# TCDC-19
## Shell-Provider Portfolio / $5/8$–Switching–Coupling Closure
### Legal-Shape Floor、Weight-Class Firewall、High-Shell Upgrade Threshold、Harmonic Portfolio Ledger 與 Switching Target

**母方法論：** TCDC — Target–Condition Dual Closure  
**對接研究線：** RG-SCP / SRBRD  
**輪次：** TCDC-19  
**日期：** 2026-10-02  
**版本：** v0.1  
**作者：** Neo.K；Aletheia（GPT-5.6 Sol）  
**性質：** Shell Provider Portfolio / Theorem-Hypothesis Ledger / Harmonic-Level Synthesis  
**前置：** TCDC-00 ～ TCDC-18；SRBRD-00 ～ SRBRD-26  
**狀態：** 將 semiprime marginal shell正式切分為低 / 中高 provider區；區分 audited upper-linear convolution floor、general well-factorable enhanced benchmark、residue-uniform balanced Type-II switching interface；建立 piecewise harmonic-level ledger；導出 low-$3/5$ 固定時 high-shell no-coupling threshold $\theta_{\rm high}>0.6571775\ldots$；證明 naive $3/5$ / $5/8$ hybrid仍需 coupled discount；未找到同時滿足 upper-linear weight、prime-product convolution、growing Goldbach residue、full-shell coverage 的單一 theorem；Goldbach仍未證明

---

# 0. 本輪目的

TCDC-18 把 aggregate semiprime marginal upper壓成：

$$
\boxed{
\kappa_{SP}
=
\frac{
2e^\gamma
}{
3\theta_H
}
}
$$

其中：

$$
\boxed{
\frac1{\theta_H}
=
\frac1{\log2}
\int_{1/3}^{1/2}
\frac{
dt
}{
t(1-t)\theta(t)
}.
}
$$

benchmark：

$$
\alpha=1,
\qquad
\kappa_{PS}=1,
\qquad
\chi=0
$$

時，

需要：

$$
\boxed{
\theta_H
>
\theta_c
=
0.641107455867\ldots
}
$$

TCDC-19 不再問：

> headline exponent是多少？

而是：

> 對每個 least-factor shell，
> 哪一個 theorem / switching architecture可以合法提供哪個 $\theta(t)$？

---

# 1. Shell Variable

least factor：

$$
\boxed{
r=X^t,
\qquad
\frac13\le t\le\frac12.
}
$$

natural shell measure：

$$
\boxed{
d\omega(t)
=
\frac{
dt
}{
t(1-t)
}.
}
$$

total：

$$
\boxed{
\omega([1/3,1/2])
=
\log2.
}
$$

---

# 2. First Structural Cut

本輪選：

$$
\boxed{
t_0=\frac38.
}
$$

理由不是任意：

Yang 2026 在 general well-factorable convolution theorem的 audited conclusion中指出，

對：

$$
\nu\ge\frac38,
$$

有：

$$
\boxed{
\mathcal L(\nu)\ge\frac58.
}
$$

但注意：

這是：

# general well-factorable weight

的 statement，

不是 upper-bound linear-sieve weights 的 full statement。

---

# 3. Shell Mass Below $3/8$

normalized weight：

$$
\boxed{
w_{\rm low}
=
\frac{
\int_{1/3}^{3/8}
dt/[t(1-t)]
}{
\log2
}.
}
$$

exact numerical value：

$$
\boxed{
w_{\rm low}
\approx
0.263034405834.
}
$$

即約：

$$
\boxed{
26.3034\%
}
$$

of the semiprime shell。

---

# 4. Shell Mass Above $3/8$

$$
\boxed{
w_{\rm high}
=
1-w_{\rm low}
\approx
0.736965594166.
}
$$

即：

$$
\boxed{
73.6966\%
}
$$

of the semiprime shell。

所以任何 high-shell theorem improvement都有很大的 harmonic influence。

---

# 5. Audited Provider A：Maynard Well-Factorable

Maynard large-moduli II establishes primes in arithmetic progressions to moduli：

$$
\boxed{
x^{3/5-\varepsilon}
}
$$

with suitably well-factorable weights。

This is a rigorously published prime-distribution benchmark。

It is directly relevant to：

- linear-sieve weight distribution；
- neutral / upper sieve architecture。

但 TCDC semiprime marginal是：

$$
rp\equiv2m\pmod q
$$

的 convolution，

不是 single-prime AP。

所以：

## Theorem-Match TCDC-19-M01

$$
\boxed{
3/5
}
$$

is a legal external exponent benchmark，

but Maynard II alone is not a complete TCDC convolution provider。

---

# 6. Audited Provider B：Yang Convolution Skeleton

Yang 2026 studies：

$$
\boxed{
\ell p\equiv a\pmod q
}
$$

with：

$$
L=x^\nu.
$$

This exactly matches the TCDC semiprime-marginal skeleton under：

$$
\boxed{
\ell\leftrightarrow r,
\qquad
p\leftrightarrow p,
\qquad
a\leftrightarrow2m.
}
$$

---

# 7. Yang General Well-Factorable Level

Yang Theorem 1.1 gives a level：

$$
\boxed{
x^{\mathcal L(\nu)-\varepsilon}
}
$$

for general well-factorable weights。

The audited paper states：

$$
\boxed{
\mathcal L(\nu)\ge\frac47
}
$$

globally，

and later records：

$$
\boxed{
\nu\ge\frac38
\Longrightarrow
\mathcal L(\nu)\ge\frac58.
}
$$

This is the strongest shell-dependent convolution shape in the currently audited source set。

---

# 8. Yang Upper-Linear Weight Level

For upper-bound well-factorable linear sieve weights，

Yang separately defines：

$$
\boxed{
\mathcal L'(\nu).
}
$$

Theorem 1.1 states only the global audited floor：

## External Fact TCDC-19-E01

$$
\boxed{
\mathcal L'(\nu)\ge\frac35.
}
$$

This distinction is critical。

---

# 9. Weight-Class Firewall

TCDC semiprime marginal upper needs positive upper-sieve weights。

Therefore：

## Research Firewall TCDC-19-R01

$$
\boxed{
\mathcal L(\nu)\ge5/8
}
$$

for general well-factorable weights cannot be silently substituted for：

$$
\boxed{
\mathcal L'(\nu)\ge5/8
}
$$

for upper-bound linear-sieve weights。

The audited theorem does not state the latter globally。

---

# 10. Consequence

The safest Yang-based shell benchmark for the actual one-stage upper-sieve architecture remains：

$$
\boxed{
\theta_Y(t)\ge3/5
}
$$

for all shell regions covered by the other theorem hypotheses。

The：

$$
5/8
$$

high-shell value is instead an：

# Enhanced Weight-Conversion Benchmark

requiring an additional argument。

---

# 11. Audited Provider C：Pascadi $5/8$

Pascadi 2025 proves primes and smooth numbers equidistributed to moduli：

$$
\boxed{
x^{5/8-o(1)}
}
$$

using：

$$
\boxed{
\text{triply-well-factorable weights}
}
$$

for the primes，

with further improvements for well-factorable linear-sieve weights。

This confirms that：

$$
5/8
$$

is a genuine modern prime-distribution frontier，

not a speculative number。

---

# 12. Pascadi Weight Firewall

But TCDC needs simultaneously：

1. prime-product convolution：
   $$
   rp;
   $$
2. upper-sieve polarity；
3. growing Goldbach residue：
   $$
   2m;
   $$
4. full shell：
   $$
   1/3\le t\le1/2.
   $$

Therefore：

## Research Firewall TCDC-19-R02

$$
\boxed{
\text{Pascadi }5/8
\text{ cannot by itself be assigned to the entire TCDC shell}.
}
$$

---

# 13. Audited Provider D：Maynard Uniform Residues

Maynard large-moduli III proves estimates beyond：

$$
x^{1/2}
$$

for moduli with conveniently sized divisors，

with complete uniformity in the residue classes in the relevant theorem family。

This directly addresses a major TCDC issue：

$$
\boxed{
a=2m
}
$$

grows with the main scale。

---

# 14. Maynard III Type-II Interfaces

The audited manuscript contains：

- Type-II estimates near：
  $$
  x^{2/5};
  $$
- a separate balanced Type-II proposition near：
  $$
  x^{1/2}.
  $$

The paper explicitly states that the ordinary earlier propositions do not handle：

$$
N,M\asymp x^{1/2},
$$

and introduces Proposition 5.3 for this balanced region with additional residue restrictions。

So：

## Theorem-Match TCDC-19-M02

$$
\boxed{
\text{Maynard III is a genuine high-shell / switching interface}.
}
$$

But it does not automatically assign a simple：

$$
\theta(t)=2/3
$$

or：

$$
5/8
$$

to the whole TCDC high shell。

---

# 15. No Legal Black-Box Stacking

We currently have different properties from different sources：

### Yang

prime-product convolution shape。

### Pascadi

headline：

$$
5/8
$$

for stronger weight classes。

### Maynard III

residue uniformity and balanced Type-II interfaces。

But no audited theorem states：

> these three properties hold simultaneously for the TCDC semiprime-marginal sequence.

Therefore：

## Research Firewall TCDC-19-R03

$$
\boxed{
\text{convolution}
+
\text{$5/8$ weight class}
+
\text{growing-residue uniformity}
}
$$

cannot be combined by citation alone。

---

# 16. Portfolio Layer A：Audited Shape Floor

Define：

## Portfolio P0

$$
\boxed{
\theta_{P0}(t)
=
3/5
}
$$

throughout：

$$
1/3\le t\le1/2.
$$

Interpretation：

> optimistic shape floor assuming the Yang upper-linear convolution estimate can be made uniform enough in the growing Goldbach residue.

This is still not a finished legal TCDC provider，

because Yang's displayed theorem permits dependence on：

$$
a.
$$

---

# 17. P0 Harmonic Level

constant portfolio：

$$
\boxed{
\theta_H(P0)=3/5.
}
$$

TCDC-18 benchmark：

$$
\boxed{
\chi_{\rm req}(P0)
\approx
0.051947.
}
$$

So a full legal：

$$
3/5
$$

convolution provider would still require about：

$$
\boxed{
5.19\%
}
$$

coupled bad-mass discount under ideal neutral and mixed providers。

---

# 18. Portfolio Layer B：Optimistic Full $5/8$

Define：

## Portfolio P1

$$
\boxed{
\theta_{P1}(t)=5/8
}
$$

throughout。

This is not currently theorem-matched，

but is useful as an optimistic frontier benchmark。

Then：

$$
\boxed{
\theta_H(P1)=5/8,
}
$$

$$
\boxed{
\chi_{\rm req}(P1)
\approx
0.019541.
}
$$

---

# 19. Portfolio Layer C：Yang-Shape Hybrid

Use：

$$
\boxed{
\theta(t)
=
\begin{cases}
3/5,
&
1/3\le t<3/8,
\\
5/8,
&
3/8\le t\le1/2.
\end{cases}
}
$$

This matches：

- upper-linear floor in low shell；
- general well-factorable $5/8$ shape benchmark in high shell。

It is **not** yet a legal upper-sieve theorem，

because of R01。

---

# 20. Hybrid Harmonic Level

By shell weights：

$$
w_{\rm low}\approx
0.263034405834,
$$

$$
w_{\rm high}\approx
0.736965594166,
$$

we get：

## Diagnostic TCDC-19-D01

$$
\boxed{
\theta_H(P2)
\approx
0.618224404627.
}
$$

and：

$$
\boxed{
\chi_{\rm req}(P2)
\approx
0.028064546504.
}
$$

So the Yang-shape hybrid still needs about：

$$
\boxed{
2.81\%
}
$$

coupled recovery。

---

# 21. Why the Hybrid Gain Is Modest

The high shell carries：

$$
73.7\%
$$

of the shell weight，

so upgrading it helps substantially。

But harmonic averaging penalizes the remaining：

$$
3/5
$$

low-shell block。

Therefore：

$$
\boxed{
3/5\text{ low}
+
5/8\text{ high}
}
$$

still cannot reach：

$$
\theta_c.
$$

---

# 22. High-Shell No-Coupling Threshold

Fix low shell：

$$
\boxed{
\theta_{\rm low}=3/5
}
$$

on：

$$
[1/3,3/8).
$$

Let high shell have constant：

$$
\theta_{\rm high}.
$$

Then：

$$
\boxed{
\frac1{\theta_H}
=
\frac{
w_{\rm low}
}{
3/5
}
+
\frac{
w_{\rm high}
}{
\theta_{\rm high}
}.
}
$$

Set：

$$
\theta_H=\theta_c.
$$

solve：

## Theorem TCDC-19-T01
### High-Shell Upgrade Threshold

$$
\boxed{
\theta_{\rm high}
>
0.657177511828
}
$$

is necessary and sufficient within this two-level benchmark for no-coupling crossing。

---

# 23. Interpretation

required：

$$
\boxed{
\theta_{\rm high}
>
0.6571775\ldots
}
$$

This is：

- above：
  $$
  5/8=0.625;
  $$
- below：
  $$
  2/3.
  $$

So the true high-shell target is quantitatively clear。

---

# 24. A $2/3$ High-Shell Target Portfolio

Define：

$$
\boxed{
\theta(t)
=
\begin{cases}
3/5,
&
1/3\le t<3/8,
\\
2/3,
&
3/8\le t\le1/2.
\end{cases}
}
$$

Then：

## Diagnostic TCDC-19-D02

$$
\boxed{
\theta_H
\approx
0.647735907817.
}
$$

which satisfies：

$$
\boxed{
\theta_H>\theta_c.
}
$$

Thus in the ideal neutral / mixed benchmark：

$$
\boxed{
\chi_{\rm req}=0.
}
$$

---

# 25. Important Firewall on $2/3$

No audited theorem in this round gives：

$$
\boxed{
\theta_{\rm high}=2/3
}
$$

for the full high shell with all TCDC conditions。

So D02 is：

# Target Portfolio

not：

# Available Provider Portfolio。

---

# 26. Switching as a Weight-Class Converter

The main purpose of switching should now be stated more precisely。

It is not merely：

> push the exponent upward。

It must potentially convert a hard positive upper-sieve term into a structure where stronger：

- general well-factorable；
- Type-II；
- complementary-divisor；

estimates apply。

So：

# Switching Goal

$$
\boxed{
\text{convert upper-linear high shell into a theorem class with effective }
\theta_{\rm high}>0.6571775.
}
$$

or compensate the remaining deficit through：

$$
\chi.
$$

---

# 27. Complementary Divisor

high shell divisibility：

$$
d\mid2m-rp
$$

can be rewritten：

$$
\boxed{
2m-rp=dk.
}
$$

When：

$$
d
$$

is large，

$$
k
$$

is short。

TCDC / SRBRD already developed：

- complement promotion；
- factorization-preserving switching；
- shift-free lattice differences。

TCDC-19 identifies their constant target：

$$
\boxed{
\theta_{\rm high}^{eff}
>
0.6571775
}
$$

if low shell stays at：

$$
3/5
$$

and coupling is not used。

---

# 28. Switching + Coupling Exchange

With coupled discount：

$$
\chi>0,
$$

critical harmonic level falls。

For each：

$$
\chi,
$$

define：

$$
\Theta_c(\chi)
=
\frac{
2e^\gamma s_*
}{
3(
1-p_*s_*+\chi
)
}.
$$

Fix low shell：

$$
3/5.
$$

Required high-shell exponent solves：

$$
\frac{
w_{\rm low}
}{
3/5
}
+
\frac{
w_{\rm high}
}{
\theta_{\rm high}
}
<
\frac1{\Theta_c(\chi)}.
$$

selected values：

| Coupled discount $\chi$ | critical harmonic level | required high-shell level |
|---:|---:|---:|
| $0.000$ | $0.641107455868$ | $0.657177511828$ |
| $0.010$ | $0.632761998398$ | $0.645338860489$ |
| $0.020$ | $0.624631018865$ | $0.633919193542$ |
| $0.030$ | $0.616706354053$ | $0.622896655183$ |
| $0.050$ | $0.601445335805$ | $0.601962886115$ |

---

# 29. Interpretation of the Trade

### No coupling

need：

$$
\theta_{\rm high}>0.65718.
$$

### Small coupling

the high-shell analytic target drops quickly。

So transport can substitute for a difficult exponent improvement。

This is exactly the TCDC analytic / geometric trade。

---

# 30. Current Portfolio Ledger

| Portfolio | harmonic level $\theta_H$ | required $\chi$ | status |
|---|---:|---:|---|
| Uniform $3/5$ benchmark | $0.600000000000$ | $0.051946894499$ | Benchmark floor; direct full-shell TCDC legality not established |
| Uniform $5/8$ optimistic benchmark | $0.625000000000$ | $0.019540567550$ | Optimistic only; current audited upper-linear convolution theorem does not grant this full-shell |
| Low $3/5$, high $5/8$ | $0.618224404627$ | $0.028064546504$ | Enhanced benchmark; needs weight-class conversion/switching |
| Low $3/5$, high $2/3$ | $0.647735907817$ | $0.000000000000$ | Target portfolio; not supplied by a single audited theorem |

The ledger makes theorem status explicit rather than blending benchmark values with legal providers。

---

# 31. What Is Actually Audited as a Theorem Today

### Maynard II

$$
3/5-\varepsilon
$$

for primes with suitably well-factorable weights。

### Maynard III

beyond：

$$
1/2
$$

for structured moduli，

with strong residue-class uniformity；

balanced Type-II near：

$$
x^{1/2}
$$

requires additional structure。

### Pascadi

$$
5/8-o(1)
$$

for primes with triply-well-factorable weights。

### Yang

prime-product convolution theorem：

$$
\ell p\equiv a\pmod q;
$$

general well-factorable：

$$
\mathcal L(\nu)\ge4/7,
$$

and for：

$$
\nu\ge3/8,
$$

the paper records：

$$
\mathcal L(\nu)\ge5/8;
$$

upper-bound well-factorable linear-sieve weights：

$$
\boxed{
\mathcal L'(\nu)\ge3/5.
}
$$

---

# 32. What Is Not Audited as a Single Theorem

We do **not** currently have one cited theorem saying：

$$
\boxed{
\theta(t)\ge5/8
}
$$

for：

- all：
  $$
  t\in[3/8,1/2];
  $$
- upper-bound TCDC semiprime sieve weights；
- convolution：
  $$
  rp;
  $$
- growing residue：
  $$
  2m;
  $$
- all required modulus factorizations。

This missing theorem is the exact external provider gap。

---

# 33. Legal-Provider Contract

Define：

# ASP
## Aggregate Shell Provider

A legal TCDC shell provider for interval：

$$
I\subseteq[1/3,1/2]
$$

must specify：

1. shell：
   $$
   t\in I;
   $$
2. coefficient class for：
   $$
   r,p;
   $$
3. sieve-weight class；
4. modulus factorization class；
5. residue dependence / uniformity；
6. exponent：
   $$
   \theta_I;
   $$
7. summable error term；
8. compatibility with neighboring providers。

Only after all eight are checked may：

$$
\theta_I
$$

enter the harmonic portfolio。

---

# 34. Provider Conversion Contract

A switching theorem may convert：

$$
\boxed{
\mathrm{ASP}_1
}
$$

to：

$$
\boxed{
\mathrm{ASP}_2
}
$$

if it proves：

- identical / dominating arithmetic count；
- new weight class；
- preserved upper polarity；
- controlled boundary terms；
- no loss larger than claimed level gain。

This prevents silently assigning a stronger theorem after a formal variable switch。

---

# 35. Low-Shell Strategy

For：

$$
1/3\le t<3/8,
$$

shell weight is only：

$$
26.3\%.
$$

Therefore it may be efficient to accept：

$$
3/5
$$

there，

rather than spend major proof effort improving it。

T11 shell-value density still matters，

but the largest strategic gain is likely in the high shell。

---

# 36. High-Shell Strategy

For：

$$
3/8\le t\le1/2,
$$

shell weight is：

$$
73.7\%.
$$

This is precisely where：

- Yang general WF reaches the $5/8$ benchmark；
- Maynard III balanced Type-II becomes structurally relevant near the top；
- complementary-divisor switching has the greatest value。

So：

# High Shell

is the primary theorem-engineering target。

---

# 37. Middle / High Split

A more realistic future partition may be：

$$
\boxed{
[1/3,3/8)
}
$$

low，

$$
\boxed{
[3/8,2/5)
}
$$

middle，

$$
\boxed{
[2/5,1/2]
}
$$

high / balanced。

Reason：

Maynard III has a technical Type-II transition around：

$$
2/5
$$

and a separate balanced treatment near：

$$
1/2.
$$

This matches the earlier SRBRD-20 ～ 25 provider architecture。

---

# 38. Why We Do Not Assign Numeric Levels Yet

Although the Maynard III propositions are powerful，

their usable ranges depend on：

- modulus factorization；
- sequence support；
- residue compatibility；
- Siegel-Walfisz conditions。

A simple：

$$
\theta=2/3
$$

label for the entire high region would overstate the audited theorem。

So TCDC-19 records：

# Structural Provider

without inventing a numeric level。

---

# 39. Numerical Target for Future High-Shell Theorem

If low shell remains：

$$
3/5,
$$

future high-shell provider only needs：

$$
\boxed{
\theta_{\rm high}
>
0.6571775\ldots
}
$$

for zero coupling under the ideal neutral / mixed benchmark。

This is below：

$$
2/3.
$$

Therefore the target is weaker than demanding a full：

$$
2/3
$$

theorem。

---

# 40. With $2\%$ Coupling

From the exchange ledger，

roughly：

$$
\chi=0.02
$$

lowers the required high-shell target substantially。

This means a legal：

$$
5/8
$$

or slightly stronger switching portfolio may already suffice when paired with TCDC transport。

The analytic target should therefore be optimized jointly with a rigorous：

$$
\chi
$$

lower bound。

---

# 41. Coupling Source Candidates

TCDC-08 ～ 13 provide：

- prime-sink LP discount；
- network doubling；
- receiver blackout；
- Hall reserve；
- mirror deficiency gain；
- center locks。

Any one of these can contribute to：

$$
\boxed{
\chi.
}
$$

So TCDC-19 does not require analytic number theory to bear the entire gap。

---

# 42. Portfolio Objective

Given shell partition：

$$
I_j
$$

with legal levels：

$$
\theta_j,
$$

compute：

$$
\boxed{
\frac1{\theta_H}
=
\sum_j
\frac{
w_j
}{
\theta_j
},
}
$$

where：

$$
\boxed{
w_j
=
\frac1{\log2}
\int_{I_j}
\frac{
dt
}{
t(1-t)
}.
}
$$

Then compute：

$$
\boxed{
\chi_{\rm req}
=
\left[
\frac{
2e^\gamma s_*
}{
3\theta_H
}
+
\kappa_{PS}p_*s_*
-
\alpha
\right]_+.
}
$$

This is the final portfolio compiler。

---

# 43. Legal Portfolio Acceptance

A future theorem portfolio is accepted only if：

1. every shell has a legal ASP；
2. shell intervals cover：
   $$
   [1/3,1/2];
   $$
3. overlaps / gaps are resolved；
4. all errors sum to：
   $$
   o(\mathfrak M);
   $$
5. common neutral normalization is consistent；
6. final:
   $$
   \chi_{\rm proven}
   >
   \chi_{\rm req}.
   $$

Only then is the bad-mass simplex crossed。

---

# 44. The Role of $5/8$ Is Now Precise

$5/8$ is not：

> the magic exponent that solves the problem。

It is：

> an exponent at which the remaining TCDC coupled discount becomes small：

$$
\boxed{
\approx1.95\%
}
$$

if granted across the whole shell under the ideal benchmark。

For the realistic Yang-shape hybrid：

$$
3/5
\text{ low}
+
5/8
\text{ high},
$$

remaining：

$$
\boxed{
\approx2.81\%.
}
$$

---

# 45. Why This Is Encouraging but Not a Proof

A few-percent constant gap is much smaller than earlier fixed-row failures。

But proving even：

$$
\chi>0.02
$$

uniformly at the correct main normalization is still a serious theorem。

Likewise a legal high-shell：

$$
0.6572
$$

convolution provider is not currently available as a black box。

So：

$$
\boxed{
\text{small numeric gap}
\neq
\text{small mathematical difficulty}.
}
$$

---

# 46. Formal Theorem Queue

```yaml
TCDC_19:

  T01:
    high_shell_level_threshold_with_low_three_fifths

  M01:
    Maynard_three_fifths_external_benchmark

  M02:
    MaynardIII_high_shell_switching_interface

  E01:
    Yang_upper_linear_global_floor_three_fifths

  P0:
    full_shell_three_fifths_shape_floor

  P1:
    optimistic_full_five_eighth_benchmark

  P2:
    low_three_fifths_high_five_eighth_hybrid

  P3:
    low_three_fifths_high_two_thirds_target

  ASP:
    legal_aggregate_shell_provider_contract
```

---

# 47. Firewall Queue

```yaml
TCDC_19_Firewalls:

  R01:
    Yang_general_WF_five_eighth
    !=
    Yang_upper_linear_five_eighth

  R02:
    Pascadi_triply_WF_five_eighth
    !=
    TCDC_full_shell_convolution_provider

  R03:
    convolution_shape
    + headline_exponent
    + residue_uniformity
    cannot_be_black_box_stacked

  R04:
    MaynardIII_structural_range
    does_not_imply_numeric_two_thirds_full_high_shell
```

---

# 48. Portfolio Benchmarks

```yaml
shell_weights:

  low_1_3_to_3_8:
    0.263034405834

  high_3_8_to_1_2:
    0.736965594166

portfolio:

  P0:
    low:
      3/5
    high:
      3/5
    theta_H:
      3/5

  P1:
    full:
      5/8
    theta_H:
      5/8

  P2:
    low:
      3/5
    high:
      5/8
    theta_H:
      0.618224404627
    chi_required:
      0.028064546504

  P3:
    low:
      3/5
    high:
      2/3
    theta_H:
      0.647735907817
    chi_required:
      0
```

---

# 49. Barrier Queue

```yaml
TCDC_19_Barriers:

  B01:
    no_uniform_growing_residue_Yang_provider

  B02:
    no_upper_linear_five_eighth_full_high_shell_theorem

  B03:
    no_weight_conversion_from_general_WF_to_required_upper_count

  B04:
    MaynardIII_factorization_hypotheses_not_compiled_into_shell_level

  B05:
    no_legal_piecewise_theta_t_covering_full_shell_above_final_threshold

  B06:
    no_uniform_coupling_discount_matching_remaining_gap

  B07:
    neutral_and_mixed_provider_losses_still_idealized_in_main benchmark
```

---

# 50. Acceptance Gate

TCDC-19 formal core前需：

1. shell weights computed with $dt/[t(1-t)]$；
2. harmonic rather than arithmetic averaging；
3. Yang general / upper-linear levels separated；
4. Pascadi weight class separated；
5. Maynard III residue-uniformity role separated from exponent assignment；
6. no black-box theorem stacking；
7. $3/5$ / $5/8$ hybrid labeled benchmark；
8. high-shell threshold algebra checked；
9. coupling-exponent trade explicit；
10. full legal ASP contract stated；
11. no claim of a complete current provider portfolio；
12. no Goldbach proof claim。

---

# 51. Verdict

本輪可以正式給：

$$
\boxed{
\mathrm{TCDC\mbox{-}19\ SHELL\mbox{-}PORTFOLIO\ LEDGER\ PASS}.
}
$$

最重要 theorem-status correction：

Yang 2026確實提供正確的 convolution skeleton：

$$
\ell p\equiv a\pmod q.
$$

而且對 general well-factorable weights，

在：

$$
\nu\ge3/8
$$

的 audited conclusion中可達：

$$
\boxed{
\mathcal L(\nu)\ge5/8.
}
$$

但對 upper-bound well-factorable linear-sieve weights，

audited global floor只有：

$$
\boxed{
\mathcal L'(\nu)\ge3/5.
}
$$

所以：

$$
\boxed{
\text{不能把 high-shell }5/8
\text{直接當成已完成的 TCDC semiprime upper provider}.
}
$$

shell mass：

$$
\boxed{
w_{\rm low}
\approx
0.263034405834,
}
$$

$$
\boxed{
w_{\rm high}
\approx
0.736965594166.
}
$$

optimistic hybrid：

$$
3/5\text{ low}
+
5/8\text{ high}
$$

gives：

$$
\boxed{
\theta_H
\approx
0.618224404627,
}
$$

still requiring：

$$
\boxed{
\chi
>
0.028064546504.
}
$$

or about：

$$
\boxed{
2.81\%
}
$$

normalized coupled recovery。

If low shell remains：

$$
3/5,
$$

then no-coupling high-shell target is：

$$
\boxed{
\theta_{\rm high}
>
0.657177511828.
}
$$

or：

$$
\boxed{
>0.6571775.
}
$$

So TCDC-19 has converted：

> find a better modern theorem

into the exact engineering target：

# Either

$$
\boxed{
\text{legal high-shell level}>0.65718
}
$$

# Or

a weaker legal level plus：

$$
\boxed{
\chi_{\rm proven}
>
\chi_{\rm req}.
}
$$

The currently audited theorem landscape does not yet supply a single legal portfolio satisfying every TCDC requirement：

- upper-sieve polarity；
- convolution structure；
- full shell；
- growing Goldbach residue；
- required harmonic level。

Therefore：

$$
\boxed{
\mathrm{NO\ LEGAL\ FULL\mbox{-}SHELL\ CLOSURE\ PORTFOLIO\ YET}.
}
$$

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

---

# 52. 下一輪建議

# TCDC-20
## High-Shell Switching Compiler / Weight-Class Conversion Closure

TCDC-19 已經指出真正缺的不是另一個 headline exponent，

而是：

> 把 high-shell positive upper-sieve problem合法轉進 stronger convolution theorem class。

下一輪直接研究：

$$
t\in[3/8,1/2].
$$

### Part A — Current Upper Object

write exact upper-sieve weighted convolution：

$$
\sum_r
\sum_p
\lambda_d^+
\mathbf1_{d\mid2m-rp}.
$$

### Part B — Factor the Sieve Weight

利用 Iwaniec / Yang 的 good factorization：

$$
d=d_1d_2d_3.
$$

### Part C — Switch

use：

$$
2m-rp=dk
$$

and factor-preserving complementary divisor。

### Part D — Provider Conversion Test

prove or disprove：

> switched coefficient belongs to a general / triply well-factorable class
> where the $5/8$-quality estimates are legal。

### Part E — Residue Uniformity

remove / control：

$$
a=2m
$$

dependence using：

- Maynard III style uniformity；
- shift-free lattice normal form；
- or explicit Goldbach-compatible residue reduction。

### Part F — Recompute Portfolio

If successful，

insert the actual high-shell exponent：

$$
\theta_{\rm high}(t)
$$

into TCDC-19 compiler，

compute：

$$
\theta_H,
\qquad
\chi_{\rm req}.
$$

This will decide whether switching is merely a formal rewrite or a genuine **weight-class lift** across the final constant barrier。

---

# 附錄 A：Shell Weights

```yaml
split:
  t0:
    3/8

low:
  interval:
    1/3_to_3/8
  weight:
    0.263034405834

high:
  interval:
    3/8_to_1/2
  weight:
    0.736965594166
```

---

# 附錄 B：Portfolio Compiler

```yaml
input:
  shell_intervals
  legal_theta_j
  shell_weights_w_j
  alpha
  kappa_PS
  chi

harmonic_level:
  1/theta_H:
    sum_j(
      w_j/theta_j
    )

required_coupling:
  chi_req:
    max(
      0,
      2*exp(gamma)*s_star/(3*theta_H)
      + kappa_PS*p_star*s_star
      - alpha
    )
```

---

# 附錄 C：Two-Level Threshold

```yaml
low_shell:
  theta_low:
    3/5

high_shell:
  theta_high:
    variable

no_coupling_target:
  theta_high:
    > 0.657177511828

full_five_eighth_high:
  theta_H:
    0.618224404627
  chi_required:
    0.028064546504

two_thirds_high:
  theta_H:
    0.647735907817
  chi_required:
    0
```

---

# 附錄 D：Provider Status

```yaml
Maynard_II:
  property:
    well_factorable_prime_distribution
  level:
    3/5-epsilon
  direct_TCDC_convolution:
    false

Pascadi_2025:
  property:
    triply_well_factorable_prime_distribution
  level:
    5/8-o(1)
  direct_TCDC_upper_convolution:
    false

Yang_2026:
  property:
    prime_product_convolution
  general_WF:
    global_floor:
      4/7
    high_shell_nu_ge_3_8:
      at_least:
        5/8
  upper_linear_WF:
    global_floor:
      3/5
  growing_residue_uniformity:
    not_established_from_audited_statement

Maynard_III:
  property:
    residue_uniformity_and_structured_large_moduli
  balanced_TypeII:
    structurally_relevant
  direct_full_high_shell_numeric_level:
    not_assigned
```

---

# 附錄 E：Benchmark Portfolio Table

| Portfolio | $\theta_H$ | required $\chi$ | status |
|---|---:|---:|---|
| Uniform $3/5$ benchmark | $0.600000000000$ | $0.051946894499$ | Benchmark floor; direct full-shell TCDC legality not established |
| Uniform $5/8$ optimistic benchmark | $0.625000000000$ | $0.019540567550$ | Optimistic only; current audited upper-linear convolution theorem does not grant this full-shell |
| Low $3/5$, high $5/8$ | $0.618224404627$ | $0.028064546504$ | Enhanced benchmark; needs weight-class conversion/switching |
| Low $3/5$, high $2/3$ | $0.647735907817$ | $0.000000000000$ | Target portfolio; not supplied by a single audited theorem |

---

# 附錄 F：Coupling Trade Table

| $\chi$ | critical $\theta_H$ | required high-shell level if low shell stays $3/5$ |
|---:|---:|---:|
| $0.000$ | $0.641107455868$ | $0.657177511828$ |
| $0.010$ | $0.632761998398$ | $0.645338860489$ |
| $0.020$ | $0.624631018865$ | $0.633919193542$ |
| $0.030$ | $0.616706354053$ | $0.622896655183$ |
| $0.050$ | $0.601445335805$ | $0.601962886115$ |

---

# 附錄 G：Commit Decision

```yaml
research_round:
  TCDC-19

decision:

  shell_provider_ledger:
    completed

  Yang_general_vs_upper_weight_firewall:
    formalized

  high_shell_mass:
    quantified

  three_fifths_five_eighth_hybrid:
    quantified

  high_shell_no_coupling_threshold:
    derived

  switching_target:
    weight_class_conversion

  full_legal_provider_portfolio:
    not_found

  Goldbach_proved:
    false

next:
  TCDC-20_HighShell_SwitchingCompiler_WeightClassConversion_Closure
```

---

# References / Audited External Sources

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

   Audited statement:
   primes of size $x$ admit mean-value estimates to moduli as large as:
   $$
   x^{3/5-\varepsilon}
   $$
   with suitably well-factorable weights.

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

   Audited statements:
   - estimates beyond $x^{1/2}$ for moduli with conveniently sized divisors;
   - complete uniformity in the residue classes in the theorem family;
   - technical Type-II estimates around $x^{2/5}$;
   - a separate balanced Type-II proposition near $x^{1/2}$ with additional residue restrictions.

3. Alexandru Pascadi, **On the exponents of distribution of primes and smooth numbers**, arXiv:2505.00653, 2025.

   Audited statement:
   primes and smooth numbers are equidistributed to moduli:
   $$
   x^{5/8-o(1)}
   $$
   with triply-well-factorable weights for the primes, with improvements also obtained for well-factorable linear-sieve weights.

4. Zhiyuan Yang, **Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications**, arXiv:2608.13299, 2026.

   Audited statements:
   - studies:
     $$
     \ell p\equiv a\pmod q;
     $$
   - for general well-factorable weights, Theorem 1.1 defines an exponent $\mathcal L(\nu)$ with:
     $$
     \mathcal L(\nu)\ge4/7;
     $$
   - the paper records:
     $$
     \nu\ge3/8
     \Longrightarrow
     \mathcal L(\nu)\ge5/8;
     $$
   - for upper-bound well-factorable linear-sieve weights, the audited global floor is:
     $$
     \mathcal L'(\nu)\ge3/5;
     $$
   - the displayed convolution estimate permits dependence on the arithmetic parameter $a$, so the TCDC growing-Goldbach-residue application is not silently assumed uniform.