# BCODR-LUC-05 — Balanced Overlap Germ Taxonomy
## Face-Constrained 與 Interior-Midpoint First Contacts 的統一分類
### Root 127 vs Root 131: Different Tangent Dimensions, the Same Second-Order Isolation Mechanism

**文件代號：** `BCODR-LUC-05-v0.1`  
**版本：** v0.1  
**日期：** 2026-09-21  
**作者：** Neo.K  
**AI 協力：** Aletheia / ChatGPT, GPT-5.6 Sol  
**機構：** EveMissLab／一言諾科技有限公司  
**Target：** `ROOT6217_BARRIER_SHAPE`  
**前置：** BCODR-LUC-03/04  
**狀態：** `GERM TAXONOMY / SECOND-ORDER CLASSIFICATION`

---

# 0. Recenter

BCODR-LUC-03 identified two selected balanced common-point germs:

$$
\boxed{
127,\ 131.
}
$$

BCODR-LUC-04 analyzed root 127 and found:

- one active root face;
- a three-dimensional linear critical tangent space;
- positive combined second-order stiffness;
- a zero-dimensional overlap birth germ.

The next question is whether root 131 is the same geometry or a different germ class.

The answer is:

$$
\boxed{
\text{different first-order class, same second-order isolation outcome}.
}
$$

---

# 1. Root 127: Face-Constrained Balanced Germ

Root 127's earliest balanced overlap is attained on the lower:

$$
x_3
$$

face of the responsibility box.

Therefore the overlap geometry includes a domain normal in addition to the two seed fields.

Define this class:

$$
\boxed{
\textbf{FCBG}
=
\text{Face-Constrained Balanced Germ}.
}
$$

---

# 2. Root 131: Interior Midpoint Balanced Germ

Root 131 is structurally different.

Its true meeting point lies strictly inside the root-131 box in all five normalized coordinates.

No root-box face is active.

The meeting point is:

$$
\boxed{
z_{131}^\star
=
\frac12
\left(
p_{131}^\star
+
q_{155}^\star
\right),
}
$$

where:

- $p_{131}^\star$ is the nearest Paper-09B analytic-ray point;
- $q_{155}^\star$ is its nearest point in prune 155.

Define this class:

$$
\boxed{
\textbf{IMBG}
=
\text{Interior Midpoint Balanced Germ}.
}
$$

---

# 3. Root 131 True Radius

The nearest ray–prune pair occurs at:

$$
\boxed{
\rho_{131}^\star
=
0.00254068891729192.
}
$$

The seed-set distance is:

$$
2T_{131}^\star.
$$

Therefore:

$$
\boxed{
T_{131}^\star
=
0.0473333498919449.
}
$$

This reproduces the BCODR-LUC-03 true common-point overlap time.

---

# 4. Midpoint Theorem

Let:

$$
S_+,\ S_-
$$

be two closed seed sets.

Assume:

1. $p^\star\in S_+$ and $q^\star\in S_-$ realize the local nearest-pair distance;
2. the midpoint:
   $$
   m^\star=\frac{p^\star+q^\star}{2}
   $$
   lies in the interior of the responsibility domain.

Then the first equal-radius overlap satisfies:

$$
\boxed{
T^\star
=
\frac12
\|p^\star-q^\star\|.
}
$$

and one common first-contact point is:

$$
\boxed{
m^\star.
}
$$

This is the canonical IMBG geometry.

---

# 5. Proof

For any common overlap point:

$$
z,
$$

with radius:

$$
T,
$$

the triangle inequality gives:

$$
\|p^\star-q^\star\|
\le
\|p^\star-z\|
+
\|z-q^\star\|
\le
2T.
$$

Hence:

$$
T\ge
\frac12\|p^\star-q^\star\|.
$$

At the midpoint:

$$
m^\star,
$$

both distances equal:

$$
\frac12\|p^\star-q^\star\|.
$$

Therefore the bound is attained.

Q.E.D.

---

# 6. Root 131 Has No Domain Normal at Birth

The meeting point has positive margin to every root-131 face.

Therefore the active first-order geometry is generated only by the two seed-set distance fields.

At the balanced midpoint:

$$
g_-
=
-g_+.
$$

So the active seed normal rank is:

$$
\boxed{
1.
}
$$

---

# 7. Root 131 Linear Critical Tangent Dimension

In five dimensions, one independent active seed normal leaves:

$$
\boxed{
5-1=4
}
$$

linear critical tangent dimensions.

Thus:

$$
\boxed{
\dim T_{131}^{\rm crit}=4.
}
$$

This is one dimension larger than root 127.

---

# 8. Root 127 vs Root 131

## Root 127

```text
type:
    FCBG

active domain face:
    yes

active normal rank:
    2

linear critical tangent dimension:
    3
```

## Root 131

```text
type:
    IMBG

active domain face:
    no

active seed normal rank:
    1

linear critical tangent dimension:
    4
```

So the two roots are genuinely different first-order contact classes.

---

# 9. Second-Order Question

A higher tangent dimension does not mean a higher-dimensional overlap birth set.

The decisive quantity is still:

$$
\boxed{
\kappa_{\rm germ}
=
\min_{\substack{v\in T_{\rm crit}\\ \|v\|=1}}
\max
\{
v^TH_+v,
v^TH_-v
\}.
}
$$

If:

$$
\kappa_{\rm germ}>0,
$$

all nonzero first-order tangent motions incur positive second-order common-overlap cost.

---

# 10. Root 131 Reduced Hessians

The inward prune-155 active set is again:

$$
(x_3,\rho_5,y_5),
$$

while:

$$
y_3,\ x_5
$$

remain locally interior.

Thus:

$$
\boxed{
H_-^{131}
=
\operatorname{diag}(1,0,1,0,1).
}
$$

The outward Hessian is obtained by eliminating the analytic-ray parameter via the same Schur-complement construction used for root 127.

---

# 11. Root 131 Second-Order Stiffness

The resulting combined stiffness is:

$$
\boxed{
\kappa_{131}
=
0.454649807778291.
}
$$

At the softest tangent direction:

$$
\boxed{
Q_+^{131}
\approx
0.454649807778291,
}
$$

$$
\boxed{
Q_-^{131}
\approx
0.454649807778291.
}
$$

The two sides are almost exactly balanced.

---

# 12. Root 131 Birth Dimension

Because:

$$
\boxed{
\kappa_{131}>0,
}
$$

every nonzero vector in the four-dimensional linear critical tangent space has positive combined second-order cost.

Therefore:

$$
\boxed{
\dim_{\rm birth}^{(2)}(131)=0.
}
$$

So the earliest IMBG is also an isolated birth point.

---

# 13. Comparison with Root 127

Root 127 had:

$$
\boxed{
\kappa_{127}
=
0.45429481027797.
}
$$

Root 131 has:

$$
\boxed{
\kappa_{131}
=
0.454649807778291.
}
$$

The ratio is:

$$
\boxed{
\frac{\kappa_{131}}{\kappa_{127}}
=
1.00078142539226.
}
$$

They differ by less than one tenth of one percent.

---

# 14. Same Second-Order Mechanism, Different First-Order Geometry

This is the central classification result.

## Root 127

A responsibility-cell face removes one first-order degree of freedom.

## Root 131

No domain face is active, so one additional first-order tangent survives.

But in both cases:

$$
\boxed{
\text{second-order seed curvatures complement each other strongly enough to isolate the first contact}.
}
$$

---

# 15. Post-Birth Radius Cost

For a unit critical tangent displacement $h$:

$$
T
-
T^\star
\gtrsim
\frac{
\kappa_{\rm germ}
}{
2T^\star
}
h^2.
$$

Define:

$$
\boxed{
C_R
=
\frac{
\kappa_{\rm germ}
}{
2T^\star
}.
}
$$

---

# 16. Root 127 Radius Cost

$$
\boxed{
C_R^{127}
=
1.78921498757354.
}
$$

---

# 17. Root 131 Radius Cost

$$
\boxed{
C_R^{131}
=
4.80263713445372.
}
$$

Root 131 has the larger coefficient because its first-contact radius is much smaller.

Therefore for the same normalized tangent displacement:

$$
\boxed{
131\text{ requires a larger relative radius increase than }127.
}
$$

---

# 18. Post-Birth Tangent Extent

At radius:

$$
T^\star+\varepsilon,
$$

the soft tangent extent scales as:

$$
|h|
\lesssim
C_H
\sqrt{\varepsilon},
$$

where:

$$
\boxed{
C_H
=
\sqrt{
\frac{
2T^\star
}{
\kappa_{\rm germ}
}
}.
}
$$

---

# 19. Root 127

$$
\boxed{
C_H^{127}
=
0.747599042164318.
}
$$

## Root 131

$$
\boxed{
C_H^{131}
=
0.456310132719225.
}
$$

Thus root 131's nascent overlap lens grows more narrowly in normalized tangent coordinates.

---

# 20. Germ Taxonomy

BCODR can now distinguish at least:

## Type I — IMBG

$$
\boxed{
\text{Interior Midpoint Balanced Germ}
}
$$

Properties:

- no responsibility-domain face active;
- midpoint of a nearest seed pair is admissible;
- first active seed normal rank generically one;
- high linear tangent dimension;
- birth dimension determined by combined second-order stiffness.

Example:

$$
131.
$$

---

## Type II — FCBG

$$
\boxed{
\text{Face-Constrained Balanced Germ}
}
$$

Properties:

- first contact is forced onto one or more active responsibility-cell faces;
- domain normals reduce linear tangent freedom;
- first-contact point need not be the unconstrained seed midpoint;
- second-order birth dimension again determined by combined stiffness.

Example:

$$
127.
$$

---

# 21. Future Types

Additional classes may appear:

## Edge-Constrained Germ

two independent domain faces active.

## Corner-Constrained Germ

several domain faces active.

## Provenance-Degenerate Germ

multiple inward seed families attain the same true overlap time.

## Flat Germ

$$
\kappa_{\rm germ}=0.
$$

Requires higher-order analysis.

## Unstable Germ

$$
\kappa_{\rm germ}<0
$$

in an appropriately defined active-set second-order model.

Would signal a local escape/counter-geometry.

---

# 22. General Birth Classification Rule

For a balanced common-point germ:

1. compute the active normal rank;
2. obtain:
   $$
   T_{\rm crit};
   $$
3. compute reduced second-order forms:
   $$
   Q_+,\ Q_-;
   $$
4. evaluate:
   $$
   \kappa_{\rm germ}.
   $$

Then:

## $\kappa_{\rm germ}>0$

$$
\boxed{
\text{second-order isolated birth}.
}
$$

## $\kappa_{\rm germ}=0$

$$
\boxed{
\text{flat birth candidate; continue to higher order}.
}
$$

## $\kappa_{\rm germ}<0$

indicates that the assumed local active-set minimum needs re-examination.

---

# 23. What BCODR Has Accomplished So Far

The original shape responsibility began as:

$$
195
$$

root boxes.

The selected local line has now become:

$$
\boxed{
\text{coarse box}
\rightarrow
\text{true common overlap}
\rightarrow
\text{germ type}
\rightarrow
\text{critical tangent space}
\rightarrow
\text{second-order birth class}.
}
$$

For 127 and 131:

$$
\boxed{
\text{both birth germs are isolated}.
}
$$

---

# 24. Consequence for Paper 09C

Neither 127 nor 131 currently presents a zero-cost BCODR tangent direction that requires shape-area $\Phi_2$ merely to resolve overlap flatness.

Therefore Paper 09C should not be resumed simply because it was next in the old sequence.

It should return only if a surviving BCODR germ has:

$$
\boxed{
\kappa_{\rm germ}=0
}
$$

or if the actual Lebesgue shape objective requires an independent second-order check after BCODR compression.

---

# 25. Next BCODR Question

The most interesting next target is no longer 127 or 131 individually.

It is:

$$
\boxed{
\textbf{Are the remaining outward-limited frontier cells structurally disposable by }A_{\rm ref}\textbf{ certificates?}
}
$$

If the tail interaction classes can all be closed by finite certificates, then the BCODR local overlap spine may collapse almost entirely to already-classified isolated germs.

That would be a much stronger global compression.

---

# 26. Suggested Next Node

$$
\boxed{
\textbf{BCODR-LUC-06 — Frontier Spine Closure Against the }A_{\rm ref}\textbf{ Tail}
}
$$

Goals:

1. route outward-limited cells to the finite certificate lane;
2. keep only balanced or provenance-degenerate germs;
3. merge germ classes by topology type, not raw root count;
4. update the root-6217 surviving responsibility spine.

---

# 27. Conclusion

Root 127 and root 131 are not the same first-contact geometry.

Root 127:

$$
\boxed{
\text{FCBG},
\quad
\dim T_{\rm crit}=3.
}
$$

Root 131:

$$
\boxed{
\text{IMBG},
\quad
\dim T_{\rm crit}=4.
}
$$

Yet both satisfy:

$$
\boxed{
\kappa_{\rm germ}\approx0.454>0.
}
$$

and therefore both have:

$$
\boxed{
\text{second-order birth dimension}=0.
}
$$

So BCODR has discovered a stronger structural statement:

> Different first-order contact topologies can collapse to the same second-order isolation class.

This is the beginning of a reusable overlap-germ taxonomy rather than a root-specific calculation.

---

# Canonical Source Declaration

本檔為 `BCODR-LUC-05-v0.1` 的 UTF-8 Markdown canonical source。

數學 delimiter 僅使用 `$...$` 與 `$$...$$`。

本篇 formalizes:

- IMBG；
- FCBG；
- linear critical tangent dimension；
- second-order birth dimension；
- combined germ stiffness；
- post-birth lens growth。

目前數值微分幾何已 fresh replay；interval certification remains open.

===END===