# TCDC-15
## Joint Buchstab Transfer / Parity-Defect Decomposition
### First-Order Liouville Defect、Second-Order Pair Correlation、Capacity-as-Correlation Lower Bound、Counterexample Parity-Reversal Budget 與 External Provider Audit

**母方法論：** TCDC — Target–Condition Dual Closure  
**對接研究線：** RG-SCP / SRBRD  
**輪次：** TCDC-15  
**日期：** 2026-10-02  
**版本：** v0.1  
**作者：** Neo.K；Aletheia（GPT-5.6 Sol）  
**性質：** Parity-Defect Kernel / Liouville Correlation Decomposition / Capacity Substitute  
**前置：** TCDC-00 ～ TCDC-14；SRBRD-00 ～ SRBRD-26  
**狀態：** 將 joint cube-root survivor 的 $2\times2$ transport matrix完整參數化為一階 Liouville mean與二階 Liouville pair correlation；建立相對於 Buchstab $1:\log2$ benchmark 的 parity-defect vector；證明 mixed-capacity upper等價於二階 correlation lower bound；建立 counterexample parity-reversal budget；外部文獻 audit顯示現有大模數分布與 logarithmic Chowla 型結果均非直接 provider；Goldbach仍未證明

---

# 0. 本輪問題

TCDC-14 得到 one-variable cube-root RootRough branch benchmark：

$$
\boxed{
P_1:P_2
=
1:\log2.
}
$$

normalize：

$$
\boxed{
p_*
=
\frac1{1+\log2},
}
$$

$$
\boxed{
s_*
=
\frac{\log2}{1+\log2}.
}
$$

numerically：

$$
\boxed{
p_*
\approx
0.590616109150,
}
$$

$$
\boxed{
s_*
\approx
0.409383890850.
}
$$

但 joint Goldbach survivor不一定繼承此 ratio。

TCDC-15不再要求直接證：

$$
\theta_m=p_*+o(1).
$$

而是問：

> joint survivor偏離 Buchstab branch benchmark多少？
> 這個偏離需要壞到什麼程度才可能支援 Goldbach failure？

---

# 1. Joint Cube-Root Survivor

令：

$$
\boxed{
\widehat{\mathcal S}_z(m)
}
$$

為 ordered states：

$$
x+y=2m,
$$

with：

$$
\operatorname{RootRough}_z(x),
$$

$$
\operatorname{RootRough}_z(y),
$$

and：

$$
z^3\ge2m-3.
$$

所以：

$$
x,y
$$

各自只能是：

- prime；
- semiprime。

總 mass：

$$
\boxed{
S_m
=
|\widehat{\mathcal S}_z(m)|.
}
$$

---

# 2. Transport Matrix

ordered branch matrix：

$$
\boxed{
M
=
\begin{pmatrix}
M_{PP}&M_{PS}\\
M_{PS}&M_{SS}
\end{pmatrix}.
}
$$

symmetry：

$$
M_{PS}=M_{SP}.
$$

total：

## Theorem TCDC-15-T01

$$
\boxed{
S_m
=
M_{PP}
+
2M_{PS}
+
M_{SS}.
}
$$

---

# 3. Liouville Coding

在 cube-root boundary：

### Prime

$$
\lambda_{\mathrm L}(x)=-1.
$$

### Semiprime

$$
\lambda_{\mathrm L}(x)=+1.
$$

所以 branch type可由：

$$
\boxed{
\lambda_{\mathrm L}
}
$$

完全編碼。

---

# 4. First-Order Joint Liouville Mean

定義：

## Definition TCDC-15-D01

$$
\boxed{
\Lambda_m
=
\frac1{S_m}
\sum_{(x,y)\in\widehat{\mathcal S}_z(m)}
\lambda_{\mathrm L}(x).
}
$$

由 symmetry，

右端 mean相同。

---

# 5. First-Order Mean Equals Coarse Branch Imbalance

prime left marginal：

$$
n_0
=
M_{PP}+M_{PS}.
$$

semiprime left marginal：

$$
n_{SP}
=
M_{PS}+M_{SS}.
$$

所以：

## Theorem TCDC-15-T02

$$
\boxed{
\Lambda_m
=
\frac{
n_{SP}-n_0
}{
S_m
}.
}
$$

equivalently：

$$
\boxed{
\theta_m
=
\frac{n_0}{S_m}
=
\frac{
1-\Lambda_m
}{2}.
}
$$

---

# 6. Second-Order Liouville Pair Correlation

定義：

## Definition TCDC-15-D02

$$
\boxed{
\mathcal C_m
=
\frac1{S_m}
\sum_{(x,y)\in\widehat{\mathcal S}_z(m)}
\lambda_{\mathrm L}(x)
\lambda_{\mathrm L}(y).
}
$$

branch signs：

- $PP$：
  $+1$；
- $PS/SP$：
  $-1$；
- $SS$：
  $+1$。

所以：

## Theorem TCDC-15-T03

$$
\boxed{
\mathcal C_m
=
\frac{
M_{PP}
-
2M_{PS}
+
M_{SS}
}{
S_m
}.
}
$$

---

# 7. Complete Matrix Reconstruction

prime indicator on the rough boundary：

$$
\boxed{
\mathbf1_P(x)
=
\frac{
1-\lambda_{\mathrm L}(x)
}{2}.
}
$$

semiprime indicator：

$$
\boxed{
\mathbf1_{SP}(x)
=
\frac{
1+\lambda_{\mathrm L}(x)
}{2}.
}
$$

therefore：

## Theorem TCDC-15-T04
### Liouville Matrix Reconstruction

$$
\boxed{
\frac{
M_{PP}
}{
S_m
}
=
\frac{
1
-
2\Lambda_m
+
\mathcal C_m
}{4},
}
$$

$$
\boxed{
\frac{
M_{PS}
}{
S_m
}
=
\frac{
1-\mathcal C_m
}{4},
}
$$

$$
\boxed{
\frac{
M_{SS}
}{
S_m
}
=
\frac{
1
+
2\Lambda_m
+
\mathcal C_m
}{4}.
}
$$

所以整個 symmetric $2\times2$ transport matrix只需要兩個 scalar parameters：

$$
\boxed{
(\Lambda_m,\mathcal C_m).
}
$$

---

# 8. Feasible Parity Region

所有 matrix entries nonnegative。

T04給：

$$
1-2\Lambda+\mathcal C\ge0,
$$

$$
1-\mathcal C\ge0,
$$

$$
1+2\Lambda+\mathcal C\ge0.
$$

所以：

## Theorem TCDC-15-T05

feasible region：

$$
\boxed{
-1\le\Lambda_m\le1,
}
$$

$$
\boxed{
-1+2|\Lambda_m|
\le
\mathcal C_m
\le1.
}
$$

這是一個 V-shaped lower boundary的 parity polygon。

---

# 9. Goldbach Failure Line

Goldbach failure：

$$
M_{PP}=0.
$$

T04 gives：

## Theorem TCDC-15-T06
### Failure Parity Line

$$
\boxed{
\mathcal C_m
=
2\Lambda_m-1.
}
$$

且由 feasible region：

$$
\boxed{
\Lambda_m\ge0.
}
$$

所以任何 failure必須位於 feasible region右半部的 lower boundary。

---

# 10. Immediate Coarse Consequence

T06 again gives：

## Corollary TCDC-15-C01

$$
\boxed{
\Lambda_m<0
\Longrightarrow
M_{PP}>0.
}
$$

這就是：

$$
n_0>n_{SP}.
$$

但現在可視為：

> failure line根本進不了 negative-$\Lambda$ half-plane。

---

# 11. Buchstab Benchmark Liouville Mean

TCDC-14 one-variable branch vector：

$$
(p_*,s_*).
$$

benchmark Liouville mean：

$$
\boxed{
\Lambda_*
=
s_*-p_*.
}
$$

因此：

## Theorem TCDC-15-T07

$$
\boxed{
\Lambda_*
=
\frac{
\log2-1
}{
\log2+1
}
=
-\delta_*,
}
$$

其中：

$$
\boxed{
\delta_*
=
\frac{
1-\log2
}{
1+\log2
}
\approx
0.181232218299.
}
$$

所以 one-variable benchmark位於：

$$
\boxed{
\Lambda_*
\approx
-18.1232\%.
}
$$

---

# 12. Buchstab Benchmark Pair Correlation

若 joint branch matrix為 rank-one independent benchmark：

$$
b_*b_*^\top,
$$

則：

$$
\boxed{
\mathcal C_*
=
\Lambda_*^2.
}
$$

numerically：

$$
\boxed{
\mathcal C_*
\approx
0.032845116950.
}
$$

這是 benchmark，

不是 joint theorem。

---

# 13. Parity-Defect Vector

定義：

## Definition TCDC-15-D03

first-order defect：

$$
\boxed{
E_{1,m}
=
\Lambda_m-\Lambda_*.
}
$$

second-order defect：

$$
\boxed{
E_{2,m}
=
\mathcal C_m-\mathcal C_*.
}
$$

所以：

$$
\boxed{
(E_{1,m},E_{2,m})
}
$$

是 joint survivor相對 rank-one Buchstab benchmark的 parity-defect vector。

---

# 14. Exact Defect Decomposition of $PP$

benchmark：

$$
\boxed{
p_*^2
=
\frac{
1-2\Lambda_*+\mathcal C_*
}{4}.
}
$$

T04 subtract benchmark：

## Theorem TCDC-15-T08

$$
\boxed{
\frac{
M_{PP}
}{
S_m
}
=
p_*^2
-
\frac12E_{1,m}
+
\frac14E_{2,m}.
}
$$

---

# 15. Exact Defect Decomposition of Mixed Branch

similarly：

## Theorem TCDC-15-T09

$$
\boxed{
\frac{
M_{PS}
}{
S_m
}
=
p_*s_*
-
\frac14E_{2,m}.
}
$$

所以 mixed branch完全不依賴 first-order defect：

$$
E_1.
$$

它只由：

$$
\boxed{
E_2
}
$$

控制。

這是本輪關鍵結構。

---

# 16. Exact Defect Decomposition of $SS$

## Theorem TCDC-15-T10

$$
\boxed{
\frac{
M_{SS}
}{
S_m
}
=
s_*^2
+
\frac12E_{1,m}
+
\frac14E_{2,m}.
}
$$

所以：

- $E_1$：
  在兩個 diagonal branches間搬 mass；
- $E_2$：
  在 mixed vs same-parity branches間搬 mass。

---

# 17. Interpretation of $E_1$

$$
E_1>0
$$

表示：

$$
\Lambda_m
$$

比 Buchstab benchmark更偏向 semiprime branch。

這會：

- decrease $PP$；
- increase $SS$；

each by：

$$
E_1/2.
$$

所以：

# First-Order Parity Defect

是：

# Prime-vs-Semiprime Marginal Shift

---

# 18. Interpretation of $E_2$

$$
E_2>0
$$

表示 same-parity coupling：

- $PP$；
- $SS$；

相對 mixed branch增加。

因 T09：

$$
M_{PS}/S
=
p_*s_*-\frac14E_2.
$$

所以：

# Second-Order Parity Defect

是：

# Mixed-vs-Diagonal Correlation Shift

---

# 19. Failure Constraint in Defect Coordinates

failure：

$$
M_{PP}=0.
$$

T08：

## Theorem TCDC-15-T11

$$
\boxed{
E_{2,m}
=
2E_{1,m}
-
4p_*^2.
}
$$

numerically：

$$
\boxed{
E_2
=
2E_1
-
1.395309553548.
}
$$

所以 counterexample必須落在 parity-defect plane的一條 affine line上。

---

# 20. First-Order Reversal Is Mandatory

failure requires：

$$
\Lambda_m\ge0.
$$

而：

$$
\Lambda_*=-\delta_*.
$$

所以：

## Theorem TCDC-15-T12

任何 failure必須：

$$
\boxed{
E_{1,m}
\ge
\delta_*.
}
$$

即：

$$
\boxed{
E_{1,m}
\ge
0.181232218299.
}
$$

所以 joint first-order branch parity至少必須相對 one-variable benchmark翻轉：

$$
\boxed{
18.1232\%
}
$$

normalized mean。

---

# 21. Capacity Reserve

TCDC-13：

$$
M_{PP}
\ge
\Delta_0
+
\Gamma_{\mathrm{row}}
+
G_{\mathrm{mir}}^*.
$$

divide by：

$$
S_m.
$$

定義：

## Definition TCDC-15-D04

$$
\boxed{
\gamma_m
=
\frac{
\Gamma_{\mathrm{row}}
}{
S_m
},
}
$$

$$
\boxed{
g_m
=
\frac{
G_{\mathrm{mir}}^*
}{
S_m
}.
}
$$

capacity reserve：

$$
\boxed{
\mathcal R_m
=
\gamma_m+g_m.
}
$$

---

# 22. Goldbach Lower in Liouville Form

因：

$$
\Delta_0/S_m
=
-\Lambda_m,
$$

TCDC-13 bound變：

## Theorem TCDC-15-T13

$$
\boxed{
\frac{
M_{PP}
}{
S_m
}
\ge
-\Lambda_m
+
\mathcal R_m.
}
$$

---

# 23. Counterexample Parity-Reversal Budget

failure需要：

$$
M_{PP}=0.
$$

T13 implies：

$$
\Lambda_m\ge\mathcal R_m.
$$

subtract：

$$
\Lambda_*=-\delta_*.
$$

得到：

## Theorem TCDC-15-T14
### Counterexample Parity-Reversal Theorem

任何 Goldbach failure必須：

$$
\boxed{
E_{1,m}
\ge
\delta_*
+
\mathcal R_m.
}
$$

即：

$$
\boxed{
E_{1,m}
\ge
0.181232218299
+
\gamma_m
+
g_m.
}
$$

---

# 24. Meaning of T14

one-variable Buchstab benchmark：

$$
\Lambda_*
\approx-0.181232.
$$

failure不只需要把它推到：

$$
0.
$$

若 capacity reserve positive，

還必須推到：

$$
\boxed{
\Lambda_m
\ge
\gamma_m+g_m.
}
$$

所以：

# Capacity Forces a Stronger Parity Reversal

這正是：

# Target–Condition Dual Closure

的 failure-rigidity版本。

---

# 25. Parity-Defect Budget Form

T14 contrapositive：

## Corollary TCDC-15-C02

如果：

$$
\boxed{
E_{1,m}
<
\delta_*
+
\mathcal R_m,
}
$$

則：

$$
\boxed{
M_{PP}>0.
}
$$

因此我們不必證：

$$
E_1=o(1).
$$

只需阻止 defect跨過：

$$
\boxed{
18.1232\%
+
\text{capacity reserve}.
}
$$

---

# 26. Capacity as Second-Order Correlation Control

T09：

$$
\frac{
M_{PS}
}{
S_m
}
=
\frac{
1-\mathcal C_m
}{4}.
$$

若有 proof-admissible mixed upper：

$$
\boxed{
M_{PS}\le U_{mix},
}
$$

define：

$$
\boxed{
\mu_m
=
\frac{
U_{mix}
}{
S_m
}.
}
$$

Then：

## Theorem TCDC-15-T15

$$
\boxed{
\mathcal C_m
\ge
1-4\mu_m.
}
$$

所以 mixed-capacity theorem等價於：

# Second-Order Liouville Correlation Lower Bound

---

# 27. Recover the Capacity Closure

T04 + T15：

$$
\frac{
M_{PP}
}{
S_m
}
\ge
\frac{
1-2\Lambda_m+1-4\mu_m
}{4}.
$$

所以：

## Theorem TCDC-15-T16

$$
\boxed{
\frac{
M_{PP}
}{
S_m
}
\ge
\frac{
1-\Lambda_m
}{2}
-
\mu_m.
}
$$

但：

$$
\frac{
1-\Lambda_m
}{2}
=
\theta_m.
$$

因此：

$$
\boxed{
\frac{
M_{PP}
}{
S_m
}
\ge
\theta_m-\mu_m.
}
$$

exactly recovering TCDC-14 Route B。

---

# 28. Unified Interpretation

Route A：

$$
\Lambda_m<0
$$

controls：

# First-Order Parity

Route B：

$$
\mathcal C_m
$$

lower bound from mixed-capacity upper controls：

# Second-Order Parity Correlation

所以 TCDC不是只在「避開 parity」。

它是：

$$
\boxed{
\text{用 transport capacity去替代直接估 second-order parity correlation}.
}
$$

---

# 29. Exact Goldbach Positivity Region

T04：

$$
M_{PP}>0
$$

iff：

## Theorem TCDC-15-T17

$$
\boxed{
\mathcal C_m
>
2\Lambda_m-1.
}
$$

failure line：

$$
\mathcal C=2\Lambda-1.
$$

因此任何 theorem只要把：

$$
(\Lambda_m,\mathcal C_m)
$$

推到此 line上方即可。

---

# 30. Capacity Lower Boundary in the Parity Plane

mixed upper：

$$
\mu_m
$$

gives：

$$
\mathcal C_m\ge1-4\mu_m.
$$

combine T17：

sufficient：

$$
1-4\mu_m
>
2\Lambda_m-1.
$$

equivalent：

## Corollary TCDC-15-C03

$$
\boxed{
\mu_m
<
\frac{
1-\Lambda_m
}{2}
=
\theta_m.
}
$$

again the capacity criterion。

geometrically：

> capacity theorem提供一條 horizontal lower barrier，
> 把 joint point推離 failure line。

---

# 31. Rank-One Benchmark Location

benchmark：

$$
\boxed{
(\Lambda_*,\mathcal C_*)
=
\left(
-\delta_*,
\delta_*^2
\right).
}
$$

numerically：

$$
\boxed{
\Lambda_*
\approx
-0.181232218299,
}
$$

$$
\boxed{
\mathcal C_*
\approx
0.032845116950.
}
$$

此 point遠離 failure half-plane：

$$
\Lambda\ge0.
$$

所以 counterexample需要巨大 structural deformation。

---

# 32. Distance-to-Failure in the First Coordinate

minimum horizontal move from benchmark到 failure-eligible region：

$$
\boxed{
\delta_*
\approx
0.181232218299.
}
$$

capacity reserve再把必要 move增加為：

$$
\boxed{
\delta_*+\mathcal R_m.
}
$$

這是最簡單的 TCDC defect distance。

---

# 33. Pair-Defect Cost at Failure

T11：

$$
E_2
=
2E_1-4p_*^2.
$$

at the minimum first-order reversal：

$$
E_1=\delta_*,
$$

failure would require：

$$
\boxed{
E_2
=
2\delta_*
-
4p_*^2.
}
$$

numerically：

$$
\boxed{
E_2
\approx
-1.032845116950.
}
$$

which corresponds：

$$
\mathcal C=-1
$$

at：

$$
\Lambda=0.
$$

所以最早可能碰到 failure line的位置，

需要：

# Maximal Mixed Anticorrelation

$$
\boxed{
\mathcal C=-1.
}
$$

這是非常 rigid 的 pair structure。

---

# 34. Failure Tradeoff

若：

$$
\Lambda_m
$$

往 semiprime side繼續增加，

failure required pair correlation：

$$
\mathcal C=2\Lambda-1
$$

也提高。

所以 failure有 tradeoff：

### Small first-order reversal

需要 extreme negative pair correlation。

### Large first-order reversal

pair correlation requirement較溫和。

這提供不同 proof technologies兩個 attack directions。

---

# 35. Attack Direction A — Bound First-Order Defect

證：

$$
\boxed{
E_{1,m}
<
\delta_*+\mathcal R_m.
}
$$

即可 closure。

這需要控制：

$$
\Lambda_m
$$

即：

# Liouville-Twisted Joint Rough Survivor Mean

---

# 36. Attack Direction B — Bound Pair Correlation

即使：

$$
E_1
$$

很大，

只要證：

$$
\boxed{
\mathcal C_m
>
2\Lambda_m-1,
}
$$

仍 closure。

mixed-capacity upper就是一種間接方式。

另一種可能是直接 Liouville pair-correlation theorem。

---

# 37. Neutral / Twisted Decomposition

定義 neutral sum：

$$
\boxed{
\mathcal N_m
=
\sum_{(x,y)}
1_{\mathrm{RR}_z}(x)
1_{\mathrm{RR}_z}(y)
=
S_m.
}
$$

first twisted：

$$
\boxed{
\mathcal T_{1,m}
=
\sum_{(x,y)}
\lambda(x)
1_{\mathrm{RR}_z}(x)
1_{\mathrm{RR}_z}(y).
}
$$

second twisted：

$$
\boxed{
\mathcal T_{2,m}
=
\sum_{(x,y)}
\lambda(x)\lambda(y)
1_{\mathrm{RR}_z}(x)
1_{\mathrm{RR}_z}(y).
}
$$

then：

$$
\boxed{
\Lambda_m
=
\mathcal T_{1,m}/\mathcal N_m,
}
$$

$$
\boxed{
\mathcal C_m
=
\mathcal T_{2,m}/\mathcal N_m.
}
$$

---

# 38. Provider Separation

### Neutral provider

needs：

$$
\mathcal N_m.
$$

ordinary sieve / convolution technology is structurally suitable。

### First parity provider

needs：

$$
\mathcal T_{1,m}.
$$

contains one Liouville sign。

### Second parity provider

needs：

$$
\mathcal T_{2,m}.
$$

contains two Liouville signs。

所以 parity problem被拆成：

# Neutral + One-Twist + Two-Twist

三個 distinct analytic objects。

---

# 39. Why Maynard-Type Large-Moduli Results Help Neutral Terms

modern large-moduli distribution results for primes provide：

- mean distribution；
- well-factorable weights；
- uniform residue classes；

which can improve：

- neutral rough survivor counts；
- mixed upper capacities；
- convolution remainder terms。

但它們本身不直接估：

$$
\lambda(n)
$$

twisted signs。

所以：

## Research Firewall TCDC-15-R01

$$
\boxed{
\text{large-moduli prime distribution technology is primarily a neutral/capacity provider,
not automatically a parity-defect provider}.
}
$$

---

# 40. Modern Liouville-Correlation Results

modern Chowla-type work proves cancellation for certain Liouville correlations。

A 2026 result gives quantitative logarithmically weighted two-point Liouville correlation bounds uniformly for shifts up to a small power of：

$$
\log x.
$$

This is strong evidence that Liouville correlations can be controlled in some growing-shift settings。

But the TCDC object is different：

$$
x+y=2m
$$

with：

- a fixed additive hyperplane；
- cube-root rough conditioning on both endpoints；
- unweighted / counting normalization；
- shifts / coefficients at arithmetic scale comparable to the main variable。

所以：

## Research Firewall TCDC-15-R02

$$
\boxed{
\text{current logarithmic Chowla-type results are not direct providers for }
\mathcal T_{1,m}
\text{ or }
\mathcal T_{2,m}.
}
$$

---

# 41. Why Arithmetic-Progression Liouville Sign Results Are Also Insufficient

recent work proves sign changes of：

$$
\lambda(n)
$$

inside arithmetic progressions at polynomial ranges。

This guarantees both parity signs occur。

But TCDC needs a quantitative weighted mean：

$$
\Lambda_m
$$

or pair correlation：

$$
\mathcal C_m
$$

inside a highly conditioned Goldbach survivor set。

So：

$$
\boxed{
\text{sign occurrence}
\neq
\text{branch proportion control}.
}
$$

---

# 42. Parity Defect as a Research Variable

instead of asking：

> solve parity problem,

define measurable defect：

$$
\boxed{
E_{1,m}.
}
$$

Goldbach only requires：

$$
E_{1,m}
<
\delta_*+\mathcal R_m
$$

through T14。

So any theorem giving：

$$
\boxed{
E_{1,m}\le c
}
$$

with：

$$
c<\delta_*+\mathcal R_m
$$

is sufficient。

This allows partial parity information to combine with capacity。

---

# 43. Capacity Buys Parity Error Tolerance

without capacity：

$$
\mathcal R_m=0.
$$

allowed defect：

$$
\boxed{
E_1<\delta_*
\approx
18.1232\%.
}
$$

with capacity：

$$
\mathcal R_m>0,
$$

allowed：

$$
\boxed{
E_1
<
18.1232\%
+
\mathcal R_m.
}
$$

所以每一單位 normalized forced internal mass，

exactly buys同量的 first-order parity-error tolerance。

---

# 44. Mirror Gain Is a Parity-Error Subsidy

TCDC-13：

$$
g_m
=
G_{\mathrm{mir}}^*/S_m.
$$

T14：

$$
E_1
<
\delta_*+\gamma_m+g_m.
$$

所以 mirror critical-layer correction可理解為：

# Parity-Error Subsidy

它不需要直接估 Liouville sign，

但讓 Liouville defect可以壞得更多仍不失 closure。

---

# 45. Bulk Capacity Is a Main-Order Parity Subsidy

若：

$$
\gamma_m
$$

有 positive main-order constant，

則 parity defect tolerance同樣增加 main-order constant。

所以：

> bulk transport不是 only alternative to parity；
> 它直接擴大可容忍的 parity-defect region。

---

# 46. Failure Rigidity Region

define：

$$
\boxed{
\mathcal F_{\mathrm{def}}
=
\left\{
(E_1,E_2):
E_1\ge\delta_*+\mathcal R_m,
\quad
E_2=2E_1-4p_*^2
\right\}
}
$$

for exact failure line plus capacity first-coordinate reserve。

actual failure must enter this very thin affine region。

This is TCDC failure-fixed-surface language in parity coordinates。

---

# 47. A More Flexible Outer Failure Region

if capacity lower is not exact，

we may only know：

$$
M_{SS}/S_m\ge\mathcal R_m.
$$

Then：

$$
\Lambda_m\ge\mathcal R_m
$$

under failure。

So outer failure region：

$$
\boxed{
\Lambda\ge\mathcal R_m,
\qquad
\mathcal C=2\Lambda-1.
}
$$

or defect：

$$
\boxed{
E_1\ge\delta_*+\mathcal R_m.
}
$$

This is enough for TCDC elimination attempts。

---

# 48. Relative Defect Budget

define：

$$
\boxed{
B_{\mathrm{par}}
=
\delta_*+\mathcal R_m-E_{1,m}.
}
$$

Then：

## Theorem TCDC-15-T18

$$
\boxed{
B_{\mathrm{par}}>0
\Longrightarrow
M_{PP}>0.
}
$$

So：

# Positive Parity Budget

is a complete sufficient certificate。

---

# 49. Pair-Correlation Budget

define：

$$
\boxed{
B_{\mathrm{pair}}
=
\mathcal C_m
-
(2\Lambda_m-1).
}
$$

From T17：

## Theorem TCDC-15-T19

$$
\boxed{
\frac{
M_{PP}
}{
S_m
}
=
\frac{
B_{\mathrm{pair}}
}{4}.
}
$$

This is exact。

So Goldbach ordered density is literally one quarter of the distance above the failure line in the parity plane。

---

# 50. Capacity Approximation to the Pair Budget

if mixed upper：

$$
\mu_m,
$$

then：

$$
\mathcal C_m\ge1-4\mu_m.
$$

therefore：

## Corollary TCDC-15-C04

$$
\boxed{
B_{\mathrm{pair}}
\ge
2
-
2\Lambda_m
-
4\mu_m.
}
$$

divide by：

$$
4:
$$

$$
\boxed{
M_{PP}/S_m
\ge
\theta_m-\mu_m.
}
$$

again showing transport as lower approximation to pair-budget distance。

---

# 51. Research Hierarchy in Parity Coordinates

### Level 0 — Neutral

estimate：

$$
S_m.
$$

### Level 1 — First Twist

bound：

$$
E_1
$$

or：

$$
\Lambda_m.
$$

### Level 2 — Capacity Substitute

upper-bound mixed branch：

$$
M_{PS}.
$$

This lower-bounds：

$$
\mathcal C_m.
$$

### Level 3 — Direct Second Twist

if available，

estimate：

$$
\mathcal C_m
$$

directly。

### Level 4 — Mirror / LP Reserve

increase：

$$
\mathcal R_m.
$$

All contribute to separating actual point from failure line。

---

# 52. Which External Technology Fits Which Level

## Maynard large-moduli / well-factorable

best fit：

- Level 0；
- Level 2。

It gives prime distribution beyond classical $1/2$ barriers for structured moduli and uniform residue classes in certain settings。

## Classical / Chen weighted sieve

best fit：

- Level 0；
- Level 2；
- almost-prime branch-neutral lower。

## Chowla / Liouville correlation

conceptually targets：

- Level 1；
- Level 3。

But currently available forms do not match the TCDC fixed-sum rough-conditioned geometry directly。

## Failure LP / mirror transport

targets：

- Level 4。

---

# 53. A Hybrid Theorem Target

Rather than prove full branch transfer，

seek constants：

$$
\boxed{
E_{1,m}\le e_1,
}
$$

$$
\boxed{
\gamma_m+g_m\ge r_0.
}
$$

Then T14 says closure if：

## Theorem TCDC-15-T20

$$
\boxed{
e_1
<
\delta_*+r_0.
}
$$

This is a clean parity-capacity hybrid theorem。

---

# 54. Pair Hybrid Target

Alternatively seek：

$$
\Lambda_m\le\lambda_0,
$$

and mixed capacity：

$$
\mu_m\le\mu_0.
$$

Then T16 gives closure if：

## Theorem TCDC-15-T21

$$
\boxed{
\mu_0
<
\frac{
1-\lambda_0
}{2}.
}
$$

This may be easier than either：

- $\Lambda<0$；
- ideal mixed asymptotic；

alone。

---

# 55. Finite Benchmark Geometry

benchmark constants：

$$
\boxed{
p_*^2
\approx
0.348827388387,
}
$$

$$
\boxed{
p_*s_*
\approx
0.241788720763,
}
$$

$$
\boxed{
s_*^2
\approx
0.167595170088,
}
$$

$$
\boxed{
\Lambda_*
\approx
-0.181232218299,
}
$$

$$
\boxed{
\mathcal C_*
\approx
0.032845116950.
}
$$

So benchmark point lies well inside Goldbach-positive region。

---

# 56. Formal Theorem Queue

```yaml
TCDC_15:

  T01:
    joint_total_matrix_mass

  T02:
    first_Liouville_mean_equals_branch_imbalance

  T03:
    pair_Liouville_correlation_matrix_expression

  T04:
    complete_matrix_reconstruction

  T05:
    feasible_parity_region

  T06:
    Goldbach_failure_line

  T07:
    Buchstab_benchmark_Liouville_mean

  T08:
    PP_defect_decomposition

  T09:
    mixed_defect_decomposition

  T10:
    SS_defect_decomposition

  T11:
    failure_affine_line_in_defect_coordinates

  T12:
    mandatory_first_order_parity_reversal

  T13:
    capacity_reserve_Liouville_lower

  T14:
    counterexample_parity_reversal_budget

  T15:
    mixed_capacity_implies_pair_correlation_lower

  T16:
    capacity_closure_recovered

  T17:
    exact_Goldbach_positive_region

  T18:
    parity_budget_certificate

  T19:
    exact_pair_budget_identity

  T20:
    hybrid_parity_capacity_closure

  T21:
    hybrid_first_second_order_closure
```

---

# 57. Barrier Queue

```yaml
TCDC_15_Barriers:

  B01:
    first_twisted_joint_sum_not_controlled

  B02:
    second_twisted_joint_sum_not_controlled

  B03:
    modern_Chowla_results_do_not_match_fixed_sum_RootRough_geometry

  B04:
    large_moduli_prime_distribution_does_not_control_Liouville_sign

  B05:
    capacity_reserve_not_uniformly_large_enough

  B06:
    no_uniform_defect_budget_crossing

  B07:
    no_direct_joint_Buchstab_transfer
```

---

# 58. Acceptance Gate

TCDC-15 formal core前需：

1. cube-root boundary only prime / semiprime；
2. ordered matrix conventions；
3. Liouville coding sign correct；
4. first and second moments separated；
5. matrix reconstruction exact；
6. feasible parity region checked；
7. failure line checked；
8. Buchstab benchmark explicitly labeled benchmark, not joint theorem；
9. defect decompositions exact relative to benchmark；
10. capacity reserve normalized correctly；
11. mixed capacity to pair-correlation direction correct；
12. external technology audit does not overclaim direct applicability；
13. logarithmic Chowla not confused with unweighted fixed-sum theorem；
14. no Goldbach proof claim。

---

# 59. Verdict

本輪可以正式給：

$$
\boxed{
\mathrm{TCDC\mbox{-}15\ PARITY\mbox{-}DEFECT\ PLANE\ PASS}.
}
$$

最重要 exact parameterization：

$$
\boxed{
\frac{
M_{PP}
}{
S_m
}
=
\frac{
1-2\Lambda_m+\mathcal C_m
}{4},
}
$$

$$
\boxed{
\frac{
M_{PS}
}{
S_m
}
=
\frac{
1-\mathcal C_m
}{4},
}
$$

$$
\boxed{
\frac{
M_{SS}
}{
S_m
}
=
\frac{
1+2\Lambda_m+\mathcal C_m
}{4}.
}
$$

Goldbach failure line：

$$
\boxed{
\mathcal C_m
=
2\Lambda_m-1,
\qquad
\Lambda_m\ge0.
}
$$

Buchstab benchmark：

$$
\boxed{
\Lambda_*
=
-\frac{
1-\log2
}{
1+\log2
}
\approx
-0.181232218299.
}
$$

所以任何 counterexample至少需要：

$$
\boxed{
E_{1,m}
\ge
0.181232218299.
}
$$

TCDC capacity reserve進一步提高要求：

$$
\boxed{
E_{1,m}
\ge
0.181232218299
+
\gamma_m
+
g_m.
}
$$

所以反例必須把 one-variable prime-favoring parity bias完全翻轉，

還要額外跨過 transport / mirror reserve。

第二個核心結果：

$$
\boxed{
M_{PS}\le\mu S_m
\Longrightarrow
\mathcal C_m\ge1-4\mu.
}
$$

因此 transport capacity可以精確理解為：

# Second-Order Liouville Correlation Substitute

而不是只是 combinatorial trick。

這讓整個 TCDC proof program在 parity plane中合流：

$$
\boxed{
\text{First-Order Parity Control}
\;\lor\;
\text{Second-Order Capacity Control}
\;+\;
\text{Transport Reserve}
}
$$

只要把 joint point：

$$
(\Lambda_m,\mathcal C_m)
$$

留在 failure line：

$$
\mathcal C=2\Lambda-1
$$

上方，

Goldbach即成立。

目前 external audit顯示：

- modern large-moduli prime distribution對 neutral / capacity terms有幫助；
- modern Liouville / Chowla correlation結果在概念上碰 parity twist；
- 但現有已審查形狀都不是 fixed-sum + cube-root RootRough-conditioned joint survivor的 direct provider。

所以：

$$
\boxed{
\mathrm{NO\ UNIFORM\ PARITY\mbox{-}DEFECT\ BOUND\ YET}.
}
$$

$$
\boxed{
\mathrm{GOLDBACH\ STILL\ OPEN}.
}
$$

---

# 60. 下一輪建議

# TCDC-16
## Failure-Line Distance / Twisted-Sum Provider Decomposition

下一輪不再只說「需要 Liouville theorem」。

直接把：

$$
\mathcal T_{1,m},
\quad
\mathcal T_{2,m}
$$

做 Buchstab / divisor decomposition。

### Part A — First Twist

展開：

$$
\lambda(x)
$$

on cube-root rough states：

- prime contribution；
- semiprime contribution。

### Part B — Second Twist

展開：

$$
\lambda(x)\lambda(y)
$$

into：

- $PP$；
- mixed；
- $SS$ shells。

### Part C — Divisor-Sum Representation

嘗試把 twisted sums轉成：

- Möbius / Liouville divisor sums；
- bilinear prime sums；
- switched convolution。

### Part D — Provider Matching

逐 term檢：

- Maynard large-moduli；
- dispersion；
- Type II；
- logarithmic Chowla；
- pretentious bounds；

到底哪一個有合法 hypothesis match。

### Part E — Failure-Line Distance

建立：

$$
\boxed{
D_{\mathrm{fail}}
=
\mathcal C_m-(2\Lambda_m-1)
}
$$

的 lower-provider ledger。

因 exact：

$$
\boxed{
M_{PP}/S_m
=
D_{\mathrm{fail}}/4.
}
$$

所以只要某組 twisted / capacity providers令：

$$
D_{\mathrm{fail}}>0,
$$

就直接 closure。

這會把下一輪變成真正的：

# Parity Provider Engineering

而不是抽象 barrier discussion。

---

# 附錄 A：Parity Plane

```yaml
coordinates:

  Lambda:
    first_Liouville_mean

  C:
    second_Liouville_pair_correlation

matrix:

  PP:
    (1-2*Lambda+C)/4

  PS:
    (1-C)/4

  SS:
    (1+2*Lambda+C)/4

failure_line:
  C:
    2*Lambda-1

failure_requires:
  Lambda:
    >= 0
```

---

# 附錄 B：Buchstab Benchmark

```yaml
L:
  log(2)

p_star:
  1/(1+L)

s_star:
  L/(1+L)

Lambda_star:
  (L-1)/(L+1)

C_star:
  Lambda_star^2

delta_star:
  (1-L)/(1+L)

values:

  p_star:
    0.590616109150

  s_star:
    0.409383890850

  delta_star:
    0.181232218299

  C_star:
    0.032845116950
```

---

# 附錄 C：Defect Matrix

```yaml
E1:
  Lambda-Lambda_star

E2:
  C-C_star

PP_over_S:
  p_star^2
  -E1/2
  +E2/4

PS_over_S:
  p_star*s_star
  -E2/4

SS_over_S:
  s_star^2
  +E1/2
  +E2/4

failure:
  E2:
    2*E1-4*p_star^2
```

---

# 附錄 D：Parity-Reversal Budget

```yaml
capacity_reserve:
  R:
    Gamma_row/S
    + G_mirror/S

failure_requires:
  Lambda:
    >= R

equivalently:
  E1:
    >= delta_star+R

closure_if:
  E1:
    < delta_star+R
```

---

# 附錄 E：Capacity as Pair Correlation

```yaml
mixed_upper:
  M_PS/S:
    <= mu

then:
  C:
    >= 1-4*mu

Goldbach_lower:
  M_PP/S:
    >= theta-mu

interpretation:
  mixed_capacity:
    second_order_Liouville_lower_provider
```

---

# 附錄 F：Commit Decision

```yaml
research_round:
  TCDC-15

decision:

  parity_plane:
    formalized

  first_order_defect:
    formalized

  second_order_defect:
    formalized

  matrix_reconstruction:
    proved

  failure_line:
    proved

  counterexample_parity_reversal_budget:
    proved

  capacity_as_pair_correlation:
    proved

  external_direct_parity_provider:
    not_found

  Goldbach_proved:
    false

next:
  TCDC-16_FailureLineDistance_TwistedSum_Provider_Decomposition
```

---

# References / External Technology Audit

1. Terence Tao, **A general parity problem obstruction**, 2014.

   Relevant point:
   sieve problems involving exact prime-factor parity encounter a structural parity obstruction. The TCDC quantity $\Lambda_m$ is directly parity-sensitive because it distinguishes prime from semiprime branches inside a rough survivor set.

2. James Maynard, **Primes in arithmetic progressions to large moduli II: Well-factorable estimates**, *Memoirs of the American Mathematical Society* 306 (1543), 2025.

   Relevant point:
   mean distribution of primes with suitably well-factorable weights is available to moduli as large as $x^{3/5-\varepsilon}$. This is highly relevant to neutral and mixed-capacity providers, but does not itself estimate Liouville-twisted branch signs.

3. James Maynard, **Primes in arithmetic progressions to large moduli III: Uniform residue classes**, *Memoirs of the American Mathematical Society* 306 (1544), 2025.

   Relevant point:
   provides large-modulus mean value estimates uniform in residue classes for suitable structured moduli. Again, this is a distribution provider rather than a direct Liouville-parity theorem.

4. Jizhou Guo, **Quantitative Logarithmic Chowla Correlations Uniformly over Growing Shifts**, arXiv:2608.23500, 2026.

   Relevant point:
   gives quantitative logarithmically weighted two-point Liouville correlation estimates uniformly for shifts up to a small power of $\log x$. This is conceptually relevant to second-order parity control, but the theorem is logarithmically weighted and concerns growing shifts of a very different scale from the TCDC fixed-sum, rough-conditioned geometry.

5. Kevin Ford and Maksym Radziwiłł, **Sign changes of the Liouville function in arithmetic progressions**, arXiv:2605.03349, 2026.

   Relevant point:
   proves occurrence of both Liouville signs in arithmetic progressions at polynomial ranges. Sign occurrence alone does not provide the quantitative branch proportion or fixed-sum correlation needed for $\Lambda_m$ or $\mathcal C_m$.