# TCDC-21
## Yang-v2 Port Audit / Special-Convolution Hypothesis Match
### Prime-Variable Blocker、Growing-Residue Blocker、Special-Class Certification、Repair Stack 與 Legal-Coverage Compiler

**母方法論：** TCDC — Target–Condition Dual Closure  
**對接研究線：** RG-SCP / SRBRD  
**輪次：** TCDC-21  
**日期：** 2026-10-03  
**版本：** v0.1  
**作者：** Neo.K；Aletheia（GPT-5.6 Sol）  
**性質：** External Theorem Type-Check / Portability Audit / Hypothesis Ledger  
**前置：** TCDC-00 ～ TCDC-20；SRBRD-00 ～ SRBRD-26  
**狀態：** 對 Yang arXiv:2608.13299v2 與 TCDC high-shell semiprime marginal逐條 type-check；確認 product-convolution skeleton、dyadic scale、prime second factor、well-factorable upper weight等結構高度匹配，但 direct port 被「第二 prime 變數」、「growing Goldbach residue」、「special $5/8$ class membership」三個核心 blocker擋住；直接丟棄第二 prime 條件會損失一整個 logarithm，故不可作主階 upper；定義 Prime-Variable Lift、Growing-Residue Uniformity、Special-Class Weight Certification 三個必要 repair operators；目前 Yang-v2 strict legal $5/8$ TCDC shell coverage為零，只有完成 repair stack 後才可進 harmonic compiler；Goldbach仍未證明

---

# 0. 本輪目的

TCDC-20 已把 high-shell upper object寫成：

$$
\boxed{
\mathcal U_H
=
\sum_{r\in\mathcal R_H}
\sum_{p\in\mathbb P}
\sum_q
\lambda_q^+
\mathbf1_{
rp\equiv2m\pmod q
}.
}
$$

其中：

$$
\boxed{
r=X^t,
\qquad
3/8\le t\le1/2.
}
$$

Yang v2 的 convolution skeleton：

$$
\boxed{
lp\equiv a\pmod q
}
$$

看起來高度相似。

本輪不再以「像不像」判斷。

直接逐條 type-check。

---

# 1. Yang-v2 Current Status

latest audited revision：

$$
\boxed{
\text{arXiv:2608.13299v2}
}
$$

submitted：

$$
13\text{ Aug }2026,
$$

revised：

$$
\boxed{
3\text{ Sep }2026.
}
$$

v2 abstract明確指出：

> 對一類 special convolution forms equipped with well-factorable weights，
> 可以使用 Pascadi 對 triply-well-factorable prime weights 的：

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

distribution level。

所以：

## External Fact TCDC-21-E01

$$
\boxed{
\text{special convolution }\to5/8
}
$$

是目前存在的技術，

不是 speculative target。

---

# 2. Yang Base Convolution Statement

accessible theorem statement contains the form：

$$
\boxed{
\sum_{q}
\lambda_q
\left[
\sum_{l\sim L}
\sum_{\substack{
p<x/l\\
lp\equiv a\pmod q
}}
1
-
\text{main term}
\right]
}
$$

with：

$$
\boxed{
L=x^\nu.
}
$$

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

Yang also states an upper-weight theorem family with a level function：

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

The accessible theorem statement records at least：

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

---

# 3. TCDC High-Shell Object

TCDC semiprime marginal shell：

$$
x=rp,
$$

where：

$$
\boxed{
r,p\in\mathbb P,
}
$$

$$
\boxed{
r=X^t,
\qquad
3/8\le t\le1/2,
}
$$

and the opposite endpoint must avoid small divisors：

$$
\boxed{
d\mid2m-rp.
}
$$

The upper-sieve distribution object is therefore：

$$
\boxed{
rp\equiv2m\pmod d.
}
$$

---

# 4. Type Map

natural map：

$$
\boxed{
l\leftrightarrow r,
}
$$

$$
\boxed{
p_{\mathrm{Yang}}
\leftrightarrow p_{\mathrm{TCDC}},
}
$$

$$
\boxed{
a\leftrightarrow2m,
}
$$

$$
\boxed{
q\leftrightarrow d.
}
$$

At pure algebraic skeleton level：

$$
\boxed{
lp\equiv a\pmod q
}
$$

matches exactly。

---

# 5. Type-Check Ledger

| Gate | Yang-v2 input | TCDC object | Status | Consequence |
|---|---|---|---|---|
| H1 Scale | $x$ global scale | $X\asymp2m$ | **PASS** | Same product scale |
| H2 Long factor | $l\sim x^\nu$ | $r\sim X^t$ | **PASS** | $\nu=t$ |
| H3 Product congruence | $lp\equiv a\bmod q$ | $rp\equiv2m\bmod d$ | **PASS** | Exact convolution skeleton |
| H4 One prime factor | $p$ prime | $p$ prime | **PASS** | Direct match |
| H5 Other factor | $l$ unrestricted in theorem statement | $r$ must be prime | **FAIL / REPAIRABLE** | Dropping $r$ primality loses one logarithm |
| H6 Upper weight | well-factorable upper linear-sieve variants admitted | Rosser–Iwaniec-type $\lambda_d^+$ | **PASS-CONDITIONAL** | Must choose an explicitly supported variant |
| H7 Special $5/8$ class | v2 special convolution class | switched TCDC upper object | **UNKNOWN / REPAIRABLE** | Membership not proved |
| H8 Residue | theorem bound depends on fixed $a$ | $a=2m\asymp X$ grows | **FAIL / REPAIRABLE** | Direct asymptotic sequence not covered |
| H9 Primitive residue / gcd | standard coprimality required in prime AP machinery | $d$ may share factors with $2m$ | **REPAIRABLE** | Separate center factors |
| H10 Dyadic support | $l\sim L$, $p<x/l$ / smooth dyadic decomposition | TCDC shells dyadicizable | **PASS / REPAIRABLE** | Standard partition loss only |
| H11 Main-term normalization | convolution AP main term | RootRough sieve main term | **REPAIRABLE** | Local factors must be matched |
| H12 Error summation | $O(x/\log^A x)$ with theorem hypotheses | all shell / sieve pieces | **UNKNOWN** | Depends on H5/H7/H8 repairs |

---

# 6. First Core Blocker — Second Prime Variable

Yang's displayed convolution theorem has：

- one generic factor：
  $$
  l;
  $$
- one prime factor：
  $$
  p.
  $$

TCDC has：

$$
\boxed{
r\in\mathbb P,
\qquad
p\in\mathbb P.
}
$$

So direct theorem lacks one prime restriction。

---

# 7. Why We Cannot Simply Drop $r$ Primality

For upper bounds one may try：

$$
\sum_{r\in\mathbb P}
\le
\sum_{r\in\mathbb N}.
$$

This is formally true，

but constant-fatal。

---

# 8. Prime-vs-Integer Shell Scale

for：

$$
r=X^t,
$$

a prime shell has density：

$$
\boxed{
\frac{
dr
}{
\log r
}
\asymp
\frac{
dr
}{
t\log X
}.
}
$$

an unrestricted integer shell has density：

$$
\boxed{
dr.
}
$$

Therefore replacing prime $r$ by unrestricted $r$ inflates the shell by：

## Diagnostic TCDC-21-D01

$$
\boxed{
\asymp
t\log X.
}
$$

This is not a fixed constant loss。

It destroys the target：

$$
\boxed{
X/\log^2X
}
$$

prime-pair normalization。

---

# 9. Logarithmic-Fatality Theorem

## Theorem TCDC-21-T01

Any direct application of a Yang-type theorem obtained only by deleting the TCDC condition：

$$
r\in\mathbb P
$$

produces an upper envelope one logarithmic order larger than the semiprime marginal main scale。

Therefore such deletion cannot be used as a main-order TCDC-17 bad-mass upper。

Symbolically：

$$
\boxed{
\text{drop second prime}
\Longrightarrow
\text{one-log fatal upper}.
}
$$

---

# 10. Symmetric Remapping Does Not Help

One might swap variables：

$$
p_{\mathrm{Yang}}
\leftrightarrow r_{\mathrm{TCDC}},
$$

$$
l_{\mathrm{Yang}}
\leftrightarrow p_{\mathrm{TCDC}}.
$$

Then $r$ primality is captured，

but $p$ primality is lost。

So：

## Corollary TCDC-21-C01

$$
\boxed{
\text{the one-prime Yang statement cannot directly encode a two-prime TCDC product by variable relabeling alone}.
}
$$

---

# 11. Repair Operator R1 — Prime-Variable Lift

define：

# PVL
## Prime-Variable Lift

Goal：

upgrade the Yang-type convolution estimate from：

$$
\sum_{l\sim L}
\sum_{p\in\mathbb P}
$$

to a form permitting：

$$
\boxed{
\sum_{l\in\mathbb P}
\sum_{p\in\mathbb P}
}
$$

or an equivalent coefficient：

$$
\alpha_l
$$

supported on primes with the correct normalization。

Possible mechanisms：

- Vaughan identity；
- Heath-Brown identity；
- von Mangoldt coefficient version；
- an explicit bilinear prime-prime extension；
- a special-convolution theorem already allowing prime-supported first coefficients。

No such extension is silently assumed from Yang's displayed theorem。

---

# 12. PVL Acceptance Condition

PVL must preserve：

1. product scale；
2. upper polarity；
3. target weight class；
4. distribution level；
5. one factor $1/\log L$ normalization；
6. summable errors。

If PVL loses a logarithm，

it has failed even if the congruence theorem remains true。

---

# 13. Second Core Blocker — Growing Residue

Yang's displayed estimate is stated with：

$$
a
$$

as arithmetic parameter，

and the error notation is of the form：

$$
\boxed{
\ll_{\varepsilon,A,a}
\frac{x}{(\log x)^A}.
}
$$

This is fixed-$a$ dependence。

TCDC requires：

$$
\boxed{
a=2m\asymp X.
}
$$

as：

$$
X\to\infty.
$$

---

# 14. Fixed-Parameter Logical Firewall

## Theorem TCDC-21-T02

A theorem asserting：

$$
E(x,a)
\ll_{a}
x(\log x)^{-A}
$$

for every fixed $a$ does **not** imply a uniform estimate along a sequence：

$$
a=a(x)\asymp x.
$$

Therefore Yang's displayed fixed-$a$ dependence cannot by itself justify the TCDC substitution：

$$
a=2m.
$$

---

# 15. Repair Operator R2 — Growing-Residue Uniformity

define：

# GRU
## Growing-Residue Uniformity

Need a theorem of schematic type：

$$
\boxed{
\sup_{|a|\le CX}
E(X,a)
\ll
\frac{
X
}{
(\log X)^A
}
}
$$

on the actual modulus / weight class，

or a weaker uniformity exactly covering：

$$
a=2m.
$$

Possible repair sources：

- Maynard-III-style uniform residue classes；
- dispersion removing $a$ from hard phases；
- shift-free lattice reduction plus uniform base-point control；
- direct refinement of Yang's argument。

---

# 16. Maynard III Is Only an Interface

Maynard III proves mean-value estimates beyond：

$$
x^{1/2}
$$

for structured moduli，

with complete uniformity in residue classes in that theorem family。

Thus：

$$
\boxed{
\text{Maynard III demonstrates that GRU-type technology exists}.
}
$$

But it does not automatically wrap Yang's convolution theorem。

The modulus and coefficient hypotheses must still be jointly matched。

---

# 17. Shift-Free Difference Is Partial GRU

TCDC-20：

for：

$$
rp+Kd=N,
$$

two solutions satisfy：

$$
\boxed{
r\Delta p
+
K\Delta d
=
0.
}
$$

so：

$$
N
$$

disappears from pair differences。

This can remove the growing shift from dispersion phases。

But：

$$
rp_0+Kd_0=N
$$

still defines the affine origin。

So：

## Corollary TCDC-21-C02

$$
\boxed{
\text{shift-free pair geometry}
\neq
\text{full growing-residue uniformity}.
}
$$

---

# 18. Third Core Blocker — Special $5/8$ Class

Yang v2 abstract proves：

$$
\boxed{
\text{special convolution class}
+
\text{well-factorable weights}
\Longrightarrow
5/8-o(1)
}
$$

through Pascadi technology。

But the currently audited public theorem statement does not establish that the TCDC switched upper object is a member of that special class。

Therefore：

## Research Firewall TCDC-21-R01

$$
\boxed{
\text{skeleton match}
\neq
\text{special-class membership}.
}
$$

---

# 19. Repair Operator R3 — Special-Class Weight Certification

define：

# SCWC
## Special-Class Weight Certification

Input：

$$
\boxed{
\sum_{r,p,d}
\lambda_d^+
\mathbf1_{rp\equiv N\pmod d}.
}
$$

After factor-first switching：

$$
\boxed{
d=d_1d_2d_3,
}
$$

$$
\boxed{
rp+k d_1d_2d_3=N.
}
$$

Need to prove the resulting coefficient system satisfies every factorization / support condition of Yang-v2's special $5/8$ convolution class。

Only then：

$$
\boxed{
5/8
}
$$

may enter the TCDC harmonic compiler。

---

# 20. Weight Class at the $3/5$ Floor

Yang's theorem statement explicitly includes：

$$
\boxed{
\lambda_q^+
}
$$

upper-bound well-factorable linear-sieve weights，

with：

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

So H6 is a genuine positive match：

the basic upper-sieve weight type is not foreign to Yang's framework。

But H5 and H8 still block a direct TCDC application even at：

$$
3/5.
$$

---

# 21. Strict Legal Yang Coverage Today

A shell may be marked legally ported only when：

- H5 prime-variable condition repaired；
- H7 special class certified for $5/8$；
- H8 growing residue repaired；
- all remaining gates pass。

At present none of these three repair theorems has been proved in the TCDC program。

Therefore：

## Verdict TCDC-21-V01

$$
\boxed{
\eta_{\mathrm{Yang},5/8}^{\mathrm{legal}}
=
0
}
$$

for the TCDC high shell at the current proof state。

This does **not** mean Yang is irrelevant。

It means no shell is yet allowed to inherit the exponent in a rigorous TCDC certificate。

---

# 22. Structural Coverage vs Legal Coverage

define：

### Structural skeleton coverage

$$
\boxed{
\eta_{\mathrm{skel}}
}
$$

fraction of shell whose scale / product form matches Yang。

For：

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

we have essentially：

$$
\boxed{
\eta_{\mathrm{skel}}=1.
}
$$

### Legal theorem coverage

$$
\boxed{
\eta_{\mathrm{legal}}=0
}
$$

until repair gates pass。

This distinction is mandatory。

---

# 23. Minimal Port Stack

A legal Yang-v2 $5/8$ port requires at minimum：

## R1

$$
\boxed{
\mathrm{PVL}
}
$$

Prime-Variable Lift。

## R2

$$
\boxed{
\mathrm{GRU}
}
$$

Growing-Residue Uniformity。

## R3

$$
\boxed{
\mathrm{SCWC}
}
$$

Special-Class Weight Certification。

plus routine / local：

## R4

$$
\boxed{
\mathrm{CFW}
}
$$

Center-Factor Firewall。

---

# 24. Port Conjunction

## Theorem TCDC-21-T03
### Port Necessity

For the currently audited Yang-v2 route，

a proof claiming legal high-shell $5/8$ coverage must provide all three nontrivial repair certificates：

$$
\boxed{
\mathrm{PVL}
\land
\mathrm{GRU}
\land
\mathrm{SCWC}.
}
$$

Absence of any one means the $5/8$ exponent cannot be inserted into the TCDC proof ledger through this port。

This is a proof-audit theorem，

not a claim those repairs are impossible。

---

# 25. Repair Priority

The three blockers are not equal。

### PVL

affects main logarithmic order。

Failure is fatal。

### GRU

affects uniform asymptotic validity。

Failure is fatal。

### SCWC

affects access to $5/8$ rather than the baseline theorem class。

Failure means one may still seek a weaker Yang-level theorem after PVL + GRU，

but not claim the v2 special $5/8$ lift。

So priority：

$$
\boxed{
\mathrm{PVL}
\sim
\mathrm{GRU}
>
\mathrm{SCWC}.
}
$$

---

# 26. A Two-Stage Port Strategy

Rather than solve all at once：

## Stage A — Legal baseline port

prove：

$$
\boxed{
\mathrm{PVL}
+
\mathrm{GRU}
}
$$

for the upper-linear Yang class。

Then attempt a legal：

$$
\boxed{
3/5
}
$$

convolution provider。

## Stage B — Weight-class upgrade

add：

$$
\boxed{
\mathrm{SCWC}
}
$$

to reach：

$$
5/8
$$

where valid。

This separates legality from optimization。

---

# 27. Why Stage A Is Already Valuable

TCDC-18：

a legal full-shell：

$$
3/5
$$

provider still needs：

$$
\boxed{
\chi\approx0.051947
}
$$

under ideal neutral / mixed benchmark。

This does not close analytically alone，

but is already vastly stronger than independent fixed-row upper architecture。

So a rigorous：

$$
3/5
$$

port is a meaningful milestone。

---

# 28. Conditional Coverage Compiler

Suppose after repairs，

a fraction：

$$
\eta
$$

of the high shell legally receives：

$$
5/8,
$$

while the remainder stays：

$$
3/5.
$$

Low shell：

$$
3/5.
$$

Then：

## Theorem TCDC-21-T04

$$
\boxed{
\frac1{\theta_H(\eta)}
=
\frac{
w_L
}{
3/5
}
+
w_H
\left(
\frac{
1-\eta
}{
3/5
}
+
\frac{
\eta
}{
5/8
}
\right).
}
$$

---

# 29. Partial-Coverage Ledger

| Legal $5/8$ coverage of high shell $\eta$ | $\theta_H$ | required coupled discount $\chi_{\rm req}$ |
|---:|---:|---:|
| $0.00$ | $0.600000000000$ | $0.051946894499$ |
| $0.25$ | $0.604454622601$ | $0.045976307501$ |
| $0.50$ | $0.608975885506$ | $0.040005720502$ |
| $0.75$ | $0.613565295374$ | $0.034035133503$ |
| $1.00$ | $0.618224404627$ | $0.028064546504$ |

Even：

$$
\eta=1
$$

leaves：

$$
\boxed{
\chi_{\rm req}
\approx
0.028064546504.
}
$$

So port completion and TCDC coupling remain complementary。

---

# 30. Strict Current Portfolio

Since：

$$
\eta_{\mathrm{legal}}=0,
$$

the current Yang-v2 port contributes no rigorously certified high-shell exponent to the TCDC harmonic proof ledger。

Therefore the current strict portfolio must label Yang as：

$$
\boxed{
\text{candidate provider / structural match}
}
$$

not：

$$
\boxed{
\text{active proof provider}.
}
$$

---

# 31. This Corrects a Common Research Failure Mode

A theorem may simultaneously have：

- correct variables；
- impressive exponent；
- matching buzzwords；
- nearby weight class；

and still be unusable because one coefficient / uniformity requirement fails。

TCDC-21 therefore enforces：

# Type-Checked Citation

instead of：

# Headline-Exponent Citation

---

# 32. Prime-Variable Repair Options

PVL candidate A：

## von Mangoldt lift

replace prime restriction by：

$$
\Lambda(r)
$$

weighted convolution，

then recover prime count by partial summation。

Needs a Yang-type theorem accepting a coefficient sequence on：

$$
l.
$$

Not present in the displayed statement。

---

# 33. PVL candidate B：Heath-Brown / Vaughan decomposition

decompose：

$$
\Lambda(r)
$$

into Type-I / Type-II convolutions。

Then each piece may fit a more general convolution theorem。

Potential benefit：

- factorization exposed；
- balanced high shell natural。

Cost：

- additional variables；
- coefficient hypotheses；
- possible loss of the special $5/8$ class。

Status：

$$
\boxed{
\text{REPAIRABLE / UNPROVED}.
}
$$

---

# 34. PVL candidate C：Symmetric prime-product theorem

seek or prove directly：

$$
\boxed{
\sum_{r,p\in\mathbb P}
\mathbf1_{
rp\equiv a\pmod q
}
}
$$

with well-factorable modulus weights。

This is the cleanest TCDC-native theorem shape，

but is stronger than the displayed Yang input schema。

---

# 35. GRU candidate A：Maynard-III adapter

Goal：

after factor-first switching，

make surviving modulus fall into a Maynard-III structured family with residue uniformity。

Need：

- convenient divisor sizes；
- coefficient hypotheses；
- prime variable range；
- compatible modulus decomposition。

Status：

$$
\boxed{
\text{REPAIRABLE / NOT COMPOSED}.
}
$$

---

# 36. GRU candidate B：Direct Yang uniformization

inspect Yang proof and prove its：

$$
\ll_{\varepsilon,A,a}
$$

dependence can be made uniform for：

$$
|a|\ll X
$$

under：

$$
(a,q)=1.
$$

This would be the cleanest repair if true。

But no such statement is taken for granted。

---

# 37. SCWC candidate A：Factor-first certification

start with an explicitly well-factorable upper-sieve weight。

factor before switching：

$$
d=d_1d_2d_3.
$$

Track：

- supports；
- coefficient convolution；
- quotient variable；
- modulus product。

Then compare line-by-line with Yang v2 special-class conditions。

This is the direct TCDC-20 compiler route。

---

# 38. SCWC candidate B：Use Yang's admitted upper-linear variant first

Theorem 1.1 explicitly recognizes upper-bound well-factorable linear-sieve weights at：

$$
\mathcal L'(\nu).
$$

One research route：

1. secure legal PVL + GRU；
2. use the guaranteed upper-linear class；
3. only afterward inspect where v2 proof upgrades selected pieces to $5/8$。

This avoids requiring a global triply-factorable identity at the outset。

---

# 39. Center-Factor Firewall

if：

$$
(d,2m)>1,
$$

primitive residue assumptions can fail。

But TCDC already has strong center arithmetic：

- some cells vanish；
- center rows have mixed capacity：
  $$
  \le1;
  $$
- divisibility locks arise。

So define：

# CFW

split moduli into：

$$
d=d_0d_1,
$$

where：

$$
d_0\mid(2m)^\infty,
$$

$$
(d_1,2m)=1.
$$

Handle：

$$
d_0
$$

arithmetically，

apply distribution theorem only to：

$$
d_1.
$$

This is likely repairable without main-order loss，

but requires proof。

---

# 40. Main-Term Compatibility

Yang's main term is an arithmetic-progression density involving：

$$
1/\varphi(q)
$$

and coprime product count。

TCDC sieve main term also carries local bad density：

$$
g_m(\ell)
=
\begin{cases}
0,&\ell\mid m,\\
1/(\ell-1),&\ell\nmid m.
\end{cases}
$$

After conditioning on prime product variables，

these structures are compatible in spirit，

but the center-prime omissions must be explicitly matched。

Status：

$$
\boxed{
\text{REPAIRABLE}.
}
$$

---

# 41. Error Budget

even after each dyadic shell theorem：

$$
O_A
\left(
X/\log^A X
\right),
$$

the full sieve uses many：

- dyadic blocks；
- weight decompositions；
- shell intervals；
- switching cases。

Need total number：

$$
\log^{O(1)}X
$$

or similarly controlled，

so choosing：

$$
A
$$

large absorbs decomposition count。

This is standard if all implied constants are uniform。

Because H8 is not uniform，

this standard summation cannot yet be invoked for TCDC。

---

# 42. Current Port Matrix

```yaml
Yang_v2_to_TCDC:

  scale:
    PASS

  dyadic_long_variable:
    PASS

  product_congruence:
    PASS

  one_prime_factor:
    PASS

  second_prime_factor:
    FAIL_REPAIRABLE:
      repair:
        PVL

  upper_well_factorable_weight:
    PASS_CONDITIONAL

  special_five_eighth_class:
    UNKNOWN_REPAIRABLE:
      repair:
        SCWC

  growing_residue:
    FAIL_REPAIRABLE:
      repair:
        GRU

  center_coprimality:
    REPAIRABLE:
      repair:
        CFW

  main_term:
    REPAIRABLE

  global_error:
    BLOCKED_BY_GRU

  current_legal_5_8_coverage:
    0
```

---

# 43. Theorem-Engineering Priority Score

PVL and GRU unlock both：

- baseline Yang upper-linear theorem；
- possible v2 $5/8$ upgrade。

SCWC only unlocks the stronger level after legality is established。

So optimal sequence：

## First

$$
\boxed{
\mathrm{PVL}
}
$$

## Second

$$
\boxed{
\mathrm{GRU}
}
$$

in parallel if possible。

## Third

$$
\boxed{
\mathrm{SCWC}.
}
$$

---

# 44. What Would Count as TCDC-21 Port Success

### Minimal success

PVL + GRU prove a legal：

$$
3/5
$$

aggregate convolution provider across a positive shell measure。

### Strong success

PVL + GRU + SCWC certify：

$$
5/8
$$

on a nonzero high-shell fraction。

### Complete high-shell success

all high shell：

$$
[3/8,1/2]
$$

receives legal：

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

Even complete success still needs：

$$
\boxed{
\chi\approx
0.028064546504
}
$$

under the current ideal benchmark if low shell remains：

$$
3/5.
$$

---

# 45. Partial Port Economics

T04 allows a port theorem to report legal coverage fraction：

$$
\eta.
$$

This immediately converts into：

- harmonic level；
- remaining coupled discount。

So an incomplete port is still valuable。

For example：

$$
\eta=1/2
$$

already reduces required coupling relative to pure：

$$
3/5.
$$

The exact table is given in the appendix。

---

# 46. Why We Do Not Ask for $5/8$ Everywhere First

The hardest repairs：

- PVL；
- GRU；

are theorem-shape issues，

not exponent issues。

Solving them at：

$$
3/5
$$

could already produce a legal TCDC provider。

Only after that does it make sense to spend effort on：

$$
5/8
$$

weight-class optimization。

This is a major research-priority correction。

---

# 47. External Source Audit — Yang v2

The latest v2 metadata states：

- v1：
  $$
  13\text{ Aug }2026;
  $$
- v2：
  $$
  3\text{ Sep }2026.
  $$

v2 abstract：

> special convolution forms equipped with well-factorable weights can use level：

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

through Pascadi's prime-distribution result with triple-well-factorable weights。

The main theorem statement also explicitly shows：

- convolution：
  $$
  lp\equiv a\pmod q;
  $$
- $L=x^\nu$；
- $\lambda_q$ well-factorable；
- upper-bound well-factorable linear-sieve variant；
- implied constant depending on：
  $$
  a.
  $$

These exact facts drive the TCDC type-check ledger above。

---

# 48. External Source Audit — Maynard III

Maynard's published Uniform Residue Classes work proves：

> mean value estimates for primes in AP to moduli larger than：

$$
x^{1/2},
$$

for structured moduli with conveniently sized divisors，

with complete uniformity in residue classes in the theorem family。

This is the strongest currently audited evidence that GRU may be repairable by a structured-modulus adapter。

It does **not** prove the Yang/TCDC composition automatically。

---

# 49. Formal Theorem Queue

```yaml
TCDC_21:

  T01:
    dropping_second_prime_is_logarithmically_fatal

  T02:
    fixed_a_theorem_does_not_imply_growing_a_uniformity

  T03:
    legal_Yang_five_eighth_port_requires_PVL_and_GRU_and_SCWC

  T04:
    partial_legal_coverage_harmonic_compiler

  C01:
    variable_swap_does_not_fix_missing_second_prime

  C02:
    shift_free_difference_does_not_equal_residue_uniformity
```

---

# 50. Repair Queue

```yaml
TCDC_21_Repairs:

  PVL:
    Prime_Variable_Lift
    priority:
      critical

  GRU:
    Growing_Residue_Uniformity
    priority:
      critical

  SCWC:
    Special_Class_Weight_Certification
    priority:
      optimization_after_legality

  CFW:
    Center_Factor_Firewall
    priority:
      local_repair
```

---

# 51. Barrier Queue

```yaml
TCDC_21_Barriers:

  B01:
    second_prime_variable_missing_from_direct_Yang_statement

  B02:
    direct_relaxation_to_integer_l_loses_one_logarithm

  B03:
    implied_constant_depends_on_fixed_a

  B04:
    a_equals_2m_grows_with_X

  B05:
    TCDC_switched_weight_not_certified_in_Yang_v2_special_class

  B06:
    center_factors_require_separate_coprimality treatment

  B07:
    full_error_budget_not_uniform_before_GRU

  B08:
    no_legal_five_eighth_shell_coverage_yet
```

---

# 52. Acceptance Gate

TCDC-21 formal core前需：

1. current Yang revision/date verified；
2. convolution skeleton extracted；
3. generic $l$ vs prime $r$ distinguished；
4. one-log loss explained；
5. fixed-$a$ dependence read from theorem；
6. growing residue logical gap explicit；
7. upper-linear weight positive match recorded；
8. special-class membership not inferred from abstract alone；
9. center gcd issue separated；
10. strict legal coverage distinguished from structural skeleton coverage；
11. partial-coverage compiler only conditional on repaired legality；
12. no claim Yang theorem already proves TCDC $3/5$ or $5/8$；
13. no Goldbach proof claim。

---

# 53. Verdict

本輪可以正式給：

$$
\boxed{
\mathrm{TCDC\mbox{-}21\ YANG\mbox{-}V2\ PORT\ AUDIT\ PASS}.
}
$$

核心 positive result：

$$
\boxed{
rp\equiv2m\pmod d
}
$$

與 Yang：

$$
\boxed{
lp\equiv a\pmod q
}
$$

具有 exact convolution skeleton match。

scale：

$$
L=X^\nu
$$

與 TCDC：

$$
r=X^t
$$

也直接匹配。

upper well-factorable linear-sieve weight類型亦在 Yang theorem family中明確出現。

但 direct port目前被三個 nontrivial blockers擋住：

$$
\boxed{
\mathrm{PVL}
}
$$

second prime-variable lift；

$$
\boxed{
\mathrm{GRU}
}
$$

growing residue：

$$
a=2m;
$$

以及：

$$
\boxed{
\mathrm{SCWC}
}
$$

v2 special $5/8$ class certification。

尤其：

$$
\boxed{
\text{直接放寬 }r\in\mathbb P\to r\in\mathbb N
}
$$

會損失一整個：

$$
\log X,
$$

所以不能拿 Yang 的 generic $l$ variable直接當主階 upper。

而：

$$
\boxed{
\ll_{\varepsilon,A,a}
}
$$

的 fixed-residue theorem也不能沿：

$$
a=2m\asymp X
$$

直接使用。

因此 current strict theorem ledger必須記：

$$
\boxed{
\eta_{\mathrm{Yang},5/8}^{legal}=0.
}
$$

while：

$$
\boxed{
\eta_{\mathrm{skel}}=1
}
$$

on the high shell。

也就是：

# Structure Match Is Complete
but
# Legal Port Is Not Yet Established

若未來 repair stack完成，

partial legal coverage：

$$
\eta
$$

可直接進 harmonic compiler：

| $\eta$ | $\theta_H$ | remaining $\chi_{\rm req}$ |
|---:|---:|---:|
| $0.00$ | $0.600000000000$ | $0.051946894499$ |
| $0.25$ | $0.604454622601$ | $0.045976307501$ |
| $0.50$ | $0.608975885506$ | $0.040005720502$ |
| $0.75$ | $0.613565295374$ | $0.034035133503$ |
| $1.00$ | $0.618224404627$ | $0.028064546504$ |

full high-shell $5/8$ port仍需：

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

所以 Yang-v2 port成功後仍須和 TCDC transport coupling接起來。

目前：

$$
\boxed{
\mathrm{YANG\mbox{-}V2\ PORT\ STATUS:
REPAIRABLE\mbox{-}BUT\mbox{-}NOT\mbox{-}LEGAL}.
}
$$

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

---

# 54. 下一輪建議

# TCDC-22
## Prime-Variable Lift / Two-Prime Convolution Adapter

TCDC-21已證明：

> 第一個 blocker不是 exponent，
> 而是 Yang input只有一個 prime variable。

所以下一輪優先攻：

$$
\boxed{
\mathrm{PVL}.
}
$$

### Part A — von Mangoldt Lift

replace：

$$
r\in\mathbb P
$$

by：

$$
\Lambda(r)
$$

and normalize：

$$
1/\log R.
$$

檢查 Yang proof是否可接受 divisor-bounded / Siegel-Walfisz first-factor coefficients。

### Part B — Heath-Brown Decomposition

若不能直接接受 prime-supported $l$，

decompose：

$$
\Lambda(r)
$$

into：

- Type I；
- Type II；
- balanced convolution pieces。

### Part C — Theorem-Class Routing

for each piece：

- Yang convolution；
- Maynard III balanced Type-II；
- trivial / classical BV；

route to strongest legal provider。

### Part D — Preserve Upper Polarity

ensure decomposition controls：

$$
\mathbf1_{r\in\mathbb P}
$$

without introducing signed lower-bound dependence that breaks the semiprime upper。

### Part E — Constant Audit

PVL不能只得到：

$$
O(X/\log X).
$$

must retain：

$$
\boxed{
X/\log^2X
}
$$

semiprime-pair normalization。

如果 PVL成功，

再進 TCDC-23 專攻 GRU。

---

# 附錄 A：Port Type Check

```yaml
scale:
  PASS

product_congruence:
  PASS

one_prime_factor:
  PASS

second_prime_factor:
  FAIL_REPAIRABLE:
    PVL

upper_linear_weight:
  PASS_CONDITIONAL

special_five_eighth_class:
  UNKNOWN_REPAIRABLE:
    SCWC

growing_residue:
  FAIL_REPAIRABLE:
    GRU

center_coprimality:
  REPAIRABLE:
    CFW

legal_five_eighth_coverage:
  0

structural_high_shell_coverage:
  1
```

---

# 附錄 B：Repair Stack

```yaml
minimal_legality:

  PVL:
    prime_variable_lift

  GRU:
    growing_residue_uniformity

five_eighth_optimization:

  SCWC:
    special_class_weight_certification

local:

  CFW:
    center_factor_firewall

legal_5_8_port_requires:
  PVL
  and GRU
  and SCWC
```

---

# 附錄 C：Partial Coverage Compiler

low shell:

$$
3/5.
$$

unconverted high shell:

$$
3/5.
$$

converted high shell:

$$
5/8.
$$

| converted high-shell fraction $\eta$ | harmonic level | remaining coupled discount |
|---:|---:|---:|
| $0.00$ | $0.600000000000$ | $0.051946894499$ |
| $0.25$ | $0.604454622601$ | $0.045976307501$ |
| $0.50$ | $0.608975885506$ | $0.040005720502$ |
| $0.75$ | $0.613565295374$ | $0.034035133503$ |
| $1.00$ | $0.618224404627$ | $0.028064546504$ |

---

# 附錄 D：Commit Decision

```yaml
research_round:
  TCDC-21

decision:

  Yang_v2_latest_status:
    audited

  convolution_skeleton:
    PASS

  second_prime_variable:
    FAIL_REPAIRABLE

  growing_residue:
    FAIL_REPAIRABLE

  upper_weight_class:
    PASS_CONDITIONAL

  special_five_eighth_membership:
    UNKNOWN_REPAIRABLE

  current_legal_five_eighth_coverage:
    0

  port_status:
    REPAIRABLE_BUT_NOT_LEGAL

  Goldbach_proved:
    false

next:
  TCDC-22_PrimeVariableLift_TwoPrimeConvolution_Adapter
```

---

# References / Audited External Sources

1. Zhiyuan Yang, **Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications**, arXiv:2608.13299v2, revised 2026-09-03.

   Audited facts used:
   - convolution form:
     $$
     lp\equiv a\pmod q;
     $$
   - $L=x^\nu$;
   - $p$ is prime while the displayed outer $l$ variable is not stated as prime-restricted;
   - well-factorable modulus weights;
   - an upper-bound well-factorable linear-sieve variant;
   - displayed error notation allows dependence on fixed arithmetic parameter $a$;
   - v2 abstract states that a special convolution class can use the:
     $$
     x^{5/8-o(1)}
     $$
     level from Pascadi's triply-well-factorable prime-distribution result.

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

   Audited fact:
   mean-value estimates beyond $x^{1/2}$ are proved for structured moduli with conveniently sized divisors, with complete uniformity in residue classes in the theorem family. This is treated here as a possible GRU interface, not as an already-composed Yang/TCDC theorem.

3. TCDC-18 through TCDC-20.

   Internal constants used:
   - low-shell weight;
   - high-shell weight;
   - harmonic compiler;
   - residual coupled-discount requirement after hypothetical full high-shell $5/8$ coverage.