# Zero Genesis Game 15｜Möbius–Divisor Anatomy、Selberg DC Normalization 與 Criticality Conservation

**Series:** SECV / Zero Genesis Experimental Playground  
**Experiment:** Zero Genesis Game 15  
**Version:** v0.1  
**Author:** Neo.K  
**AI Collaboration:** Aletheia (GPT-5.6 Sol)  
**Status:** Exploratory / Prime-to-Möbius Structural Reduction  
**Important:** This document does **not** claim a proof of RH and does **not** claim novelty for the classical Möbius, Selberg-sieve, or truncated-divisor results used below.

---

## 0. Motivation

Game 14 decomposed the prime signal into

$$
\boxed{
\Lambda
=
\text{local Ramanujan congruence shape}
+
\text{global coherent amplitude}
+
\text{connected residual}.
}
$$

The local shape can be preconditioned by the Ramanujan/Selberg template

$$
\lambda_R(n).
$$

But the RH-hard coherent mode

$$
R(N)=\psi(N)-N
$$

survives.

The natural next question is:

> If one pushes the arithmetic representation below primes, into Möbius inversion and divisor sums, does the global critical mode finally disappear?

Game 15 shows that it does not.

However, it also reveals an important distinction:

- the simplest truncated von Mangoldt sum $\Lambda_R$ **reintroduces** an RH-sensitive Möbius mean mode;
- the more refined Selberg/Ramanujan weight $\lambda_R$ **algebraically removes** that representation-induced DC mode.

This gives a concrete example of a better symbolic preconditioner removing false residual complexity without solving the genuine RH residual.

---

# 1. Exact Möbius inversion beneath $\Lambda$

The von Mangoldt function satisfies the exact identities

$$
\boxed{
\Lambda(n)
=
\sum_{d\mid n}
\mu(d)
\log\frac nd
}
$$

and equivalently

$$
\boxed{
\Lambda(n)
=
-\sum_{d\mid n}
\mu(d)\log d
}
$$

for $n>1$.

Thus the prime-power signal is already a Möbius/divisor transform.

So moving "below primes" naturally reaches the Möbius function.

---

# 2. Simple truncated von Mangoldt weight

Define

$$
\boxed{
\Lambda_R(n)
=
\sum_{\substack{d\mid n\\d\le R}}
\mu(d)
\log\frac Rd.
}
$$

This is the classical Goldston–Yıldırım truncated divisor approximation.

It is simpler than the Ramanujan/Selberg weight $\lambda_R$ and is extremely useful in sieve and pseudorandomness arguments.

But its global mass channel contains a nontrivial Möbius Riesz residual.

---

# 3. Exact first moment of $\Lambda_R$

Sum over $n\le N$:

$$
\sum_{n\le N}\Lambda_R(n)
=
\sum_{d\le R}
\mu(d)
\log\frac Rd
\left\lfloor\frac Nd\right\rfloor.
$$

Write

$$
\left\lfloor\frac Nd\right\rfloor
=
\frac Nd
-
\left\{\frac Nd\right\}.
$$

Then

$$
\boxed{
\sum_{n\le N}\Lambda_R(n)
=
N A(R)
-
F_R(N),
}
$$

where

$$
\boxed{
A(R)
=
\sum_{d\le R}
\frac{\mu(d)}d
\log\frac Rd
}
$$

and

$$
\boxed{
F_R(N)
=
\sum_{d\le R}
\mu(d)
\log\frac Rd
\left\{\frac Nd\right\}.
}
$$

---

# 4. Floor-discrepancy term is locally harmless

Trivially,

$$
|F_R(N)|
\le
\sum_{d\le R}
\log\frac Rd.
$$

By Stirling / integral comparison,

$$
\boxed{
\sum_{d\le R}
\log\frac Rd
=
R+O(\log R).
}
$$

Hence

$$
\boxed{
F_R(N)=O(R).
}
$$

So the potentially dangerous coherent term is

$$
\boxed{
N[A(R)-1].
}
$$

---

# 5. Möbius Riesz mean

The coefficient

$$
\boxed{
A(R)
=
\sum_{d\le R}
\frac{\mu(d)}d
\log\frac Rd
}
$$

is a classical smoothed Möbius / Riesz mean.

The prime number theorem implies

$$
\boxed{
A(R)\to1.
}
$$

But the rate at which it approaches $1$ is controlled by zeros of $\zeta$.

---

# 6. Perron representation

For $c>0$,

$$
\boxed{
A(R)
=
\frac{1}{2\pi i}
\int_{c-i\infty}^{c+i\infty}
\frac{1}{\zeta(s+1)}
\frac{R^s}{s^2}\,ds
}
$$

in the standard Perron/Riesz sense.

At

$$
s=0
$$

the zero of $1/\zeta(s+1)$ combines with $s^{-2}$ to produce the main residue

$$
\boxed{
1.
}
$$

---

# 7. Zeta zeros become Möbius decay modes

If

$$
\rho=\beta+i\gamma
$$

is a simple nontrivial zero of $\zeta$, then

$$
s=\rho-1
$$

is a pole of

$$
1/\zeta(s+1).
$$

Its contribution to the Riesz mean has the schematic form

$$
\boxed{
\frac{
R^{\rho-1}
}{
\zeta'(\rho)(\rho-1)^2
}.
}
$$

Higher multiplicity produces additional logarithmic factors.

Thus the decay exponent of

$$
A(R)-1
$$

is controlled by

$$
\boxed{
\beta-1.
}
$$

---

# 8. Critical Möbius normalization

Set

$$
R=e^u
$$

and define

$$
\boxed{
Z_M(u)
=
e^{u/2}
[
A(e^u)-1
].
}
$$

A zero

$$
\rho=\beta+i\gamma
$$

then contributes a mode of the form

$$
\boxed{
e^{(\beta-1/2)u}
e^{i\gamma u}
}
$$

up to coefficients and possible polynomial factors in $u$ from multiplicities.

Therefore:

### critical-line zero

$$
\beta=\frac12
$$

gives marginal oscillation;

### right off-line zero

$$
\beta>\frac12
$$

gives exponential instability.

This is exactly the same instability coordinate found in Game 12.

---

# 9. RH-level Riesz bound

Standard contour / Riesz-mean arguments give the RH-level estimate

$$
\boxed{
A(R)-1
=
O_\varepsilon(
R^{-1/2+\varepsilon}
)
\qquad
\forall\varepsilon>0.
}
$$

Conversely, such a critical bound prevents poles of

$$
1/\zeta(s+1)
$$

with real part greater than $-1/2$ and therefore has RH strength.

Thus the smoothed Möbius mean already contains the same spectral abscissa.

---

# 10. Möbius smoothing chain

Define

$$
\boxed{
M(x)=\sum_{n\le x}\mu(n)
}
$$

and

$$
\boxed{
m(x)=\sum_{n\le x}\frac{\mu(n)}n.
}
$$

Partial summation gives

$$
\boxed{
m(x)
=
\frac{M(x)}x
+
\int_1^x
\frac{M(t)}{t^2}\,dt.
}
$$

Also,

$$
\log\frac Rd
=
\int_d^R\frac{du}{u},
$$

so

$$
\boxed{
A(R)
=
\int_1^R
\frac{m(u)}u\,du.
}
$$

Thus

$$
\boxed{
M
\rightarrow
m
\rightarrow
A-1
}
$$

is a successive smoothing chain.

Smoothing reduces local roughness but does not change the underlying spectral instability exponent.

---

# 11. Mertens criticality is the same RH fixed point

The classical RH-level Möbius statement is

$$
\boxed{
M(x)
=
O_\varepsilon(
x^{1/2+\varepsilon}
)
\qquad
\forall\varepsilon>0.
}
$$

This is equivalent to RH.

Therefore moving from primes to Möbius inversion does not escape the RH core.

It merely moves the same criticality from the logarithmic derivative of $\zeta$ to the reciprocal $1/\zeta$.

---

# 12. First-moment error of the simple truncation

From the exact decomposition,

$$
\boxed{
\sum_{n\le N}\Lambda_R(n)-N
=
N[A(R)-1]
-
F_R(N).
}
$$

Under an RH-level separate estimate,

$$
A(R)-1
=
O_\varepsilon(R^{-1/2+\varepsilon}),
$$

and

$$
F_R(N)=O(R),
$$

so the natural independent-error envelope is

$$
\boxed{
O_\varepsilon(
N R^{-1/2+\varepsilon}
+
R
).
}
$$

---

# 13. Naive optimization gives a $2/3$ scale

Ignoring the small $\varepsilon$ in the exponent and taking

$$
R=N^\alpha,
$$

the two exponents are

$$
1-\frac{\alpha}{2}
$$

and

$$
\alpha.
$$

Balancing them gives

$$
\boxed{
\alpha=\frac23.
}
$$

The resulting separate-term envelope is approximately

$$
\boxed{
N^{2/3+\varepsilon}.
}
$$

This is **not** a theorem that $\Lambda_R$ can never do better through cancellation between the two terms.

It only shows that straightforward independent control of its Möbius mean error and floor discrepancy does not naturally reach the RH square-root scale.

---

# 14. At the square-root major-arc scale

If one insists on

$$
R\asymp\sqrt N,
$$

then the same separate estimates give

$$
\boxed{
N[A(R)-1]
=
O_\varepsilon(
N^{3/4+\varepsilon}
),
}
$$

while

$$
\boxed{
F_R(N)=O(\sqrt N).
}
$$

So the simple truncation carries an RH-sensitive coherent mean error much larger than the desired square-root prime residual.

---

# 15. Refined Selberg/Ramanujan weight

Now define

$$
\boxed{
\lambda_R(n)
=
\sum_{r\le R}
\frac{\mu(r)^2}{\varphi(r)}
\sum_{d\mid(r,n)}
d\mu(d).
}
$$

This weight arises from circle-method / Selberg-sieve considerations.

In the relevant class of divisor approximations it has a least-squares optimal interpretation, and its correlation errors are often substantially smaller than those of the simpler $\Lambda_R$.

---

# 16. Divisor reorganization of $\lambda_R$

Interchange the $r$ and $d$ sums.

For squarefree $d$,

$$
r=dm,
\qquad
(m,d)=1.
$$

Then

$$
\boxed{
\lambda_R(n)
=
\sum_{\substack{d\mid n\\d\le R}}
\frac{
d\mu(d)
}{
\varphi(d)
}
L_d(R/d),
}
$$

where

$$
\boxed{
L_d(X)
=
\sum_{\substack{m\le X\\(m,d)=1}}
\frac{\mu(m)^2}{\varphi(m)}.
}
$$

---

# 17. Hildebrand asymptotic

A standard estimate used by Goldston–Yıldırım is

$$
\boxed{
L_d(X)
=
\frac{\varphi(d)}d
[
\log X+c+v(d)
]
+
O\!\left(
\frac{w(d)}{\sqrt X}
\right),
}
$$

where

$$
\boxed{
v(d)
=
\sum_{p\mid d}
\frac{\log p}{p}.
}
$$

Hence schematically

$$
\boxed{
\lambda_R
=
\Lambda_R
+
\text{constant/divisor-local corrections}
+
\text{smaller truncation error}.
}
$$

So $\Lambda_R$ is only the first main layer of the more refined preconditioner.

---

# 18. Exact first moment of $\lambda_R$

Now sum $\lambda_R$ over $n\le N$ directly:

$$
\sum_{n\le N}\lambda_R(n)
=
\sum_{r\le R}
\frac{\mu(r)^2}{\varphi(r)}
\sum_{d\mid r}
d\mu(d)
\left\lfloor\frac Nd\right\rfloor.
$$

Insert

$$
\left\lfloor\frac Nd\right\rfloor
=
\frac Nd+O(1).
$$

The main term is

$$
N
\sum_{r\le R}
\frac{\mu(r)^2}{\varphi(r)}
\sum_{d\mid r}\mu(d).
$$

But

$$
\boxed{
\sum_{d\mid r}\mu(d)
=
\begin{cases}
1,&r=1,\\
0,&r>1.
\end{cases}
}
$$

Therefore the main term collapses exactly to

$$
\boxed{
N.
}
$$

No prime number theorem is used.

No RH is used.

---

# 19. Error term is only local size $O(R)$

The floor error is bounded by

$$
\sum_{r\le R}
\frac{\mu(r)^2}{\varphi(r)}
\sum_{d\mid r}d
=
\sum_{r\le R}
\mu(r)^2
\frac{\sigma_1(r)}{\varphi(r)}.
$$

For squarefree $r$,

$$
\frac{\sigma_1(r)}{\varphi(r)}
=
\prod_{p\mid r}
\frac{p+1}{p-1}.
$$

This multiplicative factor has bounded mean.

Indeed, write

$$
\prod_{p\mid r}
\left(
1+\frac{2}{p-1}
\right)
$$

as a positive divisor convolution; the corresponding Euler product for its mean contains local corrections of size $O(p^{-2})$ and converges.

Hence

$$
\boxed{
\sum_{r\le R}
\mu(r)^2
\frac{\sigma_1(r)}{\varphi(r)}
=
O(R).
}
$$

Therefore

$$
\boxed{
\sum_{n\le N}\lambda_R(n)
=
N+O(R).
}
$$

---

# 20. Algebraic DC Normalization

This is the central structural difference.

For $\Lambda_R$:

$$
\boxed{
\text{DC gain}
=
A(R)
=
1+\text{RH-sensitive Möbius residual}.
}
$$

For $\lambda_R$:

$$
\boxed{
\text{DC gain}
=
1
}
$$

at the main-term level because of the exact algebraic identity

$$
\boxed{
\sum_{d\mid r}\mu(d)=\delta_{r1}.
}
$$

Thus $\lambda_R$ contains an intrinsic

$$
\boxed{
\textbf{Algebraic DC Normalizer}.
}
$$

---

# 21. Why the lower-order corrections matter

The additional constant/divisor-local terms in the Hildebrand expansion of $\lambda_R$ may appear lower-order relative to

$$
\log(R/d).
$$

But globally they are not semantically irrelevant.

Collectively they convert a template whose mean contains the smoothed reciprocal-zeta residual into one whose leading mean is exactly normalized.

So the refinement from

$$
\Lambda_R
$$

to

$$
\lambda_R
$$

removes a **representation-induced coherent Möbius mode**.

---

# 22. Square-root scale becomes available

At

$$
R\asymp\sqrt N,
$$

the refined template satisfies

$$
\boxed{
\sum_{n\le N}\lambda_R(n)
=
N+O(\sqrt N).
}
$$

Thus its own mass error is already at square-root scale **unconditionally**.

This does not prove RH, because the actual prime mass is

$$
\psi(N)
=
N+R(N).
$$

The difference is

$$
\boxed{
\sum_{n\le N}
[
\Lambda(n)-\lambda_R(n)
]
=
R(N)+O(R).
}
$$

At $R\asymp\sqrt N$,

$$
\boxed{
\sum_{n\le N}
[
\Lambda(n)-\lambda_R(n)
]
=
R(N)+O(\sqrt N).
}
$$

So the true global prime residual is isolated rather than absorbed into the template.

---

# 23. Coherent-mode isolator

This motivates the interpretation:

$$
\boxed{
\textbf{Selberg/Ramanujan Coherent-Mode Isolator}
}
$$

The refined local template:

1. captures modular/congruence structure;
2. has the correct global equilibrium mass algebraically;
3. leaves the actual PNT deviation in the residual.

It does **not** predict that global deviation.

It isolates it.

---

# 24. Comparison of the two truncations

### Simple Möbius truncation

$$
\boxed{
\Lambda_R:
\quad
\sum\Lambda_R
=
N A(R)+O(R).
}
$$

The template itself contains an RH-sensitive reciprocal-zeta mode.

### Selberg/Ramanujan truncation

$$
\boxed{
\lambda_R:
\quad
\sum\lambda_R
=
N+O(R).
}
$$

The template DC mode is algebraically normalized.

Thus the second representation is better conditioned for studying the actual coherent prime residual.

---

# 25. Criticality has not disappeared

Although $\lambda_R$ removes the template's own Möbius instability,

$$
\boxed{
\sum_{n\le N}
[
\Lambda(n)-\lambda_R(n)
]
}
$$

still contains

$$
\boxed{
R(N)=\psi(N)-N.
}
$$

Therefore the RH core survives.

A local divisor preconditioner can remove artificial coherent modes introduced by its own truncation, but it cannot remove the genuine global prime amplitude without importing equally global information.

---

# 26. Amplitude correction would be circular

One could define an amplitude-corrected template

$$
\boxed{
\frac{\psi(N)}N
\lambda_R(n).
}
$$

Its total mass would track $\psi(N)$ by construction.

But the required amplitude

$$
\boxed{
\frac{\psi(N)}N
=
1+\frac{R(N)}N
}
$$

already contains the unknown global residual.

So such a correction is not a proof mechanism.

It simply feeds the target answer back into the model.

---

# 27. Criticality Conservation under Local Inversion

This motivates the main methodological principle of Game 15:

$$
\boxed{
\textbf{Criticality Conservation under Local Inversion}
}
$$

> Möbius inversion, divisor truncation, Ramanujan projection, and Selberg preconditioning may relocate or remove representation-induced coherent modes, but any local transformation that does not already encode the global prime-counting amplitude cannot eliminate the genuine RH-critical coherent mode.

The criticality moves between:

- template normalization;
- residual mass;
- reciprocal-zeta Riesz modes;
- prime-counting error.

It does not disappear.

---

# 28. Möbius and prime criticality are the same instability exponent

The exact Möbius inversion layer is governed by

$$
\frac1{\zeta(s)},
$$

while the prime residual layer is governed by

$$
-\frac{\zeta'(s)}{\zeta(s)}.
$$

Both have singularities at the same nontrivial zeros.

Therefore the spectral abscissa

$$
\boxed{
\Theta_\zeta
=
\sup_\rho\Re\rho
}
$$

controls both:

- prime residual growth;
- Möbius/Riesz residual decay.

The representation changes.

The instability coordinate does not.

---

# 29. Representation cycle

The arithmetic reduction has now formed the cycle

$$
\boxed{
\Lambda
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{Ramanujan / divisor template}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\mu
}
$$

$$
\Downarrow
$$

$$
\boxed{
1/\zeta
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{same zero spectral abscissa}.
}
$$

This is another strong representation-fixed-point signal.

---

# 30. What Game 15 actually adds

The useful new structural clarification is not that Möbius sums are related to RH; that is classical.

The useful distinction is:

$$
\boxed{
\text{Möbius criticality caused by a naive template}
}
$$

versus

$$
\boxed{
\text{genuine global prime criticality}.
}
$$

The refined Selberg/Ramanujan template proves that the former can be algebraically removed without touching the latter.

That is a concrete example of SECV doing exactly what it is supposed to do:

$$
\boxed{
\text{remove representation-induced degrees of freedom
while preserving the true residual}.
}
$$

---

# 31. Current smallest arithmetic residual

After Games 13–15, the prime-side structure is

$$
\boxed{
\Lambda
=
\text{local modular/divisor skeleton}
+
\text{large incoherent residual}
+
\text{small RH-critical coherent mode}.
}
$$

The local skeleton can be heavily preconditioned.

Its mean can be normalized algebraically.

The residual can still have very large $L^2$ energy.

The remaining question is only:

$$
\boxed{
\text{why can the coherent cumulative mode not grow supercritically?}
}
$$

This is the same Prime Residual Criticality Problem from Game 12, now stripped of additional Möbius/divisor artifacts.

---

# 32. Game 16 candidate

The next meaningful step should not descend into another classical inversion identity.

Two possible directions remain.

## A. Coherent-mode dynamics

Study the cumulative residual

$$
\boxed{
Q_R(N)
=
\sum_{n\le N}
[
\Lambda(n)-\lambda_R(n)
]
}
$$

as a scale-dependent state, with $R$ chosen near $\sqrt N$.

Ask whether varying $R$ and $N$ yields a renormalization equation whose coherent mode has a stability invariant.

## B. Möbius Riesz flow as a diagnostic

Use

$$
\boxed{
A(R)-1
}
$$

not as the main template, but as an independent sensor of the same instability exponent.

Compare the prime residual and Möbius residual phases to see whether a cross-domain coupling can constrain a supercritical mode more strongly than either representation alone.

The key requirement is to avoid merely restating the same zero pole twice.

---

# 33. Monster status

After Game 15:

$$
\boxed{
\text{Simple Truncated-Divisor Möbius Mode: IDENTIFIED}
}
$$

$$
\boxed{
\text{Möbius Riesz Criticality: IDENTIFIED}
}
$$

$$
\boxed{
\text{Selberg/Ramanujan DC Normalizer: IDENTIFIED}
}
$$

$$
\boxed{
\text{Exact }\sum\lambda_R=N+O(R)\text{ Mean Law: IDENTIFIED}
}
$$

$$
\boxed{
\text{Representation-Induced Critical Mode: REMOVED}
}
$$

$$
\boxed{
\text{Genuine Prime Coherent Mode: SURVIVES}
}
$$

The surviving monster remains

$$
\boxed{
\textbf{Global Coherent Criticality}.
}
$$

---

# 34. Final diagram

$$
\boxed{
\begin{aligned}
\Lambda
&\xrightarrow{\text{simple truncation}}
\Lambda_R
\\
&\rightarrow
A(R)-1
\\
&\xrightarrow{\text{Mellin}}
1/\zeta
\\
&\rightarrow
\text{RH-critical Möbius mode},
\end{aligned}
}
$$

while

$$
\boxed{
\begin{aligned}
\Lambda
&\xrightarrow{\text{Selberg/Ramanujan}}
\lambda_R
+
\epsilon_R
\\
\sum\lambda_R
&=
N+O(R)
\\
\sum\epsilon_R
&=
\psi(N)-N+O(R).
\end{aligned}
}
$$

Thus the refined preconditioner removes its own coherent artifact and isolates the real global residual.

---

# 35. Conclusion

Game 15 pushes the prime route one level deeper into Möbius and divisor structure.

The simplest truncated von Mangoldt function

$$
\Lambda_R(n)
=
\sum_{\substack{d\mid n\\d\le R}}
\mu(d)\log(R/d)
$$

has first moment

$$
\sum_{n\le N}\Lambda_R(n)
=
N A(R)+O(R),
$$

where

$$
A(R)
=
\sum_{d\le R}
\frac{\mu(d)}d\log(R/d).
$$

The factor $A(R)-1$ is a smoothed Möbius residual whose Perron transform contains $1/\zeta(s+1)$, so its critical modes are generated by the same zeta zeros.

Thus naive descent from primes to Möbius simply recreates the RH fixed point.

The refined Selberg/Ramanujan weight behaves differently.

Because

$$
\sum_{d\mid r}\mu(d)=\delta_{r1},
$$

its main first moment is algebraically normalized:

$$
\boxed{
\sum_{n\le N}\lambda_R(n)
=
N+O(R).
}
$$

At $R\sim\sqrt N$, the template's own mass error is already at critical scale without invoking RH.

This shows that the smoothed Möbius mode in $\Lambda_R$ is not the genuine RH residual; it is partly a representation artifact that a better preconditioner can eliminate.

But the true prime residual remains:

$$
\boxed{
\sum_{n\le N}
[
\Lambda(n)-\lambda_R(n)
]
=
\psi(N)-N+O(\sqrt N).
}
$$

So the final problem does not live in local congruence structure, simple Möbius inversion, or template normalization.

It lives in the global coherent amplitude of the actual prime signal.

That is the current arithmetic fixed point.

---

# References

1. Goldston, D. A., & Yıldırım, C. Y. *Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem*. arXiv:math/0412366.
2. Goldston–Yıldırım literature on the truncated divisor sums $\Lambda_R$ and $\lambda_R$ and their sieve/correlation properties.
3. Green, B., & Tao, T. *The primes contain arbitrarily long arithmetic progressions*, for the standard truncated von Mangoldt weight $\Lambda_R$.
4. Montgomery, H. L., & Vaughan, R. C. *Multiplicative Number Theory I: Classical Theory*, for Möbius inversion, Riesz means, PNT equivalences, and contour methods.
5. Classical equivalence:
   $$
   \mathrm{RH}
   \iff
   M(x)=O_\varepsilon(x^{1/2+\varepsilon})
   $$
   for every $\varepsilon>0$.
6. Classical Riesz mean:
   $$
   \sum_{n\le x}
   \frac{\mu(n)}n
   \log(x/n),
   $$
   with Perron transform
   $$
   \frac{1}{\zeta(s+1)s^2}.
   $$