> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # TCDC-06 ## Strict-Pressure Extra-Cost Ledger / Branch-Neutral Two-Sink Closure ### Generalized Half-Weight, Coarse-Cutoff Optimization, Prime Strictness Tax, Separate-Upper No-Go, and TCDC Branch-Neutral Capacity Reformulation **Parent methodology:** TCDC — Target–Condition Dual Closure **Linked research lines:** RG-SCP / SRBRD **Round:** TCDC-06 **Date:** 2026-10-02 **Version:** v0.1 **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Nature:** Weighted-Sieve Optimization / Parity-Tax Audit / Failure-Capacity Reformulation **Prerequisites:** TCDC-00 through TCDC-05; SRBRD-00 through SRBRD-26 **Status:** Generalizes the TCDC-05 half-weight to any coarse cutoff $w Any composite coarse survivor, > unless it is the desired: > $$ > P_2 > \text{ with all factors }>z, > $$ > has at least one prime factor in: > $$ > (w,z]. > $$ This uses only: $$ z^3\ge X. $$ Thus, quarter root is not necessary for the validity of the pointwise weight. --- # 1. General Coarse Cutoff Take: $$ \boxed{ 2\le ww. $$ Also, no factor lies in: $$ (w,z], $$ so all prime factors are actually: $$ >z. $$ If: $$ y $$ has only two prime factors, then it is precisely the desired: $$ P_2^{>z}, $$ contradicting the assumption. If it has at least three prime factors, then: $$ y>z^3\ge X, $$ again contradicting: $$ y\le X $$ as displayed above. Thus: $$ \nu_{w,z}(y)\ge1. $$ This completes the proof. --- # 4. General Strict-Semiprime Half-Weight Define: $$ \boxed{ \omega_{w,z}^{strict}(y) = \mathbf1_{\operatorname{RootRough}_w(y)} \left[ 1 - \mathbf1_{\mathbb P}(y) - \frac12\nu_{w,z}(y) - \frac12E_1(y) \right]. } $$ Desired indicator: $$ \boxed{ H_z(y) = \mathbf1_{\{ y\text{ semiprime}, \operatorname{RootRough}_z(y) \}}. } $$ --- # 5. Generalized Pointwise Theorem ## Theorem TCDC-06-T02 for all: $$ 3\le y\le X, $$ $$ \boxed{ \omega_{w,z}^{strict}(y) \le H_z(y). } $$ This holds for any: $$ 2\le wz}$ $$ \nu=0, $$ $$ E_1=0, $$ Thus: $$ \omega=1=H_z. $$ ### Any Other Composite By T01: $$ \nu\ge1. $$ If: $$ \nu=1, $$ then: $$ E_1=1, $$ Thus: $$ \omega=0. $$ If: $$ \nu\ge2, $$ then: $$ 1-\frac12\nu\le0. $$ This completes the proof. --- # 7. Branch-Neutral Cube-Root Target Define: $$ \boxed{ G_z(y) = \mathbf1_{\operatorname{RootRough}_z(y)}. } $$ Since: $$ z^3\ge X, $$ For: $$ 3\le y\le X, $$ we have: $$ \boxed{ G_z(y) = \mathbf1_{\mathbb P}(y) + H_z(y). } $$ Thus: # Branch-Neutral $P_2$ Mass is: $$ \boxed{ \text{prime branch} + \text{strict semiprime branch}. } $$ --- # 8. General Branch-Neutral Half-Weight Define: $$ \boxed{ \omega_{w,z}^{neutral}(y) = \mathbf1_{\operatorname{RootRough}_w(y)} \left[ 1 - \frac12\nu_{w,z}(y) - \frac12E_1(y) \right]. } $$ --- # 9. Branch-Neutral Pointwise Theorem ## Theorem TCDC-06-T03 $$ \boxed{ \omega_{w,z}^{neutral}(y) \le G_z(y). } $$ The proof is the same as T02, except that a prime is no longer treated as a bad branch. For a prime: - target: $$ G_z=1; $$ - weight: $$ 1-\nu/2 \le1. $$ Thus, it is admissible. --- # 10. Strict Weight = Neutral Weight − Prime Tax Exactly: ## Theorem TCDC-06-T04 $$ \boxed{ \omega_{w,z}^{strict}(y) = \omega_{w,z}^{neutral}(y) - \mathbf1_{\operatorname{RootRough}_w(y)} \mathbf1_{\mathbb P}(y). } $$ But primes are always RootRough$_w$, Thus: $$ \boxed{ \omega_{w,z}^{strict} = \omega_{w,z}^{neutral} - \mathbf1_{\mathbb P}. } $$ This fully isolates the strictness cost as: # Prime Tax --- # 11. Prime-Tax Coefficient Is Unavoidable Locally Consider any local weight: $$ \boxed{ \mathbf1_{\operatorname{RootRough}_w} \left[ 1 - c_P\mathbf1_{\mathbb P} - \text{other nonnegative penalties} \right] } $$ where, for a prime: $$ y>z, $$ all other medium penalties are: $$ 0. $$ To lower-bound the strict semiprime indicator pointwise: $$ H_z(y)=0, $$ one must have: ## Theorem TCDC-06-T05 $$ \boxed{ c_P\ge1. } $$ Thus: $$ \boxed{ \text{A strict-semiprime local selector must pay the full prime-branch coefficient }1. } $$ This is the: # Local Strictness Tax --- # 12. Sum-Level Notation On the high-suffix base: $$ \mathcal A_\lambda, $$ Define: $$ A_1(w) = \#\{ (r,p)\in\mathcal A_\lambda: \operatorname{RootRough}_w(y) \}. $$ Medium multiplicity: $$ A_2(w,z) = \sum_{\substack{ (r,p)\in\mathcal A_\lambda\\ \operatorname{RootRough}_w(y) }} \nu_{w,z}(y). $$ Exactly-one correction: $$ E(w,z) = \sum E_1(y). $$ Prime branch: $$ A_P=P_\lambda. $$ --- # 13. Neutral Weighted Lower Summing T03: ## Theorem TCDC-06-T06 $$ \boxed{ S_\lambda(z) \ge A_1 - \frac12A_2 - \frac12E. } $$ This lower-bounds: $$ \boxed{ P_\lambda+\Pi_\lambda. } $$ There is no need to distinguish the parity branches within the sieve. --- # 14. Strict Weighted Lower Summing T02: ## Theorem TCDC-06-T07 $$ \boxed{ \Pi_\lambda \ge A_1 - A_P - \frac12A_2 - \frac12E. } $$ Thus: $$ \boxed{ \text{strict route} = \text{neutral route} - A_P. } $$ --- # 15. Parameterize the Coarse Cutoff Let: $$ \boxed{ w=X^{1/u}, } $$ where: $$ u>3. $$ Current distribution benchmark: $$ \boxed{ \theta=3/5. } $$ Coarse lower-sieve parameter: $$ \boxed{ s_0=u\theta=\frac{3u}{5}. } $$ A positive lower linear sieve requires: $$ s_0>2, $$ That is: $$ \boxed{ u>10/3. } $$ --- # 16. Medium Upper-Sieve Endpoint Condition Medium prime: $$ \ell=X^v, $$ $$ 1/u\le v\le1/3. $$ Remaining distribution level: $$ D/\ell. $$ Relative to: $$ w=X^{1/u}, $$ Upper-sieve parameter: $$ \boxed{ s_\ell = u(\theta-v). } $$ The minimum occurs at: $$ v=1/3. $$ Thus: $$ s_\ell>1 $$ throughout the full medium interval requires: ## Theorem TCDC-06-T08 $$ \boxed{ u>\frac{15}{4}. } $$ This matches the sign threshold of TCDC-05 exactly. --- # 17. Explicit Constant Window When: $$ \boxed{ 15/40 $$ iff: $$ \frac{ 4(3u-5) }{ 25 } >1. $$ Thus: ## Corollary TCDC-06-C01 $$ \boxed{ C_{\mathrm{core}}(u)>0 \iff u>15/4 } $$ inside the explicit window. This is exactly the same as T08. Thus: > When the upper-sieve endpoint first enters the admissible regime, > the half-weight core also turns positive. --- # 22. Explicit Constant Table | $u$ | $C_0(u)$ | $C_{\mathrm{med}}(u)$ | $C_{\mathrm{core}}(u)$ | |---:|---:|---:|---:| | $4$ | $0.499401$ | $0.830597$ | $0.084103$ | | $5$ | $0.823030$ | $1.087987$ | $0.279037$ | | $20/3$ | $0.978354$ | $1.177071$ | $0.389818$ | Thus: $$ \boxed{ \text{quarter-root }u=4 \text{ is far from optimal even before extra costs}. } $$ --- # 23. Monotonicity on the Explicit Window The derivative sign in T11 is controlled by: $$ \frac{ 3u }{ 3u-5 } - \log \left( \frac{ 4(3u-5) }{ 25 } \right) $$ as displayed above. This quantity, on: $$ [15/4,20/3] $$ decreases monotonically, and at: $$ u=20/3 $$ is still positive. Thus: ## Theorem TCDC-06-T12 $$ \boxed{ C_{\mathrm{core}}(u) \text{ strictly increases on } (15/4,20/3]. } $$ --- # 24. Beyond $u=20/3$ When: $$ u>20/3, $$ The linear-sieve functions enter subsequent delay-differential segments. The pointwise T03 remains fully admissible; only: $$ F_1, \quad f_1 $$ are no longer represented by the elementary closed forms of the first two segments. Numerical evaluation using the classical delay-differential equations gives: as follows: ## Computational Diagnostic TCDC-06-D01 $$ \boxed{ C_{\mathrm{core}}(u) } $$ near: $$ u\approx7.806 $$ attains approximately: $$ \boxed{ 0.406677. } $$ --- # 25. The $u=8$ Benchmark Take: $$ \boxed{ u=8, \qquad w=X^{1/8}. } $$ Standard linear-sieve numerical values: $$ \boxed{ C_0(8) \approx 0.997028, } $$ $$ \boxed{ C_{\mathrm{med}}(8) \approx 1.181542, } $$ Thus: ## Diagnostic TCDC-06-D02 $$ \boxed{ C_{\mathrm{core}}(8) \approx 0.406257. } $$ This is very close to the D01 optimum. --- # 26. Relation to Classical Chen's $x^{1/8}$ Cutoff Classical expositions of Chen often choose: $$ z_{\mathrm{Chen}} = x^{1/8}. $$ This round does not directly copy Chen's cutoff: $$ 1/8. $$ Instead, after re-optimizing the cutoff using: - the current: $$ \theta=3/5; $$ - generalized strict / neutral half-weight; - standard linear-sieve functions; as listed above, the numerical optimum naturally lies at: $$ \boxed{ u\approx7.8, } $$ that is, very close to the exponent: $$ 1/8 $$ as displayed above. This is a strong structural-consistency signal, not a theorem identification. --- # 27. Prime Branch: Standard Separate Upper The strict route requires an upper bound on: $$ A_P $$ as displayed above. If the same level: $$ D=X^{3/5} $$ is used, then, to isolate a prime complement: $$ y\le X, $$ one can apply the upper sieve up to: $$ \boxed{ Z_P=X^{1/2}. } $$ because every composite: $$ y\le X $$ has a prime divisor: $$ \le\sqrt X. $$ --- # 28. Prime Upper Sieve Parameter $$ \boxed{ s_P = \frac{ \log D }{ \log Z_P } = \frac{ 3/5 }{ 1/2 } = \frac65. } $$ Thus: $$ \boxed{ F_1(6/5) = \frac{ 5e^\gamma }{3} \approx 2.968454. } $$ --- # 29. Euler-Product Ratio Relative to the coarse cutoff: $$ w=X^{1/u}, $$ the dimension-1 Euler product obeys: $$ W(t) \asymp C/\log t. $$ Thus: ## Diagnostic TCDC-06-D03 $$ \boxed{ \frac{ W(X^{1/2}) }{ W(X^{1/u}) } \to \frac{ 2 }{u}. } $$ --- # 30. Standard Separate Prime-Cost Constant Thus, the normalized main coefficient of the standard separate prime upper bound: ## Diagnostic TCDC-06-D04 $$ \boxed{ C_P^{\mathrm{std}}(u) = \frac{ 2 }{u} F_1(6/5) = \frac{ 10e^\gamma }{ 3u }. } $$ at: $$ u=8, $$ $$ \boxed{ C_P^{\mathrm{std}}(8) \approx 0.742114. } $$ --- # 31. Standard Separate-Upper No-Go at $u=8$ Branch-neutral core: $$ \boxed{ C_{\mathrm{core}}(8) \approx0.406257. } $$ Standard prime upper bound: $$ \boxed{ C_P^{\mathrm{std}}(8) \approx0.742114. } $$ Thus: ## Diagnostic TCDC-06-D05 $$ \boxed{ C_{\mathrm{core}}(8) - C_P^{\mathrm{std}}(8) \approx -0.335857. } $$ This has not paid anything at all for: $$ E_1. $$ Therefore: ## Provider No-Go TCDC-06-N01 $$ \boxed{ \text{generalized half-weight} + \text{standard separate linear-sieve prime upper} } $$ cannot close strict pressure at: $$ u=8. $$ This is not a mathematical impossibility theorem; it rejects this specific provider stack. --- # 32. Ideal Prime Density Is Much Smaller Than the Standard Upper If the prime branch had precisely the ideal sieve multiplier: $$ 1 $$ rather than: $$ F_1(6/5), $$ its normalized density ratio would be only approximately: $$ \boxed{ C_P^{\mathrm{ideal}}(u) = 2/u. } $$ at: $$ u=8: $$ $$ \boxed{ C_P^{\mathrm{ideal}}(8) = 1/4. } $$ Thus: $$ \boxed{ C_{\mathrm{core}}(8) - C_P^{\mathrm{ideal}}(8) \approx 0.156257. } $$ before: $$ E_1. $$ --- # 33. Interpretation The strict route is not purely combinatorially impossible. The actual problem: $$ \boxed{ \text{available / standard prime upper multiplier is far above ideal}. } $$ Thus, for the strict route to remain viable, it needs: - a near-asymptotic upper bound for balanced prime product + prime; - a transport-assisted prime cost; - or a different treatment of the branches. --- # 34. Classical Chen Triple Benchmark The triple-factor term of the classical Chen $x^{1/8}$ treatment has a normalized integral constant of approximately: $$ 0.363 $$ in Tao's exposition. If, very crudely, one takes: $$ \frac12\times0.363 \approx0.1815 $$ as an architectural benchmark, then even the ideal: $$ u=8 $$ window after the prime tax: $$ 0.1563 $$ may be insufficient. However: ## Research Firewall TCDC-06-R01 $$ \boxed{ \text{classical Chen triple constant is not automatically the TCDC high-suffix }E_1\text{ constant}. } $$ The base sequences differ: - classical Chen: prime + complement; - here: balanced prime product + complement. Thus, this serves only as a warning, not as a no-go theorem. --- # 35. Strict Local Route Is Parity-Taxed T05 shows that the prime coefficient: $$ 1 $$ is unavoidable. Meanwhile: $$ P_\lambda $$ and: $$ \Pi_\lambda $$ are precisely the cube-root RootRough boundary's: - odd-parity branch; - even-parity branch. Thus: $$ \boxed{ \text{The strict local pressure selector explicitly puts the parity split back into the proof ledger}. } $$ This is the essence of the prime tax. --- # 36. TCDC Allows a Different Strategy TCDC failure geometry does not require the sieve itself to distinguish: $$ P_\lambda \text{ vs } \Pi_\lambda. $$ We can let the sieve prove only that: $$ \boxed{ S_\lambda(z) = P_\lambda+\Pi_\lambda } $$ is large. Then failure transport determines: - where the prime branch can flow; - where the semiprime branch can flow. This is: # Branch-Neutral Sieve + Branch-Sensitive Capacity --- # 37. Two Sink Channels For the high suffix: ### Prime Branch $$ P_\lambda $$ flows into: $$ \boxed{ \text{prime sink}. } $$ ### Semiprime Branch $$ \Pi_\lambda $$ flows into: $$ \boxed{ \text{mirror semiprime receiver}. } $$ Therefore: $$ S_\lambda=P_\lambda+\Pi_\lambda $$ is the total sender mass. --- # 38. Prime-Sink Capacity Let: $$ \boxed{ C_P^{fail}(\lambda) } $$ to be any proof-admissible failure upper bound satisfying: $$ \boxed{ P_\lambda \le C_P^{fail}(\lambda). } $$ It may come from: - row mixed capacities: $$ \sum_{r\in B_\lambda}U_r^P; $$ - prime-sink saturation cuts; - center-divisor locks; - balanced-convolution upper bounds; - an LP-dual combination. --- # 39. Mirror-Sink Capacity TCDC-02 defines: $$ \boxed{ C_{\mathrm{mir}}^{fail}(\lambda) } $$ such that: $$ \boxed{ \Pi_\lambda \le C_{\mathrm{mir}}^{fail}(\lambda) } $$ under failure. Sources: - a mirror-prefix receiver; - cross-cell capacities; - prime usage in the prefix; - recursion; - MRC analytic compression. --- # 40. Two-Sink Failure Inequality Since: $$ S_\lambda = P_\lambda+\Pi_\lambda, $$ failure must satisfy: ## Theorem TCDC-06-T13 $$ \boxed{ S_\lambda(z) \le C_P^{fail}(\lambda) + C_{\mathrm{mir}}^{fail}(\lambda). } $$ This is an exact branch-capacity consequence. --- # 41. Branch-Neutral Closure Theorem Suppose there is a branch-neutral lower bound: $$ \boxed{ L_{P_2}(\lambda) \le S_\lambda(z). } $$ If: $$ \boxed{ L_{P_2}(\lambda) > C_P^{fail}(\lambda) + C_{\mathrm{mir}}^{fail}(\lambda), } $$ then: ## Theorem TCDC-06-T14 $$ \boxed{ \text{Goldbach failure impossible}. } $$ Thus: $$ \boxed{ M_{PP}>0. } $$ --- # 42. Two-Sink Deficit Define: $$ \boxed{ \Delta_{2S}(\lambda) = L_{P_2}(\lambda) - C_P^{fail}(\lambda) - C_{\mathrm{mir}}^{fail}(\lambda). } $$ If: $$ \boxed{ \Delta_{2S}(\lambda)>0, } $$ then the failure polytope is empty. This is a: # Branch-Neutral TCDC Certificate --- # 43. Why This Is Not Just Moving Terms Around Strict weighted route: $$ \Pi \ge L_{P_2}-U_P. $$ It requires: $$ U_P $$ as an isolated analytic upper bound, directly consuming the sieve margin. The two-sink TCDC route instead allows: $$ C_P^{fail} $$ to be obtained jointly from: - multiple row constraints; - global prime-sink saturation; - the transport dual; - center locks; as listed above. Thus: $$ \boxed{ C_P^{fail} } $$ need not equal an independent analytic upper bound. This is the additional freedom genuinely provided by the failure-polytope method. --- # 44. LP-Dual Interpretation In the TCDC-01 failure LP: - mixed edges: $$ M_{r,0}; $$ - internal edges: $$ M_{r,s}. $$ A branch-neutral lower bound: $$ S_\lambda $$ merely adds a lower-mass constraint to selected sender rows. The LP dual can automatically supply: - a prime-sink price; - a mirror-sink price; - a row price. Thus: $$ \boxed{ \text{branch-neutral weighted sieve} + \text{LP dual transport} } $$ is a natural combined proof architecture. --- # 45. A Parametric Branch Price More generally, Take: $$ 0\le\eta\le1. $$ Define: $$ \boxed{ S_{\lambda,\eta} = \Pi_\lambda + \eta P_\lambda. } $$ Pointwise weight: $$ \boxed{ \omega_{\eta} = \omega_{neutral} - (1-\eta)\mathbf1_{\mathbb P}. } $$ Thus: ## Theorem TCDC-06-T15 $$ \boxed{ S_{\lambda,\eta} \ge A_1 - (1-\eta)A_P - \frac12A_2 - \frac12E. } $$ --- # 46. Dual Capacity for Branch Price Under failure: $$ \Pi_\lambda \le C_{\mathrm{mir}}, $$ $$ P_\lambda \le C_P. $$ Thus: ## Theorem TCDC-06-T16 $$ \boxed{ S_{\lambda,\eta} \le C_{\mathrm{mir}} + \eta C_P. } $$ If the lower bound exceeds the right side, failure is impossible. --- # 47. $\eta$ Is a TCDC Dual Variable ### $\eta=0$ Strict semiprime pressure route. ### $\eta=1$ Fully branch-neutral $P_2$ route. ### Intermediate The prime branch receives partial credit. Thus: $$ \boxed{ \eta } $$ can be jointly optimized using: - analytic prime-upper quality; - transport prime-sink capacity; - mirror capacity; as listed above. This is not merely a sieve parameter, but a: # Failure-Polytope Dual Price --- # 48. Relation to Classical Chen Classical Chen target: $$ P_2 $$ naturally corresponds to: $$ \boxed{ \eta=1. } $$ because primes and semiprimes are both acceptable. Goldbach strict pressure: $$ \boxed{ \eta=0. } $$ TCDC instead allows: $$ \boxed{ 0<\eta<1 } $$ because, although the prime branch is not the pressure target, it still consumes failure-transport capacity. This is a new degree of freedom of this framework relative to classical Chen. --- # 49. Remaining Medium-$P_2$ Cost The branch-neutral choice: $$ \eta=1 $$ eliminates the prime tax, but: $$ E = A_{2,1} + A_{3,1} +\text{higher-factor }\nu=1\text{ states} $$ still exists. The target still requires: $$ \operatorname{RootRough}_z. $$ A medium semiprime: $$ \ell q, \quad w<\ell\le z, $$ does not belong to the cube-root transport boundary. Thus: $$ \boxed{ \text{medium-}P_2\text{ cost remains a genuine cutoff-descent cost}. } $$ --- # 50. Lowering $w$ Increases $E_1$ Complexity The generalized T02 permits: $$ u>4, $$ but coarse RootRough$_w$ may contain: $$ P_4,P_5,\ldots $$ Although the pointwise weight remains admissible, $$ E_1 $$ includes: > exactly one medium factor, > plus multiple factors $>z$. However, if: $$ z^3\ge X, $$ after exactly one medium factor, there cannot be three additional factors: $$ >z. $$ Since: $$ w\cdot z^3>X $$ for any: $$ w>1. $$ Thus: ## Theorem TCDC-06-T17 $$ \boxed{ E_1 } $$ still contains only: - medium $P_2$; - medium $P_3$. even if: $$ w $$ is far below quarter root. This is an important reason why the generalized cutoff remains analytically manageable. --- # 51. Proof of T17 If: $$ \nu=1, $$ there is a unique medium prime factor: $$ \ell>w. $$ All other prime factors are: $$ >z. $$ If there are at least three other factors, then: $$ y>\ell z^3>wX>X, $$ a contradiction. Thus, there are at most two other prime factors. It is composite and has at least one other factor, Hence: - one other: $$ P_2; $$ - two others: $$ P_3. $$ This completes the proof. --- # 52. Why $u\approx8$ Is Especially Attractive at: $$ u=8, $$ we simultaneously obtain: 1. a coarse lower multiplier almost equal to: $$ 1; $$ 2. the half-medium core: $$ \approx0.406; $$ 3. an exactly-one correction still requiring only: - medium $P_2$; - medium $P_3$; 4. a cutoff at the same scale as classical Chen: $$ x^{1/8} $$ as displayed above. Thus: $$ \boxed{ u=8 } $$ becomes a more natural TCDC-06 baseline than: $$ u=4 $$ as displayed above. --- # 53. Updated Strict Extra-Cost Gate at $u=8$ Strict route: $$ \Pi \ge \text{core} - A_P - \frac12E. $$ Normalized: $$ \boxed{ C_P + \frac12C_E < 0.406257 } $$ is required for positive pressure. This is much larger than TCDC-05's: $$ 0.0841 $$ as displayed above. However, the standard separate prime upper bound: $$ C_P^{std} \approx0.7421 $$ already exceeds the budget. --- # 54. Updated Neutral Gate at $u=8$ Branch-neutral route: $$ S_z \ge \text{core} - \frac12E. $$ Thus, one needs only: $$ \boxed{ \frac12C_E < 0.406257 } $$ for positive $P_2$ mass. Then leave branch allocation to the two-sink capacity. This is clearly more consistent with classical Chen polarity. --- # 55. Provider Priorities After This Round The priority is no longer: > sharply upper-bound $A_P$, then subtract it outright from the strict weight. Instead: ### Priority 1 lower-bound the branch-neutral: $$ S_\lambda(z). $$ ### Priority 2 upper-bound / dual-certify: $$ C_P^{fail}+C_{\mathrm{mir}}^{fail}. $$ ### Priority 3 optimize: $$ \eta. $$ ### Priority 4 only if needed, return to strict: $$ \eta=0. $$ --- # 56. Formal Theorem Queue ```yaml TCDC_06: T01: generalized_bad_composite_has_medium_factor T02: generalized_strict_half_weight T03: generalized_branch_neutral_half_weight T04: strict_equals_neutral_minus_prime T05: local_prime_tax_coefficient_at_least_one T06: summed_neutral_lower T07: summed_strict_lower T08: full_medium_upper_legal_requires_u_gt_15_4 T09: explicit_coarse_lower_constant T10: explicit_medium_integral T11: explicit_half_core_constant T12: core_monotone_on_closed_form_window T13: two_sink_failure_inequality T14: branch_neutral_failure_closure T15: eta_weighted_lower T16: eta_dual_capacity_upper T17: exactly_one_medium_remains_only_P2_or_P3 ``` --- # 57. Diagnostic Queue ```yaml TCDC_06_Diagnostics: quarter_root: u: 4 core: 0.084102748153 u_5: core: 0.279036833468 closed_form_endpoint: u: 20/3 core: 0.389818305228 numerical_optimum: u: 7.805950 core: 0.406676567891 u_8: coarse: 0.997027667435 medium: 1.181541540743 core: 0.406256897064 standard_prime_upper: 0.742113507496 ideal_prime_density: 0.250000000000 ``` --- # 58. Barrier Queue ```yaml TCDC_06_Barriers: B01: strict_local_prime_tax: coefficient_at_least_one B02: standard_prime_upper_multiplier_too_large B03: exact_one_medium_cost_not_closed B04: branch_neutral_prime_sink_capacity_not_closed B05: mirror_capacity_not_closed B06: eta_dual_optimization_not_done B07: no_positive_two_sink_deficit_yet ``` --- # 59. Acceptance Gate Before the TCDC-06 formal core, the following are required: 1. a generalized pointwise theorem independent of the factor-depth cutoff; 2. the desired $P_2$ branch exact; 3. a lower bound on the prime coefficient; 4. the $u$ parameter domain; 5. lower / upper sieve parameters separated; 6. closed-form constants used only within their valid $s$ ranges; 7. $u\approx7.8$ labeled a numerical diagnostic; 8. the classical $1/8$ comparison labeled structural, not a derivational identity; 9. the standard prime-upper no-go labeled a provider-stack no-go; 10. ideal prime density labeled diagnostic; 11. the branch-neutral two-sink theorem exact; 12. the $\eta$ dual price exact; 13. no positive two-sink-deficit claim; 14. no Goldbach-proof claim. --- # 60. Verdict This round can formally give: $$ \boxed{ \mathrm{TCDC\mbox{-}06\ BRANCH\mbox{-}NEUTRAL\ DUAL\mbox{-}SINK\ KERNEL\ PASS}. } $$ The greatest combinatorial improvement: $$ \boxed{ \text{TCDC-05 half-weight does not require }w=X^{1/4}. } $$ It holds for any: $$ w0. } $$ This avoids requiring the sieve itself to cross the prime / semiprime parity split. More generally, there is also: $$ \boxed{ \eta\in[0,1] } $$ as a prime-branch dual price, connecting classical Chen: $$ \eta=1 $$ and strict pressure: $$ \eta=0 $$ into a single family of TCDC certificates. However, currently: $$ \boxed{ \mathrm{NO\ POSITIVE\ TWO\mbox{-}SINK\ DEFICIT\ YET}. } $$ Therefore: $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 61. Recommendations for the Next Round # TCDC-07 ## Two-Sink Capacity Ledger / $\eta$-Dual Optimization Closure The next round should no longer attempt a separate analytic subtraction of the prime branch. Directly establish: $$ \boxed{ \Delta_\eta = L_\eta - C_{\mathrm{mir}} - \eta C_P. } $$ ### Part A — Weighted Lower Using: $$ u\approx8 $$ as a baseline, establish: $$ L_\eta. $$ ### Part B — Prime Sink Do not use only single-row BT. Combine: - prime-sink saturation; - center locks; - suffix arithmetic progressions; - the transport LP dual; to form: $$ C_P^{fail}. $$ ### Part C — Mirror Sink Use TCDC-02 / MRC: $$ C_{\mathrm{mir}}^{fail}. $$ ### Part D — Optimize $\eta$ Find: $$ \boxed{ \sup_{0\le\eta\le1} \Delta_\eta. } $$ If positive, the failure polytope is immediately empty. This will be the first genuine combination of: # Chen Weighted Sieve and: # TCDC Failure Transport Dual within one optimized certificate. --- # Appendix A: Generalized Pointwise Weights ```yaml strict: omega: RR_w * ( 1 - prime - nu/2 - E1/2 ) lower_bounds: strict_semiprime_lpf_gt_z neutral: omega: RR_w * ( 1 - nu/2 - E1/2 ) lower_bounds: RootRough_z: prime_or_semiprime conditions: w < z z^3 >= X ``` --- # Appendix B: Closed-Form $3/5$ Core ```yaml u_range: 15/4 < u <= 20/3 theta: 3/5 C0: (10*exp(gamma)/(3u)) * log((3u-5)/5) Cmed: (10*exp(gamma)/(3u)) * log((3u-5)/4) Ccore: (5*exp(gamma)/(3u)) * log(4*(3u-5)/25) positive_iff: u > 15/4 ``` --- # Appendix C: $u=8$ Diagnostic ```yaml u: 8 coarse: 0.997027667435 medium: 1.181541540743 half_core: 0.406256897064 standard_separate_prime_upper: 0.742113507496 ideal_prime_density: 0.250000000000 strict_with_standard_prime_upper: negative_before_E1: true neutral_before_E1: positive: true ``` --- # Appendix D: Two-Sink Closure ```yaml sender: S_lambda: prime_branch + semiprime_branch failure_sinks: prime: capacity: C_P_fail mirror_semiprime: capacity: C_mir_fail failure_requires: S_lambda <= C_P_fail + C_mir_fail if_lower: L_P2 > C_P_fail + C_mir_fail then: failure_polytope_empty: true Goldbach_success: true ``` --- # Appendix E: $\eta$ Dual Price ```yaml eta: range: 0_to_1 weighted_sender: S_eta: Pi + eta*P weighted_lower: A1 -(1-eta)*A_P -A2/2 -E/2 failure_capacity: C_mir + eta*C_P special_cases: eta_0: strict_pressure eta_1: branch_neutral_P2 ``` --- # Appendix F: Commit Decision ```yaml research_round: TCDC-06 decision: half_weight_generalized: true quarter_root_necessity: false cutoff_optimization: true near_eighth_root_optimum: computationally_identified strict_prime_tax: formalized standard_separate_prime_upper_closure: false branch_neutral_two_sink_route: formalized eta_dual_price: formalized positive_two_sink_deficit: not_proved Goldbach_proved: false next: TCDC-07_TwoSink_CapacityLedger_EtaDual_Optimization_Closure ``` --- # References / Audited External Sources 1. Terence Tao, **254A, Supplement 5: The linear sieve and Chen's theorem**, 2015. Relevant facts: - classical dimension-$1$ linear sieve functions and their delay-differential continuation; - Chen's proof uses a coarse cutoff $z=x^{1/8}$, half-weighting of medium divisors, and switching / triple-factor control; - Chen's target is $P_2$, hence branch-neutral with respect to prime versus semiprime. 2. James Maynard, **Primes in arithmetic progressions to large moduli II: Well-factorable estimates**, *Memoirs AMS* 306 (2025), no. 1543. Relevant benchmark: mean distribution with suitably well-factorable weights to level $x^{3/5-\varepsilon}$. 3. James Maynard, **Primes in arithmetic progressions to large moduli III: Uniform residue classes**, *Memoirs AMS* 306 (2025), no. 1544. Relevant as a possible source of sharper prime-sink / balanced-convolution upper estimates than the standard isolated linear-sieve prime upper. 4. Zhiyuan Yang, **Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications**, arXiv:2608.13299 (2026). Relevant as a modern convolution / well-factorable-weight technology source. No direct Goldbach-shift application is asserted here.