> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # TCDC-05 ## Quarter-Root Weighted Pressure / Chen-Lemma Adaptation ### Strict-Semiprime Half-Weight, Medium-Divisor Multiplicity, Extra Prime-Branch Cost, Constant Gate and Balanced-Convolution Provider Map **Parent methodology:** TCDC — Target–Condition Dual Closure **Connected research lines:** RG-SCP / SRBRD **Round:** TCDC-05 **Date:** 2026-10-02 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Pointwise Weighted-Sieve Lemma / Chen-Type Adaptation / Constant Audit **Prerequisites:** TCDC-00 ～ TCDC-04; SRBRD-00 ～ SRBRD-26 **Status:** Establishes a quarter-root Chen-type pointwise lower weight for strict semiprime pressure; proves that the half-weight coefficient is minimal within the natural linear ansatz; the standard $3/5$ linear-sieve main constant retains a small positive margin after subtracting the half medium-divisor term, but the prime branch and exactly-one-medium correction remain unclosed; Goldbach remains unproved --- # 0. Goals of This Round TCDC-04 has written high-suffix pressure as: $$ \boxed{ \Pi_\lambda = S_\lambda(z)-P_\lambda, } $$ where: - $S_\lambda(z)$： the complement $y=2m-rp$ passes cube-root RootRough; - $P_\lambda$： those among them for which $y$ is prime; - $\Pi_\lambda$： the strict semiprime branch. Directly using: $$ z=X^{1/3} $$ the lower linear sieve requires: $$ \theta>2/3, $$ Thus the current: $$ 3/5 $$ and even: $$ 5/8 $$ cannot directly carry the lower sieve all the way through. TCDC-04 therefore instead uses: $$ \boxed{ w=X^{1/4} } $$ as a coarse cutoff, such that: $$ \operatorname{RootRough}_w(y) $$ allowing only: $$ P_1,\ P_2,\ P_3. $$ The goal of TCDC-05: > Within this finite factor-depth domain, > establish a genuinely pointwise Chen-type weight > that directly lower-bounds: > $$ > \mathbf1_{\{ > y\text{ strict semiprime}, > \operatorname{lpf}(y)>z > \}}. > $$ --- # 1. Basic Scales Let: $$ X=2m-3. $$ Define: $$ \boxed{ w=X^{1/4}, } $$ and the cube-root cutoff: $$ \boxed{ z=z_3(m), \qquad z^3\ge X. } $$ High-suffix base: $$ \boxed{ \mathcal A_\lambda = \left\{ (r,p): r,p\in\mathbb P, \quad r\ge\lambda\sqrt m, \quad p\ge r, \quad rp\le X \right\}. } $$ Complement: $$ \boxed{ y=2m-rp. } $$ --- # 2. Desired Pressure Indicator Define: $$ \boxed{ H_z(y) = \mathbf1_{\{ y\text{ semiprime}, \operatorname{RootRough}_z(y) \}}. } $$ For composite: $$ y, $$ this is equivalent to: $$ \boxed{ y=q_1q_2, \qquad q_1,q_2\in\mathbb P, \qquad zz. $$ Thus: ## Theorem TCDC-05-T01 $$ \boxed{ H_z(y)=1 \Longrightarrow y>z^2. } $$ Therefore: $$ y\le z^2 $$ cannot contribute to strict semiprime pressure at all. This is structurally compatible with the classical Chen lemma often working in the: $$ n>x^{2/3} $$ region. --- # 4. Quarter-Root Factor Depth If: $$ \operatorname{RootRough}_w(y) $$ and: $$ y $$ is composite, then: $$ \operatorname{lpf}(y)>w. $$ If: $$ \Omega_{\mathrm{mult}}(y)\ge4, $$ then: $$ y>w^4=X, $$ a contradiction. Thus: ## Theorem TCDC-05-T02 $$ \boxed{ \operatorname{RootRough}_w(y) \Longrightarrow y\in \mathbb P \cup P_2 \cup P_3. } $$ --- # 5. Medium-Prime Multiplicity Define: $$ \boxed{ \nu_{w,z}(y) = \sum_{\substack{ \ell\in\mathbb P\\ w<\ell\le z }} v_\ell(y), } $$ where: $$ v_\ell(y) $$ is the prime-factor multiplicity of $\ell$ in $y$. Since: $$ \Omega_{\mathrm{mult}}(y)\le3 $$ in the coarse composite domain, Thus: $$ \boxed{ 0\le\nu_{w,z}(y)\le3. } $$ For prime: $$ y, $$ If: $$ wz, $$ then: $$ y>z^3\ge X, $$ a contradiction. Thus: $$ \nu_{w,z}(y)\ge1. $$ ### $\nu=1$ $$ E_1=1, $$ Thus: $$ \omega=0. $$ ### $\nu=2$ $$ \omega=0. $$ ### $\nu=3$ $$ \omega=-1/2. $$ Therefore: $$ \boxed{ P_3 \text{ never receives positive pressure weight}. } $$ This completes the proof of T03. --- # 13. Why the Half Coefficient Is Forced Consider the natural linear ansatz: $$ \boxed{ 1 - a\nu - b\mathbf1_{\{\nu=1\}} } $$ Considering only composite coarse states, we want all bad states: $$ \nu=1,2,3 $$ to have weight: $$ \le0. $$ Constraints: ### $\nu=1$ $$ 1-a-b\le0. $$ ### $\nu=2$ $$ 1-2a\le0. $$ ### $\nu=3$ $$ 1-3a\le0. $$ Thus: ## Theorem TCDC-05-T04 ### Half-Weight Optimality $$ \boxed{ a\ge1/2. } $$ If we take the minimum: $$ a=1/2, $$ then: $$ \boxed{ b\ge1/2. } $$ Therefore: $$ \boxed{ (a,b)=(1/2,1/2) } $$ is the componentwise minimal admissible choice for this two-statistic linear ansatz. Thus half-weight is not an arbitrary imitation of Chen. --- # 14. Structure of $E_1$ Since: $$ E_1 $$ requires compositeness and medium multiplicity exactly: $$ 1, $$ within the coarse: $$ P_2/P_3 $$ domain it splits into exactly: ## Medium-$P_2$ Branch $$ \boxed{ y=\ell q, } $$ $$ w<\ell\le z T05 is a proof-friendly lower relaxation, > whereas TCDC-04 is an exact least-factor identity. --- # 18. Exact Medium-Divisor Inclusion–Exclusion For composite RootRough$_w$: $$ \nu=\nu_{w,z}(y)\in\{0,1,2,3\}. $$ we have the exact identity: $$ \boxed{ \mathbf1_{\{\nu\ge1\}} = \nu - \binom{\nu}{2} + \binom{\nu}{3}. } $$ Thus shell loss can also be written as a finite hyperintersection expansion. This gives another route: $$ \boxed{ B = D_1-D_2+D_3. } $$ However: - $D_1$ requires an upper bound; - $D_2$ has a positive sign and requires a lower bound; - $D_3$ again requires an upper bound. The half-weight in T05 avoids directly requesting a: $$ D_2 $$ lower bound, at the cost of adding an: $$ E_1 $$ upper cost. --- # 19. Half-Weight Identity Behind T05 For: $$ \nu=0,1,2,3, $$ we have: $$ \boxed{ \mathbf1_{\{\nu\ge1\}} = \frac12\nu + \frac12\mathbf1_{\{\nu=1\}} - \frac12\mathbf1_{\{\nu=3\}}. } $$ Thus: $$ \boxed{ B \le \frac12D_1 + \frac12E_1. } $$ dropping the favorable: $$ +\frac12E_3 $$ recovery. T05 is precisely this pointwise relaxation. --- # 20. Difference from Classical Chen's Lemma The classical Chen lower-weight lemma for: $$ 1_{\mathcal P_2}(n) $$ uses: - coefficient： $$ 1/2 $$ on the medium-prime divisor term; - a triple-factor correction; - a square-factor correction. Its target: $$ \mathcal P_2 $$ includes: - primes; - semiprimes. By contrast, this TCDC target is: $$ \boxed{ \text{strict semiprime with lpf}>z. } $$ Thus we must additionally pay for: ## Extra Cost I $$ \boxed{ A_P } $$ the prime branch. ## Extra Cost II $$ \boxed{ A_{2,1} } $$ the medium-$P_2$ branch. These two costs are not bad states for the classical Chen target. Therefore: $$ \boxed{ \text{strict-pressure problem is combinatorially stricter than Chen's }P_2\text{ problem}. } $$ --- # 21. Square Correction Is Absorbed by Multiplicity The classical Chen formulation needs an explicit: $$ p^2\mid n $$ correction, because it uses divisor indicators rather than multiplicity bookkeeping. This round defines: $$ \nu_{w,z}(y) = \sum v_\ell(y). $$ Thus: - $\ell^2$ automatically contributes: $$ 2; $$ - $\ell^3$ contributes: $$ 3. $$ Thus square / cube multiplicity is already built into: $$ A_2. $$ This simplifies the TCDC-tailored weight. --- # 22. Coarse Lower-Sieve Constant at Current $3/5$ Assume a proof-admissible dimension-1 lower-sieve provider with: $$ D=X^{3/5}, $$ $$ w=X^{1/4}. $$ then: $$ \boxed{ s_0 = \frac{\log D}{\log w} = \frac{12}{5}. } $$ The classical lower function: $$ f_1(s) = \frac{ 2e^\gamma }{s} \log(s-1) $$ for: $$ 2\le s\le4. $$ Thus: ## Diagnostic TCDC-05-D01 $$ \boxed{ C_{\mathrm{coarse}} = f_1(12/5) = \frac{ 5e^\gamma }{6} \log\frac75 } $$ $$ \boxed{ C_{\mathrm{coarse}} \approx 0.499401183388. } $$ --- # 23. Medium-Divisor First-Order Upper Integral Let: $$ \ell=X^u, $$ $$ \frac14\le u\le\frac13. $$ Conditioning on: $$ \ell\mid y $$ reduces the available sieve level approximately to: $$ D/\ell. $$ Relative to: $$ w=X^{1/4}, $$ the parameter is: $$ \boxed{ s_\ell = 4 \left( \frac35-u \right). } $$ Its range: $$ \boxed{ 16/15 \le s_\ell \le 7/5. } $$ The upper linear sieve: $$ F_1(s) = \frac{ 2e^\gamma }{s} $$ for: $$ 10. } $$ That is: $$ \boxed{ 0.0841\text{ normalized units} } $$ a positive window of approximately this size. --- # 27. The New Strict-Pressure Constant Gate T05 still needs to pay: $$ A_P + \frac12A_{2,1} + \frac12A_{3,1}. $$ Thus if all terms have constants on the same normalized base scale: $$ C_P, $$ $$ C_{2,1}, $$ $$ C_{3,1}, $$ we require: ## Theorem Template TCDC-05-T07 $$ \boxed{ C_P + \frac12C_{2,1} + \frac12C_{3,1} < C_* } $$ to obtain positive weighted pressure. The current diagnostic: $$ \boxed{ C_* \approx 0.0841. } $$ This is a very thin constant budget. --- # 28. Why This Is Harder Than Classical Chen The classical Chen $P_2$ ledger need not subtract: $$ A_P $$ because primes are acceptable, and need not subtract the medium: $$ P_2 $$ branch: $$ A_{2,1}. $$ Our strict pressure ledger needs both. Thus: $$ \boxed{ \text{the same half-weight that is effective for Chen leaves a much tighter residual budget here}. } $$ This is the most important quantitative warning of this round. --- # 29. Balanced Nature of the Base Variables High suffix: $$ r\ge\lambda\sqrt m, $$ and: $$ p\ge r, $$ $$ rp\le2m. $$ Thus: $$ \boxed{ r,p \asymp X^{1/2} } $$ up to constants depending on: $$ \lambda. $$ Therefore: $$ \boxed{ rp\equiv2m\pmod d } $$ is a balanced prime-product convolution. --- # 30. Maynard-III Balanced Type-II Shape Match Maynard-III Proposition 5.3 specifically treats balanced convolutions: $$ N,M\approx x^{1/2} $$ with modulus: $$ q_1q_2 $$ and residue compatibility: $$ a\equiv b\pmod{q_1}. $$ For the global Goldbach shift: $$ A=2m, $$ Take: $$ a=A\bmod q_1q_2, $$ $$ b=A\bmod q_1, $$ compatibility is automatic. Thus: ## Theorem-Match TCDC-05-M01 $$ \boxed{ \text{the high-suffix base has the correct balanced-convolution shape for Maynard-III Proposition 5.3}. } $$ However, this is still not a direct sieve provider. --- # 31. Why Proposition 5.3 Is Not Yet $A_1$ The coarse lower bound: $$ A_1=S_\lambda(w) $$ requires: - lower linear-sieve weights; - squarefree moduli up to: $$ D\approx X^{3/5}; $$ - a suitable factorization decomposition; - a remainder sum with the correct sign / norm. Proposition 5.3 provides only a mean-value discrepancy for certain factorized moduli. Thus: $$ \boxed{ \text{balanced convolution shape PASS} \neq \text{coarse lower-sieve provider PASS}. } $$ --- # 32. Medium Divisor Term as Convolution-Sieve Sequence The condition: $$ \ell\mid y $$ is equivalent to: $$ \boxed{ rp\equiv2m\pmod\ell. } $$ with: $$ w<\ell\le z. $$ If we additionally sieve the remaining complement by small primes: $$ d, $$ the modulus becomes: $$ \ell d. $$ Therefore: $$ A_2 $$ requires mean distribution for: $$ \boxed{ rp\equiv2m \pmod{\ell d}. } $$ This is precisely a: # Balanced Prime-Product Convolution with a Medium Forced Factor --- # 33. Yang 2026 Shape Interface Yang 2026 proves convolution-type Bombieri–Vinogradov estimates with well-factorable / upper linear-sieve weights for expressions of the form： $$ lp\equiv a\pmod q. $$ Thus in: # convolution shape and: # upper-weight polarity it is very close to: $$ A_2 $$ as displayed above. However, in the audited theorem the residue: $$ a $$ remains a fixed parameter, whereas in this problem: $$ a=2m\asymp X. $$ Thus: ## Theorem-Match TCDC-05-M02 $$ \boxed{ \text{Yang 2026 is a medium-term technology template, not a legal uniform Goldbach-shift provider as currently stated}. } $$ --- # 34. The $A_{2,1}$ Term $$ A_{2,1} $$ counts: $$ \boxed{ 2m = rp + \ell q, } $$ with four prime variables: $$ r,p,\ell,q, $$ and ranges: $$ r,p\asymp\sqrt X, $$ $$ X^{1/4}<\ell\le X^{1/3}, $$ $$ q>X^{1/3}. $$ This is not the classical Chen triple-factor cost. It is a: # Two-Product Additive Correlation and is a new term introduced by the strict-pressure adaptation. --- # 35. The $A_{3,1}$ Term $$ A_{3,1} $$ counts: $$ \boxed{ 2m = rp + \ell q_2q_3, } $$ where: $$ w<\ell\le z maximize the estimated analytic main constant. Thus: # Chen-Weight Design Itself Becomes an LP This fits TCDC / AI proof-operator synthesis very well. --- # 44. T03 Weight as the First LP Solution Restrict the features to: $$ \boxed{ 1, \mathbf1_{\mathbb P}, \nu, \mathbf1_{\{\text{composite},\nu=1\}} } $$ and fix the target coefficient: $$ 1, $$ then T04 proves that: $$ \boxed{ 1/2,\ 1/2 } $$ are the minimal penalty coefficients. Thus T03 can be viewed as the: # Minimal Four-Feature Pressure Weight --- # 45. Finite Computational Calibration This serves only as finite evidence. Take: $$ \lambda=1.1. $$ Define the weighted lower bound: $$ \boxed{ W_\lambda = A_1 - A_P - \frac12A_2 - \frac12E_1. } $$ and the crude bound: $$ \boxed{ W_\lambda^{simple} = A_1-A_P-A_2. } $$ Selected exact enumerations: | $m$ | $\lambda$ | $A_1$ | $A_P$ | $\Pi_\lambda$ | $A_2$ | $E_1$ | $W_\lambda$ | $W_\lambda^{simple}$ | |---:|---:|---:|---:|---:|---:|---:|---:|---:| | 1000 | 1.10 | 5 | 4 | 0 | 1 | 1 | 0.0 | 0 | | 5000 | 1.10 | 9 | 5 | 2 | 2 | 2 | 2.0 | 2 | | 10000 | 1.10 | 11 | 9 | 0 | 2 | 2 | 0.0 | 0 | | 50000 | 1.10 | 51 | 32 | 7 | 15 | 10 | 6.5 | 4 | All sampled cases satisfy: $$ \boxed{ W_\lambda \le \Pi_\lambda. } $$ For the example: $$ m=50000 $$ the weighted bound: $$ W_\lambda=6.5 $$ is closer to: $$ \Pi_\lambda=7, $$ than the crude bound: $$ 4 $$ as displayed above. Finite evidence cannot be promoted to a uniform theorem. --- # 46. Why Fractional Weight Is Fine $$ W_\lambda $$ can be a half-integer. This causes no problem. As long as: $$ \boxed{ W_\lambda>0, } $$ And: $$ W_\lambda\le\Pi_\lambda, $$ we obtain: $$ \Pi_\lambda>0. $$ Thus a weighted certificate need not be integer-valued. --- # 47. Correct Polarity of Every Term T05： $$ \Pi \ge A_1 - A_P - \frac12A_2 - \frac12A_{2,1} - \frac12A_{3,1}. $$ Thus the provider directions are: ### $A_1$ requires a: $$ \boxed{ \text{lower bound}. } $$ ### $A_P$ requires an: $$ \boxed{ \text{upper bound}. } $$ ### $A_2$ requires an: $$ \boxed{ \text{upper bound}. } $$ ### $A_{2,1}$ requires an: $$ \boxed{ \text{upper bound}. } $$ ### $A_{3,1}$ requires an: $$ \boxed{ \text{upper bound}. } $$ Apart from the coarse survivor, all the others have upper polarity. This is a very favorable proof architecture. --- # 48. Why This Still Does Not Automatically Beat Parity The prime branch: $$ A_P $$ and the strict pressure: $$ \Pi $$ compete on the same: $$ m/\log^3m $$ scale. If the upper sieve for: $$ A_P $$ carries the classical factor: $$ 2, $$ it may directly consume the entire: $$ 0.0841 $$ window. Thus: $$ \boxed{ \text{the strict-prime subtraction is the new parity-sensitive constant bottleneck}. } $$ --- # 49. TCDC Interpretation ### Target $$ \Pi_\lambda>0. $$ ### Enlarged Domain Quarter-root RootRough: $$ P_1/P_2/P_3. $$ ### Pointwise Proof Operator $$ \omega_{w,z}(y). $$ ### Bad Branches - prime; - medium $P_2$; - $P_3$. ### Weighted Closure If the weighted total: $$ W_\lambda>0, $$ then target pressure exists. This is a complete instance of TCDC-00: # Reverse Domain Enlargement → Weighted Closure to Target as displayed above. --- # 50. Formal Theorem Queue ```yaml TCDC_05: T01: desired_semiprime_above_z_squared T02: quarter_root_factor_depth_at_most_3 T03: strict_semiprime_Chen_weight_pointwise T04: half_weight_optimality_in_two_statistic_ansatz T05: summed_weighted_pressure_lower_bound T06: medium_divisor_integrated_constant T07: strict_pressure_constant_gate ``` --- # 51. External Match Queue ```yaml External_Match: MaynardIII_Prop5_3: balanced_prime_product_shape: PASS direct_lower_sieve_provider: NOT_YET Yang_2026: convolution_shape_for_medium_divisor: NEAR_MATCH growing_Goldbach_residue: MISMATCH_IN_AUDITED_STATEMENT Chen_classical: half_weight_philosophy: MATCH direct_strict_semiprime_target: NO extra_prime_cost: absent_classically extra_medium_P2_cost: absent_classically ``` --- # 52. Barrier Queue ```yaml TCDC_05_Barriers: B01: full_medium_subtraction_negative_constant B02: half_weight_margin_only_about_0_0841 B03: prime_branch_upper_cost_not_closed B04: exactly_one_medium_P2_cost_not_closed B05: triple_branch_switching_not_closed B06: balanced_convolution_lower_sieve_weight_match_not_closed B07: Goldbach_shift_uniform_convolution_provider_incomplete ``` --- # 53. Acceptance Gate Before the TCDC-05 formal core, we need: 1. the $H_z$ strict target definition; 2. multiplicity, not distinct-factor count; 3. the coarse factor-depth proof; 4. all prime/P2/P3 cases checked; 5. the half-weight optimality proof; 6. $E_1$ split into medium P2 / P3; 7. square multiplicity handling; 8. weighted sum polarity; 9. the linear-sieve constant audit labeled diagnostic unless the provider is proved; 10. the difference from classical Chen stated explicitly; 11. Maynard/Yang only shape-matched where legal; 12. finite calibration labeled computational; 13. no HSP proof claim; 14. no Goldbach proof claim. --- # 54. Verdict This round can formally give: $$ \boxed{ \mathrm{TCDC\mbox{-}05\ STRICT\mbox{-}SEMIPRIME\ HALF\mbox{-}WEIGHT\ PASS}. } $$ The most important pointwise lemma: $$ \boxed{ H_z(y) \ge \mathbf1_{\operatorname{RootRough}_w(y)} \left[ 1 - \mathbf1_{\mathbb P}(y) - \frac12\nu_{w,z}(y) - \frac12E_1(y) \right]. } $$ where: $$ \boxed{ w=X^{1/4}, \qquad z=X^{1/3+o(1)}. } $$ The half-weight: $$ \boxed{ 1/2 } $$ is the minimal admissible coefficient within the natural two-statistic ansatz. Standard: $$ \theta=3/5 $$ constant audit: $$ \boxed{ C_{\mathrm{coarse}} \approx 0.499401183388, } $$ $$ \boxed{ C_{\mathrm{med}} \approx 0.830596870470. } $$ Full subtraction: $$ \boxed{ C_{\mathrm{coarse}}-C_{\mathrm{med}} \approx -0.331195687082 <0, } $$ but half-weight: $$ \boxed{ C_{\mathrm{coarse}} - \frac12C_{\mathrm{med}} \approx 0.084102748153 >0. } $$ Thus: $$ \boxed{ \text{Chen-style half weighting genuinely rescues the sign}. } $$ However, strict pressure must still pay: $$ \boxed{ A_P + \frac12A_{2,1} + \frac12A_{3,1}, } $$ and the remaining normalized window is only: $$ \boxed{ \approx0.0841. } $$ Therefore: $$ \boxed{ \mathrm{HSP\ STILL\ NOT\ PROVED}. } $$ The actual next bottleneck has been reduced to a sharp upper ledger for: # Prime Cost + Medium-$P_2$ Cost + Triple Cost as displayed above. --- # 55. Recommendation for the Next Round # TCDC-06 ## Strict-Pressure Extra-Cost Ledger / Prime–Medium–Triple Closure The next round no longer changes the pointwise weight. It directly handles: $$ A_P, \quad A_{2,1}, \quad A_{3,1}. $$ ### Part A — Prime Branch $$ 2m=rp+q. $$ Establish the sharpest upper constant, rather than using only one-dimensional BT. ### Part B — Medium-$P_2$ $$ 2m=rp+\ell q, $$ with： $$ w<\ell\le z how far strict HSP remains from closure. --- # Appendix A: Pointwise State Table ```yaml coarse_domain: RootRough_w: true max_Omega: 3 states: prime: target: 0 weight: <= 0 desired_P2: medium_multiplicity: 0 target: 1 weight: 1 medium_P2_nu1: target: 0 weight: 0 medium_P2_nu2: target: 0 weight: 0 P3_nu1: target: 0 weight: 0 P3_nu2: target: 0 weight: 0 P3_nu3: target: 0 weight: -1/2 ``` --- # Appendix B: Weighted Pressure Ledger ```yaml Pi_lower: positive: A1: coarse_RootRough_w negative: AP: prime_complement coefficient: 1 A2: medium_prime_multiplicity coefficient: 1/2 A2_1: exactly_one_medium_P2 coefficient: 1/2 A3_1: exactly_one_medium_P3 coefficient: 1/2 ``` --- # Appendix C: Current $3/5$ Constant Audit ```yaml theta: 3/5 w: X^(1/4) coarse_s: 12/5 C_coarse: 0.499401183388 C_medium_full: 0.830596870470 C_medium_half: 0.415298435235 full_subtraction_margin: -0.331195687082 half_weight_margin_before_extra_costs: 0.084102748153 ``` --- # Appendix D: Finite Calibration ```yaml lambda: 1.1 samples: pointwise_weight: verified: true weighted_total_le_actual_pressure: verified: true status: computational_evidence_only: true ``` --- # Appendix E: Commit Decision ```yaml research_round: TCDC-05 decision: strict_semiprime_pointwise_weight: proved half_weight_optimality: proved_within_selected_ansatz square_correction_absorbed_by_multiplicity: true half_weight_sign_rescue: diagnostic_positive strict_extra_costs: unresolved HSP_proved: false Goldbach_proved: false next: TCDC-06_StrictPressure_ExtraCostLedger_PrimeMediumTriple_Closure ``` --- # References / Audited External Sources 1. Terence Tao, **254A, Supplement 5: The linear sieve and Chen's theorem**, 2015. Relevant facts: - classical Chen lower-weight lemma uses a $1/2$ coefficient on medium-prime divisibility together with triple-factor and square corrections; - Chen's target is $P_2$, so primes and semiprimes are both acceptable; - the proof proceeds by lower-bounding a rough term and upper-bounding medium-factor / triple-factor terms, with switching used on the difficult triple term. 2. James Maynard, **Primes in arithmetic progressions to large moduli III: Uniform residue classes**, *Memoirs of the American Mathematical Society* 306 (2025), no. 1544. Relevant fact: Proposition 5.3 treats balanced bilinear convolutions near $x^{1/2}$ under a residue compatibility condition. The high-suffix base $r,p\asymp X^{1/2}$ has the correct broad convolution shape, but no direct linear-sieve provider claim is made here. 3. Zhiyuan Yang, **Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications**, arXiv:2608.13299 (2026). Relevant fact: provides a modern convolution / well-factorable-weight template potentially relevant to medium-divisor terms. The audited statement retains fixed-residue dependence, so no direct Goldbach-shift application is asserted. 4. Classical dimension-$1$ linear sieve functions: $$ F_1(s)=\frac{2e^\gamma}{s} $$ for $1