> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # TCDC-14 ## Bulk Mixed-Deficiency / Coarse-Balance Constant Closure ### Cube-Root Buchstab Branch Constants, Prime–Semiprime Bias, Joint-Transfer Parity Barrier, Bulk Capacity Rescue and Main-Order Closure Hierarchy **Parent methodology:** TCDC — Target–Condition Dual Closure **Connected research lines:** RG-SCP / SRBRD **Round:** TCDC-14 **Date:** 2026-10-02 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Main-Order Constant Audit / Buchstab Branch Decomposition / Parity-Transfer Barrier / Bulk Capacity Closure **Prerequisites:** TCDC-00 ～ TCDC-13; SRBRD-00 ～ SRBRD-26 **Status:** Determines the exact leading ratio $1:\log2$ of the one-variable prime / semiprime branches of cube-root RootRough; establishes a joint survivor branch-transfer condition, a coarse prime-share threshold and a Liouville parity formulation; derives a rank-one Buchstab branch matrix benchmark and bulk mixed-capacity closure; proves that the TCDC-13 mirror gain is a critical-layer correction after the main order; does not prove joint branch-transfer; Goldbach remains unproved --- # 0. Purpose of This Round TCDC-13 proves: $$ \boxed{ M_{PP} \ge \Delta_0 + \Gamma_{\mathrm{row}} + G_{\mathrm{mir}}^* } $$ where: $$ \Delta_0 = n_0-n_{SP}. $$ and points out: - $\Delta_0$： coarse branch balance; - $\Gamma_{\mathrm{row}}$： bulk mixed-capacity deficiency; - $G_{\mathrm{mir}}^*$： mirror critical-layer correction. Thus TCDC-14 returns to the main order. The first question: > Within the cube-root RootRough boundary, > what exactly are the one-variable natural constants of the prime and semiprime branches? Answer: $$ \boxed{ 1:\log2. } $$ --- # 1. One-Variable Cube-Root Boundary Let: $$ \boxed{ z=X^{1/3}. } $$ Consider: $$ \operatorname{RootRough}_z(x) $$ for: $$ 1X^{1/3} \right\}. } $$ PNT: ## Theorem TCDC-14-T02 $$ \boxed{ \mathcal P_1(X) \sim \frac{X}{\log X}. } $$ Since: $$ \pi(X^{1/3}) = o \left( X/\log X \right). $$ --- # 3. Semiprime Branch Define: $$ \boxed{ \mathcal P_2(X) = \#\left\{ pq\le X: p,q\in\mathbb P, \quad X^{1/3}1. $$ The Buchstab function: $$ u\omega(u)=1 $$ for: $$ 1\le u\le2, $$ and: $$ (u\omega(u))' = \omega(u-1). $$ at: $$ u=3, $$ ## Theorem TCDC-14-T05 $$ \boxed{ 3\omega(3) = 1+\log2. } $$ Thus: $$ \boxed{ \Phi(X,X^{1/3}) \sim (1+\log2) \frac{ X }{ \log X }. } $$ exactly matching: $$ P_1+P_2. $$ --- # 8. One-End Branch Vector Define: $$ \boxed{ L=\log2. } $$ Unnormalized branch vector: $$ \boxed{ b= \begin{pmatrix} 1\\ L \end{pmatrix}. } $$ Normalized: $$ \boxed{ p_* = \frac1{1+L}, } $$ $$ \boxed{ s_* = \frac{L}{1+L}. } $$ Numerically: $$ \boxed{ p_* \approx 0.590616109150, } $$ $$ \boxed{ s_* \approx 0.409383890850. } $$ --- # 9. One-Variable Prime Bias The prime share exceeds: $$ 1/2 $$ by: ## Theorem TCDC-14-T06 $$ \boxed{ \beta_* = p_*-\frac12 = \frac{ 1-L }{ 2(1+L) } } $$ $$ \boxed{ \beta_* \approx 0.090616109150. } $$ Thus the one-variable cube-root survivor has an absolute prime-share buffer of about: $$ \boxed{ 9.06\% } $$ above: $$ 1/2. $$ --- # 10. Relative Branch Gap Prime-minus-semiprime fraction: $$ \boxed{ p_*-s_* = \frac{ 1-L }{ 1+L } } $$ Numerically: $$ \boxed{ 0.181232218299. } $$ That is, within the total rough survivor, a normalized branch imbalance of about: $$ \boxed{ 18.12\% } $$ as displayed above. --- # 11. Joint Goldbach Survivor Return to SRBRD / TCDC. Let: $$ \boxed{ \widehat{\mathcal S}_z(m) } $$ be the ordered Goldbach cube-root RootRough survivor states: $$ x+y=2m, $$ $$ x,y\text{ odd}, $$ $$ \operatorname{RootRough}_z(x), $$ $$ \operatorname{RootRough}_z(y). $$ At the cube-root boundary: $$ x,y\in P_1\cup P_2. $$ --- # 12. Joint Marginals Define: $$ \boxed{ S_m = |\widehat{\mathcal S}_z(m)| } $$ and: $$ \boxed{ n_0 = n_P = M_{PP}+M_{PS}, } $$ $$ \boxed{ n_{SP} = M_{PS}+M_{SS}. } $$ Thus: ## Theorem TCDC-14-T07 $$ \boxed{ S_m = n_0+n_{SP}. } $$ --- # 13. Coarse Prime Share Define: $$ \boxed{ \theta_m = \frac{ n_0 }{ S_m }. } $$ then: $$ \boxed{ 1-\theta_m = \frac{ n_{SP} }{ S_m }. } $$ Coarse deficit: $$ \boxed{ \Delta_0 = n_0-n_{SP} = (2\theta_m-1)S_m. } $$ --- # 14. Exact Coarse Closure SRBRD-14： $$ M_{PP} = \Delta_0+M_{SS}. $$ Since: $$ M_{SS}\ge0, $$ we obtain: ## Theorem TCDC-14-T08 ### Coarse Branch-Bias Closure If: $$ \boxed{ \theta_m>\frac12, } $$ then: $$ \boxed{ M_{PP}>0. } $$ Equivalently: $$ \boxed{ n_0>n_{SP}. } $$ --- # 15. Buchstab Branch-Transfer Hypothesis The one-variable T02–T06 cannot automatically be applied to the joint survivor. Define: ## Research Target TCDC-14-H01 ### Buchstab Branch-Ratio Transfer We hope to prove: $$ \boxed{ \theta_m = \frac1{1+\log2} + o(1). } $$ Equivalently: $$ \boxed{ \frac{ n_{SP} }{ n_0 } = \log2 + o(1). } $$ --- # 16. Conditional Coarse Closure If H01 holds, then: $$ \theta_m \to p_* > 1/2. $$ Thus: ## Theorem TCDC-14-T09 H01 implies: $$ \boxed{ M_{PP}>0 } $$ for all sufficiently large: $$ m. $$ Thus H01 is itself already an asymptotic Goldbach-level theorem. It cannot be treated as an ordinary sieve technicality. --- # 17. We Do Not Need Full Asymptotic Transfer T08 only needs: $$ \theta_m>1/2. $$ Predicted: $$ \theta_* = 0.590616109150. $$ Thus the branch-transfer estimate can allow an absolute error: ## Theorem TCDC-14-T10 ### Transfer Tolerance If: $$ \boxed{ \theta_m \ge p_*-\varepsilon } $$ and: $$ \boxed{ \varepsilon<\beta_*, } $$ then Goldbach success follows. where: $$ \boxed{ \beta_* \approx 0.090616109150. } $$ Thus we need not determine the branch ratio precisely to: $$ o(1) $$ as displayed above. We need only preserve an absolute buffer of approximately: $$ 9.06\% $$ as displayed above. --- # 18. Ratio Tolerance Equivalently, if we can prove: $$ \boxed{ \frac{ n_{SP} }{ n_0 } <1, } $$ that is enough. Predicted ratio: $$ \boxed{ \log2 \approx 0.693147. } $$ Thus, relative to the prediction, we can tolerate a substantial constant loss, as long as the ratio does not exceed: $$ 1. $$ The problem is not that the margin is small, but that the parity branch itself is difficult to prove. --- # 19. Liouville Formulation Within the cube-root survivor: ### Prime $$ \lambda_{\mathrm L}(p)=-1. $$ ### Semiprime $$ \lambda_{\mathrm L}(pq)=+1. $$ Define the joint survivor left-end Liouville mean: $$ \boxed{ \Lambda_m = \frac1{S_m} \sum_{(x,y)\in\widehat{\mathcal S}_z} \lambda_{\mathrm L}(x). } $$ then: ## Theorem TCDC-14-T11 $$ \boxed{ \Lambda_m = \frac{ n_{SP}-n_0 }{ S_m } = 1-2\theta_m. } $$ Therefore: $$ \boxed{ \Delta_0>0 \iff \Lambda_m<0. } $$ --- # 20. Predicted Liouville Bias H01 predicts: $$ \boxed{ \Lambda_m \to \frac{ L-1 }{ L+1 } } $$ Numerically: $$ \boxed{ -0.181232218299. } $$ Thus coarse Goldbach closure is equivalent to proving that: > the left-end Liouville mean of the joint cube-root rough survivor > has a strictly negative bias. This is a parity-sensitive statement. --- # 21. Parity Firewall Plain sieve theory is very good at estimating the: $$ \boxed{ P_1\cup P_2 } $$ total rough mass. However, distinguishing: $$ P_1 $$ from: $$ P_2 $$ exactly tracks the parity of the prime-factor count. Therefore: ## Research Firewall TCDC-14-R01 $$ \boxed{ \text{the one-variable Buchstab ratio }1:\log2 \text{cannot automatically be transferred through ordinary positive sieve weights to the joint Goldbach branch ratio.} } $$ This is precisely the form of the classical parity problem. --- # 22. Rank-One Buchstab Branch Matrix Benchmark Although this cannot be treated as a theorem, if Goldbach joint roughness introduces no additional leading-order bias in branch type, then the normalized branch probabilities are: $$ \boxed{ (p_*,s_*). } $$ Joint $2\times2$ transport matrix benchmark: ## Heuristic TCDC-14-H02 $$ \boxed{ \frac1{S_m} \begin{pmatrix} M_{PP}&M_{PS}\\ M_{PS}&M_{SS} \end{pmatrix} \approx \begin{pmatrix} p_*^2&p_*s_*\\ p_*s_*&s_*^2 \end{pmatrix}. } $$ This is the rank-one outer product: $$ \boxed{ \begin{pmatrix} p_*\\s_* \end{pmatrix} \begin{pmatrix} p_*&s_* \end{pmatrix}. } $$ --- # 23. Benchmark Constants Numerically: $$ \boxed{ p_*^2 \approx 0.348827388387, } $$ $$ \boxed{ p_*s_* \approx 0.241788720763, } $$ $$ \boxed{ s_*^2 \approx 0.167595170088. } $$ Thus the benchmark is: - $PP$： about $34.88\%$; - each mixed orientation: about $24.18\%$; - $SS$： about $16.76\%$. --- # 24. Benchmark Coarse Gap $$ \boxed{ \frac{\Delta_0}{S_m} \approx p_*-s_* = 0.181232218299. } $$ And: $$ \boxed{ \frac{M_{SS}}{S_m} \approx s_*^2 = 0.167595170088. } $$ The exact identity: $$ M_{PP} = \Delta_0+M_{SS} $$ becomes the benchmark: $$ \boxed{ p_*^2 = (p_*-s_*)+s_*^2. } $$ This is an algebraically exact identity under: $$ p_*+s_*=1. $$ --- # 25. Why Bulk Deficiency Matters Even If Coarse Bias Is Unknown Joint branch-transfer H01 is parity-sensitive. However, the mixed prime-sink branch: $$ M_{PS} $$ requires only an: # Upper Bound which is a friendlier sieve polarity. Thus the TCDC alternative is: > not to prove directly: > $$ > n_0>n_{SP}, > $$ > but to prove that semiprime rows cannot send enough mass to the prime sink. --- # 26. Rowwise Bulk Mixed Capacity TCDC-08： $$ \boxed{ P_{\mathrm{row}} = \sum_r \min( n_r,U_r^P ). } $$ Row deficiency: $$ \boxed{ \Gamma_{\mathrm{row}} = n_{SP}-P_{\mathrm{row}}. } $$ Goldbach lower bound: ## Theorem TCDC-14-T12 $$ \boxed{ M_{PP} \ge \Delta_0+\Gamma_{\mathrm{row}} = n_0-P_{\mathrm{row}}. } $$ Thus the closure condition is: $$ \boxed{ P_{\mathrm{row}}1/2. $$ --- # 28. Two Main-Order Closure Routes Thus the coarse layer has two logically independent routes. ## Route A — Branch Bias $$ \boxed{ \theta_m>\frac12. } $$ ## Route B — Prime-Sink Capacity $$ \boxed{ \mu_m<\theta_m. } $$ Route A needs parity-sensitive branch lower/upper bounds. Route B can make extensive use of upper-sieve / transport capacities. --- # 29. Benchmark Mixed Fraction Under H02: Actual mixed orientation: $$ \boxed{ \frac{ M_{PS} }{ S_m } \approx p_*s_*. } $$ Numerically: $$ \boxed{ p_*s_* \approx 0.241788720763. } $$ And: $$ \theta_* \approx 0.590616109150. $$ Thus the ideal actual mixed mass is very far from the closure threshold: $$ \theta_* $$ as displayed above. The real difficulty is: > the rigorous upper capacity may be much looser than the actual mixed branch. --- # 30. Benchmark Bulk Internal Fraction H02 predicts: $$ \boxed{ \frac{ M_{SS} }{ S_m } \approx s_*^2 \approx 0.167595170088. } $$ Thus the actual semiprime network has a main-order fraction of: $$ \boxed{ S_m } $$ as displayed above, rather than a boundary correction. This shows that: # Bulk Mixed Deficiency can indeed be a main-order mechanism. --- # 31. Standard Upper Constant Problem Reappears TCDC-11 has already observed that the: standard separate-prime upper multiplier: $$ \approx2.97 $$ is too loose relative to the ideal density. Similarly, if the bulk mixed upper bound only achieves: $$ \kappa \times \text{ideal mixed density}, $$ then we need: $$ \boxed{ \kappa p_*s_* Any apparently simple argument claiming that the joint survivor branch ratio is still $1:\log2$ > must be held to the review standard that it is already almost proving Goldbach. --- # 36. Bulk Route Is More TCDC-Native TCDC need not directly determine which is more numerous: $$ P_1 $$ versus: $$ P_2 $$ as displayed above. It can ask: > Can semiprime rows send all the requisite mass into the prime sink? This is precisely the strength of the: - capacity; - LP; - shadow-price; - transport; framework. Thus even if Route A is blocked by parity, Route B still has a different proof polarity. --- # 37. Liouville vs Capacity Duality Route A: $$ \boxed{ \Lambda_m<0 } $$ is a branch parity statement. Route B: $$ \boxed{ P_{\mathrm{row}}0. $$ The bulk capacity only needs: $$ \boxed{ \mu_m\le\mu_+ } $$ with: $$ \boxed{ \mu_+<\theta_-. } $$ Then: ## Theorem TCDC-14-T15 ### Relative Coarse-Bulk Closure $$ \boxed{ M_{PP} \ge (\theta_--\mu_+)S_m >0. } $$ This avoids directly requiring: $$ \theta_->1/2. $$ --- # 40. Relative Route Can Be Strictly Easier For example, if we can only prove: $$ \theta_m\ge0.45, $$ that is entirely insufficient for Route A. However, if we can also prove: $$ \mu_m\le0.40, $$ then Route B immediately closes. Thus: $$ \boxed{ \text{a capacity theorem can compensate for the shortfall of a branch-bias theorem}. } $$ --- # 41. Benchmark Relative Numbers Rank-one benchmark: $$ \boxed{ \theta_* = 0.590616109150, } $$ $$ \boxed{ \mu_*^{actual} = p_*s_* = 0.241788720763. } $$ Gap: $$ \boxed{ \theta_*-\mu_*^{actual} = p_*^2 = 0.348827388387. } $$ This is exactly the predicted $PP$ branch fraction. Thus the T13 lower bound: $$ \theta-\mu $$ is not a loose bound in the ideal matrix, but is exactly: $$ \boxed{ M_{PP}/S_m. } $$ --- # 42. This Explains Why Transport Matrix Is Natural Matrix identity: $$ n_0=M_{PP}+M_{PS}. $$ if: $$ P_{\mathrm{row}} $$ could equal the actual: $$ M_{PS}, $$ then: $$ \boxed{ n_0-P_{\mathrm{row}} = M_{PP}. } $$ Thus the prime-sink maximum / capacity approach essentially consists of: > approaching the mixed branch from above, > until the exact $PP$ mass is exposed. --- # 43. Finite Calibration Selected exact finite data: | $m$ | $n_0$ | $n_{SP}$ | prime share $\theta_m$ | ratio $n_{SP}/n_0$ | |---:|---:|---:|---:|---:| | 96 | 29 | 9 | 0.763158 | 0.310345 | | 255 | 78 | 20 | 0.795918 | 0.256410 | | 783 | 137 | 47 | 0.744565 | 0.343066 | | 870 | 186 | 48 | 0.794872 | 0.258065 | | 969 | 153 | 49 | 0.757426 | 0.320261 | | 1000 | 101 | 35 | 0.742647 | 0.346535 | One-variable Buchstab benchmark: $$ \boxed{ \theta_* \approx 0.590616109150, } $$ $$ \boxed{ n_{SP}/n_0 \approx \log2 \approx 0.693147180560. } $$ The selected finite cases show a stronger prime bias, However: $$ \boxed{ \text{finite data is computational evidence only}. } $$ --- # 44. Earlier Scan Context An earlier finite scan: $$ 3\le m\le5000 $$ observed: $$ \boxed{ n_0>n_{SP} } $$ throughout the sampled range. This agrees in direction with the one-variable branch bias. However, it cannot be extrapolated into: $$ \boxed{ \forall m } $$ or an asymptotic theorem. --- # 45. Why Finite Bias Can Exceed $p_*$ The finite: $$ \theta_m $$ need not approach: $$ p_*. $$ Possible reasons: - endpoint interval geometry; - finite cube-root cutoff; - center-divisor local factors; - additive correlation; - small-prime singular effects; - parity correlation itself. Therefore: $$ \boxed{ p_* } $$ is only a one-variable asymptotic benchmark, not a finite target value. --- # 46. A More Conservative Research Target Rather than directly proving H01, we can define: # JPB ## Joint Prime Bias Seek: $$ \boxed{ \delta>0 } $$ such that: $$ \boxed{ n_0 \ge \left( \frac12+\delta \right) S_m } $$ for sufficiently large: $$ m. $$ Any: $$ \delta>0 $$ is enough. Predicted available buffer: $$ \delta\approx0.0906. $$ --- # 47. Capacity-Assisted Joint Prime Bias If we can only obtain: $$ \delta\le0, $$ we can still define a mixed-capacity gain: $$ \boxed{ \gamma_m = \frac{ n_{SP}-P_{\mathrm{row}} }{ S_m }. } $$ then: $$ \boxed{ \frac{M_{PP}}{S_m} \ge 2\theta_m-1+\gamma_m. } $$ closure iff: $$ \boxed{ 2\theta_m-1+\gamma_m>0. } $$ This is the: # Bias + Capacity joint main-order gate. --- # 48. Mirror Correction Added Last TCDC-13： $$ \boxed{ \frac{M_{PP}}{S_m} \ge 2\theta_m-1 + \gamma_m + \frac{ G_{\mathrm{mir}}^* }{ S_m }. } $$ Thus the full hierarchy is: $$ \boxed{ \text{Branch Bias} + \text{Bulk Capacity} + \text{Mirror Critical Correction}. } $$ --- # 49. Main-Order Target Should Be Relative Absolute quantities: $$ m/\log^2m $$ carry many singular factors. The relative normalized gate: $$ \boxed{ 2\theta_m-1+\gamma_m>0 } $$ is better suited to: - theorem comparison; - LP normalization; - finite-to-asymptotic pattern mining. This should become the coarse benchmark after TCDC-14. --- # 50. Formal Theorem Queue ```yaml TCDC_14: T01: cube_root_RootRough_only_P1_or_P2 T02: prime_branch_main_term T03: semiprime_integral_equals_log2 T04: semiprime_branch_main_term T05: Buchstab_u3_consistency T06: one_variable_prime_bias T07: joint_survivor_partition T08: coarse_branch_bias_closure T09: branch_transfer_implies_large_Goldbach T10: branch_transfer_tolerance T11: joint_Liouville_bias_identity T12: bulk_capacity_Goldbach_lower T13: normalized_bulk_capacity_closure T14: benchmark_bulk_multiplier_ceiling T15: relative_coarse_bulk_closure ``` --- # 51. Research Targets ```yaml TCDC_14_Targets: H01: Buchstab_Branch_Ratio_Transfer: n_SP/n_0: log(2)+o(1) JPB: Joint_Prime_Bias: n_0/S_m: > 1/2+delta Bulk: mixed_capacity: P_row/S_m: < n_0/S_m Relative_Master: 2*theta_m-1+gamma_m: > 0 ``` --- # 52. Barrier Queue ```yaml TCDC_14_Barriers: B01: one_variable_branch_ratio_does_not_transfer_automatically_to_joint_survivor B02: joint_prime_vs_semiprime_split_is_parity_sensitive B03: no_joint_Liouville_negative_bias_theorem B04: bulk_two_prime_capacity_not_sharp_enough_uniformly B05: relative_prime_share_lower_not_proved B06: relative_mixed_capacity_upper_not_proved_below_prime_share B07: no_positive_uniform_main_order_gap ``` --- # 53. Acceptance Gate Before the TCDC-14 formal core, we need: 1. to distinguish the one-variable rough count from the Goldbach joint survivor; 2. semiprimes counted once with $p\le q$; 3. the cube-root branch integral checked; 4. the Buchstab formula used only as one-variable consistency; 5. $1:\log2$ never silently promoted to a joint theorem; 6. branch-transfer labeled a research target; 7. the parity firewall explicit; 8. the heuristic rank-one matrix labeled heuristic; 9. the exact transport lower bound separated from the heuristic matrix; 10. the bulk capacity route proof-admissible; 11. the mirror layer kept lower-order / conditional where appropriate; 12. finite calibration labeled computational; 13. no large-Goldbach proof claim; 14. no Goldbach proof claim. --- # 54. Verdict This round can formally give: $$ \boxed{ \mathrm{TCDC\mbox{-}14\ COARSE\mbox{-}BULK\ BRANCH\ KERNEL\ PASS}. } $$ The most important one-variable exact asymptotic: $$ \boxed{ P_1:P_2 = 1:\log2 } $$ at the cube-root rough boundary. Normalized benchmark: $$ \boxed{ p_* = \frac1{1+\log2} \approx 0.590616109150, } $$ $$ \boxed{ s_* = \frac{\log2}{1+\log2} \approx 0.409383890850. } $$ so the prime branch has a: $$ \boxed{ 9.0616\% } $$ absolute buffer above: $$ 1/2. $$ However, transferring this bias to the Goldbach joint survivor would require: $$ \boxed{ \Lambda_m<0, } $$ that is, proving strictly negative bias in the Liouville parity within the joint rough survivor. This is a substantive parity-sensitive correlation, not something automatically obtainable from an ordinary rough-number sieve. Thus main-order closure now formally splits into two routes: ### Branch Route $$ \boxed{ \theta_m>\frac12. } $$ ### Capacity Route $$ \boxed{ \mu_m<\theta_m. } $$ And the full relative master inequality: $$ \boxed{ \frac{M_{PP}}{S_m} \ge 2\theta_m-1 + \gamma_m + \frac{ G_{\mathrm{mir}}^* }{ S_m }. } $$ where: - branch bias： main order; - bulk mixed deficiency： main order; - mirror gain： critical-layer correction. Thus the TCDC proof program formally becomes: $$ \boxed{ \text{Coarse Branch Bias} \;\lor\; \text{Bulk Capacity Closure} \;\to\; \text{Mirror Critical Layer}. } $$ Currently: $$ \boxed{ \mathrm{NO\ JOINT\ BRANCH\ TRANSFER\ THEOREM}. } $$ $$ \boxed{ \mathrm{NO\ UNIFORM\ BULK\ CAPACITY\ GAP}. } $$ Therefore: $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 55. Recommendation for the Next Round # TCDC-15 ## Joint Buchstab Transfer / Parity-Defect Decomposition The next round does not directly try to prove: $$ n_0/n_{SP} = 1/\log2. $$ but instead decomposes the joint branch-transfer error. ### Part A — Neutral Joint Survivor First estimate: $$ S_m = n_0+n_{SP}. $$ This is parity-neutral. ### Part B — Parity Defect Define: $$ \boxed{ \mathcal E_{\mathrm{par}} = n_{SP}-n_0 } $$ or normalized: $$ \Lambda_m. $$ ### Part C — Buchstab Prediction + Defect Write: $$ \boxed{ \theta_m = p_* + \varepsilon_{\mathrm{par}}(m). } $$ Goldbach only needs: $$ \boxed{ \varepsilon_{\mathrm{par}} > -\beta_*. } $$ ### Part D — Capacity Substitute If we cannot directly control: $$ \varepsilon_{\mathrm{par}}, $$ seek: $$ \gamma_m $$ such that: $$ 2\varepsilon_{\mathrm{par}} + \gamma_m > -2\beta_*. $$ That is: > bulk capacity allows the parity defect not to be completely eliminated. ### Part E — External Technology Audit Check: - dispersion; - Type-II / Type-III; - Liouville-twisted convolution; - parity-breaking bilinear forms; which of these can genuinely reach: $$ \Lambda_m. $$ This would turn the parity barrier from a warning into a quantitatively bounded defect variable. --- # Appendix A: Cube-Root Branch Constants ```yaml L: log(2) prime: constant: 1 semiprime: constant: log(2) total: constant: 1+log(2) normalized: p_star: 1/(1+log(2)) s_star: log(2)/(1+log(2)) prime_bias_above_half: (1-log(2))/(2*(1+log(2))) ``` --- # Appendix B: Rank-One Benchmark ```yaml branch_vector: - p_star - s_star matrix_fraction: PP: p_star^2 PS: p_star*s_star SP: p_star*s_star SS: s_star^2 warning: heuristic_joint_transfer_only: true ``` --- # Appendix C: Relative Closure ```yaml S_m: n_0+n_SP theta: n_0/S_m mu: P_row/S_m gamma: (n_SP-P_row)/S_m exact_lower: M_PP/S_m: >= theta-mu equivalent: M_PP/S_m: >= 2*theta-1+gamma closure_if: theta > mu ``` --- # Appendix D: Finite Calibration | $m$ | $n_0$ | $n_{SP}$ | prime share $\theta_m$ | $n_{SP}/n_0$ | |---:|---:|---:|---:|---:| | 96 | 29 | 9 | 0.763158 | 0.310345 | | 255 | 78 | 20 | 0.795918 | 0.256410 | | 783 | 137 | 47 | 0.744565 | 0.343066 | | 870 | 186 | 48 | 0.794872 | 0.258065 | | 969 | 153 | 49 | 0.757426 | 0.320261 | | 1000 | 101 | 35 | 0.742647 | 0.346535 | benchmark: ```yaml prime_share: 0.590616109150 semiprime_to_prime_ratio: 0.693147180560 ``` All finite statements are computational evidence only. --- # Appendix E: Commit Decision ```yaml research_round: TCDC-14 decision: cube_root_prime_semiprime_ratio: proved_one_variable Buchstab_consistency: verified joint_transfer: not_proved parity_defect: formalized rank_one_branch_matrix: heuristic_benchmark bulk_capacity_route: formalized main_order_hierarchy: formalized uniform_positive_main_gap: not_proved Goldbach_proved: false next: TCDC-15_JointBuchstabTransfer_ParityDefect_Decomposition ``` --- # References / External Benchmarks 1. Buchstab rough-number asymptotic. For fixed $u>1$ and: $$ y=x^{1/u}, $$ the rough-number count satisfies: $$ \Phi(x,y) \sim \omega(u)\frac{x}{\log y}. $$ The Buchstab function obeys: $$ u\omega(u)=1 $$ on $1\le u\le2$, and: $$ (u\omega(u))'=\omega(u-1) $$ for $u>2$. At $u=3$ this yields: $$ 3\omega(3)=1+\log2. $$ 2. Prime Number Theorem. Used for: $$ \mathcal P_1(X) \sim X/\log X $$ and for the partial-summation evaluation of the cube-root semiprime shell. 3. Terence Tao, **Open question: The parity problem in sieve theory**, 2007, and **A general parity problem obstruction**, 2014. Relevant conceptual fact: ordinary sieve methods can efficiently count parity-neutral almost-prime sets, but distinguishing numbers with odd versus even prime-factor parity is the classical parity obstruction. The TCDC joint prime-versus-semiprime branch-transfer target is of this parity-sensitive type.