# Monotonicity Obstruction in RH Axis-Notch Co-Design

## Subspace Failure, External Dimensional Lift Saturation, and the Paley–Wiener Pivot

Version: v0.5  
Date: 2026-07-24  
Research Mode: Semi-AI Autonomous Mathematical Research  
Technical Research Lead: OpenAI Codex  
On-site Research Authorization and Review: Neo.K / EveMissLab

## Abstract

This node continues the support–prime dual frontier from v0.4. The parent node preserved joint-dual witnesses with $\alpha_{\rm safe}>1$ on the sampled hard sub-rectangles for $R=10.25,12,14,16$, and discovered that the coarse axis grid $\alpha<1$ at $R=16$ was a false escape. Therefore, instead of expanding the support radius, v0.5 attempts to reverse-engineer real-axis value/derivative notches, external spectral directions, and local bump geometries from the witness support peaks.

The equally-weighted aggregated peak atlas of the 12 parent witnesses yields the following on the five axis bands:

$$
17.83,\quad20.38,\quad23.24,\quad42.18,\quad83.05.
$$

Among these, $20.38$ falls within the target real interval $[20,20.5]$, while the far-band peaks are located at approximately $2.07$ and $4.08$ times the scale. This supports a natural Taylor proposition: if $G(x_0)=0$ and $G'(x_0)\ne0$, then

$$
G(x_0+iy)^2=-y^2G'(x_0)^2+O(|y|^3),
$$

so real-axis notches might simultaneously suppress axis values and preserve the off-axis negative direction.

The critical correction in this node is: if a notch merely adds homogeneous linear constraints to the existing test function space, the new space is simply a subspace of the parent space. The PSD Gram of the parent node has already searched the complete parent space; therefore, the subspace cannot produce a primal feasible point that does not exist in the parent problem. This is an exact feasible-set inclusion within a finite model, not just a numerical failure.

To bypass this monotonicity obstruction, this node introduces compact spectral-slope atoms outside the parent dictionary:

$$
\psi_{\omega,p}(t)
=t\left(1-\frac{t^2}{R^2}\right)_+^p\sin(\omega t).
$$

They improve the uniform/core screen at $R=16$ by at most about $3.10\%$, but the joint raw dual lower bound only drops from $1.189562$ to $1.176230$, and the safe lower bound remains $1.088115>1$. Another sweep of 27 polynomial-bump geometries yielded the optimal `d12_w2_p5`, which improved the raw joint dual by $3.87\%$, but

$$
\alpha_{\rm safe}=1.071761>1,
$$

and the minimum tail eigenvalue is only about $1.15\times10^{-3}$. A densified complementary audit also shows that the far-band peaks primarily migrate rather than vanish, and the $A_1$ charge still dominates.

Consequently, this node does not initiate a primal Gram search and halts three branch lines: pure homogeneous notches, the current spectral-slope lift family, and further polynomial-bump scaling. The next node pivots to the continuous Paley–Wiener axis/core extremal, attempting to interpret the recurring discrete dual measures as approximate supports for a reproducing-kernel inequality, thereby establishing an analytic lower bound, a continuous extremizer, or Galerkin convergence with error control.

This manuscript is neither a proof nor a disproof of the RH, nor does it contain continuous-axis or interval-certified transfers.

## 1. Problem Context

### 1.1 Decision State Inherited from the Parent Node

The research chain v0.1–v0.4 has progressively rewritten the original idea into a finite PSD Gram and joint-dual gate. The key states from the parent node are:

1. The 18 original patches were subdivided into 288 sub-rectangles;
2. Sparse dual witnesses were preserved on 12 hard sub-rectangles across four radii;
3. Each sampled radius has at least one reconstructed safe lower bound greater than $1$;
4. The $\alpha<1$ that appeared at $R=16$ with an axis step of $0.25$ vanished after densifying the step to $0.025$;
5. The support truncation cost grows as $e^{2R}$, making the decision value of further increasing the radius too low.

The parent node thus hands over a very specific problem: can we identify recurring peaks from the active axis supports, and then design test functions to reduce the axis burden without destroying the off-axis core negative direction?

### 1.2 Fixed Finite Model

This node fixes the target rectangle

$$
\mathcal R=[20,20.5]\times[-0.2,-0.1]
$$

and the five real-axis bands

$$
A_0=[14,18],\quad A_1=[18,23],\quad A_2=[23,35],
$$

$$
A_3=[35,70],\quad A_4=[70,145].
$$

The primary joint pilot uses the parent node's hard sub-rectangle `x4_Y3__r3_3` at $R=16$:

$$
\mathcal P
=[20.395,20.42]\times[-0.10625,-0.1].
$$

The test functions are generated by the Fourier transform of a real-even compactly supported $\psi$

$$
G(z)=\int_{-R}^{R}\psi(t)e^{izt}\,dt
$$

preserving the structural constraints of the parent node

$$
G(0)=0,\qquad G(i/2)=0.
$$

### 1.3 Gate-First Principle

For the PSD Gram variable $A\succeq0$, the axis-side quadratic matrices are denoted as $P_x$, the off-axis core matrices as $C_z$, and the tail matrix as $T$. If there exist five normalized non-negative axis measures $\mu_j$, a normalized core measure $\nu$, and $\alpha>1$ such that

$$
W
=T+\sum_{j=0}^{4}\underline N_j\int_{A_j}P_x\,d\mu_j(x)
+\alpha\int_{\mathcal P}C_z\,d\nu(z)
\succeq0,
$$

then for the specified finite primal constraints, trace pairing yields

$$
J(A)\ge\alpha.
$$

Therefore, as long as the reconstructed $\alpha_{\rm safe}>1$, the branch is not worth initiating an expensive primal search. This gate saves not just a little time, but avoids continuing to build large-scale prime matrices on finite branches that are already known to be infeasible.

## 2. Building an Axis Peak Atlas from Witnesses

### 2.1 Aggregation Rules

The parent node preserved 12 sparse witnesses. If all support weights were directly superimposed, witnesses with more supports or larger numerical mass would dominate the atlas. This node therefore adopts the following:

1. Normalize each witness and each band individually;
2. Assign the same total mass to each witness in each band;
3. Aggregate support locations using a fixed-bandwidth Gaussian KDE;
4. Treat only the KDE maxima as dictionary-design diagnostics.

These rules do not claim to estimate any true zero distribution, nor do they use known zero ordinates.

### 2.2 Aggregation Results

The primary peaks for the five bands are:

| band | weighted mean | weighted standard deviation | KDE primary peak |
|---|---:|---:|---:|
| $A_0$ | $16.8735$ | $1.0222$ | $17.83$ |
| $A_1$ | $20.4385$ | $0.1772$ | $20.38$ |
| $A_2$ | $23.7274$ | $0.6168$ | $23.24$ |
| $A_3$ | $42.1315$ | $0.1640$ | $42.18$ |
| $A_4$ | $83.0129$ | $0.2636$ | $83.05$ |

The most notable feature is not the near-harmonics in the far bands, but that the $A_1$ peak falls directly into the target real interval. This indicates that "excavating the axis peak" and "generating off-axis negative values near the same real part" might share the same set of degrees of freedom. Local and global are not two separable optimization problems.

The far-band ratios are

$$
\frac{42.18}{20.38}\approx2.069676,\qquad
\frac{83.05}{20.38}\approx4.075074.
$$

These are sufficient to support harmonic-notch ablation, but insufficient to claim a true analytic harmonic law; the active supports of the parent witnesses are themselves discrete optimization outputs.

## 3. Taylor Notch: Correct Intuition and Incorrect Inference

### 3.1 Local Negative Squares

Because $\psi$ is real-even, $G(x)$ is real-valued on the real axis. If we design $G(x_0)=0$ at $x_0$, then

$$
G(x_0+iy)
=iyG'(x_0)-\frac{y^2}{2}G''(x_0)+O(|y|^3).
$$

Squaring and taking the leading term:

$$
G(x_0+iy)^2
=-y^2G'(x_0)^2+O(|y|^3).
$$

Thus, when a single value zero preserves the slope, it indeed has the potential to generate the required negative direction. If we further require $G'(x_0)=0$, this second-order negative term vanishes, and the core effect is delayed to higher orders.

This Taylor assessment is correct, but it only describes the local behavior near a specific function direction; it does not imply that "adding a notch constraint to the existing complete Gram space will improve the optimal value."

### 3.2 Subspace Monotonicity Proposition

**Proposition 3.1.** Let $V$ be a finite-dimensional test function space, and $V'\subseteq V$ be the subspace after adding arbitrary homogeneous value/derivative constraints. Let $\mathcal F(V)$ and $\mathcal F(V')$ be the Gram feasible sets formed under the same PSD, core, and axis rules, respectively. Then

$$
\mathcal F(V')\subseteq\mathcal F(V).
$$

Consequently, for the same minimization objective $J$,

$$
\inf_{A\in\mathcal F(V)}J(A)
\le
\inf_{A\in\mathcal F(V')}J(A).
$$

**Proof.** Take a basis of $V'$ and embed it into $V$ via the inclusion map. Every PSD Gram form on $V'$ can be embedded as a PSD Gram form on $V$ via congruence, and the generated functions and all evaluation quadratic forms remain unchanged. Therefore, every feasible point in $V'$ is also a feasible point in $V$. Taking the infimum yields the conclusion. $\square$

### 3.3 Research Consequences

The parent node already allowed arbitrary PSD Gram mixing on the complete $V$. If a dual witness already proves that $J(A)\ge1$ in this complete space, then none of the following actions can rescue it:

$$
G(a)=0,\qquad G'(a)=0,\qquad
G(a_k)=0\ \text{for finitely many }a_k,
$$

as long as these conditions do not add new directions outside the parent space.

This conclusion is stronger, yet narrower, than "a certain set of notch parameters failed to run":

- Stronger, because it simultaneously rules out all pure homogeneous subspace notches;
- Narrower, because it does not rule out affine normalization, nonlinear construction, external atoms, or a larger continuous function space.

## 4. Experimental Verification of Subspace Notches

This node still ran ten sets of code, not to numerically prove the proposition, but to check whether the Taylor predictions align with the program coordinates. The code includes:

- patch center value zero;
- center plus $A_3$ or $A_4$ primary peaks;
- center plus double, quadruple harmonic zeros;
- five-band atlas zeros;
- patch left/right edge pair;
- center simultaneous value/derivative zero.

At $R=16$:

| code | dimension | optimized-core / uniform-axis threshold |
|---|---:|---:|
| baseline | $158$ | $0.251927$ |
| `anchor1` | $157$ | $0.252055$ |
| `anchor_A3` | $156$ | $0.258737$ |
| `anchor_A4` | $156$ | $0.278994$ |
| `harmonic3` | $155$ | $0.253755$ |
| `edge_pair` | $156$ | $32.487862$ |
| `anchor_flat` | $156$ | $33.845656$ |

`anchor1` barely changes the threshold, but the direction slightly worsens. Far-band zeros progressively impair core efficiency. The clearest is `anchor_flat`: its anchor derivative Frobenius norm is approximately

$$
1.07\times10^{-12},
$$

as predicted by the Taylor analysis, the second-order negative direction is almost entirely eliminated.

At $R=10.25$, the `anchor_flat` threshold even reaches $691.837880$. This is not because $R=10.25$ has some mysterious pathology, but because when fewer dimensions are available and the core is already close to the gate, close zeros or a flat zero can more easily destroy the core normalization.

## 5. External Spectral-Slope Lift

### 5.1 Why Dimensional Lift is Necessary

Proposition 3.1 indicates: for the notch idea to retain decision value, we must add

$$
V_{\rm new}\not\subseteq V_{\rm parent}.
$$

This node selects

$$
\psi_{\omega,p}(t)
=tq_{R,p}(t)\sin(\omega t),
$$

where

$$
q_{R,p}(t)
=\left(1-\frac{t^2}{R^2}\right)_+^p,\qquad p\ge3.
$$

$t\sin(\omega t)$ is even, and after multiplying by a real-even compact window, it remains a real-even compact atom. Its Fourier transform provides slope-like modulation near $\omega$, rather than deleting parent space directions via hard constraints.

### 5.2 Implementation Details

The program analytically computes

$$
\psi_{\omega,p}''(t),
$$

for each atom, and then builds the derivative/tail quadratic form together with the local bump basis. The new columns first undergo $L^2$ normalization, and are then constrained-whitened together with

$$
G(0)=G(i/2)=0.
$$

Because some frequency directions become correlated after constraints and numerical rank determination, the 21 candidate atoms ultimately add 15 effective dimensions.

### 5.3 Screen Results

At $R=10.25$:

$$
0.999424\longrightarrow0.979093.
$$

At $R=16$:

$$
0.251927\longrightarrow0.245526
$$

corresponding to the first round of six-direction lift.

Next, scaling is performed at $R=16$:

| lift | effective added dimension | threshold | improvement |
|---|---:|---:|---:|
| `grid5_p4` | $5$ | $0.246645$ | $2.10\%$ |
| `grid9_p4` | $8$ | $0.245755$ | $2.45\%$ |
| `grid13_p4` | $10$ | $0.245248$ | $2.65\%$ |
| `grid21_p4` | $15$ | $0.244123$ | $3.10\%$ |
| `grid13_p46` | $12$ | $0.244687$ | $2.87\%$ |

Multi-power variants did not break past the 21-frequency single-power grid, indicating that this family is beginning to saturate under the current metric.

### 5.4 Joint Dual

The true joint gate results are:

| model | raw $\alpha$ | safe $\alpha$ | safe $\lambda_{\min}$ |
|---|---:|---:|---:|
| baseline | $1.189562$ | $1.094781$ | $0.114990$ |
| `grid21_p4` | $1.176230$ | $1.088115$ | $0.114990$ |

The raw improvement is

$$
1-\frac{1.176230}{1.189562}
\approx1.1208\%.
$$

This is much smaller than the $3.10\%$ from the uniform/core screen. The reason is not a miscalculation in the screen, but that the joint dual can concentrate the axis measure at locations not fully controlled by the lift, and can also readjust the core measure. A single averaged screen cannot substitute for an adversarial joint measure.

## 6. Polynomial-Bump Geometry Sweep

### 6.1 Sweep Design

To check whether the local dictionary itself is too narrow, this node sweeps

$$
d\in\{10,12,14\},\quad
w\in\{1.2,1.5,2.0\},\quad
p\in\{3,4,5\},
$$

for a total of 27 sets. Here, $d$ controls the local basis density per unit radius, $w$ controls the bump width relative to spacing, and $p$ controls the polynomial edge smoothness.

### 6.2 Optimal Screen Values

Among the top five sets, the top three are:

| geometry | dimension | threshold | tail $\lambda_{\min}$ |
|---|---:|---:|---:|
| `d12_w2_p5` | $190$ | $0.236986$ | $0.001150$ |
| `d14_w2_p5` | $222$ | $0.237848$ | $0.002317$ |
| `d10_w2_p4` | $158$ | $0.237886$ | $0.005438$ |

Compared to the baseline $0.251927$, the screen improves by about $5.9\%$. However, the optimal values are accompanied by extremely small tail eigenvalues, indicating that certain directions are approaching tail-null. This makes floating optimization more sensitive and implies that the added density is primarily exploiting narrow, weak directions.

### 6.3 Joint Results

Select `d10_w2_p4` and `d12_w2_p5` for joint pilots:

| geometry | raw $\alpha$ | safe $\alpha$ | safe $\lambda_{\min}$ | raw improvement |
|---|---:|---:|---:|---:|
| baseline | $1.189562$ | $1.094781$ | $0.114990$ | — |
| `d10_w2_p4` | $1.146055$ | $1.073027$ | $0.005438$ | $3.66\%$ |
| `d12_w2_p5` | $1.143522$ | $1.071761$ | $0.001151$ | $3.87\%$ |

Both show substantial improvement, but remain explicitly greater than $1$. Therefore

$$
\texttt{primal\_search\_started}=\texttt{false}.
$$

This is not overly conservative: the safe bound has been renormalized by serialized measures and the PSD matrix reconstructed. The maximum reconstruction difference of the minimum-eigenvalue across the four joint objects is approximately

$$
5.15\times10^{-16}.
$$

## 7. Dense Complementary Audit and Peak Migration

To avoid looking solely at the dual objective, this node takes the complementary rank-one direction of the joint witness, scales it to a 4,941-point core grid, and then computes the complete finite objective with an axis step of $0.025$.

The four scaled objectives are

$$
1.275147,\quad1.254665,\quad1.265481,\quad1.263246.
$$

None pass $1$.

The maxima for each band show:

- baseline far-band maxima are around $42.3$ and $82.9$;
- `grid21_p4` shifts to approximately $41.275$ and $81.875$;
- `d10_w2_p4` is around $43.35$ and $83.1$;
- `d12_w2_p5` shifts to approximately $36.325$ and $73.575$.

The optimal geometry suppresses the maximum values of $A_3, A_4$ very low, but the maxima migrate toward the left ends of the bands; meanwhile, the $A_1$ charge is about $0.8675$, remaining the dominant axis-side term. This shows that the core of the problem is not merely eliminating the far-band harmonic-looking peaks in isolation, but the structural competition between the target-near $A_1$ and the off-axis core normalization.

## 8. What This Node Accomplished

### 8.1 Positive Outcomes

1. Converted 12 sparse dual witnesses into a reproducible peak atlas.
2. Identified and proved homogeneous-notch subspace monotonicity.
3. Corrected the notch design from "deleting directions" to "must add external directions."
4. Implemented a compact spectral lift family with analytic second derivatives.
5. Completed 27 sets of local geometry and two joint pilots.
6. Established a peak migration stop audit.
7. Avoided primal and prime matrix computations lacking decision value by using a safe reconstructed dual gate.

### 8.2 Negative but Usable Results

This node did not find a finite-model gate crossing. However, it ruled out three vague claims:

- "Placing a few more zeros on the peaks should make it work";
- "Adding more similar Fourier atoms should eventually cross the threshold";
- "As long as the bumps are denser, wider, and smoother, the axis energy can be suppressed."

The first claim is ruled out by exact inclusion in the pure homogeneous subspace case; the latter two claims only show saturation and peak migration within the tested range, and can no longer be used as justification for infinitely adding parameters.

## 9. Next Steps: Continuous Paley–Wiener Extremal

### 9.1 Problem Rewrite

Fixing $R$, consider on an appropriate real-even compact-support Hilbert domain

$$
G(z)=\int_{-R}^{R}\psi(t)e^{izt}\,dt.
$$

Define

$$
\mathcal J_R(G)
=\mathcal T_R(G)
+\sum_{j=0}^{4}\underline N_j
  \sup_{x\in A_j}G(x)^2,
$$

and study

$$
\Lambda_R
=\inf_G
\left\{
\mathcal J_R(G):
\sup_{z\in\mathcal P}2\operatorname{Re}G(z)^2\le-1,\ 
G(0)=G(i/2)=0
\right\}.
$$

The first task here is not to compute $\Lambda_R$ directly, but to first determine:

- Which Sobolev/domain $\psi$ should reside in to make the tail functional coercive;
- The reproducing kernels for complex evaluation and derivative evaluation;
- The measure dual of the five-band supremum;
- The conditions for existence, strong duality, and complementary slackness;
- Whether finite bump spaces form a Galerkin sequence with error control.

### 9.2 New Role of Discrete Witnesses

The measures of v0.5 are no longer viewed merely as "numerical outputs blocking a certain dictionary," but can be treated as candidate approximations for continuous KKT supports. If the support locations are stable in the densified Galerkin space and the dual values converge, this may lead to an analytic reproducing-kernel inequality; if the values drop below $1$, one can reverse-engineer a continuous extremizer.

This is more discerning than continuing a blind dictionary search: whether the result is an obstruction or a construction, it demands a continuous object that can account for convergence and error.

### 9.3 v0.6 Success Gate

The next node should accomplish at least one of the following:

1. An analytic lower bound with explicit assumptions that can explain the persistent dual block;
2. An upper-bound construction with convergence/error control, showing that some $R\le16$ might have $\Lambda_R<1$;
3. A rigorous separating inequality that resolves the simplified one-band, one-point core problem.

Until one of these three is achieved, no large prime matrices will be built, nor will any new coarse-grid $\alpha<1$ be treated as progress.

## 10. Trust Boundaries and Prohibited Inferences

### 10.1 Evidence Levels

- E0: Subspace feasible-set inclusion and specified finite conic dual algebra.
- E1: Automated checks of files, schemas, dimensions, row counts, reconstructions, and false global flags.
- E2: Fourier quadrature, KDE, optimization, eigenvalues, and dense finite audits.
- E3: Continuous analytic or interval-certified transfer; this node has no E3.

### 10.2 Prohibited Inferences

The following cannot be inferred from this node:

1. All external notch dictionaries fail;
2. The continuous Paley–Wiener extremal value must be greater than $1$;
3. The finite dual obstruction proves or disproves the RH;
4. A branch not blocked by the current witnesses is necessarily primal feasible;
5. Checking only the target patch completes the global leakage budget for unknown off-axis zeros;
6. The existence of zeros in a patch can be asserted without the argument principle.

### 10.3 Formal Conclusion of This Node

The strongest and legitimate conclusion is:

> In the explicitly specified parent finite space, pure homogeneous value/derivative notches cannot improve primal feasibility due to subspace monotonicity; among the newly added spectral-slope family and the 27 sets of polynomial-bump geometries, the optimal joint raw dual improvements are $1.12\%$ and $3.87\%$, respectively, but all reconstructed safe lower bounds remain greater than $1$. Therefore, these three branch lines are halted, pivoting to the continuous Paley–Wiener axis/core extremal formulation.

This conclusion is sufficient as a clean handover for the next round of research, but it does not cross the analytic trust boundary of the RH.