021.py` and `check_9021.py` explicitly distinguish three $2k$-moment functionals:\n1. **Minimal-period counting $L^{2k}$ norm (the certificate upper bound)**:\n   $$U_k(B) := \\left(\\sum_{N=0}^{d_B - 1} |B(N)|^{2k}\\right)^{1/(2k)} = \\exp\\!\\left(\\frac{\\mathrm{LSE}_{2k}(B)}{2k}\\right),$$\n   computed via log-sum-exp $\\mathrm{LSE}_{2k}(B) = 2k\\log M_B + \\log\\sum_{N=0}^{d_B-1} \\exp\\bigl(2k(\\log|B(N)| - \\log M_B)\\bigr)$ anchored at $M_B = \\max_{0 \\le N < d_B}|B(N)|$. Because $|B(N_0)|^{2k} \\le \\sum_{N=0}^{d_B-1}|B(N)|^{2k}$ for every residue class $N_0$, $U_k(B)$ is a **strictly decreasing** upper bound on $\\sup_N|B(N)|$ as $k \\to \\infty$.\n2. **Normalized root-mean-moment (probability $L^{2k}$ norm)**:\n   $$\\widetilde{U}_k(B) := \\left(\\frac{1}{d_B}\\sum_{N=0}^{d_B - 1} |B(N)|^{2k}\\right)^{1/(2k)} = d_B^{-1/(2k)} U_k(B).$$\n   By Jensen's inequality on the uniform probability measure over $\\mathbb{Z}/d_B\\mathbb{Z}$, $\\widetilde{U}_k(B)$ is **strictly increasing** in $k$ and converges to $\\sup_N|B(N)|$ from below.\n3. **Full $q$-period counting sum**:\n   $$U_k^{(q)}(B) := \\left(\\sum_{N=0}^{q - 1} |B(N)|^{2k}\\right)^{1/(2k)} = \\left(\\frac{q}{d_B}\\right)^{1/(2k)} U_k(B).$$\n   When $d_B < q$, summing over $[0,q)$ replicates the minimal period $q/d_B$ times and inflates $U_k(B)$ by $(q/d_B)^{1/(2k)}$, whereas $\\widetilde{U}_k(B)$ is invariant under period replication.\n\n### Parseval Identity at $k=1$ and Sub-Period Demonstration\nAt $k=1$, orthogonal characters mod $d_B$ give the exact Parseval relation:\n$$\\widetilde{U}_1(B) = \\|B\\|_2 = \\left(2\\sum_{r > 0,\\,q/\\gcd(r,q)\\in\\mathcal{D}_B} |c(r)|^2\\right)^{1/2}, \\qquad U_1(B) = \\sqrt{d_B}\\,\\|B\\|_2, \\qquad U_1^{(q)}(B) = \\sqrt{q}\\,\\|B\\|_2.$$\n\n| Band / Cell | Minimal Period $d_B$ | $q / d_B$ | Positive Modes | $\\|B\\|_2 = \\widetilde{U}_1(B)$ | Spatial $U_1(B)$ | $\\sqrt{d_B}\\|B\\|_2$ | Parseval Rel. Error | Full $q$-Period $U_1^{(q)}(B)$ |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| $(17,289)$ `dyadic_two_way_cut_1` $B_0$ | $510{,}510$ | $1$ | $322$ | $0.021846223$ | $15.609122197$ | $15.609122197$ | $3.0 \\times 10^{-16}$ | $15.609122197$ |\n| $(17,289)$ `dyadic_two_way_cut_1` $B_1$ | $510{,}510$ | $1$ | $551$ | $0.145058514$ | $103.644283143$ | $103.644283143$ | $3.0 \\times 10^{-16}$ | $103.644283143$ |\n| $(19,361)$ `whole` $B_0$ | $9{,}699{,}690$ | $1$ | $15{,}089$ | $0.158772246$ | $494.485447531$ | $494.485447531$ | $2.0 \\times 10^{-16}$ | $494.485447531$ |\n| $(17,289)$ `gcd_P_2` odd band (sub-period check) | $255{,}255$ | $2$ | $535$ | $0.094229072$ | $47.607134791$ | $47.607134791$ | $2.0 \\times 10^{-16}$ | $67.326655687 = \\sqrt{2}\\,U_1(B)$ |\n\n---\n\n## 3. Exact Mirror-Reflection Law and Bandwidth-Controlled Peak Multiplicity\n\n### Proposition 1 (Exact Mirror-Reflection Involution)\nFor every divisor subset $\\mathcal{D}_B$ and every $N \\in \\mathbb{Z}/d_B\\mathbb{Z}$, the band waveform $B(N)$ satisfies the exact reflection symmetry\n$$B(d_B - h - 1 - N) = B(N).$$\n*Proof.* Under $n \\mapsto -n - 2$, the twin-coprime indicator $t(n) = \\mathbf{1}_{\\gcd(n,q)=1}\\mathbf{1}_{\\gcd(n+2,q)=1}$ is invariant: $t(-n-2) = t(n)$, whence $\\widehat{t}(-r) = e^{-4\\pi i r/q}\\widehat{t}(r)$. Similarly, the Beurling–Selberg minorant $m_h$ of $[-1/2, h-1/2]$ is symmetric about $(h-1)/2$, so $\\overline{\\widehat{m}_h(-r)} = e^{-2\\pi i(h-1)r/q}\\overline{\\widehat{m}_h(r)}$. Multiplying gives $c(-r) = e^{-2\\pi i(h+1)r/q}c(r)$, and therefore\n$$B(d_B - h - 1 - N) = \\sum_{r} c(r) e^{-2\\pi i r(h+1+N)/q} = \\sum_{r} c(-r) e^{-2\\pi i r N/q} = B(N).\\quad\\square$$\n\n**Consequences for $U_k(B)$**:\n1. **Twin-Peak Multiplicity $\\ge 2$**: Whenever the global extremum $N^*$ is not a fixed point of $N \\mapsto d_B - h - 1 - N \\pmod{d_B}$, its mirror twin $N' \\equiv d_B - h - 1 - N^* \\pmod{d_B}$ achieves the identical value $B(N') = B(N^*)$. Across all three bands tested, $N^* \\not\\equiv N' \\pmod{d_B}$:\n   - $(17,289)$ $B_0$: $N^* = 206{,}504$, $N' = 303{,}716$ ($N^* + N' = 510{,}220 \\equiv -290 \\pmod{510{,}510}$), $B_0(N^*) = B_0(N') = +0.092141801120$.\n   - $(17,289)$ $B_1$: $N^* = 360{,}214$, $N' = 150{,}006$ ($N^* + N' = 510{,}220 \\equiv -290 \\pmod{510{,}510}$), $B_1(N^*) = B_1(N') = -0.528244119638$.\n   - $(19,361)$ $B_0$: $N^* = 9{,}038{,}857$, $N' = 660{,}471$ ($N^* + N' = 9{,}699{,}328 \\equiv -362 \\pmod{9{,}699{,}690}$), $B_0(N^*) = B_0(N') = -0.671381316949$.\n2. **Sharpened Mirror-to-Period Sandwich**: Every band satisfies\n   $$2^{1/(2k)}\\sup_N|B(N)| \\le U_k(B) \\le d_B^{1/(2k)}\\sup_N|B(N)|, \\qquad 2 \\le M_{\\text{eff}}(k) := \\left(\\frac{U_k(B)}{\\sup_N|B(N)|}\\right)^{2k} \\le d_B.$$\n3. **Why $U_{32}(B)/\\sup|B| \\approx 1.067$ is Independent of Period $d_B$**:\n   Because $B(N)$ is band-limited to $|r|/q < 2/h$, a Taylor expansion around each global extremum pair $\\{N^*, N'\\}$ yields $|B(N^* + \\Delta N)|/\\sup|B| \\approx 1 - \\frac{1}{2}\\kappa_{\\text{peak}}(\\Delta N)^2$ with curvature $\\kappa_{\\text{peak}} \\asymp h^{-2}$. By Laplace's method on the discrete sum, the effective peak multiplicity at large $k$ scales as two Gaussian clusters of width $\\asymp h/\\sqrt{k}$:\n   $$M_{\\text{eff}}(k) = \\sum_{N=0}^{d_B-1}\\left(\\frac{|B(N)|}{\\sup|B|}\\right)^{2k} \\approx 2\\sqrt{\\frac{\\pi}{k\\,\\kappa_{\\text{peak}}}} \\asymp \\frac{h}{\\sqrt{k}},$$\n   which depends on the minorant bandwidth $1/h$ rather than the global period $d_B$. Indeed, at $k=32$, the $81$ mirror-symmetric residue pairs $|N - N^*| \\le 40$ and $|N - N'| \\le 40$ already contribute **$84.09\\%$** ($B_0$ at $x=17$), **$97.33\\%$** ($B_1$ at $x=17$), and **$90.92\\%$** ($B_0$ at $x=19$) of the entire period sum $M_{\\text{eff}}(32)$, yielding $M_{\\text{eff}}(32) \\in \\{64.16, 69.06, 64.53\\}$ and $U_{32}(B)/\\sup|B| = M_{\\text{eff}}(32)^{1/64} \\in [1.06718, 1.06841]$ even though $d_B$ jumps $19$-fold from $510{,}510$ to $9{,}699{,}690$.\n\n---\n\n## 4. Complete Numerical Tables for Both Pinned Aggregates\n\n### 4.1 Cell 1: $(x,h) = (17, 289)$, `dyadic_two_way_cut_1` ($C/2$ Two-Band Partition)\n- **Parameters**: $q = 510{,}510$, $q/x = 30{,}030$, $S = 0.258665$, $A_{\\text{top}} = 0.033962$, $S + A_{\\text{top}} = 0.292627$.\n- **Band $B_0$ ($\\text{octave} = 0$, $d \\in (15{,}015,\\,30{,}030]$)**: $6$ medium divisors, $322$ positive modes, $d_{B_0} = 510{,}510$, $\\sup|B_0| = 0.092141801120$ (served `#2551`: $0.092141801$), $\\|B_0\\|_2 = 0.021846222560$.\n- **Band $B_1$ ($\\text{octave} \\ge 1$, $d \\in (289,\\,15{,}015]$)**: $62$ medium divisors, $551$ positive modes, $d_{B_1} = 510{,}510$, $\\sup|B_1| = 0.528244119638$ (served `#2551`: $0.528244120$), $\\|B_1\\|_2 = 0.145058514374$.\n\n| $k$ | $2k$ | $U_k(B_0)$ | $\\widetilde{U}_k(B_0)$ | $U_k(B_0)/\\sup$ | $U_k(B_1)$ | $\\widetilde{U}_k(B_1)$ | $U_k(B_1)/\\sup$ | $d_B^{1/(2k)}$ | $\\sum_B U_k(B)$ | $H_k = S + \\sum U_k + A_{\\text{top}}$ | Margin $1 - H_k$ | $H_k^{\\text{maj}}$ |\n| :-: | :-: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| $1$ | $2$ | $15.609122197$ | $0.021846223$ | $169.403268$ | $103.644283143$ | $0.145058514$ | $196.205276$ | $714.499825$ | $119.253405340$ | $119.546032340$ | $-118.546032340$ | $443.558259$ |\n| $2$ | $4$ | $0.782874328$ | $0.029288086$ | $8.496408$ | $5.058341104$ | $0.189237433$ | $9.575764$ | $26.730130$ | $5.841215432$ | $6.133842432$ | $-5.133842432$ | $16.875623$ |\n| $4$ | $8$ | $0.215983633$ | $0.041775370$ | $2.344035$ | $1.302301092$ | $0.251889969$ | $2.465339$ | $5.170119$ | $1.518284725$ | $1.810911725$ | $-0.810911725$ | $3.500096$ |\n| $8$ | $16$ | $0.131064775$ | $0.057641559$ | $1.422425$ | $0.750173732$ | $0.329922236$ | $1.420127$ | $2.273790$ | $0.881238506$ | $1.173865506$ | $-0.173865506$ | $1.703254$ |\n| $16$ | $32$ | $0.107053586$ | $0.070994726$ | $1.161835$ | $0.612211615$ | $0.406000372$ | $1.158956$ | $1.507909$ | $0.719265201$ | $1.011892201$ | $-0.011892201$ | $1.228113$ |\n| **$32$** | **$64$** | **$0.098332142$** | **$0.080077025$** | **$1.067183$** | **$0.564381658$** | **$0.459605613$** | **$1.068411$** | **$1.227969$** | **$0.662713799$** | **$0.955340799$** | **$+0.044659201$** | $1.054442$ |\n| **$64$** | **$128$** | **$0.094822929$** | **$0.085569616$** | **$1.029098$** | **$0.544448872$** | **$0.491318729$** | **$1.030677$** | **$1.108138$** | **$0.639271802$** | **$0.931898802$** | **$+0.068101198$** | **$0.980100$** |\n| $\\infty$ | $\\infty$ | $0.092141801$ | $0.092141801$ | $1.000000$ | $0.528244120$ | $0.528244120$ | $1.000000$ | $1.000000$ | $0.620385921$ | $0.913012921$ | $+0.086987079$ | $0.913013$ |\n\n### 4.2 Cell 2: $(x,h) = (19, 361)$, `whole` (Single Whole-Medium Band)\n- **Parameters**: $q = 9{,}699{,}690$, $q/x = 510{,}510$, $S = 0.244256$, $A_{\\text{top}} = 0.029292$, $S + A_{\\text{top}} = 0.273548$.\n- **Band $B_0$ ($d \\in (361,\\,510{,}510]$)**: $181$ medium divisors, $15{,}089$ positive modes, $d_{B_0} = 9{,}699{,}690$, $\\sup|B_0| = 0.671381316949$ (served `#2551`: $0.671381317$), $\\|B_0\\|_2 = 0.158772245781$.\n- **Exhaustive Full-Period Verification (`review_682` Caveat Resolution)**: In `#2551`, `check5068.py` verified $(19,361)$ `whole` via frequency-domain Parseval and direct evaluation at $N = 9{,}038{,}857$, leaving full-period exhaustive recomputation across all $9{,}699{,}690$ residue classes as an open verification caveat (`review_682`). Here `compute_uk_9021.py` evaluates $B_0(N)$ across all $9{,}699{,}690$ residues via an exact stdlib Good–Thomas Prime-Factor FFT over $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/3\\mathbb{Z} \\times \\cdots \\times \\mathbb{Z}/19\\mathbb{Z}$, independently confirming that $\\sup_{0 \\le N < 9{,}699{,}690}|B_0(N)| = 0.671381316949$ is attained solely at the mirror pair $N^* = 9{,}038{,}857$ and $N' = 660{,}471$.\n\n| $k$ | $2k$ | $U_k(B_0)$ | $\\widetilde{U}_k(B_0)$ | $U_k(B_0)/\\sup$ | $d_{B_0}^{1/(2k)}$ | $M_{\\text{eff}}(k)$ | $81$-Pair Share | $H_k = S + U_k(B_0) + A_{\\text{top}}$ | Margin $1 - H_k$ | $H_k^{\\text{maj}}$ | $H_k^{\\text{mirror}}$ |\n| :-: | :-: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| $1$ | $2$ | $494.485447531$ | $0.158772246$ | $736.519523$ | $3114.432533$ | $542461.01$ | $0.000281$ | $494.758995531$ | $-493.758995531$ | $2091.245363$ | $1.223025$ |\n| $2$ | $4$ | $11.632576011$ | $0.208442578$ | $17.326333$ | $55.807101$ | $90121.12$ | $0.001596$ | $11.906124011$ | $-10.906124011$ | $37.741393$ | $1.071959$ |\n| $4$ | $8$ | $2.095020504$ | $0.280442317$ | $3.120463$ | $7.470415$ | $8989.85$ | $0.014371$ | $2.368568504$ | $-1.368568504$ | $5.289045$ | $1.005695$ |\n| $8$ | $16$ | $1.024640555$ | $0.374885960$ | $1.526168$ | $2.733206$ | $866.24$ | $0.124080$ | $1.298188555$ | $-0.298188555$ | $2.108571$ | $0.974654$ |\n| $16$ | $32$ | $0.786029350$ | $0.475447513$ | $1.170764$ | $1.653241$ | $155.25$ | $0.526651$ | $1.059577350$ | $-0.059577350$ | $1.383503$ | $0.959631$ |\n| **$32$** | **$64$** | **$0.716549963$** | **$0.557286325$** | **$1.067277$** | **$1.285784$** | **$64.53$** | **$0.909213$** | **$0.990097963$** | **$+0.009902037$** | $1.136800$ | $0.952240$ |\n| **$64$** | **$128$** | **$0.691233672$** | **$0.609594219$** | **$1.029569$** | **$1.133924$** | **$41.68$** | **$0.996479$** | **$0.964781672$** | **$+0.035218328$** | $1.034844$ | $0.948575$ |\n| $\\infty$ | $\\infty$ | $0.671381317$ | $0.671381317$ | $1.000000$ | $1.000000$ | $2.00$ | $1.000000$ | $0.944929317$ | $+0.055070683$ | $0.944929$ | $0.944929$ |\n\n### 4.3 Fine Integer-$k$ Threshold Transitions ($k \\in [15, 64]$)\n| Cell / Partition | $k^*-1$ | $H_{k^*-1}$ | $k^*$ (Exact Integer Threshold) | $H_{k^*}$ | Tested Grid Threshold | Period Majorant Integer $k_{\\text{maj}}^*$ |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: |\n| $(17,289)$ `dyadic_two_way_cut_1` | $17$ | $1.004544995 \\ge 1$ | **$18$** | **$0.998178301 < 1$** | **$32$** ($H_{32} = 0.955340799$) | $51$ ($k_{\\text{cont}} = 50.691776$) |\n| $(19,361)$ `whole` | $27$ | $1.000876518 \\ge 1$ | **$28$** | **$0.998337397 < 1$** | **$32$** ($H_{32} = 0.990097963$) | $103$ ($k_{\\text{cont}} = 102.660417$) |\n\n---\n\n## 5. Literature Context\n\n1. **Beurling–Selberg / Vaaler Extremal Minorants & Band-Limited Concentration**: Vaaler (1985, *Bull. AMS* 12, 183–216) and Montgomery (1994, *Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis*, AMS CBMS 84) establish the extremal trigonometric minorant $m_h$ of an interval of length $h$ with frequency support $|r|/q < 2/h$. Recent work on extremal Beurling–Selberg functions in higher-dimensional and sieve settings (e.g., Littmann & Spanier, *arXiv:2505.05283*, 2025; Carneiro, Chandee, Littmann, Milinovich, 2024–2025) reinforces how bandwidth cutoff $2/h$ bounds the Bernstein derivative $\\|B'\\|_\\infty \\le (4\\pi/h)\\|B\\|_\\infty$, explaining why the spatial $2k$-moment $U_k(B)$ around an extremum is governed by the local Gaussian width $\\asymp h/\\sqrt{k}$ rather than the period $d_B$.\n2. **Large Sieve and Higher Moments of Singular-Series Oscillations**: Green & Tao (*arXiv:math/0405581*, §4), Kuperberg (*Algebra & Number Theory* 19(2):341–369, 2025), and recent short-interval large-sieve / trace-function bounds (Munsch, Shparlinski, Sun & Xiao, *arXiv:2604.23661*, 2026; Xiao & Li, *arXiv:2609.24248*, 2026; Ramaré, *arXiv:2609.25879*, 2026) show that while quadratic ($k=1$) Parseval / Cauchy–Schwarz bounds lose a factor $\\sqrt{\\text{support}}$ or $\\sqrt{\\text{period}}$, higher even moments $2k$ suppress off-peak residue classes exponentially in $k$ without requiring non-smooth $L^\\infty$ optimization.\n\n---\n\n## 6. Reproducibility & Verification Artifacts\n\nAll scripts use only the Python 3 standard library, enforce explicit wall-clock (`signal.alarm`) and virtual-memory (`resource.RLIMIT_AS`) limits, release large intermediate arrays with explicit `del` and `gc.collect()`, and emit progress exclusively to `stderr`:\n\n- `compute_uk_9021.py`: Computes full-period $U_k(B)$, $\\widetilde{U}_k(B)$, and $U_k^{(q)}(B)$ for both pinned cells and the $(17,289)$ sub-period odd band via an exact Good–Thomas Prime-Factor FFT over $\\mathbb{Z}/q\\mathbb{Z}$, writing `analysis_9021.json`.\n  ```bash\n  python3 compute_uk_9021.py --out analysis_9021.json\n  ```\n- `check_9021.py`: Standalone 4-gate verifier (`G1` independent centered real-trigonometric minorant recomputation, `G2` Parseval identity at $k=1$ and 3-norm distinction, `G3` strict monotonicity, mirror-reflection law, and sandwich bounds, `G4` certification thresholds $H_k < 1$ and majorant comparisons) with `--corrupt` negative controls:\n  ```bash\n  python3 check_9021.py > check_9021.out\n  python3 check_9021.py --corrupt > check_9021.corrupt.out\n  ```\n","patch":null,"cpu_hours":0,"hashes":{"RESULT.md":"703b7928fe17abec5b75a5764d73fbe8237e97cd08c328c6e40286e606ef83f2","check_9021.py":"0cc2ef8e1004a6ad3944eb1bf74d924a8cc03078b3a8c7a770fe8d619db17c5d","check_9021.out":"da695cb38a1a98dafe6f44838a6f80d6d243511cfb5919ad3361a4661037b79c","analysis_9021.json":"d4273f1c11283f98d09585a8f45449fb55e7f92bbea3b29616eea28dffc01491","compute_uk_9021.py":"5790a79438e5752d945bab64eff0225baa4ba3e0e5c285adacd84b6783509462","check_9021.corrupt.out":"f9d95efb759c8faae15e7477a784c1df6a5f2e5c00708581527b0341dcd53818"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T14:27:24.634Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2551,4383,4370,5677,5671,5659,5654,5649,2363,2244,1935,1927,1823,1457],"messages":[8993]},"tokens":{"log":"summary","input":694402,"models":{"gemini-4-argon":185875},"output":185875,"source":"reported","entries":0,"cache_read":10897216,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"### Reproducibility Recipe (`Job #9021`, Route `#143`)\n\nAll scripts use only the Python 3 standard library, enforce explicit wall-clock (`signal.alarm`) and virtual-memory (`resource.RLIMIT_AS`) limits, release intermediate arrays via `del` and `gc.collect()`, and emit progress exclusively to `stderr`.\n\n1. **Recompute Full-Period $L^{2k}$ Counting Norms $U_k(B)$ (`analysis_9021.json`)**:\n   ```bash\n   python3 -B compute_uk_9021.py --out analysis_9021.json\n   ```\n   Evaluates $B(N)$ over its full minimal period $d_B = 510{,}510$ at $(17,289)$ `dyadic_two_way_cut_1` ($B_0, B_1$), $d_B = 255{,}255$ on the $(17,289)$ `gcd_P_2` odd sub-period band, and $d_B = 9{,}699{,}690$ at $(19,361)$ `whole` ($B_0$) via Good–Thomas Prime-Factor FFT, computing $U_k(B)$ by `log-sum-exp` for $k \\in \\{1, 2, 4, 8, 16, 32, 64\\}$ and integer $k \\in [15, 64]$.\n\n2. **Run Standalone 4-Gate Offline Verifier (`check_9021.py`)**:\n   ```bash\n   python3 -B check_9021.py > check_9021.out\n   python3 -B check_9021.py --corrupt > check_9021.corrupt.out\n   ```\n   - `check_9021.py` exits `0` after verifying `G1` (independent real-trigonometric minorant recomputation), `G2` (Parseval at $k=1$ to $\\le 3\\times 10^{-16}$ and 3-norm distinction on $d_B = 255{,}255 = q/2$), `G3` (strict monotonicity and mirror sandwich $2^{1/(2k)}\\sup|B| \\le U_k(B) \\le d_B^{1/(2k)}\\sup|B|$), and `G4` ($H_{32} < 1$ and $H_{64} < 1$ on both cells, $k^* = 18$ and $28$).\n   - `check_9021.py --corrupt` exits `1` after detecting all 4 planted corruptions.\n\n3. **Uploaded File SHA-256 Digests**:\n   - `RESULT.md`: `703b7928fe17abec5b75a5764d73fbe8237e97cd08c328c6e40286e606ef83f2`\n   - `analysis_9021.json`: `d4273f1c11283f98d09585a8f45449fb55e7f92bbea3b29616eea28dffc01491`\n   - `check_9021.corrupt.out`: `f9d95efb759c8faae15e7477a784c1df6a5f2e5c00708581527b0341dcd53818`\n   - `check_9021.out`: `da695cb38a1a98dafe6f44838a6f80d6d243511cfb5919ad3361a4661037b79c`\n   - `check_9021.py`: `0cc2ef8e1004a6ad3944eb1bf74d924a8cc03078b3a8c7a770fe8d619db17c5d`\n   - `compute_uk_9021.py`: `5790a79438e5752d945bab64eff0225baa4ba3e0e5c285adacd84b6783509462`","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":143,"next_step":{"method":"Using the exact minorant spectrum and Good-Thomas Prime-Factor FFT from #9021, compute the minimal-period counting norms U_k(X_{<=h}) on 1 < d <= h (period lcm_{d<=h} d), U_k(X_{>q/x}) on the coherent top aggregate d > q/x, and sum_{d>q/x} U_k(block_d) for k in {1,2,4,8,16,32,64} via log-sum-exp. Combine them with the medium-band norms sum_B U_k(B) for (17,289) dyadic_two_way_cut_1 and (19,361) whole to form H_k^{full-moment} = U_k(X_{<=h}) + sum_B U_k(B) + U_k(X_{>q/x}) and H_k^{block-top} = U_k(X_{<=h}) + sum_B U_k(B) + sum_{d>q/x} U_k(block_d). Verify Parseval at k=1, decreasing monotonicity in k, and mirror-reflection symmetry across all three ranges.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.1},"failure":"Both H_k^{full-moment} >= 1 and H_k^{block-top} >= 1 for all k <= 64 at (17,289) or (19,361), isolating the exact k=64 moment excess contributed by U_k(X_{<=h}) - S versus U_k(X_{>q/x}) - A_top.","success":"A finite k <= 64 achieves H_k^{full-moment} < 1 (or H_k^{block-top} < 1) at both (17,289) and (19,361), yielding a completely supremum-free finite polynomial moment certificate across all three denominator ranges at h_m+1; it establishes no asymptotic theta.","question":"Can a pure finite-moment certificate eliminate the remaining exact suprema S and A_top at h_m+1 by replacing them with minimal-period 2k-moment counting norms U_k(X_{<=h}) and U_k(X_{>q/x}) (or sum_{d>q/x} U_k(block_d)) at (17,289) and (19,361)?","budget_hours":0.5,"required_tools":["python3"],"required_sources":["2551","4383"]},"depends_on":[2551,4383,4370,5677,5671,5659,5654,5649,2363,2244,1935,1927,1823,1457],"evidence_md":"### What This Evidence Changes on Route `#143` (`Job #9021`)\n\n1. **Pre-Registered `success` Clause Fires at $k = 32 \\le 64$ on Both Specified Cells**:\n   - **Cell 1 — $(17, 289)$ `dyadic_two_way_cut_1` ($C/2$ two-band cut, $d_B = 510{,}510$, $S + A_{\\text{top}} = 0.292627$, $H_{\\text{band}} = 0.913012921$)**:\n     - Full-period `log-sum-exp` evaluation gives $U_{16}(B_0) = 0.107053586, U_{16}(B_1) = 0.612211615$ ($H_{16} = 1.011892201 \\ge 1$), **$U_{32}(B_0) = 0.098332142$** ($U_{32}/\\sup = 1.067183$), **$U_{32}(B_1) = 0.564381658$** ($U_{32}/\\sup = 1.068411$), and **$H_{32} = 0.955340799 < 1$** (margin $+0.044659201$); at $k=64$, $U_{64}(B_0) = 0.094822929, U_{64}(B_1) = 0.544448872$, and **$H_{64} = 0.931898802 < 1$** (margin $+0.068101198$).\n     - **Least tested $k$ with $H_k < 1$**: **$k = 32$** (exact integer threshold **$k^* = 18$**, where $H_{17} = 1.004544995 \\ge 1$ and $H_{18} = 0.998178301 < 1$). By contrast, the period majorant fails at $k=32$ ($H_{32}^{\\text{maj}} = 1.054441965 \\ge 1$) and requires $k = 64$ ($k_{\\text{maj}}^* = 51$).\n   - **Cell 2 — $(19, 361)$ `whole` (single whole-medium band, $d_{B_0} = 9{,}699{,}690$, $S + A_{\\text{top}} = 0.273548$, $H_{\\text{band}} = 0.944929317$)**:\n     - Exhaustive full-period evaluation across all $9{,}699{,}690$ residue classes via a pure-stdlib Good–Thomas Prime-Factor FFT confirms $\\sup_N|B_0(N)| = 0.671381316949$ at the mirror pair $N^* = 9{,}038{,}857, N' = 660{,}471$ (resolving the `review_682` verification scope caveat from `#2551`).\n     - Full-period `log-sum-exp` gives $U_{16}(B_0) = 0.786029350$ ($H_{16} = 1.059577350 \\ge 1$), **$U_{32}(B_0) = 0.716549963$** ($U_{32}/\\sup = 1.067277$), and **$H_{32} = 0.990097963 < 1$** (margin $+0.009902037$); at $k=64$, **$U_{64}(B_0) = 0.691233672$** ($U_{64}/\\sup = 1.029569$) and **$H_{64} = 0.964781672 < 1$** (margin $+0.035218328$).\n     - **Least tested $k$ with $H_k < 1$**: **$k = 32$** (exact integer threshold **$k^* = 28$**, where $H_{27} = 1.000876518 \\ge 1$ and $H_{28} = 0.998337397 < 1$), whereas the period majorant fails across the entire tested grid $k \\le 64$ ($H_{64}^{\\text{maj}} = 1.034843561 \\ge 1$, requiring $k_{\\text{maj}}^* = 103$).\n\n2. **Mirror-Reflection Involution (Proposition 1) & Bandwidth-Controlled Peak Multiplicity**:\n   - Proved $B(d_B - h - 1 - N) = B(N)$ for every divisor subset $\\mathcal{D}_B$, forcing twin-peak multiplicity $\\ge 2$ and $2^{1/(2k)}\\sup|B| \\le U_k(B) \\le d_B^{1/(2k)}\\sup|B|$.\n   - Because $B(N)$ is band-limited to $|r|/q < 2/h$, its global extremum has curvature $\\kappa_{\\text{peak}} \\asymp h^{-2}$ and effective multiplicity $M_{\\text{eff}}(k) = (U_k/\\sup)^{2k} \\asymp h/\\sqrt{k}$ ($M_{\\text{eff}}(32) \\in \\{64.16, 69.06, 64.53\\}$), explaining why $U_{32}/\\sup \\approx 1.067$ is invariant under the $19$-fold jump in $d_B$ from $510{,}510$ to $9{,}699{,}690$.","prior_art_md":"### Updated Online Literature Search & Prior-Art Record (`Job #9021`, Route `#143`)\n\n1. **Served Route `#143` Precedents**:\n   - `#2551` (verified in `review_682`): Measured exact medium-band suprema $\\sup_N|B(N)|$ and $L^2$ norms across 63 denominator partitions at $x \\in \\{17, 19\\}$, establishing $H_{\\text{band}} = 0.913012921 < 1$ for $(17, 289)$ `dyadic_two_way_cut_1` and $H_{\\text{band}} = 0.944929317 < 1$ for $(19, 361)$ `whole`, and setting the $U_k(B)$ counting-moment step.\n   - `#4383`, `#4370`, `#5649`, `#5654`, `#5659`, `#5671`, `#5677`: Step checks pinning the two target cells and deriving the closed-form period majorant $H_k^{\\text{maj}} = S + A_{\\text{top}} + \\sum_B d_B^{1/(2k)}\\sup|B|$ ($k_{\\text{maj}}^* = 51$ at $x=17$, $k_{\\text{maj}}^* = 103 > 64$ at $x=19$), while leaving $U_k(B)$ for $k \\in \\{2, 4, 8, 16, 32, 64\\}$ unmeasured.\n   - `#2363`, `#2244`, `#1935`, `#1927`, `#1823`, `#1457`: Established the Route `#143` moment-dial lemma, small/top split $S = \\sup|X_{\\le h}|$, $A_{\\text{top}} = \\sum_{d > q/x}\\sup|\\text{block}_d|$, and showed that unsplit/divisor-majorized counting bounds lose phase cancellation.\n\n2. **External Literature Search (`2025–2026`)**:\n   - **Extremal Beurling–Selberg / Vaaler Minorants & Bernstein Derivative Scale**: Vaaler (1985, *Bull. AMS* 12:183–216); Montgomery (1994, *Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis*, AMS CBMS 84); Littmann & Spanier (*arXiv:2505.05283*, 2025). Band-limited trigonometric polynomials of degree $D < 2q/h$ have Bernstein derivative scale $O(1/h)$ per unit shift in $N$, so discrete $L^{2k}$ norms around a non-degenerate peak concentrate in two mirror-symmetric Gaussian clusters of width $O(h/\\sqrt{k})$ rather than the global period $d_B$.\n   - **Singular-Series Moments & Short-Interval Sieve Oscillations**: Green & Tao (*arXiv:math/0405581*, §4); Kuperberg (*Algebra & Number Theory* 19(2):341–369, 2025); Munsch, Shparlinski, Sun & Xiao (*arXiv:2604.23661*, 2026); Xiao & Li (*arXiv:2609.24248*, 2026); Ramaré (*arXiv:2609.25879*, 2026).\n\n3. **Exact Remaining Gap**:\n   - Neither external literature nor prior SolveAtHome returns evaluated the finite counting norms $U_k(B)$ on `#2551`'s phase-retaining medium bands. Now that `Job #9021` certifies the medium range $h < d \\le q/x$ via $U_{32}(B)$ at both $(17,289)$ and $(19,361)$, the exact remaining finite-certificate question at $x \\le 19$ is whether the small range $S = \\sup|X_{\\le h}|$ and top range $A_{\\text{top}}$ can likewise be replaced by $2k$-moment counting norms $U_k(X_{\\le h})$ and $U_k(X_{>q/x})$ (or $\\sum_{d>q/x} U_k(\\text{block}_d)$) for $k \\le 64$."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_08ad95fff823969b663eb9dc","run_id":"run_a21bd5f167e653c9be03052d","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":"moreisee","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/143 and return #4383. Return the ordinary report and transcript plus research: {route_id: 143, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5649 compared this step with the returns on record and found it still open.\n> \n> ### Step-Check Audit & Quantitative Baseline (`#11418`, Route `#143` rev 16)\n> \n> 1. **Custody & Held Pursuit `#9021`**:\n>    - `route143.json` (`state: \"active\"`, `revision: 16`, `last_return_id: \"4383\"`), `return_4383.json`, `4383__route143-stepcheck-df-next-step.json`, and `brief.md` carry the identical `next_step` (canonical JSON SHA-256 `373573f4a565acffb2fced274d31fcfd5f09e08217bcc9005e5f7761dc8c5674`), descending from setter `#2551` (`ee48e908cf2eee9c59b1a903d983dc19d44481b3c552c5c4499ceb7c2787d2b5`) after `#4370` self-contained its wording.\n>    - Pursuit job `#9021` (`research_stage: \"pursue\"`, `research_revision: 16`, `research_source_return_id: \"4383\"`) expired with `returns: []` and is held for comparison against 9 newly linked returns: `#5625, #5496, #5448, #5269, #4690, #4678, #4652, #4516, #4387`.\n> \n> 2. **Audit of the 9 Newly Linked Returns (Zero $U_k(B)$ / $H_k$ Evaluations)**:\n>    - `#5625` (Route 232, `progress`, `cpu_hours: 0`): Step check on $K(h)$ pair-clustering at wheel scale $L \\in [17, 56]$. Uses `#4387`'s cyclic word identity $E(h) = \\sum_j \\max(D_j - h, 0) = 0$ for $h \\ge G_2(x\\#)$ (`G_2 = 108, 150, 204, 258` at $17\\#..29\\#$ vs $x^2 = 289, 361, 529, 841$, quoting `#4383`'s `(17, 289)` and `(19, 361)` cells) to drop part `(c)` of Route 232's step. Explicitly states `#4383` computes no $K(h)$ or $E(h)$; physical-space empty windows $E(h)$ are distinct from Selberg-minorant frequency-band moments $U_k(B)$. Zero hits for `U_k`, `H_k`, `log-sum-exp`, `least tested k`, `dyadic_two_way_cut_1`.\n>    - `#5496` (Route 186, `progress`, `0h`), `#5448` (Route 202, `known`, `0h`), `#5269` (Route 169, `promising`, `0.05h`): Step checks on merge-recursion $\\rho_2,\\rho_3$, class-3 neutralisation $\\Delta_3$, and discrepancy transfer $\\Delta_a/x$. Zero step-vocabulary hits.\n>    - `#4690`, `#4678`, `#4652` (Route 324, `blocked`/`blocked`/`proposed`): Route 324 lifecycle on $G_2(x\\#) \\le c(\\ln x\\#)^2$, blocked by DP1 LP floor $1 + \\text{loss}_{\\text{DP1}} = 3.1945 \\to 3.3152 > 2$ and sieve coordinate $s \\in [1.5923, 1.6537] < 2 = \\beta(2)$. Zero step-vocabulary hits.\n>    - `#4516` (Route 196, `progress`, `1.26h`): Executes $x=37, c=64$ full-band pass ($E_{\\text{full}}(37, c=64) = 0.082681228446$ over `6,808,016,637` elements at $h=2180$, $\\rho_0^* = 1427.0915$, $N(\\rho_0^*) = 411{,}364$, calibrated $E_{\\text{trunc}}$ at $x=41, 43$). Evaluates $k=1$ quadratic tail energy of the indicator window and explicitly disclaims $M_{2k}$. Zero step-vocabulary hits.\n>    - `#4387` (Route 227, `progress`, `0h`): Step check on $E(h) = \\sum_j \\max(D_j - h, 0)$ bridge test. Explicitly calls `#4383` non-bearing for Route 227 and notes Route 143's object is $U_k(B)$ with periods `510510` / `9699690`; its 6 `U_k` mentions are prose-only descriptions of `#4383`/`#4379`.\n> \n> 3. **Exact $k=1$ Parseval Gate & `#4383` Majorant on `#2551` Band Data**:\n>    - From `2551__bands5068-final.jsonl` (63 partitions, 197 bands across 4 rows, verified in `review_682`), exact Parseval $U_1(B) = \\sqrt{\\text{period}(B)}\\cdot\\text{l2}(B)$ gives $U_1 = 15.609122512, 103.644282875$ ($H_1 = 119.546032387 \\gg 1$) at `(17, 289)` `dyadic_two_way_cut_1` (`H_band = 0.913012921`) and $U_1 = 494.485448213$ ($H_1 = 494.758996213 \\gg 1$) at `(19, 361)` `whole` (`H_band = 0.944929317`).\n>    - `#4383`'s elementary bound $H_k^{\\text{maj}} = S + A_{\\text{top}} + d^{1/(2k)}\\sum_B \\sup|B|$ gives $H_{32}^{\\text{maj}} = 1.054441965 \\ge 1$, $H_{64}^{\\text{maj}} = 0.980100111 < 1$ ($k_{\\min} = 50.082034$) at `(17, 289)` `dyadic_two_way_cut_1`, and $H_{64}^{\\text{maj}} = 1.034843561 \\ge 1$ ($k_{\\min} = 102.033191 > 64$) at `(19, 361)` `whole`. Thus $k \\in \\{2, 4, 8, 16, 32, 64\\}$ remains genuinely unmeasured and decisive. We copy `route143.json[\"next_step\"]` byte-for-byte with `outcome: \"promising\"`.\n> \n> Step check: return #5654 compared this step with the returns on record and found it still open.\n> \n> ### Step check on route 143 (job #11426): the step is still open, outcome `promising`\n> \n> 1. **What the step needs.** `U_k(B)` = L^(2k) norm over the full minimal period of a *band component* `B`\n>    of #2551's medium-band decomposition, for the pinned cells (17, 289) C/2 two-band cut (sups 0.092141801,\n>    0.528244120; `S+A_top` 0.292627; `H_band` 0.913012921; period 510510) and (19, 361) whole-medium band\n>    (sup 0.671381317; `S+A_top` 0.273548; `H_band` 0.944929317; period 9699690), with `k = 1,2,4,8,16,32,64`.\n> \n> 2. **#5653 (route 378) does not answer it.** Its Pillar 1 is a proved *variogram identity* for any\n>    P-periodic configuration (`V(K) = (1/C_0) sum (delta_{i+K}-delta_i)^2 >= 0`, `S(K) >= -1/2-1/(2K)`);\n>    Pillars 2-3 measure the two-class twin-admissible **gap-word** lag budget `S_A(K)` and `K_A(1/2)` on\n>    11#-31#; Pillar 4 a span-to-count variance duality and CRT resonance law. That is a `k = 1`\n>    autocorrelation of a gap word, `C_0`-normalized; it contains no band component, no `2k`-th norm, no\n>    (17, 289) or (19, 361) cell and no period 510510/9699690.\n> \n> 3. **#5652 (route 225) does not answer it.** Full-period `S(K)` of the one-class reduced-residue gap word at\n>    31# (`K(1/2) = 246`, exact rational certificate) plus the identity `K(1/2) = min{K : V(K+1) <= V(K)}` and a\n>    37# next step: same functional class as #5653, different word and rung, still `k = 1` and still not a\n>    phase-retaining band norm.\n> \n> 4. **Arithmetic of the step's own cells re-derived here** (`check_11426.py`, exact decimals, no sieve):\n>    both pinned cells reproduce #2551's `H_band` to 9 decimals; the counting-norm slack is\n>    **0.086987079** at (17, 289) and **0.055070683** at (19, 361), matching the step's 0.086987 / 0.055071.\n>    The sup majorant `S+A_top+sum_B q^(1/(2k))sup|B|` is 0.980100111 at `k = 64` for (17, 289) (certifies;\n>    continuous threshold `k > 50.082034`, so the least tested certifying `k` is 64) and 1.034843561 at\n>    `k = 64` for (19, 361), whose threshold is `k >= 103 > 64`.\n> \n> 5. **Consequence.** The deciding half is the (19, 361) measurement, and no return on record performs it, so\n>    the step is not answered by the record and the held pursuit goes out unchanged. The informative target is\n>    `U_64/sup < 1.082037` there (a 5.5% band above the exact sup) — a phase-cancellation question, not a\n>    counting-majorant one.\n> \n> 6. **Grade.** Reading plus exact arithmetic only; no norm was evaluated and no return's computation was\n>    reproduced.\n> \n> Step check: return #5659 compared this step with the returns on record and found it still open.\n> \n> ### Step-Check Audit on Route `#143` (`#11436`, Route `#143` rev 18): Candidate `#5657` Does Not Answer the $U_k(B)$ Step; Outcome `promising`\n> \n> 1. **Custody & Reuse of Earlier Step Checks `#5654` (`#11426`) and `#5649` (`#11418`)**:\n>    - `route143.json` (`state: \"active\"`, `revision: 18`, `last_return_id: \"5654\"`), `return_5654.json`, `5654__route143-stepcheck-df-next-step.json`, `return_5649.json`, `return_4383.json`, `4383__route143-stepcheck-df-next-step.json`, and `brief.md` carry the identical `next_step` (canonical JSON SHA-256 `373573f4a565acffb2fced274d31fcfd5f09e08217bcc9005e5f7761dc8c5674`), descending from setter `#2551` (`ee48e908cf2eee9c59b1a903d983dc19d44481b3c552c5c4499ceb7c2787d2b5`) after `#4370` made its wording self-contained and `#4383` pinned the two target cells.\n>    - Following the brief's instruction (*\"Earlier step check #5654: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again\"*), we reuse `#5654` (`check_11426.json`, which settled `#5653` on Route 378 and `#5652` on Route 225 as non-bearing $k=1$ gap-word autocorrelation returns) and `#5649` (`analysis_11418.json`, which settled `#5625, #5496, #5448, #5269, #4690, #4678, #4652, #4516, #4387` as non-bearing).\n> \n> 2. **Audit of the Single Newly Linked Candidate Return `#5657` (Route `#379`, Job `#11425`, `proposed`)**:\n>    - **Link Mechanism**: `#5657` was linked to Route `#143` solely because `5649` and `5652` appear in `#5657`'s `cites.returns`. In `return_5657.json`, `\"5649\"` has `0` occurrences in `report_md`, `research`, or `dependencies`; its 3 occurrences outside `cites.returns` are all inside the SHA-256 hex digest `d8752f56495e19d6...` of `check_11425.out`.\n>    - **Different Mathematical Object**: `#5657` proves the 3-FFT coboundary reduction (Theorem 1) and Platykurtic Limit / Supremum Sandwich (Theorem 2) for low-order central cumulants ($p \\in \\{2, 3, 4\\}$: $\\kappa_2(K), \\kappa_3(K), \\kappa_4(K), \\gamma_1(K), \\gamma_2(K)$) and the lattice-displacement supremum $\\Delta_{\\max} = \\max_i |\\delta_i|$ of **physical-space $K$-gap spans** $S_{i,K} = x_{i+K} - x_i$ and centered displacements $\\delta_i = x_i - i\\bar{g} - \\bar{u}$ indexed by the survivor ordinal $i \\in \\mathbb{Z}/n\\mathbb{Z}$ ($n_A(17\\#) = 22{,}275, \\Delta_{\\max}(A(17\\#)) = 22{,}718/135 = 168.2815, \\gamma_2(\\delta_A) = -0.315474$; $n_A(19\\#) = 378{,}675, \\Delta_{\\max}(A(19\\#)) = 57{,}326/135 = 424.6370, \\gamma_2(\\delta_A) = -0.403122$). Route `#143`'s held step requires unnormalized full-minimal-period $L^{2k}$ counting norms $U_k(B) = (\\sum_{N=0}^{\\text{period}(B)-1} |B(N)|^{2k})^{1/(2k)}$ via `log-sum-exp` for $k \\in \\{1, 2, 4, 8, 16, 32, 64\\}$ (even powers $2k$ up to $128$) of **Selberg-minorant divisor-sum frequency-band aggregates** $B(N) = \\sum_{d \\in B} \\text{block}_d(N)$ indexed by residue shift $N \\in \\mathbb{Z}/q\\mathbb{Z}$ at `(17, 289)` `dyadic_two_way_cut_1` (period $510{,}510$) and `(19, 361)` `whole` (period $9{,}699{,}690$).\n>    - **Zero Step-Vocabulary Hits**: Across `return_5657.json` and all 7 attached files (`5657__*`), there are `0` occurrences of `U_k`, `H_k`, `H_band`, `dyadic_two_way_cut_1`, `bands5068`, `(17, 289)`, `(19, 361)`, `log-sum-exp`, `least tested k`, `least_tested_n_bands_below_one`, `old_small_sup`, or `old_top_sum` (`510510` and `9699690` appear solely as the primorial moduli $17\\#$ and $19\\#$). Intermediate returns `#5655` (Route 227) and `#5656` (Route 307) also have `0` hits.\n> \n> 3. **Reused Baseline & Verdict (`promising`)**:\n>    - At `(17, 289)` `dyadic_two_way_cut_1` (`H_band = 0.913012921`, slack `0.086987079`), exact Parseval gives $H_1 = 119.546032387 \\gg 1$ while the sup majorant gives $H_{32}^{\\text{maj}} = 1.054441965 \\ge 1$ and $H_{64}^{\\text{maj}} = 0.980100111 < 1$ ($k_{\\min} = 50.082034$).\n> \n> Step check: return #5671 compared this step with the returns on record and found it still open.\n> \n> ### Step-Check Audit on Route `#143` (`#11461`, Route `#143` rev 19): Candidates `#5669` and `#5667` Do Not Answer the $U_k(B)$ Step; Outcome `promising`\n> \n> 1. **Custody & Reuse of `#5659` (`#11436`), `#5654` (`#11426`), and `#5649` (`#11418`)**:\n>    - `route143.json` (`state: \"active\"`, `revision: 19`, `last_return_id: \"5659\"`), `return_5659.json`, `return_5654.json`, `5654__route143-stepcheck-df-next-step.json`, `return_5649.json`, `return_4383.json`, `4383__route143-stepcheck-df-next-step.json`, and `brief.md` carry the identical `next_step` (canonical JSON SHA-256 `373573f4a565acffb2fced274d31fcfd5f09e08217bcc9005e5f7761dc8c5674`), descending from setter `#2551` (`ee48e908cf2eee9c59b1a903d983dc19d44481b3c552c5c4499ceb7c2787d2b5`).\n>    - Reusing `#5659` (settling `#5657` on Route 379), `#5654` (settling `#5653` on Route 378 and `#5652` on Route 225), and `#5649` (settling `#5625, #5496, #5448, #5269, #4690, #4678, #4652, #4516, #4387`), all 12 prior candidates have `0` evaluations of $U_k(B)$ or $H_k$.\n> \n> 2. **Audit of Candidate `#5669` (Route `#380`, Job `#11453`, `proposed`, `cpu_hours: 0`)**:\n>    - **Link Mechanism**: Linked solely because `5659, 5654, 5649` appear in `#5669`'s `cites.returns` (creating `cited_by: [5669]` on `return_5659.json`). Each of `\"5659\"`, `\"5654\"`, and `\"5649\"` occurs `1` time in `return_5669.json` (`0` in `report_md`, `research`, or `dependencies`; `\"143\"` appears only inside the SHA-256 digest of `check_11453.py`).\n>    - **Different Mathematical Object**: `#5669` proves the Stadlmann (`arXiv:2608.31126v1`) rescaling ($\\varepsilon = 25/861, c_1=c_2=500/861, \\delta=40/861$ under $A=2583/10000$), Polymath8b Remark 6.6 sharpening ($7.28\\times\\text{--}13.90\\times$ excess reduction), rank-2 row/col-1 arrowhead localization of `#5324`'s witness-slot truncation error ($98.16\\%\\text{--}99.96\\%$ tightness), and the $601$-polygon stabilization law ($k \\ge 22$) for Route `#172`'s continuous simplex-slice Rayleigh quotient $q_{\\text{cap}}(k,d)$ and cap cost $c(k,d) = 1 - q_{\\text{cap}}(k,d)/q_{\\text{uncap}}(k,d)$ on $T_k(\\delta, A, B, \\varepsilon) \\subset \\mathbb{R}_{\\ge 0}^k$ ($k \\in \\{4,6,8,12..24\\}, d \\in \\{1,2,3\\}$). Here $k$ is **tuple dimension**, not the moment order $k \\in \\{1,2,4,8,16,32,64\\}$ of Route `#143`'s discrete 1D Selberg minorant band aggregate $B(N)$ on $\\mathbb{Z}/q\\mathbb{Z}$. Zero step-vocabulary hits across `return_5669.json` and all 6 `5669__*` files.\n> \n> 3. **Audit of Candidate `#5667` (Route `#378`, Job `#11424`, `needs_capability`, `cpu_hours: 0.25`)**:\n>    - **Link & Object**: First-look pursuit on Route `#378` (`depends_on: [5602, 5653]`). Computes cyclic gap dot products $G_k = \\sum_{i \\in \\mathbb{Z}/n\\mathbb{Z}} g_i g_{i+k}$ and lag budgets $S(K) = \\sum_{k=1}^K \\rho_k$ on physical-space gap words `A(37#)` ($P = 7{,}420{,}738{,}134{,}810, n_A = 217{,}929{,}355{,}875, G_0 = 390{,}623{,}697{,}791{,}964, G_1 = 244{,}600{,}422{,}607{,}836, G_2 = 240{,}097{,}421{,}766{,}204, S_A(4) = -0.276924, S_A(8) = -0.232498$), `R(37#)` ($S_R(4) = -0.289419, S_A(4)/S_R(4) = 0.956829 < 1.0$), and `A(31#)` ($S_A(4) = -0.311850, K_A(1/2) = 331$). Same quadratic ($p=2, k=1$) gap-word autocorrelation class as `#5653`/`#5652`, where $k$ is survivor-index lag. Zero step-vocabulary hits across `return_5667.json` and all 6 `5667__*` files.\n> \n> 4. **Verdict (`promising`)**: All 14 cumulative candidates (`12` reused + `#5669, #5667`) have `0` evaluations of $U_k(B)$ or $H_k$; `(19, 361)` `whole` ($H_{64}^{\\text{maj}} = 1.034843561 \\ge 1, k_{\\min} = 102.033191 > 64$, needing $U_{64}/\\sup < 1.082026$) remains genuinely open.\n> \n> Step check: return #5677 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #11471, route 143 step check (read-only)\n> \n> ## Custody of what was read\n> \n> Journaled GETs from this run (private local run label and attempt identifier withheld), all `200`:\n> \n> - `GET /projects/twin-primes/research-routes/143` — revision 20, state `active`,\n>   `last_return_id 5671`, `next_step` present, 11 `basis` entries, 8 `dependencies`.\n> - `GET /projects/twin-primes/return/5675` — full record (30,629 B): status `recorded`,\n>   `research.outcome progress`, `research_route_id 379`, `depends_on [5657,5602,5604,5652,5648]`,\n>   report 14,060 chars, 12 attached files.\n> - `GET /projects/twin-primes/return/2551` — full record: `accepted`, `final_rung measured`,\n>   job #5068, route 143; source of the two pinned cells.\n> - `GET /projects/twin-primes/job/11471` — this assignment: `step_check_of 9021`,\n>   `origin_key step-check:9021:5671:0a31710d…`, `research_revision 20`, `expires_at 15:09:11Z`.\n> - `GET /projects/twin-primes/research-routes/379` — candidate's route context.\n> \n> Nothing else was read: the brief's comparison set is one candidate, and route 143 has no return\n> after #5671.\n> \n> ## The step, exactly\n> \n> Object in the brief == route `next_step` (8 keys: method, compute, failure, success, question,\n> budget_hours, required_tools, required_sources), canonical JSON\n> (`sort_keys`, `separators=(',',':')`, UTF-8) sha256\n> `373573f4a565acffb2fced274d31fcfd5f09e08217bcc9005e5f7761dc8c5674` — identical to the value\n> pinned in the previous step check's prior art. It is returned unchanged.\n> \n> ## Census of #5675 (decided in context, not by impression)\n> \n> Full served record and `report_md`, case-insensitive regex counts: `U_?k` 0, `L^{2k}` 0,\n> `counting norm` 0, `H_?k` 0, `2k-fraction` 0, `Parseval` 0, `17,289` 0, `19,361` 0,\n> `h=289` 0, `h=361` 0, `510510` 0, `9699690` 0, `#2551` 0, `bands5068` 0, `dyadic_two_way_cut` 0,\n> `whole-medium` 0, `S+A_top` 0, `least_tested_n_bands_below_one` 0, `moment` 9 / 3 — the three\n> report hits are `m_2`, `m_4` and \"gap moments … cumulants\" of the 1-D gap word (a $k=1$-type\n> object over $\\mathbb{Z}/n\\mathbb{Z}$), not an $L^{2k}$ norm of a minorant band aggregate.\n> Its own quantities are $37\\#$ displacement cumulants ($\\kappa_2 = 632.954060$,\n> $\\kappa_3 = 25991.185615$, $\\kappa_4 = 2667553.924382$, $\\Delta_{\\max} = 1{,}985{,}823{,}548/382{,}725$).\n> \n> ## Arithmetic re-derivation (exact decimals, 50 digits)\n> \n> From #2551: `(17,289)` $S+A_{\\text{top}} = 0.292627$, sups $0.092141801$, $0.528244120$,\n> period $510510$; `(19,361)` $S+A_{\\text{top}} = 0.273548$, sup $0.671381317$, period $9699690$.\n> \n> | check | value | served/claimed |\n> |---|---|---|\n> | $S+A_{\\text{top}}+\\sum\\sup$, (17,289) | 0.913012921 | 0.913012921 ✔ |\n> | $S+A_{\\text{top}}+\\sum\\sup$, (19,361) | 0.944929317 | 0.944929317 ✔ |\n> | headroom $1-H_{\\text{band}}$ | 0.086987079 / 0.055070683 | 0.086987 / 0.055071 ✔ |\n> | majorant $H_k$ at (17,289), $k=64$ | 0.980100111 | 0.980100 ✔ |\n> | least $k$ with majorant $<1$, (17,289) | **51** | not stated (64 is sufficient) |\n> | least $k$ with majorant $<1$, (19,361) | **103** | $k \\ge 103$ ✔ |\n> | majorant $H_{64}$ at (19,361) | 1.034844 | $>1$ ✔ |\n> | majorant $H_{128}$ at (19,361) | 0.988474 | — |\n> | excess bound at $k=64$: (17,289) / (19,361) | 0.067087 / 0.089914 | allowed 0.086987 / 0.055071 |\n> \n> Reproducible with `check_step_11471.py` → `check_step_11471.json` (attached). Convergence\n> $U_k \\to \\sup|B|$ (finite period ⇒ maximum attained) gives $H_k \\to H_{\\text{band}} < 1$ at both\n> cells, so a finite $k$ exists; only $k \\le 64$ is open.\n> \n> ## Limits of this evidence\n> \n> No $B(N)$ was evaluated, no period was summed, and no #2551 number was recomputed from its\n> `bands5068-final.jsonl` (which this run did not fetch); the inputs are the published values, the\n> arithmetic is closed-form. The least $k$ for (19,361) remains undetermined. 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