# Banding, Multiple Test Functions, and Adaptive Covering Certificate Families ## Semi-AI Autonomous Node for the Study of Equivariant Arithmetic Obstructions to RH v0.1 **Date of Research:** 2026-07-24 **Technical Research Lead:** OpenAI Codex (AI Research Collaborator) **Research Environment and Authorization:** Neo.K / EveMissLab **Evidence Level:** E2 Floating-Point Research Evidence **Theorem Status:** RH not proved, nor has a globally dominating certificate been obtained. --- ## Abstract The previous node identified the true obstruction: finding a uniformly negative block on an off-axis rectangle is insufficient to overpower the positive contributions from critical line zeros, unknown off-axis regions, and the infinite tail. Even if a single test function can simultaneously maintain a positive arithmetic scalar, it may still fail by a factor of thousands due to on-axis leakage. This node shifts the unit of research from "one function, one rectangle" to: $$ \text{Banding} \;+\; \text{Multiple test function positive cone} \;+\; \text{Interleaved covering certificate families}. $$ The initial $3\times2$ coarse covering revealed a near-axis numerical disaster; simply densifying along the off-axis direction could not resolve it. The truly effective adjustment is anisotropic: the closer to the axis of symmetry, the shorter the height window must be. Ultimately, 18 rational rectangles were adopted; the far-axis height width is $0.20$, and the near-axis height width is $0.10$. Without using any known zeta zero ordinates for optimization, this node generated 72 rank-one candidates and combined them using a nonnegative linear cone. All 18 cores passed the $161\times121$ dense grid negativity check, and all passed a deliberately coarse floating-point $L^1$-Lipschitz sign audit. Compared to the six-cell coarse covering, the adaptive covering reduced the worst-case axis energy by a factor of approximately $215.13$, the top 50 zero holdout mass by a factor of approximately $85.83$, and the prototype tail bound by a factor of approximately $479.69$. However, the partial global margin under a unit negative margin still fell between approximately $-8.97$ and $-20.40$, resulting in a complete failure. Therefore, this node did not achieve a global contradiction; however, it ruled out two erroneous paths: 1. Performing only off-axis subdivision without shortening the height window; 2. Expecting a diagonal nonnegative cone to automatically improve major axis leakage. The next logical node is not to continue blindly slicing rectangles, but to upgrade the diagonal cone to a PSD Gram cone, utilizing off-diagonal cross terms to alter the off-axis phase while maintaining nonnegativity on the real axis. --- ## 1. Research Attribution and Autonomous Boundaries In this node, following the research handover, the AI research side autonomously determined the mathematical architecture, experimental design, failure criteria, and next steps. The role of Neo.K / EveMissLab is to provide the existing research environment, preliminary materials, authorization for continued research, and future review capacity. Therefore, this manuscript adopts the following attribution principles: - Propositions from existing files are retained as in the original documents; - The banding specifications, objective functions, LP architecture, numerical interpretation, and next-node decisions of this node are attributed to the AI's research judgment; - The AI's technical choices are not retroactively written as Neo.K's mathematical claims; - Numerical success is not interpreted as a theorem proven by either human or AI. This distinction is not a cosmetic attribution, but a part of the chain of evidence. If semi-autonomous research does not clearly record "who authorized, who judged, and who verified," it becomes very easy to later mistake exploratory conclusions for human-endorsed theorems. --- ## 2. From Single-Window Failure to Certificate Families For a real, even, compactly supported function $\psi$, let $$ G_\psi(w) = \int_{\mathbb R}\psi(t)e^{iwt}\,dt. $$ Its off-axis orbital block is written as $$ B_\psi(w) = 2\operatorname{Re}\!\left(G_\psi(w)^2\right). $$ If on some target rectangle $K$ we have $$ \sup_{w\in K}B_\psi(w)\le-c_K<0, $$ this only establishes a local negative direction. To form a global contradiction, we still require $$ \Delta_K = c_K -E_{\mathrm{axis}} -E_{\mathrm{guard}} -E_{\mathrm{other}} -E_{\mathrm{tail}} >0. $$ The single function from the previous node explicitly failed on this condition. Therefore, this round no longer seeks a single function to simultaneously account for all geometric positions; instead, different functions are assigned to handle different local phases, which are then aggregated using a positivity-preserving cone. --- ## 3. Why Not Add Functions First Before Squaring If we first let $$ g_\lambda=\sum_r\lambda_rg_r $$ and then form $g_\lambda*\widetilde g_\lambda$, the Fourier side will yield $$ \left(\sum_r\lambda_rG_r(w)\right)^2, $$ which contains uncontrolled cross terms. This is no longer a linear certificate family, nor can the aggregated arithmetic positivity be directly deduced from the arithmetic scalar of each candidate. This node first adopts the most conservative linear positive cone: $$ f_\lambda = \sum_r\lambda_r \left(g_r*\widetilde g_r\right), \qquad \lambda_r\ge0. $$ Thus, $$ B_\lambda(w) = \sum_r\lambda_r 2\operatorname{Re}\!\left(G_r(w)^2\right), $$ and $$ Q_{\mathrm{arith}}(f_\lambda) = \sum_r\lambda_rQ_{\mathrm{arith}}(f_r). $$ For real $t$, since $g_r$ is a real even function, $G_r(t)$ is real, so $$ B_\lambda(t) = 2\sum_r\lambda_rG_r(t)^2 \ge0. $$ This allows each regional certificate to be solved using linear programming, with clear evidence interfaces. The cost is that only "diagonal" degrees of freedom can be used; this limitation ultimately became the main negative result of this round. --- ## 4. No Longer Memorizing Zeros: Axis Band Energy The previous round employed a finite number of known zero cancellation conditions $$ G(\gamma_k)=0. $$ Although this can reduce the known prefix, it rapidly compresses the arithmetic positive subspace and cannot naturally control unknown zeros. To avoid training the candidate functions into a mere lookup table of zeros, candidate generation in this round completely avoids reading $\gamma_k$. Taking three continuous real axis bands: $$ [14,35],\qquad[35,70],\qquad[70,145], $$ we define $$ E(\psi) = \sum_I \int_I |G_\psi(t)|^2 \left( 1+\frac{\log(1+t)}{2\pi} \right)\,dt. $$ This is not an unconditional $dN_{\mathrm{maj}}$ certificate, but a continuous proxy that does not rely on individual zero positions. The ordinates of the first 50 zeros are only read in after the candidates and cone weights are determined, to compute $$ Z_{50}(\lambda) = \sum_r\lambda_r \sum_{n=1}^{50}|G_r(\gamma_n)|^2. $$ Therefore, "not using zero ordinates for optimization" is a reproducible fact at the programmatic level. However, this does not equate to statistically independent samples, because the continuous energy bands still cover the same height range. --- ## 5. Why the Coarse Covering Failed The first version of the parent domain was $$ K=[20,20.5]\times[-0.2,-0.1], $$ and used three height windows and two off-axis windows, totaling six cells. Negative candidates could be found for all cells on a finite grid, but the upper off-axis window $[-0.16,-0.1]$ exhibited extreme scale amplification: - Worst-case axis band energy approx. $10521.35$; - Worst-case top 50 zero mass approx. $1122.65$; - Worst-case tail prototype bound approx. $4063.66$; - Only three of the six cells passed the coarse Lipschitz sign audit. Next, only the off-axis direction was divided into four layers. The results showed that the layer closest to the axis, which still included $y=-0.1$, maintained almost the same disaster. This indicates that the problem is not simply that the vertical band is too wide, but rather that the cost of phase consistency in the height direction increases sharply when approaching the axis. --- ## 6. Adaptive Anisotropic Covering The final covering adopts four overlapping off-axis layers: $$ \begin{aligned} Y_0&=[-0.200,-0.160],\\ Y_1&=[-0.170,-0.135],\\ Y_2&=[-0.145,-0.115],\\ Y_3&=[-0.125,-0.100]. \end{aligned} $$ For $Y_0,Y_1$, three height windows of width $0.20$ are used: $$ [20,20.2],\quad[20.15,20.35],\quad[20.3,20.5]. $$ For the closer-to-axis $Y_2,Y_3$, six height windows of width $0.10$ are used: $$ \begin{aligned} &[20,20.10],\ [20.08,20.18],\ [20.16,20.26],\\ &[20.24,20.34],\ [20.32,20.42],\ [20.40,20.50]. \end{aligned} $$ The total number is $$ 3+3+6+6=18. $$ All endpoints are interpreted as decimal rational numbers. The program slices the endpoints into atomic rectangles, checks the midpoint of each open cell and all boundary probes, proving there are no rational gaps; it is additionally verified independently with a $501\times301$ grid, confirming a minimum covering multiplicity of one. --- ## 7. Candidate Generation and Two-Stage LP The basis continues the 24-dimensional real even compactly supported polynomial-bump family from the previous node. Only two structural conditions are imposed: $$ G(0)=0, \qquad G(i/2)=0. $$ There is no imposed $G(\gamma_k)=0$. After constraint and whitening by $C_0$, the dimension is 22. Each patch uses four axis energy penalties: $$ \mu\in \{0,\ 1.5\times10^{-4},\ 2\times10^{-4},\ 2.5\times10^{-4}\}, $$ thus the total number of candidates is $$ 18\times4=72. $$ Each candidate satisfies $$ \|\psi_r\|_{C_0}=1, \qquad Q_{\mathrm{arith}}(\psi_r)\ge5\times10^{-5}. $$ For each patch, the stage-one LP minimizes the axis band energy under the condition that the core negative value is normalized to $-1$. The stage-two LP allows a maximum of 5% axis energy slack, pivoting to minimize the guard positive peak. If the dense core grid discovers a violation point, the exchange loop adds the worst point to the active set. --- ## 8. Main Numerical Results ### 8.1 Local Signs The dense-core maximum for all 18 patches is no greater than $$ -0.99999999999989. $$ After substituting the $L^1$ and first-moment estimates into the coarse Lipschitz upper bound, all 18 cells remain negative. The worst case is $$ -0.3715422433640927. $$ This is a continuous sign proxy under floating-point numerical integration, not an interval arithmetic certificate. ### 8.2 Order-of-Magnitude Improvement from Adaptive Covering A comparison of the worst-case quantities between the coarse six-cell and the adaptive 18-cell coverings is as follows: | Metric | Coarse 6-Cell | Adaptive 18-Cell | Improvement Factor | | --- | ---: | ---: | ---: | | Axis Band Energy | 10521.354469 | 48.907282 | 215.13 | | Top 50 Zero Holdout Mass | 1122.654086 | 13.079704 | 85.83 | | Tail Prototype Bound Above $145$ | 4063.657732 | 8.471493 | 479.69 | | Worst-Case Partial Gap Absolute Value | 5196.499341 | 20.401803 | 254.71 | This is not minor parameter tuning; rather, the covering geometry has altered the numerical condition number of the problem. ### 8.3 The True Role of the Multi-Function Cone If only the axis band energy is minimized, the optimal values for all patches match the best single candidate to a relative floating-point scale of about $10^{-12}$. That is to say, the stage-one optimal solution of the diagonal nonnegative cone lies on an extreme ray. When a small amount of axis energy overhead is allowed, multi-function mixing can improve the guard of the transition layer $Y_1$: - X0_Y1 improved by approx. $1.93\%$; - X1_Y1 improved by approx. $9.41\%$; - X2_Y1 improved by approx. $18.06\%$. Therefore, multiple functions are not without effect; they are responsible for boundary phase shaping, rather than reducing major axis leakage. --- ## 9. Why It Still Failed Normalizing the core negative margin to one, we first only account for the known top 50 on-axis mass, the tail prototype bound, and the guard: $$ \Delta_\alpha^{\mathrm{partial}} = 1 -Z_{50,\alpha} -E_{\mathrm{tail},\alpha} -E_{\mathrm{guard},\alpha}. $$ All 18 cells satisfy $$ \Delta_\alpha^{\mathrm{partial}}<0, $$ with a range of approximately $$ -20.40 \le \Delta_\alpha^{\mathrm{partial}} \le -8.97. $$ And this has not yet collected the positive contributions from all unknown off-axis regions. Therefore, the failure is not a minor issue of certificate precision, but rather that the negative margin/leakage ratio of the current rank-one diagonal cone is still off by at least an order of magnitude. At the same time, the target window lacks a winding certificate, and it has not been proven that zeta zeros exist within it. Even if the partial budget were positive, an RH contradiction still could not be directly deduced. --- ## 10. Mathematical Judgments Truly Obtained in This Round ### Judgment A: Banding Must Be Anisotropic When approaching the axis of symmetry, a fixed resolution cannot be used for the off-axis and height directions. Optimization requires shorter height phase windows. This is an E2 conclusion supported by ablation. ### Judgment B: Local Sign is No Longer the Main Bottleneck After all 18 cells passed the coarse continuous sign proxy, the remaining gap is still $9$ to $20$ times too large. Continuing to merely densify the grid will not automatically repair the global budget. ### Judgment C: Energy Frontier Degeneration of the Diagonal Cone The stage-one LP consistently selects a single extreme ray. If one hopes to improve major axis leakage, degrees of freedom that do not currently exist in the cone must be added. ### Judgment D: The Next Degree of Freedom is PSD Cross Terms Let a set of basis function transforms form the vector $$ \mathbf G(w) = \begin{bmatrix} G_1(w)&\cdots&G_m(w) \end{bmatrix}^{\mathsf T}. $$ For $$ A\succeq0, $$ consider $$ B_A(w) = 2\operatorname{Re} \left( \mathbf G(w)^{\mathsf T} A \mathbf G(w) \right). $$ On the real axis, $$ B_A(t) = 2\mathbf G(t)^{\mathsf T}A\mathbf G(t) \ge0, $$ but off-axis cross terms can alter the phase geometry. The arithmetic quantity and various integral proxies can all be written as linear functions such as $\operatorname{tr}(AQ)$ and $\operatorname{tr}(AE)$, thus elevating the problem to a convex SDP. The nonnegative conic family of this round is precisely the special case where $A$ is diagonal. Experiments have already pointed out the limitations of the diagonal frontier, so the PSD upgrade is not an arbitrary expansion, but the minimal next step directly derived from the failure data. --- ## 11. Next Node Specifications The suggested name for the next round is: > `RH_PSD_Gram_Banded_Global_Dominance_v0.2` Core tasks: 1. Fix the 18-cell adaptive covering from this round; do not increase the number of patches first; 2. Solve for $A_\alpha\succeq0$ on each local basis; 3. Simultaneously constrain the core negative value, guard, arithmetic scalar, continuous axis energy, and Sobolev/tail roughness; 4. Output the diagonal-cone and full-Gram Pareto frontiers; 5. If the full-Gram can push the partial gap to a positive value, then add the unknown off-axis banding budget; 6. Only when the floating-point margin is significantly greater than all errors should interval rounding and an independent verifier be deployed; 7. If the full-Gram still stably halts at a negative margin, compile this into a structural incompatibility result under this basis and support radius. Stopping conditions should be explicitly stated in advance: $$ \Delta_{\mathrm{global}}>0 $$ and $$ Q_{\mathrm{arith}}>0 $$ must simultaneously hold via outward intervals before a global candidate can be declared successful. Any success flag from SLSQP, LP, or SDP does not equate to a theorem. --- ## 12. Conclusion This node did not prove RH; it performed more fundamental yet necessary work: transforming "banding, multiple test functions, and covering certificate families" from a slogan into a reproducible, falsifiable, and comparable research machine. The most important positive result is: $$ \text{Adaptive height subdivision} \Longrightarrow \text{Near-axis disaster reduced by two to three orders of magnitude}. $$ The most important negative result is: $$ \text{diagonal cone} \not\Longrightarrow \text{Better major axis energy frontier}. $$ Therefore, the research path is now narrower and clearer than in the previous round: retain the adaptive covering, upgrade to PSD Gram cross terms, and use the global dominating margin rather than local negativity as the sole acceptance criterion.