# RH-W-10: Prime Boundary Local Mode and Seventh-Order Soft Activation v0.1

- Project: Riemann Hypothesis GAP Engineering
- Node: `RH-W-10-PRIME-BOUNDARY-LOCAL-MODE`
- Date: 2026-07-23
- Original Research Concept: Neo.K
- Mathematical Engineering, Derivation & Implementation: Aletheia (GPT-5.6 Thinking)
- Status: Local activation law closed; 15-dimensional positive margins before and after the boundary obtained; does not constitute a proof of the RH

---

## 0. Problem of the Current Round

The adaptive continuation in the previous round pushed the 15-dimensional lowest mode to the vicinity of

$$
\log 3=d+4h
$$

where

$$
h=\frac{87}{400},\qquad N=15.
$$

This boundary indicates that the left endpoint of the lag-$1$ correlation function is about to hit the sample

$$
-\log3.
$$

This round needs to answer:

1. At what order does the prime-$3$ matrix block appear?
2. Does it raise or lower the lowest mode?
3. Does the boundary cause a kink, eigenvalue crossing, or avoided crossing?
4. Can exact certificates be obtained using the full explicit formula both before and after the boundary?

---

## 1. Local Boundary Parameters

Let

$$
\mu=d+4h-\log3.
$$

Then:

- $\mu<0$: $-\log3$ lies outside the lag-$1$ correlation support;
- $\mu=0$: the sample is exactly at the left endpoint of the support;
- $\mu>0$: the sample enters the support.

The lag-$1$ correlation kernel is written as

$$
f_1(x)=\beta_7\!\left(\frac{x+d}{h}\right),
$$

where $\beta_7$ is the centralized degree-$7$ cardinal B-spline.

At $x=-\log3$:

$$
\frac{-\log3+d}{h}
=-4+\frac{\mu}{h}.
$$

When

$$
0<\mu<h
$$

the sample falls into the first polynomial segment of the B-spline, hence

$$
\boxed{
\beta_7\!\left(-4+\frac{\mu}{h}\right)
=\frac1{7!}\left(\frac{\mu}{h}\right)^7
}.
$$

---

## 2. Exact Soft Activation Law of Prime-$3$

In this local interval, the positive sample $+\log3$ is still outside the support, so the newly added lag-$1$ prime-$3$ element is exactly

$$
\boxed{
p_3(\mu)
=-\frac{\log3}{\sqrt3}\,
\frac{(\mu_+/h)^7}{7!}
},
$$

where

$$
\mu_+=\max(\mu,0).
$$

Therefore:

$$
p_3(\mu)=0\qquad(\mu\le0),
$$

and

$$
p_3(\mu)<0\qquad(0<\mu<h).
$$

### 2.1 Regularity

From $\mu_+^7$, we can directly obtain:

$$
\boxed{
p_3\in C^6\ \text{but}\ p_3\notin C^7
\quad\text{at }\mu=0
}.
$$

The one-sided derivatives from order zero to six are all zero; the jump in the seventh derivative is

$$
\boxed{
p_3^{(7)}(0^+)-p_3^{(7)}(0^-)
=-\frac{\log3}{\sqrt3\,h^7}
}.
$$

Thus, this is not a sharp switch, but a seventh-order soft activation.

---

## 3. Action Sign on Arbitrary Finite-Dimensional Directions

Let $A_1$ be the matrix whose first subdiagonal and first superdiagonal are equal to $1$. The newly added prime-$3$ block is

$$
P_3(\mu)=p_3(\mu)A_1.
$$

For any real coefficient vector

$$
c=(c_1,\ldots,c_N)^T,
$$

we have

$$
\boxed{
\Delta Q_3(c)
=2p_3(\mu)
\sum_{i=1}^{N-1}c_ic_{i+1}
}.
$$

Since $p_3(\mu)<0$:

- If the adjacent correlation is positive, prime-$3$ lowers the energy in that direction;
- If the adjacent correlation is negative, prime-$3$ raises the energy in that direction;
- If the adjacent correlation is zero, that direction is invisible on the first-order matrix block.

This is more precise than saying "primes make the matrix more positive or more negative": the action sign depends on the mode geometry.

---

## 4. Current Lowest Mode

Near the boundary, the exploratory generalized lowest mode is even-symmetric and has a positive adjacent correlation:

$$
\sum_{i=1}^{14}v_iv_{i+1}
\approx1.04087>0.
$$

Therefore, the direct action of the newly entering $P_3$ on this mode is to lower the energy.

However, it is extremely small at very shallow penetration depths. For the strict post-boundary parameters of this round,

$$
\mu\approx1.03336\times10^{-4},
$$

we obtain

$$
\boxed{
p_3(\mu)
\approx-6.87717\times10^{-28}
}.
$$

Thus, the direct change to the lowest mode is only on the order of $10^{-27}$, which is far smaller than the current spectral bottom scale of $10^{-9}$.

Conclusion:

> The searcher approaching the $\log3$ boundary is a real geometric phenomenon, but the current near-zero spectral values are not caused by the instantaneous effect of prime-$3$ just entering.

---

## 5. Three States Around the Boundary

### 5.1 $\theta_-$: Before the Boundary

Take

$$
d_-=\frac{117}{512}.
$$

Strict interval proof:

$$
\log3-(d_-+4h)
>9.66636\times10^{-5}.
$$

Thus, the lag-$1$ prime-$3$ sample is not yet activated.

The full 15-dimensional explicit formula matrix has been proven to satisfy:

$$
\boxed{
Q_-(c)>10^{-9}c^TG_-c
\qquad(c\ne0)
}.
$$

### 5.2 $\theta_0$: On the Boundary

Symbolically take

$$
d_0=\log3-4h.
$$

At this point,

$$
\mu=0,
\qquad
p_3(0)=0.
$$

Since $d_0$ is a transcendental constant expression, this round does not generate a rational input certificate for the entire 15-dimensional matrix; the zero value of the local prime-$3$ block on the boundary is an exact identity.

### 5.3 $\theta_+$: After the Boundary

Take

$$
d_+=\frac{117}{512}+\frac1{5000}.
$$

Strict proof:

$$
(d_++4h)-\log3
>1.03336\times10^{-4}.
$$

The prime-$3$ lag-$1$ element is strictly negative and falls into the first B-spline segment.

The full 15-dimensional explicit formula matrix similarly proves:

$$
\boxed{
Q_+(c)>10^{-9}c^TG_+c
\qquad(c\ne0)
}.
$$

Therefore, in both rational chambers before and after this local boundary, the lowest generalized margin is strictly higher than $10^{-9}$.

---

## 6. Does an Avoided Crossing Exist?

The floating-point local scan is for exploration only, with the range

$$
-8\times10^{-4}
\le\mu\le
8\times10^{-4}.
$$

Observations:

1. The gap between the first and second generalized eigenvalues is always greater than approximately
   $$
   4.82\times10^{-8};
   $$
2. The lowest mode remains even-symmetric;
3. The lowest mode overlap between adjacent sampling points is greater than approximately $0.9998$;
4. The lowest eigenvalue smoothly rises from approximately
   $$
   9.85\times10^{-10}
   $$
   to
   $$
   2.00\times10^{-9}.
   $$

Therefore, no avoided crossing or mode exchange was observed in this round.

This is a numerical diagnosis, not a full parameter interval theorem.

---

## 7. Intuition Corrected in This Round

The original intuition might have been:

> As soon as a prime power enters, the matrix will immediately exhibit sharp spectral changes.

This round proves that this intuition does not hold for the current B-spline dictionary. In fact:

$$
\text{Geometric support crosses the threshold}
\quad\not\Rightarrow\quad
\text{Low-order non-smooth spectral jump}.
$$

The smoothness of the threshold is determined by the vanishing order of the test function at the support boundary. The degree-$7$ autocorrelation kernel brings a seventh-order activation, so the prime layer can come "online" in a combinatorial sense, yet remain numerically almost invisible for a long time.

---

## 8. GAP Status

| GAP ID | Status | Result |
|---|---|---|
| `RH-W-10-BOUNDARY-PARAMETER` | `CLOSED` | Fixed $\mu=d+4h-\log3$ |
| `RH-W-10-ACTIVATION-ORDER` | `CLOSED` | prime-$3$ activates with $\mu_+^7$ |
| `RH-W-10-REGULARITY` | `CLOSED` | $C^6$, not $C^7$ |
| `RH-W-10-MODE-SIGN` | `CLOSED_FORMULA` | Action sign determined by adjacent correlation |
| `RH-W-10-THETA-MINUS` | `CERTIFIED_POSITIVE` | 15D margin $>10^{-9}$ |
| `RH-W-10-THETA-PLUS` | `CERTIFIED_POSITIVE` | 15D margin $>10^{-9}$ |
| `RH-W-10-THETA-ZERO-FULL` | `NOT_BUILT` | Only fixed symbolic local block |
| `RH-W-10-AVOIDED-CROSSING` | `NOT_OBSERVED_LOCALLY` | Not globally excluded |
| `RH-W-10-OTHER-KERNEL-ORDERS` | `OPEN` | Compare different boundary vanishing orders |

---

## 9. Next Node

This round points out a new engineering degree of freedom: **the test dictionary itself determines the analytic sensitivity of the prime-power threshold.**

The next node is set as:

$$
\boxed{
\texttt{RH-W-11-KERNEL-SENSITIVITY-ORDER}
}
$$

Compare:

- Different B-spline degrees;
- Compactly supported kernels with lower boundary vanishing orders;
- Trade-offs between smoothness and numerical sensitivity;
- Strict tail bound costs;
- The visibility speed of the same prime boundary to the lowest mode.

The goal is not to deliberately create negative values, but to determine whether the current degree-$7$ kernel smooths out valuable arithmetic boundary effects to the point of being excessively weak.

---

## 10. Scope Statement

This round did NOT:

- Prove or disprove the RH;
- Find a true Weil negative direction;
- Prove the absence of eigenvalue crossings near all parameters;
- Elevate the 15-dimensional positivity to infinite-dimensional positivity.

What this round accomplished is a local GAP: the exact appearance order of the prime-$3$ support boundary in the specified B-spline dictionary, its action sign, and strict finite-dimensional certificates before and after the boundary.

## 11. Reference Benchmarks

- Bombieri, *Problems of the Millennium: The Riemann Hypothesis*, Clay Mathematics Institute: Weil explicit formula and criterion benchmarks.
- Masatoshi Suzuki, *Weil's quadratic form via the screw function*, arXiv:2606.09096: Continuous kernel and operator framework for the Weil form on a fixed finite interval.
- A. Groskin, *A finite Guinand–Weil dictionary and archimedean tail order for the truncated Weil quadratic form*, arXiv:2607.02828: Finite dictionary, cutoff-free assembly, and archimedean tail bound background.