> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# DMSD Phase 3D — Scale-Normalized Density Model

**Date:** 2026-09-30  
**Status:** COMPLETE  
**Exact prime bound:** \(10^7\)  
**Bases:** 2 / 8 / 10 / 16

---

## 1. Question for This Round

Phases 3B / 3C observed:

\[
\text{normalized center bias}<0
\]

and increasing similarity among normalized histograms at higher scales.

However, these phenomena may largely result from a known first-order effect:

\[
\boxed{
\text{prime density decreases approximately like }1/\log x.
}
\]

If a multiplicative scale block:

\[
[a,b)
\]

has a higher prime density at its left endpoint than at its right endpoint, its normalized local-position histogram will naturally lean left.

This round therefore no longer asks directly:

> Does the raw histogram have an attractive pattern?

Instead, it asks:

> **After first subtracting the \(1/\log x\) baseline, what remains?**

---

# 2. Baseline

For the block:

\[
B=[a,b),\qquad a>1,
\]

divide normalized positions into \(R=100\) bins:

\[
[x_j,x_{j+1}).
\]

Use logarithmic-integral mass:

\[
M_j
=
\int_{x_j}^{x_{j+1}}\frac{dt}{\log t}
=
\operatorname{li}(x_{j+1})
-
\operatorname{li}(x_j).
\]

Then normalize:

\[
Q_j
=
\frac{M_j}{\sum_r M_r}.
\]

Thus:

\[
Q=(Q_0,\ldots,Q_{99})
\]

is the block-internal reference profile "when only the first-order \(1/\log x\) density gradient is considered".

This is not an exact prime law, only a baseline for a known asymptotic structure.

---

# 3. Observed and Residual

The exact prime histogram:

\[
P_j
=
\frac{H_j}{\sum_r H_r}.
\]

Define the residual:

\[
R_j=P_j-Q_j.
\]

Also use:

\[
E_j=Q_j\cdot N_{\mathbb P}(B)
\]

to define a diagnostic standardized residual:

\[
Z_j
=
\frac{H_j-E_j}{\sqrt{E_j}}.
\]

Here, \(\sqrt{E_j}\) is only a normalization.

**This round does not assume that primes form an independent Poisson process.**

---

# 4. Center Bias Decomposition

Raw center bias:

\[
B_{\mathrm{raw}}
=
\sum_jP_j\lambda_j-\frac12.
\]

PNT baseline bias:

\[
B_{\mathrm{PNT}}
=
\sum_jQ_j\lambda_j-\frac12.
\]

Residual bias:

\[
\boxed{
B_{\mathrm{res}}
=
B_{\mathrm{raw}}
-
B_{\mathrm{PNT}}.
}
\]

Thus:

\[
B_{\mathrm{raw}}
=
B_{\mathrm{PNT}}
+
B_{\mathrm{res}}.
\]

---

# 5. Most Important Result: The Baseline Explains Most of the High-Scale Negative Bias

## base 10

The highest three levels:

### \(k=4\)

\[
B_{\mathrm{raw}}=-0.0150000
\]

\[
B_{\mathrm{PNT}}=-0.0156482
\]

\[
B_{\mathrm{res}}=+6.48\times10^{-4}.
\]

The baseline explains approximately **95.68%** of the raw-bias magnitude.

### \(k=5\)

\[
B_{\mathrm{raw}}=-0.0129272
\]

\[
B_{\mathrm{PNT}}=-0.0128147
\]

\[
B_{\mathrm{res}}=-1.13\times10^{-4}.
\]

Approximately **99.13%** is explained.

### \(k=6\)

\[
B_{\mathrm{raw}}=-0.0109271
\]

\[
B_{\mathrm{PNT}}=-0.0108514
\]

\[
B_{\mathrm{res}}=-7.57\times10^{-5}.
\]

Approximately:

\[
\boxed{99.31\%}
\]

is explained by the first-order density baseline.

---

# 6. The Cross-Base Results Are Even More Pronounced

Near the highest complete scale in each base:

### base 2, \(k=22\)

\[
B_{\mathrm{raw}}
\approx
-3.6467\times10^{-3}
\]

\[
B_{\mathrm{res}}
\approx
-3.08\times10^{-6}.
\]

Approximately:

\[
99.92\%
\]

is explained by the baseline.

### base 8, \(k=6\)

Residual bias:

\[
+3.35\times10^{-5},
\]

Approximately:

\[
99.70\%
\]

is explained by the baseline.

### base 10, \(k=6\)

Residual bias:

\[
-7.57\times10^{-5},
\]

Approximately:

\[
99.31\%
\]

is explained.

### base 16, \(k=4\)

Residual bias:

\[
-1.99\times10^{-5},
\]

Approximately:

\[
99.86\%
\]

is explained.

The cross-base negative bias from Phase 3C
therefore now has a very direct reinterpretation:

\[
\boxed{
\text{It is mainly a scale projection of the }1/\log x\text{ density gradient.}
}
\]

At least, this is the case in the current exact finite domain.

---

# 7. This Is a Good Result, Not "The Pattern Disappeared, So Nothing Remains"

If our only goal were to find attractive patterns,
this might look like removing what we had found.

For research, however, it is better.

Because DMSD has now successfully carried out:

\[
\boxed{
\text{Raw Structure}
=
\text{Known Baseline}
+
\text{Residual}.
}
\]

This is much cleaner than renaming known density decay as "new geometry".

---

# 8. Residual JS Also Decreases Rapidly

Base 10:

\[
k=3:
\quad
JS(P,Q)
\approx4.71\times10^{-3}
\]

\[
k=4:
\quad
4.51\times10^{-4}
\]

\[
k=5:
\quad
4.78\times10^{-5}
\]

\[
k=6:
\quad
\boxed{
4.94\times10^{-6}
}
\]

At the current exact scales,
the observed normalized histogram gets increasingly close to the \(1/\log x\) baseline.

The same direction is observed at high scales in the other bases.

This is finite evidence,
not a convergence theorem as:

\[
k\to\infty
\]

tends to infinity.

---

# 9. Standardized Residual

The RMS standardized residual in the high-scale tail is approximately:

```text
base 2:  ~0.45–0.52
base 8:  ~0.38–0.45
base 10: ~0.40–0.46
base 16: ~0.40–0.43
```

It does not explode.

But this also cannot be directly interpreted as:

> Primes follow a Poisson stochastic model.

Because:

- Residue restrictions induce dependence;
- Bin counts are not independent;
- \(\sqrt E\) is used only for scale normalization;
- The current bound is only \(10^7\).

This round therefore uses it only as a residual-amplitude diagnostic.

---

# 10. Residual Center Bias No Longer Has a Fixed Sign

The raw center biases in Phase 3C's high-scale tail were all negative.

After subtracting the PNT baseline:

- Some scale residuals are positive;
- Some are negative;
- Their magnitudes are much smaller than the raw bias.

This is important.

Because:

\[
\boxed{
\text{raw negative sign}
}
\]

itself is no longer worth treating as a candidate new invariant.

What should actually be studied is whether:

\[
\boxed{
B_{\mathrm{res}}
}
\]

still has reproducible structure across scales / across bases.

The current data is insufficient to say that it does.

---

# 11. A Third Observation Layer Emerges in DMSD

A property view can now be separated into:

## Raw Layer

Direct exact observation.

## Baseline Layer

A reference model generated from known mathematical structure.

## Residual Layer

\[
\text{Observed}-\text{Baseline}.
\]

Thus:

\[
\boxed{
\text{DMSD Observation}
=
\text{Raw}
+
\text{Baseline}
+
\text{Residual}.
}
\]

This is a more mature research workflow than visualization alone.

---

# 12. The Baseline Itself Also Needs Provenance

A baseline cannot be chosen by intuition alone.

This round registers:

```text
baseline family:
  normalized logarithmic-integral mass

formula:
  li(x_{j+1}) - li(x_j)

purpose:
  remove first-order 1/log(x) density gradient

status:
  asymptotic reference model
  NOT exact prime law
```

The meaning of a residual is therefore always relative to:

\[
\boxed{
\text{registered baseline}.
}
\]

Changing the baseline also changes the residual.

---

# 13. Implications for the Original Pyramid / Staircase Idea

Initially, we wanted only:

> Change scale, look upward, look downward, look rightward.

Now an additional layer has been introduced:

> **First inspect the raw observation, then ask how much of it is merely a projection of known scale laws.**

The pyramid can therefore show not only:

\[
\text{prime density by scale},
\]

but also:

\[
\text{raw bias}
\]

\[
\text{baseline bias}
\]

\[
\text{residual bias}.
\]

This gives scale visualization the beginnings of an ability to "analyze known effects".

---

# 14. Correction to the Previous Two Rounds

The observations of Phases 3B / 3C were not wrong.

The error would be:

> Directly treating a raw observation as a new prime invariant.

Phase 3D shows:

\[
\boxed{
\text{Raw Pattern}
\neq
\text{Novel Structure}.
}
\]

Only after baseline subtraction
can one discuss whether any additional residual structure remains.

This is in the same spirit as the earlier Representation Audit:

\[
\boxed{
\text{Exclude known sources before discussing new ones.}
}
\]

---

# 15. Next Step

The most worthwhile next round is:

# DMSD Phase 3E — Residual Structure Audit

But it must be approached very conservatively.

Rather than looking for a new law directly in the residual,
first check whether it is mainly caused by known arithmetic constraints, such as:

- parity exclusion；
- mod 3；
- mod 5；
- small-prime wheel structure。

That is, establish:

\[
\text{PNT Baseline}
\rightarrow
\text{Wheel/Residue-Corrected Baseline}
\rightarrow
\text{Residual}.
\]

If small-prime divisibility structure explains away another large part of the residual,
continue subtracting it.

Only what genuinely remains warrants further study.

---

# 16. Conclusion

The most important aspect of Phase 3D is not discovering a new pattern.

Quite the contrary.

It successfully demonstrates the necessity of a research workflow:

\[
\boxed{
\text{Observe}
\rightarrow
\text{Model Known Effects}
\rightarrow
\text{Subtract}
\rightarrow
\text{Audit Residual}.
}
\]

In the current exact domain:

\[
p<10^7,
\]

the cross-base high-scale negative center bias can be explained almost entirely by the first-order density gradient of:

\[
1/\log x
\]

that baseline.

The most reasonable current conclusion is therefore:

\[
\boxed{
\text{The raw left bias of Phases 3B/3C is mainly not new prime geometry.}
}
\]

And this is precisely where DMSD's value lies:

> It not only helps us see patterns, but also helps us separate known patterns from new ones.

---

```text
DMSD Phase 3D: COMPLETE
Validation: PASS

Next:
DMSD Phase 3E — Residual Structure Audit
```