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

# DMSD Phase 2B — Residue Layers

**Date:** 2026-09-30  
**Status:** COMPLETE  
**Primary layers:** mod 3 / mod 5 / mod 10  
**Representation negative control:** mod 7

---

## 1. Purpose of this round

Phase 2A used parity to verify:

\[
\text{Property computation}
\neq
\text{Element materialization}.
\]

Phase 2B extends a binary property into a multiclass residue vector.

For any modulus \(m\), define the block summary:

\[
V_m(B)
=
(C_0(B),C_1(B),\ldots,C_{m-1}(B)),
\]

where:

\[
C_r([a,b))
=
\#\{n\in[a,b):n\equiv r\pmod m\}.
\]

What we actually want to verify is whether:

\[
\boxed{
V_m(B)
=
\sum_i V_m(B_i)
}
\]

can remain exact during multiscale lazy refinement.

---

## 2. An exact residue vector without element-by-element enumeration

For the prefix interval \([0,N)\), let:

\[
N=qm+s,\qquad 0\le s<m.
\]

Then the count for residue \(r\) is:

\[
P_r(N)
=
q+\mathbf 1_{r<s}.
\]

Thus, for any:

\[
[a,b)
\]

we can directly obtain:

\[
C_r([a,b))
=
P_r(b)-P_r(a).
\]

The computational cost therefore depends mainly on:

\[
m
\]

rather than being proportional to:

\[
b-a
\]

as the interval length.

Even for:

\[
B=[10^{100},10^{101}),
\]

we can directly obtain the exact residue vector.

---

## 3. Validation results

This round checked:

```text
Scale-level checks:          404
Hierarchy conservation:      28
Nested refinement trials:   300
Primary moduli:            3,5,10
Representation control:       7
```

Result:

```text
ALL PASS
```

Including:

\[
0\le k\le100.
\]

No test required enumeration of the entire scale interval.

---

## 4. Residue-Vector Conservation

If:

\[
B=\bigsqcup_iB_i,
\]

then:

\[
C_r(B)=\sum_iC_r(B_i)
\]

holds for every:

\[
r=0,\ldots,m-1
\]

residue.

Therefore:

\[
\boxed{
V_m(B)=\sum_iV_m(B_i)
}
\]

is a natural generalization of the parity scalar/vector invariant.

It can directly serve as a refinement certificate.

---

## 5. The first truly interesting representation audit

The decimal scale boundaries are:

\[
10^k.
\]

So we examine:

\[
\phi_m(k)=10^k\bmod m.
\]

### mod 5

For \(k\ge1\):

\[
10^k\equiv0\pmod5.
\]

The scale boundaries are therefore completely locked to the mod 5 residue.

### mod 10

Likewise:

\[
10^k\equiv0\pmod{10},
\qquad k\ge1.
\]

This is the most direct base-10 coupling.

### mod 3

Because:

\[
10\equiv1\pmod3,
\]

we have:

\[
10^k\equiv1\pmod3
\]

for every \(k\).

It also exhibits a fixed phase, but not because \(3\mid10\); rather, base 10 equals the identity element modulo 3.

### mod 7 — Negative control

\[
10^k\bmod7
\]

exhibits:

\[
1,3,2,6,4,5,1,\ldots
\]

with a period of 6.

Thus, the decimal scale skeleton does not phase-lock to mod 7; instead, it produces cyclic relative alignment.

---

## 6. What does this mean?

For the first time, this round turns “decimal notation is only a representation” into a directly observable runtime phenomenon.

The same:

\[
L_k=[10^k,10^{k+1})
\]

exhibits different scale-boundary phases for different moduli.

Therefore, whenever DMSD encounters an attractive pattern in later work, we must ask:

\[
\boxed{
\text{Is this mathematical structure, or coupling between scale grammar and property?}
}
\]

In particular, the attractive alignments for mod 5 / mod 10 cannot directly be elevated to base-independent structure.

---

## 7. Scale Phase Signature

We can define:

\[
\Phi_m
=
(\phi_m(0),\phi_m(1),\phi_m(2),\ldots)
\]

where:

\[
\phi_m(k)=10^k\bmod m.
\]

This sequence describes the phase relationship between the property layer and decimal scale grammar.

This round gives:

\[
\Phi_3=(1,1,1,\ldots)
\]

\[
\Phi_5=(1,0,0,0,\ldots)
\]

\[
\Phi_{10}=(1,0,0,0,\ldots)
\]

and:

\[
\Phi_7=(1,3,2,6,4,5)^\omega.
\]

We may provisionally call this:

\[
\boxed{
\text{Scale–Property Phase Signature}
}
\]

---

## 8. A new classification for DMSD

After Phase 2A, we roughly classified property summaries as analytic / recursive / computed / approximate.

Phase 2B adds another dimension:

### Scale-Neutral Property
No special phase locking to the chosen scale grammar.

### Scale-Coupled Property
The algebra of the property has fixed alignment or short-period alignment with the scale grammar.

### Representation-Sensitive Observation
The visual pattern changes with the base / scale grammar.

These three must not be conflated.

---

## 9. MVP

`DMSD_Phase2B_Residue_MVP.html` supports:

- Kmax up to 100;
- Pyramid / Staircase;
- Small→large / large→small traversal;
- Real-time switching among mod 3 / 5 / 10 / 7;
- exact residue vector;
- decimal boundary phase;
- lazy block refinement;
- residue-vector conservation check;
- BigInt large-scale intervals.

We can now directly overlay a scale on the order of:

\[
10^{100}
\]

and a residue property in the same runtime, without materializing the enormous interval.

---

## 10. The most important progress

Phase 2A demonstrated that the prototype can preserve:

\[
\text{exact scalar-like property summary}.
\]

Phase 2B further demonstrated that the prototype can preserve:

\[
\boxed{
\text{exact multistate vector summary}.
}
\]

Mathematical LOD is therefore not limited to “a single density.”

It can be a composable state vector.

Formally:

\[
S(B)\in\mathbb R^d
\]

In the future, it could even be a:

- matrix;
- histogram;
- certificate object;
- local model;
- operator state.

---

## 11. Next step

The suggested next round is:

# DMSD Phase 2C — Square Layer

The reason is that squares, unlike residues, no longer have a fixed finite-state periodic partition.

Yet interval square counts still have an exact analytic summary:

\[
\#\{n^2\in[a,b)\}
\]

which can be obtained directly from square-root boundaries.

This places them precisely between:

```text
finite periodic property
```

and:

```text
prime-like irregular property
```

in that progression.

This will test whether DMSD can handle:

\[
\boxed{
\text{a property without a fixed period but still with an exact multiscale summary}.
}
\]

---

## 12. Conclusion

The results of this round support:

\[
\boxed{
\text{DMSD Core}
+
\text{Exact Vector Property}
+
\text{Lazy Refinement Conservation}
}
\]

The representation audit also shows, for the first time:

\[
\boxed{
\text{Scale Pattern}
\neq
\text{Automatically Base-Independent Mathematics}.
}
\]

There is phase coupling between decimal scales and the property, which can be formalized as:

\[
\Phi_m(k)=10^k\bmod m
\]

in this setting.

DMSD therefore provides not only visualization, but also a direct mechanism for identifying representation artifacts.

---

```text
DMSD Phase 2B: COMPLETE
Validation: PASS
Next: Phase 2C — Square Layer
```