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

# DMSD Phase 2C — Square Layer

**Date:** 2026-09-30  
**Status:** COMPLETE  
**Property class:** Non-periodic / exact analytic block summary

---

## 1. Purpose

Phase 2A validated parity; Phase 2B validated finite-state residue vectors.

Phase 2C deliberately switches to:

\[
Q(n)=\mathbf 1_{\exists r\in\mathbb N:r^2=n}.
\]

Perfect squares are not a fixed-period property, so a finite residue cycle can no longer provide a complete description.

Yet they still have an exact block summary.

This makes squares an excellent intermediate test between the following:

\[
\boxed{
\text{finite-period property}
\quad\longrightarrow\quad
\text{irregular sparse property}
}
\]

two classes.

---

## 2. Exact square count

For:

\[
B=[a,b),\qquad 1\le a<b,
\]

the squares in the block satisfy:

\[
r^2\in[a,b).
\]

Therefore:

\[
\boxed{
C_{\square}(a,b)
=
\lfloor\sqrt{b-1}\rfloor
-
\lfloor\sqrt{a-1}\rfloor.
}
\]

This formula requires only an integer square root.

Thus, even for:

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

we do not need to enumerate the interval's elements.

---

## 3. SquareSummary

This round does not merely preserve the count; it defines:

\[
S_{\square}(B)
=
(
C,
r_{\min},
r_{\max},
q_{\min},
q_{\max}
),
\]

where:

- \(C\): the number of perfect squares;
- \(r_{\min}\): the first square root;
- \(r_{\max}\): the last square root;
- \(q_{\min}=r_{\min}^2\);
- \(q_{\max}=r_{\max}^2\).

The summary therefore consists of:

```text
count
first_root
last_root
first_square
last_square
```

and each block requires only \(O(1)\) summary state.

---

## 4. Validation

This round completed:

```text
Scale checks k=0..100:            101
Hierarchy merge checks:             7
Nested refinement trials:          300
Independent brute-force oracles:   500
```

All:

```text
PASS
```

For 500 small intervals, element-by-element enumeration was used as an independent oracle,
then compared with the integer-sqrt formula.

This round therefore did not merely have “the formula validating itself.”

---

## 5. Composable Summary

If:

\[
B=\bigsqcup_iB_i
\]

and the children are ordered along the number line, then:

### Count

\[
C(B)=\sum_iC(B_i).
\]

### First square

Take the first square of the first non-empty child.

### Last square

Take the last square of the last non-empty child.

There is therefore a merge operator:

\[
\boxed{
S_{\square}(B)
=
\bigoplus_iS_{\square}(B_i).
}
\]

This goes a step beyond Phase 2A / 2B:

> A DMSD summary need not merely be an additive scalar/vector; it can also be a structured object with a custom merge algebra.

---

## 6. The first Summary Algebra

We can provisionally write the DMSD property layer as:

\[
(\mathcal S_Q,\oplus_Q)
\]

where:

- \(\mathcal S_Q\): property summary space;
- \(\oplus_Q\): disjoint ordered block merge operator.

For squares:

\[
\mathcal S_{\square}
=
(C,r_{\min},r_{\max},q_{\min},q_{\max}).
\]

This gives DMSD a new direction:

\[
\boxed{
\text{Property Layer}
=
\text{Summary State}
+
\text{Merge Algebra}
+
\text{Local Generator}.
}
\]

---

## 7. Non-periodic, yet still exact at coarse scales

This is the most important conceptual result of this round.

The square property does not require:

```text
fixed period
finite residue state
element enumeration
```

to be exact on a coarse block.

Therefore:

\[
\boxed{
\text{Exact coarse summary}
\not\Rightarrow
\text{finite periodicity}.
}
\]

What may actually be needed is:

\[
\boxed{
\text{boundary-computable summary}.
}
\]

For squares, the two boundaries:

\[
\sqrt a,\sqrt b
\]

determine the exact count for the entire block.

---

## 8. Scale geometry

For the decade:

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

the number of squares is approximately:

\[
C_k
\sim
(\sqrt{10}-1)10^{k/2}.
\]

but the block size is:

\[
|L_k|=9\cdot10^k.
\]

Square density therefore decreases roughly according to:

\[
10^{-k/2}
\]

this rate.

In other words:

\[
\boxed{
\text{absolute square count grows}
}
\]

but:

\[
\boxed{
\text{relative square density shrinks}.
}
\]

This is a phenomenon that multiscale visualization is well suited to present directly.

---

## 9. Even / Odd Scale Phase

Taking the square root turns the exponent:

\[
k
\]

into:

\[
k/2.
\]

Decimal decades in the square layer therefore naturally have two-phase scale geometry governed by:

\[
\boxed{k\bmod2}
\]

this parity.

This differs from Phase 2B's:

\[
10^k\bmod m
\]

phase relation.

The residue phase comes from the relationship between the scale base and modular algebra;

the square phase instead comes from:

\[
\text{scale exponent}
\mapsto
\frac{1}{2}\text{ scale exponent}.
\]

DMSD thus begins to distinguish different kinds of scale coupling.

---

## 10. No direct claim of base independence

Even though “perfect squares” are themselves a base-independent property,

the following visual patterns:

- decade boundary;
- even/odd \(k\) phase;
- the visual width of each layer;
- normalized location;

still depend on decimal scale grammar.

We should therefore distinguish:

\[
\boxed{
\text{Property Ontology}
\neq
\text{Scale Projection Geometry}.
}
\]

The ontology of squares is base-independent;

but their shape in the DMSD decimal projection still has a representation component.

---

## 11. The second level of Mathematical LOD

At least two exact property mechanisms have now emerged:

### Class E1 — Finite-state / periodic exact summary

For example:

```text
parity
mod m
```

### Class E2 — Boundary-analytic exact summary

For example:

```text
perfect squares
```

E2 does not require the property to have a finite period.

DMSD Mathematical LOD has therefore moved from:

```text
periodic counting
```

to:

```text
non-periodic analytic counting.
```

---

## 12. MVP

`DMSD_Phase2C_Square_MVP.html` supports:

- Kmax up to 100;
- Pyramid / Staircase;
- upward / downward traversal;
- exact square count;
- exact first/last root;
- exact first/last square;
- lazy refinement;
- structured-summary merge certificate;
- BigInt integer sqrt.

Thus, the square property at a scale of:

\[
10^{100}
\]

can also be browsed directly and interactively.

---

## 13. Why this step is closer to primes than residues are

Squares already have:

- sparsity;
- density that decreases with scale;
- gaps that increase with scale;
- no fixed period;
- local positions not fully determined by a short residue tuple.

Yet they are simpler than primes, because:

\[
\boxed{
\text{square membership}
\leftrightarrow
\text{integer sqrt boundary / equality}
}
\]

gives us an exact generator and summary.

The Square Layer is therefore a good control:

> If the future prime layer struggles to be exact at coarse scales, this does not mean DMSD has failed; it may simply mean that the prime property lacks a compressed summary operator like the square-root boundary.

---

## 14. Next round

We can now move on to:

# DMSD Phase 3A — Prime Layer Baseline

But the first round of the Prime Layer should be very restrained.

Only finite work with explicit certificate boundaries:

- exact prime count for materialized / computed blocks;
- scale density;
- normalized local positions;
- lazy refinement;
- computation-state labels;
- **Unknown blocks must not masquerade as exact summaries**.

The crucial difference will be:

```text
Parity / Residue / Square:
coarse block can be exact from compact rule.

Prime:
coarse block may require real computation or external certificate.
```

This is precisely the most worthwhile aspect of our DMSD to examine.

---

## 15. Conclusion

Phase 2C gives:

\[
\boxed{
S_{\square}(B)
=
\bigoplus_i S_{\square}(B_i)
}
\]

and:

\[
\boxed{
\text{non-periodic property}
+
\text{exact O(1) block summary}
}
\]

is indeed feasible.

The core of DMSD is therefore no longer merely:

\[
\text{lazy visual rendering}.
\]

but is gradually becoming:

\[
\boxed{
\text{multiscale mathematical state algebra}.
}
\]

---

```text
DMSD Phase 2C: COMPLETE
Validation: PASS

Next:
DMSD Phase 3A — Prime Layer Baseline
```