# TCDC-20
## High-Shell Switching Compiler / Weight-Class Conversion Closure
### Factor-First Switching、No-Promotion Firewall、Yang-v2 $5/8$ Special-Class Update、Shift-Free Residue Reduction 與 High-Shell Constant Compiler

**母方法論：** TCDC — Target–Condition Dual Closure  
**對接研究線：** RG-SCP / SRBRD  
**輪次：** TCDC-20  
**日期：** 2026-10-03  
**版本：** v0.1  
**作者：** Neo.K；Aletheia（GPT-5.6 Sol）  
**性質：** Weight-Class Compiler / Complementary-Divisor Switching / Provider-Hypothesis Audit  
**前置：** TCDC-00 ～ TCDC-19；SRBRD-00 ～ SRBRD-26  
**狀態：** 修正 TCDC-19 對 Yang $5/8$ 技術狀態的過度保守敘述；formalize upper-sieve high-shell term的 factor-first switching compiler；證明 naive complementary-divisor reindexing不會自動升級 weight class；建立 factor-preservation、upper-domination、residue-uniformity、boundary-loss四大 compiler gates；Yang v2 證明特殊 convolution class確可將 well-factorable convolution接到 Pascadi $5/8-o(1)$ triply-well-factorable prime distribution，但 TCDC high-shell object尚未完成同類 hypothesis match；即使 high shell完整升到 $5/8$，低 shell維持 $3/5$ 時仍需約 $2.806\%$ coupled recovery，或等價約 $4.896\%$ high-shell constant saving；Goldbach仍未證明

---

# 0. 本輪問題

TCDC-19 將 semiprime least-factor shell：

$$
\frac13
\le
t
\le
\frac12
$$

切成：

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

$$
\boxed{
I_H
=
[3/8,1/2].
}
$$

低 shell 權重：

$$
\boxed{
w_L
\approx
0.263034405834.
}
$$

高 shell：

$$
\boxed{
w_H
\approx
0.736965594166.
}
$$

上一輪得到：

若低 shell保持：

$$
3/5,
$$

高 shell做到：

$$
5/8,
$$

則 harmonic level：

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

仍需要：

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

coupled bad-mass discount。

TCDC-20 不再只問：

> high shell可以拿哪個 exponent？

而是：

> 原本的 positive upper-sieve weight，
> 能不能經 switching 後合法進入更強的 theorem weight class？

---

# 1. High-Shell Upper Object

對：

$$
N=2m,
$$

高 shell least factor：

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

semiprime marginal upper的典型 sifted object可寫：

$$
\boxed{
\mathcal U_H
=
\sum_{r\in\mathcal R_H}
\sum_{p\in\mathbb P}
\sum_{d\le D}
\lambda_d^+
\mathbf1_{
d\mid N-rp
}.
}
$$

其中：

$$
\boxed{
\lambda_d^+
}
$$

是 proof-admissible upper-sieve weight。

---

# 2. Positive Upper Polarity

我們需要：

$$
\boxed{
\mathbf1_{\operatorname{RootRough}_z(N-rp)}
\le
\sum_{d\mid N-rp}
\lambda_d^+
}
$$

或等價 dyadic / weighted form。

所以 switching後必須保留：

# Upper Domination

若新 weight不再 pointwise / aggregate dominate original bad set，

就不能當 semiprime marginal upper。

---

# 3. Complementary Divisor Identity

divisibility：

$$
d\mid N-rp
$$

等價：

$$
\boxed{
N-rp=dk
}
$$

for：

$$
k\ge1.
$$

所以：

$$
\mathcal U_H
=
\sum_{r,p,d,k}
\lambda_d^+
\mathbf1_{
rp+dk=N
}.
$$

這只是 exact variable enlargement。

---

# 4. Naive Switch

solve：

$$
d
=
\frac{
N-rp
}{
k
}.
$$

Then weight becomes：

$$
\boxed{
\lambda_{(N-rp)/k}^+.
}
$$

This may look like a new coefficient in：

$$
k.
$$

But no factorability theorem follows from this algebraic rewrite。

---

# 5. No-Promotion Principle

## Theorem TCDC-20-T01
### Algebraic Reindexing Does Not Promote Weight Class

The identity：

$$
N-rp=dk
$$

alone does not imply that the induced coefficient：

$$
k
\mapsto
\lambda_{(N-rp)/k}^+
$$

is：

- well-factorable；
- doubly well-factorable；
- triply-well-factorable；
- or any other stronger factorization class。

Therefore：

$$
\boxed{
\text{switching identity}
\not\Rightarrow
\text{weight-class upgrade}.
}
$$

This is a logical theorem about what has and has not been established：

membership in a stronger class requires an explicit factorization proof。

---

# 6. Why Naive Reindexing Fails Structurally

well-factorability concerns decompositions in the modulus index：

$$
d.
$$

After substituting：

$$
d=(N-rp)/k,
$$

the coefficient depends simultaneously on：

$$
N,
r,
p,
k.
$$

So the new $k$ coefficient is not an autonomous arithmetic sequence。

Thus：

# Weight Factorability

and：

# Variable Solvability

must not be conflated。

---

# 7. Factor-First Principle

The only safe compiler order is：

## Step 1

factor the original sieve weight。

## Step 2

keep factor variables explicit。

## Step 3

then perform complementary-divisor switching。

Symbolically：

$$
\boxed{
\text{Factor}
\rightarrow
\text{Switch}
}
$$

not：

$$
\boxed{
\text{Switch}
\rightarrow
\text{Assume Factorability}.
}
$$

---

# 8. Two-Factor Well-Factorable Expansion

If：

$$
\lambda_d^+
$$

is well-factorable of level：

$$
D,
$$

then for an admissible split：

$$
D=D_1D_2,
$$

one has a representation of the schematic form：

$$
\boxed{
\lambda_d^+
=
\sum_{\substack{
d=d_1d_2
}}
\alpha_{d_1}
\beta_{d_2},
}
$$

with bounded factor coefficients and prescribed supports。

The precise definition depends on the theorem's weight class。

---

# 9. Triple-Factor Target

Pascadi-type stronger prime-distribution theorems use：

# Triply-Well-Factorable

weight structures。

A schematic target representation is：

$$
\boxed{
\lambda_d
=
\sum_{\substack{
d=d_1d_2d_3
}}
\alpha_{d_1}
\beta_{d_2}
\gamma_{d_3},
}
$$

with the required uniformity over admissible three-way level decompositions。

Ordinary well-factorability does not by definition allow us to silently assert this stronger property。

---

# 10. Factor-First Switched Equation

keep：

$$
d=d_1d_2d_3
$$

explicit before switching。

Then：

## Definition TCDC-20-D01

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

This is the canonical：

# Factor-Preserving Complementary-Divisor Form

---

# 11. Partial Modulus Promotion

Fix：

$$
K
=
k d_2d_3.
$$

Then：

$$
\boxed{
rp
\equiv
N
\pmod K.
}
$$

and：

$$
\boxed{
d_1
=
\frac{
N-rp
}{
K
}.
}
$$

So one original modulus factor can be converted into a quotient variable while：

$$
d_2d_3
$$

remains inside the active modulus。

This is the algebraic core of factor-preserving switching。

---

# 12. Alternative Partial Switch

One may instead keep：

$$
d_1d_2
$$

in the modulus and solve for：

$$
d_3,
$$

or any permutation。

Thus the compiler may choose which original factors remain available to the analytic theorem。

This is exactly why factorization must be exposed before switching。

---

# 13. Factor-Preservation Gate

## Compiler Gate G1

The switched expression is accepted only if the coefficient attached to the surviving modulus variables can be proved to lie in the weight class required by the target theorem。

Formally：

$$
\boxed{
\mathrm{WeightClass}_{out}
\supseteq
\mathrm{RequiredClass}_{provider}.
}
$$

A visual factorization of the integer：

$$
d=d_1d_2d_3
$$

is not enough；

the coefficient sequence must factor appropriately。

---

# 14. Upper-Domination Gate

## Compiler Gate G2

Switching must preserve a rigorous upper relation：

$$
\boxed{
\mathcal U_H^{original}
\le
\mathcal U_H^{switched}
+
\mathcal E_{sw}
}
$$

or exact equality plus controlled truncation。

Required：

$$
\boxed{
\mathcal E_{sw}
=
o(\mathfrak M)
}
$$

or an explicitly budgeted constant loss。

This prevents a weight-class conversion from silently losing bad configurations。

---

# 15. Residue Gate

The target congruence after switching still has：

$$
\boxed{
rp
\equiv
N
\pmod K.
}
$$

where：

$$
N=2m
$$

grows with：

$$
X.
$$

If target theorem has constants depending arbitrarily on a fixed residue parameter：

$$
a,
$$

then setting：

$$
a=N
$$

is not automatically uniform。

So：

## Compiler Gate G3

need either：

1. theorem uniform in the residue class；
2. a proof that the theorem's dependence on $a=N$ is harmless；
3. or a transformation eliminating $N$ from the analytic phase。

---

# 16. Boundary / Support Gate

switching introduces：

- $k$ range；
- divisibility endpoints；
- coprimality exclusions；
- possible repeated factor allocations。

Therefore：

## Compiler Gate G4

all boundary terms must satisfy：

$$
\boxed{
\mathcal E_{boundary}
=
o(\mathfrak M)
}
$$

or enter the TCDC constant ledger explicitly。

---

# 17. Coprimality Firewall

For：

$$
rp+Kd_1=N,
$$

if：

$$
(K,N)=1,
$$

then：

$$
(rp,K)=1.
$$

But if：

$$
(K,N)>1,
$$

degenerate local configurations can appear。

So the switching compiler must separately track：

$$
\boxed{
(K,N)
}
$$

and center-divisor factors。

These may be：

- favorable zero-density cells；
- exact locks；
- or exceptional moduli。

---

# 18. Shift-Free Lattice Form

Fix：

$$
r,
K.
$$

solutions：

$$
rp+Kd=N
$$

form an affine lattice。

For two solutions：

$$
(p_1,d_1),
\qquad
(p_2,d_2),
$$

subtract：

## Theorem TCDC-20-T02

$$
\boxed{
r(p_1-p_2)
+
K(d_1-d_2)
=
0.
}
$$

The large Goldbach shift：

$$
N
$$

vanishes from the difference equation。

---

# 19. What Shift-Free Lattice Actually Solves

T02 means：

> once two configurations are compared inside dispersion / second-moment arguments，
> the large shift need not remain in the oscillatory difference phase。

This is valuable for：

- dispersion；
- Cauchy-Schwarz；
- pair correlation；
- lattice spacing。

---

# 20. What Shift-Free Lattice Does Not Solve

the base solution：

$$
(p_0,d_0)
$$

still satisfies：

$$
rp_0+Kd_0=N.
$$

So the residue class / affine origin still depends on：

$$
N.
$$

Therefore：

## Research Firewall TCDC-20-R01

$$
\boxed{
\text{shift-free differences do not by themselves prove full residue uniformity}.
}
$$

They remove $N$ from difference geometry，

not necessarily from theorem constants。

---

# 21. Latest Yang v2 Update

TCDC-19 was intentionally conservative about the status of：

$$
5/8.
$$

The latest audited Yang revision：

$$
\boxed{
\text{arXiv:2608.13299v2,
revised 2026-09-03}
}
$$

states in its abstract that：

> for a special class of convolution forms equipped with well-factorable weights，
> one may use the level：

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

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

---

# 22. Correction to TCDC-19

## Correction TCDC-20-C01

The TCDC-19 phrasing：

> $5/8$ is only an external benchmark pending any demonstrated weight-class conversion

was too strong。

It should be replaced by：

$$
\boxed{
\text{a special convolution weight-class lift to }5/8
\text{ is now explicitly demonstrated by Yang v2};
}
$$

however：

$$
\boxed{
\text{TCDC high-shell upper object has not yet been proved to belong to that special class}.
}
$$

So the missing task is：

# Compiler Porting

not：

# Inventing the Conversion Technology from Nothing

---

# 23. Why Yang v2 Is Highly Relevant

Yang's convolution object is of the form：

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

TCDC：

$$
\boxed{
rp
\equiv
2m
\pmod q.
}
$$

So the multiplicative skeleton is exactly aligned：

$$
\ell
\leftrightarrow
r.
$$

This is currently the closest audited external theorem family to the TCDC semiprime marginal。

---

# 24. But Skeleton Match Is Not Full Match

Need additionally verify：

1. shell support for $r$；
2. prime / rough coefficient assumptions；
3. exact well-factorable weight produced by TCDC sieve；
4. modulus factorization；
5. residue dependence：
   $$
   a=2m;
   $$
6. dyadic smoothing；
7. error uniformity；
8. endpoint truncations。

Therefore：

## Research Firewall TCDC-20-R02

$$
\boxed{
\text{Yang-v2 special-class success}
\neq
\text{automatic TCDC provider}.
}
$$

---

# 25. Maynard III Residue Interface

Maynard III proves mean-value results for primes in arithmetic progressions beyond：

$$
x^{1/2}
$$

for moduli with suitable divisor structure，

with the key feature of uniformity in residue classes in the theorem family。

Thus it supplies exactly the kind of residue-uniformity interface that TCDC needs。

But its modulus / convolution factorization hypotheses must be satisfied after switching。

---

# 26. Compiler View of Maynard III

The compiler should try to manufacture moduli：

$$
\boxed{
K=q_1q_2
}
$$

or：

$$
\boxed{
K=q_1q_2q_3
}
$$

with factor sizes lying in the relevant Type-II admissible windows。

So Maynard III is not simply：

> another exponent。

It is：

# A Target Modulus Shape for the Compiler

---

# 27. Compiler Acceptance Contract

A high-shell switched provider is legally accepted only if all are proved：

```yaml
G1_factorability:
  switched_coefficients:
    satisfy_required_weight_class

G2_upper_polarity:
  switched_count:
    dominates_original_bad_count
  loss:
    o(main)_or_budgeted

G3_residue:
  a_equals_2m:
    uniform_or_removed

G4_support:
  dyadic_ranges:
    inside_provider_hypotheses

G5_coprimality:
  exceptional_center_factors:
    separated

G6_coefficients:
  divisor_bounded_or_SW_conditions:
    verified

G7_errors:
  shell_sum:
    o(main)

G8_overlap:
  no_uncontrolled_double_count
```

---

# 28. Weight-Class Conversion Efficiency

Define：

## Definition TCDC-20-D02

$$
\boxed{
\eta_{\mathrm{conv}}
\in[0,1]
}
$$

as the normalized high-shell mass fraction for which the compiler successfully places the upper object into the stronger provider class。

This is not an arithmetic invariant；

it is a theorem-coverage metric。

---

# 29. Partial Conversion Portfolio

Suppose high shell：

- fraction：
  $$
  \eta_{\mathrm{conv}}
  $$
  receives level：
  $$
  \theta_+;
  $$
- remainder receives：
  $$
  \theta_0.
  $$

low shell stays：

$$
3/5.
$$

Then：

## Theorem TCDC-20-T03

$$
\boxed{
\frac1{\theta_H}
=
\frac{
w_L
}{
3/5
}
+
w_H
\left[
\frac{
1-\eta_{\mathrm{conv}}
}{
\theta_0
}
+
\frac{
\eta_{\mathrm{conv}}
}{
\theta_+
}
\right].
}
$$

This compiles theorem coverage directly into the TCDC harmonic ledger。

---

# 30. Pure $5/8$ Conversion Cannot Close Without Coupling

Even if：

$$
\eta_{\mathrm{conv}}=1,
$$

$$
\theta_+=5/8,
$$

$$
\theta_0=3/5,
$$

TCDC-19 gives：

$$
\boxed{
\theta_H
=
0.618224404627.
}
$$

which is below：

$$
\boxed{
\theta_c
=
0.641107455868.
}
$$

Therefore：

## Theorem TCDC-20-T04
### $5/8$ Conversion Ceiling

With low shell fixed at：

$$
3/5,
$$

a compiler whose strongest high-shell output is only：

$$
5/8
$$

cannot close the benchmark simplex by exponent lift alone。

---

# 31. Required Coupled Discount after Perfect $5/8$ Conversion

T04 portfolio leaves：

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

or about：

$$
\boxed{
2.806\%
}
$$

normalized bad-mass recovery。

So weight-class conversion and TCDC transport must still be joined。

---

# 32. Alternative：Direct High-Shell Constant Saving

Switching may do more than change a formal level。

It may improve the actual high-shell upper constant relative to the one-stage：

$$
F(3\theta).
$$

Let：

$$
\boxed{
\sigma_H
}
$$

be a fractional saving on the high-shell bad-mass contribution after the：

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

portfolio。

---

# 33. High-Shell Contribution

under that portfolio，

high-shell semiprime bad cost is：

$$
\boxed{
B_H
=
s_*
w_H
F(15/8).
}
$$

numerically：

$$
\boxed{
B_H
\approx
0.573176351881.
}
$$

---

# 34. Constant-Saving Threshold

To replace the entire remaining coupling requirement by high-shell constant saving alone，

need：

## Theorem TCDC-20-T05

$$
\boxed{
\sigma_H
>
\frac{
\chi_{\rm req}
}{
B_H
}
}
$$

numerically：

$$
\boxed{
\sigma_H
>
0.048963196776.
}
$$

or about：

$$
\boxed{
4.896\%
}
$$

relative saving on the high-shell upper contribution。

---

# 35. Meaning of T05

Thus a switching theorem need not necessarily reach：

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

It can instead：

- remain effectively near：
  $$
  5/8;
  $$
- but save about：
  $$
  4.9\%
  $$
  on the high-shell upper constant。

This is a second escape route beyond exponent lift。

---

# 36. Three Compiler Success Modes

TCDC-20 therefore identifies three legitimate high-shell success modes。

## Mode A — Exponent Lift

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

with low shell：

$$
3/5,
$$

and no coupling。

## Mode B — $5/8$ + Transport

legal high shell：

$$
5/8
$$

plus：

$$
\boxed{
\chi>0.0280645\ldots
}
$$

## Mode C — $5/8$ + Constant Saving

legal high shell：

$$
5/8
$$

plus high-shell upper saving：

$$
\boxed{
\sigma_H>0.0489632\ldots
}
$$

under ideal neutral / mixed benchmark。

Hybrid combinations are allowed。

---

# 37. General Compiler Closure Ledger

Define：

- harmonic level：
  $$
  \theta_H;
  $$
- upper-constant saving：
  $$
  \sigma;
  $$
- coupled discount：
  $$
  \chi;
  $$
- neutral efficiency：
  $$
  \alpha;
  $$
- mixed multiplier：
  $$
  \kappa_{PS}.
  $$

Then final bad-mass gate can be written schematically：

## Definition TCDC-20-D03

$$
\boxed{
\mathfrak G_{\mathrm{comp}}
=
\alpha
-
\kappa_{PS}p_*s_*
-
\mathcal K_{SP}(\theta_H,\sigma)
+
\chi.
}
$$

Compiler closure requires：

$$
\boxed{
\mathfrak G_{\mathrm{comp}}>0.
}
$$

---

# 38. Residue-Removal Compiler

Suppose after factor-first switching we obtain：

$$
rp+Kd=N.
$$

A dispersion proof compares pairs：

$$
(r,p_1,d_1),
\qquad
(r,p_2,d_2).
$$

T02：

$$
r(p_1-p_2)+K(d_1-d_2)=0.
$$

This suggests：

# Shift-Free Pair Geometry

can be used to eliminate：

$$
N
$$

from the hard oscillatory difference。

---

# 39. Residue-Removal Acceptance Condition

A successful residue compiler must prove：

## Gate G3b

after all Cauchy-Schwarz / dispersion operations，

the remaining dependence on：

$$
N
$$

is confined to：

- harmless coprimality indicators；
- bounded phases；
- or residue classes covered uniformly by a Maynard-III-type theorem。

Without this proof，

shift-free lattice algebra alone is insufficient。

---

# 40. Goldbach Shift Compatibility

One favorable property：

all congruences descend from the same global：

$$
N=2m.
$$

So reductions modulo factors：

$$
q_1,
q_1q_2,
\ldots
$$

are automatically compatible：

$$
\boxed{
N\bmod q_1q_2
\equiv
N\bmod q_1.
}
$$

This matches the type of nested residue compatibility appearing in balanced Type-II frameworks。

But theorem-specific coprimality and factor-size conditions remain。

---

# 41. Center-Divisor Firewall

If a modulus factor：

$$
q\mid N,
$$

then residue：

$$
N\bmod q
$$

is zero，

while prime-product variables may be units。

Such factors can produce：

- automatic zero contributions；
- degenerate residues；
- exact center locks。

So compiler should strip：

$$
(N,q)>1
$$

factors before invoking primitive-residue theorems。

This may actually reduce bad mass，

but must be handled explicitly。

---

# 42. Compiler Normal Form

A desirable final switched term has shape：

$$
\boxed{
\sum_{\mathbf q}
\omega_{\mathbf q}
\left[
\sum_{r\sim R}
\sum_{p\sim P}
\alpha_r\beta_p
\mathbf1_{
rp\equiv N\pmod{Q(\mathbf q)}
}
-
\mathrm{MT}
\right].
}
$$

Required：

- $\omega_{\mathbf q}$ in target factorability class；
- $Q(\mathbf q)$ in target modulus range；
- $\alpha_r,\beta_p$ meet theorem coefficient hypotheses；
- residue $N$ legal；
- main term MT matches the sieve normalization。

This is the compiler's canonical output IR。

---

# 43. Weight-Class Compiler as a Typed Transformation

We can formalize：

$$
\boxed{
\mathsf{Switch}:
\mathsf{UpperSieveWF}
\longrightarrow
\mathsf{ConvolutionProviderClass}
}
$$

only when gates：

$$
G1\text{--}G8
$$

pass。

If any gate fails，

the transformation is only algebraic，

not proof-preserving。

This typed viewpoint prevents theorem-class confusion。

---

# 44. Latest Yang v2 Changes the Feasibility Assessment

Yang v2 explicitly demonstrates that：

> special convolution forms with well-factorable weights can be reorganized so Pascadi's triply-well-factorable prime-distribution level：

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

becomes usable。

Therefore：

$$
\boxed{
\text{weight-class lift by structural convolution is a real technique}.
}
$$

The TCDC question is now：

> does factor-first Goldbach switching land in Yang's special class or a close variant?

This is substantially more concrete than TCDC-19's generic search。

---

# 45. But $5/8$ Alone Remains Below the Constant Barrier

even if the answer is yes for all high shell，

T04–T05 say：

$$
\boxed{
\text{conversion success}
\neq
\text{closure}.
}
$$

Need at least one further resource：

- transport discount；
- constant saving；
- stronger high-shell level；
- improved low shell；
- improved neutral / mixed providers。

So compiler engineering must report both：

# Type Success

and：

# Constant Success。

---

# 46. Compiler Output Record

Each switched theorem candidate should produce：

```yaml
shell:
  [t_min,t_max]

input_weight:
  class
  level

factorization:
  d1
  d2
  d3

switch:
  complementary_variable_range

output_modulus:
  factor_sizes

output_weight:
  proved_class

residue:
  fixed
  uniform
  shift_free_partial

provider:
  theorem

effective_level:
  theta

constant_saving:
  sigma

boundary_loss:
  epsilon

status:
  PASS_or_FAIL
```

Only PASS records enter TCDC-19 harmonic compiler。

---

# 47. Current Audited External State

## Yang v2 — 2026-09-03

Latest abstract states：

- special convolution forms；
- well-factorable weights；
- uses Pascadi triply-well-factorable prime-distribution technology；
- usable level：
  $$
  x^{5/8-o(1)}.
  $$

This is the strongest direct evidence of a successful convolution weight-class lift in the currently audited source set。

## Pascadi 2025

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

for primes with triply-well-factorable weights。

## Maynard III 2025

large-moduli mean values with the key feature of residue-class uniformity for the stated structured-modulus families。

---

# 48. What Is Still Missing

No audited statement in this round yet proves：

$$
\boxed{
\mathcal U_H^{TCDC}
}
$$

satisfies the exact special-class hypotheses of Yang v2。

Specifically unresolved：

1. upper-sieve weight decomposition after Goldbach shell partition；
2. factor support matching；
3. growing：
   $$
   N=2m
   $$
   residue；
4. center factors；
5. $k$-range after switching；
6. full high-shell coverage；
7. uniform main-term normalization。

---

# 49. Corrected External-Technology Hierarchy

After Yang v2：

### Tier 1 — Known underlying prime technology

Pascadi：

$$
5/8.
$$

### Tier 2 — Known special convolution lift

Yang v2：

special convolution forms can exploit：

$$
5/8.
$$

### Tier 3 — TCDC port

not yet proved：

$$
\boxed{
\text{Goldbach high-shell upper}
\to
\text{Yang special class}.
}
$$

This is now the exact external-provider bottleneck。

---

# 50. Proof-Resource Priority

Given T05，

if the TCDC-to-Yang port can deliver high-shell：

$$
5/8
$$

then analytic effort should not immediately chase a harder exponent。

Instead compare costs：

### Option A

prove：

$$
\chi>0.0281.
$$

### Option B

extract：

$$
4.90\%
$$

high-shell constant saving。

### Option C

upgrade high shell above：

$$
0.65718.
$$

Whichever theorem is cheapest should be pursued。

This is TCDC proof-economics in a concrete final-barrier setting。

---

# 51. Coupling / High-Level Trade Table

Low shell fixed：

$$
3/5.
$$

Selected coupled discounts give required high-shell level：

| Coupled discount $\chi$ | critical harmonic level | required high-shell level |
|---:|---:|---:|
| $0.000000$ | $0.641107455868$ | $0.657177511828$ |
| $0.010000$ | $0.632761998398$ | $0.645338860489$ |
| $0.020000$ | $0.624631018865$ | $0.633919193542$ |
| $0.028065$ | $0.618224404627$ | $0.625000000000$ |
| $0.030000$ | $0.616706354053$ | $0.622896655183$ |
| $0.050000$ | $0.601445335805$ | $0.601962886115$ |

At：

$$
\chi
\approx
0.0280645,
$$

high-shell target drops essentially to：

$$
5/8.
$$

So proving a roughly：

$$
2.81\%
$$

coupled discount and porting Yang-v2-quality：

$$
5/8
$$

are naturally matched objectives。

---

# 52. Formal Theorem Queue

```yaml
TCDC_20:

  T01:
    algebraic_reindexing_does_not_prove_weight_class_promotion

  T02:
    shift_free_lattice_difference

  T03:
    partial_conversion_harmonic_level

  T04:
    five_eighth_conversion_ceiling_without_coupling

  T05:
    high_shell_constant_saving_threshold

  C01:
    correction_to_TCDC_19_Yang_status

  D01:
    factor_preserving_switched_form

  D02:
    conversion_coverage_metric

  D03:
    compiler_gap
```

---

# 53. Compiler Gate Queue

```yaml
TCDC_20_Compiler_Gates:

  G1:
    factorability_preserved_or_proved

  G2:
    upper_domination_preserved

  G3:
    growing_residue_uniform_or_removed

  G4:
    boundary_loss_controlled

  G5:
    coprimality_and_center_factors_separated

  G6:
    coefficient_hypotheses_matched

  G7:
    total_error_o_main_or_budgeted

  G8:
    overlap_and_shell_coverage_exact
```

---

# 54. Barrier Queue

```yaml
TCDC_20_Barriers:

  B01:
    TCDC_upper_weight_not_yet_proved_in_Yang_v2_special_class

  B02:
    naive_switching_does_not_upgrade_factorability

  B03:
    growing_Goldbach_residue_not_fully_removed

  B04:
    factor_first_switching_output_class_not_verified

  B05:
    full_high_shell_coverage_not_verified

  B06:
    five_eighth_alone_still_below_no_coupling_constant_gate

  B07:
    no_uniform_two_point_eight_percent_coupled_discount

  B08:
    no_uniform_four_point_nine_percent_high_shell_constant_saving
```

---

# 55. Acceptance Gate

TCDC-20 formal core前需：

1. original upper-sieve object explicit；
2. complementary divisor exact；
3. naive reindexing vs factorability separated；
4. factor-first principle explicit；
5. stronger weight class not assumed；
6. upper polarity retained；
7. growing-residue issue explicit；
8. shift-free difference not overclaimed；
9. Yang v2 date / latest revision distinguished from earlier v1 statements；
10. TCDC-19 correction explicit；
11. $5/8$ conversion constant gate recomputed；
12. constant-saving alternative quantified；
13. no claim TCDC satisfies Yang special-class hypotheses；
14. no Goldbach proof claim。

---

# 56. Verdict

本輪可以正式給：

$$
\boxed{
\mathrm{TCDC\mbox{-}20\ SWITCHING\mbox{-}COMPILER\ FIREWALL\ PASS}.
}
$$

最重要 algebraic firewall：

$$
\boxed{
N-rp=dk
}
$$

本身不會把：

$$
\lambda_d^+
$$

升級成更強 factorability class。

合法順序必須是：

$$
\boxed{
\text{Factor First}
\rightarrow
\text{Switch Second}.
}
$$

canonical switched form：

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

並逐項驗證 factor coefficients。

最重要 external update：

Yang 最新：

$$
\boxed{
\text{v2，2026-09-03}
}
$$

已明確證明：

> 對一類特殊 convolution forms with well-factorable weights，
> 可調用 Pascadi triply-well-factorable prime distribution，
> 使用：

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

level。

所以：

$$
\boxed{
\text{convolution-induced weight-class lift確實存在於當代技術中}.
}
$$

TCDC 現在缺的是：

$$
\boxed{
\text{port proof}
}
$$

而不是證明這種技術概念上可行。

然而即使完整 high shell成功 port到：

$$
5/8,
$$

low shell維持：

$$
3/5,
$$

仍只有：

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

並需：

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

約：

$$
\boxed{
2.806\%
}
$$

coupled recovery。

若不使用 coupling，

等價 high-shell constant saving至少：

$$
\boxed{
\sigma_H
>
0.048963196776
}
$$

約：

$$
\boxed{
4.896\%.
}
$$

或者把 high-shell effective level真正推到：

$$
\boxed{
>0.6571775.
}
$$

因此 switching compiler有三種成功終點：

1. stronger exponent；
2. $5/8$ + TCDC coupling；
3. $5/8$ + direct constant saving。

目前：

$$
\boxed{
\mathrm{YANG\mbox{-}V2\ PORT\ TO\ TCDC\ NOT\ YET\ PROVED}.
}
$$

$$
\boxed{
\mathrm{NO\ FINAL\ COMPILER\ GAP\ CROSSING}.
}
$$

因此：

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

---

# 57. 下一輪建議

# TCDC-21
## Yang-v2 Port Audit / Special-Convolution Hypothesis Match

下一輪停止抽象 switching。

直接逐條拿 Yang v2 的 special convolution theorem對 TCDC high-shell object做 type checking。

### Part A — Exact Yang-v2 Input Schema

抽出：

- coefficient support；
- factorability；
- prime / rough sequence hypotheses；
- modulus range；
- residue dependence；
- special convolution condition。

### Part B — TCDC Type Map

map：

$$
r
\to
\ell,
$$

$$
p
\to
p,
$$

$$
d_i
\to
q_i,
$$

$$
2m
\to
a.
$$

### Part C — Fail Fast

對每 hypothesis：

```yaml
PASS
FAIL
REPAIRABLE
UNKNOWN
```

不要先假設能用。

### Part D — Repair Operators

若 weight-class fail：

factor-first decomposition。

若 residue fail：

shift-free / Maynard-III uniformity。

若 support fail：

shell trim。

### Part E — Ported Exponent

只有所有 gates PASS 後，

才允許把：

$$
5/8
$$

放入 TCDC harmonic compiler。

若只能覆蓋部分 high shell，

計：

$$
\eta_{\mathrm{conv}},
\quad
\theta_H,
\quad
\chi_{\rm req}.
$$

這會真正回答：

> Yang v2 的最新 $5/8$ convolution breakthrough，
> 距離 TCDC semiprime marginal到底還差哪一條 hypothesis？

---

# 附錄 A：High-Shell Upper Object

```yaml
object:
  sum_r_prime
  sum_p_prime
  sum_d:
    lambda_plus_d
    * 1[d divides 2m-rp]

shell:
  3/8 <= log_X(r) <= 1/2

polarity:
  upper
```

---

# 附錄 B：Compiler Normal Form

```yaml
factor:
  d:
    d1*d2*d3

switch:
  2m-rp:
    k*d1*d2*d3

keep_modulus:
  k*d2*d3

solve:
  d1:
    (2m-rp)/(k*d2*d3)

required:
  coefficient_factorability
  residue_uniformity
  boundary_control
```

---

# 附錄 C：Constant Ledger

```yaml
low_shell:
  level:
    3/5

high_shell_after_full_5_8_conversion:
  level:
    5/8

harmonic_level:
  0.618224404627

coupling_required:
  0.028064546504

high_shell_bad_contribution:
  0.573176351881

relative_high_shell_constant_saving_if_no_coupling:
  0.048963196776
```

---

# 附錄 D：Switching Success Modes

```yaml
Mode_A:
  high_shell_level:
    > 0.6571775
  coupling:
    0

Mode_B:
  high_shell_level:
    5/8
  coupling:
    > 0.028064546504

Mode_C:
  high_shell_level:
    5/8
  high_shell_constant_saving:
    > 0.048963196776

Hybrid:
  exponent
  + coupling
  + constant_saving
```

---

# 附錄 E：Commit Decision

```yaml
research_round:
  TCDC-20

decision:

  naive_weight_promotion:
    rejected

  factor_first_switching:
    formalized

  shift_free_difference:
    proved

  residue_uniformity:
    partial_only

  Yang_v2_special_5_8_conversion:
    externally_verified_as_existing_technology

  TCDC_to_Yang_port:
    not_proved

  full_high_shell_five_eighth:
    not_yet_legal_for_TCDC

  constant_gap_after_hypothetical_port:
    quantified

  Goldbach_proved:
    false

next:
  TCDC-21_YangV2_PortAudit_SpecialConvolution_HypothesisMatch
```

---

# References / External Technology Audit

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

   Latest audited abstract states that for a special class of convolution forms equipped with well-factorable weights, the proof can use level:
   $$
   x^{5/8-o(1)}
   $$
   from Pascadi's prime-distribution theorem with triple-well-factorable weights.

   This materially upgrades the feasibility assessment relative to Yang v1 / the earlier TCDC-19 audit.

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

   Audited abstract states that primes and smooth numbers are equidistributed to moduli:
   $$
   x^{5/8-o(1)}
   $$
   using triply-well-factorable weights for primes, with additional improvements for well-factorable linear-sieve weights.

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

   Audited abstract states that mean value estimates beyond:
   $$
   x^{1/2}
   $$
   are obtained for moduli with conveniently sized divisors, with the key feature of uniformity in the residue classes in the stated theorem family.

4. TCDC-18 / TCDC-19.

   Internal constants used here:
   - benchmark critical harmonic level:
     $$
     \theta_c
     =
     0.641107455868;
     $$
   - low-$3/5$ / high-$5/8$ harmonic level:
     $$
     0.618224404627;
     $$
   - residual coupled discount:
     $$
     0.028064546504.
     $$