# TCDC-17 ## Bad-Mass Simplex / Semiprime-Marginal Upper Constant Closure ### Exact Bad-Mass Geometry、Provider-Loss Budget、Fixed-Row Semiprime Upper No-Go、Aggregate Level-Lift Target 與 Coupled-Discount Requirement **母方法論：** TCDC — Target–Condition Dual Closure **對接研究線：** RG-SCP / SRBRD **輪次：** TCDC-17 **日期：** 2026-10-02 **版本：** v0.1 **作者：** Neo.K；Aletheia（GPT-5.6 Sol） **性質：** Bad-Mass Simplex / Upper-Sieve Constant Audit / Provider-Loss Geometry **前置：** TCDC-00 ～ TCDC-16；SRBRD-00 ～ SRBRD-26 **狀態：** 將 exact identity $M_{PP}=S_m-n_{SP}-M_{PS}$ 正規化成二維 bad-mass simplex；建立 neutral-lower loss、semiprime-upper loss、mixed-upper loss共享的 provider budget；證明在 rank-one Buchstab benchmark下，standard fixed-row linear upper即使給予整個 row-scale 的最佳 distribution level，semiprime marginal multiplier仍約 $2.043896$，與 ideal mixed upper相加仍超出 simplex；因此 semiprime marginal必須使用 aggregate convolution / switching 或 coupled bad-mass discount；未證明 uniform positive simplex gap；Goldbach仍未證明 --- # 0. 本輪目標 TCDC-16 exact： $$ \boxed{ M_{PP} = S_m - n_{SP} - M_{PS}. } $$ 這一式把 Goldbach target改寫成： # Neutral Mass − Bad Mass 其中 bad mass有兩類： 1. semiprime left marginal： $$ n_{SP}; $$ 2. mixed prime-semiprime branch： $$ M_{PS}. $$ TCDC-17 要做： > 把這三個量放進同一 normalization， > 研究各種 proof providers 可以容許多少 constant loss。 --- # 1. Exact Bad-Mass Coordinates 定義： $$ \boxed{ a_m = \frac{ n_{SP} }{ S_m }, } $$ $$ \boxed{ b_m = \frac{ M_{PS} }{ S_m }. } $$ 因： $$ S_m = n_0+n_{SP}, $$ 有： $$ \boxed{ \frac{ n_0 }{ S_m } = 1-a_m. } $$ --- # 2. Exact Matrix in $(a,b)$ Coordinates 由： $$ n_{SP} = M_{PS}+M_{SS}, $$ 得到： $$ \boxed{ \frac{ M_{SS} }{ S_m } = a_m-b_m. } $$ 由 TCDC-16 bad-mass identity： $$ \boxed{ \frac{ M_{PP} }{ S_m } = 1-a_m-b_m. } $$ 所以： ## Theorem TCDC-17-T01 $$ \boxed{ \frac1{S_m} \begin{pmatrix} M_{PP}&M_{PS}\\ M_{PS}&M_{SS} \end{pmatrix} = \begin{pmatrix} 1-a-b&b\\ b&a-b \end{pmatrix}. } $$ --- # 3. Feasible Bad-Mass Simplex matrix entries nonnegative要求： $$ b\ge0, $$ $$ a-b\ge0, $$ $$ 1-a-b\ge0. $$ 因此： ## Theorem TCDC-17-T02 ### Bad-Mass Simplex $$ \boxed{ 0\le b\le a, \qquad a+b\le1. } $$ 所以所有 exact joint transport都落在： $$ (a,b) $$ plane 的 triangular feasible region。 --- # 4. Goldbach Failure Face $$ M_{PP}=0 $$ iff： $$ 1-a-b=0. $$ 因此： ## Theorem TCDC-17-T03 Goldbach failure正是 bad-mass simplex的斜邊： $$ \boxed{ a+b=1. } $$ 且因： $$ b\le a, $$ failure edge滿足： $$ \boxed{ a\ge1/2. } $$ --- # 5. Failure-Line Distance exact good mass： ## Theorem TCDC-17-T04 $$ \boxed{ d_{\mathrm{bad}} = 1-a_m-b_m = \frac{ M_{PP} }{ S_m }. } $$ 所以 ordered Goldbach density就是： # Bad-Mass Point 到 Failure Face 的 Linear Slack --- # 6. Buchstab Rank-One Benchmark Point TCDC-14： $$ \boxed{ p_* = \frac1{1+\log2}, } $$ $$ \boxed{ s_* = \frac{\log2}{1+\log2}. } $$ rank-one benchmark： $$ \boxed{ a_*=s_*, } $$ $$ \boxed{ b_*=p_*s_*. } $$ numerically： $$ \boxed{ a_* \approx 0.409383890850, } $$ $$ \boxed{ b_* \approx 0.241788720763. } $$ --- # 7. Benchmark Distance to Failure T04： $$ 1-a_*-b_* = 1-s_*-p_*s_*. $$ 因： $$ p_*+s_*=1, $$ 得到： ## Theorem TCDC-17-T05 $$ \boxed{ 1-a_*-b_* = p_*^2. } $$ numerically： $$ \boxed{ p_*^2 \approx 0.348827388387. } $$ 所以： $$ \boxed{ 0.348827\ldots } $$ 就是 benchmark point到 failure face的 exact bad-mass slack。 --- # 8. Common Main-Scale Normalization 令： $$ \boxed{ \mathfrak M_m>0 } $$ 為某 proof / asymptotic common neutral scale。 假設： $$ \boxed{ S_m \ge \alpha \mathfrak M_m, } $$ $$ \boxed{ n_{SP} \le \kappa_{SP} s_* \mathfrak M_m, } $$ $$ \boxed{ M_{PS} \le \kappa_{PS} p_*s_* \mathfrak M_m. } $$ 其中： - $\alpha$： neutral lower efficiency； - $\kappa_{SP}$： semiprime marginal upper multiplier； - $\kappa_{PS}$： mixed upper multiplier。 --- # 9. Separate-Provider Simplex Gate TCDC-16： $$ M_{PP} \ge L_S-U_{SP}-U_{PS}. $$ 所以： ## Theorem TCDC-17-T06 ### Separate Bad-Mass Gate $$ \boxed{ \frac{ M_{PP} }{ \mathfrak M_m } \ge \alpha - \kappa_{SP}s_* - \kappa_{PS}p_*s_*. } $$ 因此 sufficient： $$ \boxed{ \kappa_{SP}s_* + \kappa_{PS}p_*s_* < \alpha. } $$ --- # 10. Provider-Loss Budget subtract ideal benchmark： $$ 1 = p_*^2 + s_* + p_*s_*. $$ T06 rearranges： ## Theorem TCDC-17-T07 ### Provider-Loss Budget closure iff： $$ \boxed{ (1-\alpha) + s_* (\kappa_{SP}-1) + p_*s_* (\kappa_{PS}-1) < p_*^2. } $$ left side： # Total Provider Loss right side： # Benchmark Failure-Face Distance --- # 11. Interpretation of the Loss Budget three independent losses all consume the same： $$ \boxed{ p_*^2 \approx 0.348827388387 } $$ budget。 ### Neutral loss $$ 1-\alpha. $$ ### Semiprime-upper excess $$ s_*(\kappa_{SP}-1). $$ ### Mixed-upper excess $$ p_*s_*(\kappa_{PS}-1). $$ 所以： > 不能只看某一個 upper multiplier是否「看起來還行」； > 必須看全部 provider loss的加總。 --- # 12. Minimum Neutral Efficiency even if both upper providers are ideal： $$ \kappa_{SP} = \kappa_{PS} = 1, $$ T06 requires： ## Corollary TCDC-17-C01 $$ \boxed{ \alpha > s_*+p_*s_* = 1-p_*^2. } $$ numerically： $$ \boxed{ \alpha > 0.651172611613. } $$ 所以 neutral lower若丟掉超過約： $$ \boxed{ 34.88\% } $$ main mass， separate stack甚至在 perfect uppers下也不可能 closure。 --- # 13. One Ideal Upper at a Time ## Semiprime upper ideal if： $$ \kappa_{SP}=1, $$ T06 gives： ## Corollary TCDC-17-C02 for： $$ \alpha=1, $$ $$ \boxed{ \kappa_{PS} < \frac{ 1-s_* }{ p_*s_*} = \frac1{s_*} } $$ numerically： $$ \boxed{ \kappa_{PS} < 2.442695040889. } $$ --- # 14. Mixed upper ideal if： $$ \kappa_{PS}=1, $$ then： ## Corollary TCDC-17-C03 for： $$ \alpha=1, $$ $$ \boxed{ \kappa_{SP} < \frac{ 1-p_*s_* }{ s_*} } $$ numerically： $$ \boxed{ \kappa_{SP} < 1.852078931739. } $$ --- # 15. Equal-Multiplier Gate if： $$ \kappa_{SP} = \kappa_{PS} = \kappa, $$ then： ## Corollary TCDC-17-C04 $$ \boxed{ \kappa < \frac{ \alpha }{ s_*+p_*s_*} } $$ for general： $$ \alpha. $$ selected values： | Neutral efficiency $\alpha$ | max equal multiplier $\kappa$ | |---:|---:| | $1.000000$ | $1.535691$ | | $0.950000$ | $1.458907$ | | $0.900000$ | $1.382122$ | | $0.800000$ | $1.228553$ | | $0.700000$ | $1.074984$ | | $0.651173$ | $1.000000$ | at： $$ \alpha=1, $$ $$ \boxed{ \kappa < 1.535691124237. } $$ --- # 16. Coupled Bad-Mass Discount separate upper： $$ U_{SP}+U_{PS} $$ may double-count failure resources。 define a coupled bad-mass support： $$ \boxed{ B_{\max} } $$ with： $$ \boxed{ B_{\max} \le U_{SP}+U_{PS}. } $$ define normalized coupling discount： ## Definition TCDC-17-D01 $$ \boxed{ \chi = \frac{ U_{SP}+U_{PS}-B_{\max} }{ \mathfrak M_m }. } $$ --- # 17. Coupled Simplex Gate if： $$ B_{\max} \le \left( \kappa_{SP}s_* + \kappa_{PS}p_*s_* - \chi \right) \mathfrak M_m, $$ then： ## Theorem TCDC-17-T08 $$ \boxed{ \frac{ M_{PP} }{ \mathfrak M_m } \ge \alpha - \kappa_{SP}s_* - \kappa_{PS}p_*s_* + \chi. } $$ closure if： $$ \boxed{ \chi > \kappa_{SP}s_* + \kappa_{PS}p_*s_* - \alpha. } $$ --- # 18. Required Coupling Discount define： ## Definition TCDC-17-D02 $$ \boxed{ \chi_{\mathrm{req}} = \left( \kappa_{SP}s_* + \kappa_{PS}p_*s_* - \alpha \right)_+. } $$ Then any coupled LP / Hall / mirror improvement only needs： $$ \boxed{ \chi>\chi_{\mathrm{req}}. } $$ This turns： > coupling might help into： # Exact Discount Target --- # 19. Coupling Extends the Provider-Loss Budget T07 + T08： closure iff： $$ \boxed{ (1-\alpha) + s_*(\kappa_{SP}-1) + p_*s_*(\kappa_{PS}-1) < p_*^2+\chi. } $$ 所以 coupled transport直接增加 usable budget： $$ \boxed{ p_*^2 \to p_*^2+\chi. } $$ --- # 20. Semiprime Marginal Least-Factor Shell exact： $$ \boxed{ n_{SP} = \sum_{r>z}n_r. } $$ write least factor： $$ \boxed{ r=X^t, \qquad \frac13\le t\le\frac12. } $$ prime cofactor： $$ p $$ lives at： $$ \boxed{ X^{1-t}. } $$ ideal semiprime shell density is weighted by： $$ \boxed{ \frac{ dt }{ t(1-t) }. } $$ and total shell integral： $$ \boxed{ \int_{1/3}^{1/2} \frac{ dt }{ t(1-t) } = \log2. } $$ --- # 21. Fixed-Row Opposite-Rough Upper for fixed： $$ r=X^t, $$ row mass： $$ n_r $$ requires： - $p$ prime； - opposite endpoint RootRough to： $$ z=X^{1/3}. $$ Treating the prime cofactor sequence as the base， a standard dimension-$1$ upper linear sieve has parameter： $$ \boxed{ s_r = \frac{ \log D_r }{ \log z }. } $$ --- # 22. Absolute Row-Level Distribution Ceiling the prime cofactor interval has scale： $$ X^{1-t}. $$ even under an idealized full row-scale distribution： $$ \boxed{ D_r \le X^{1-t+o(1)}. } $$ therefore： ## Theorem TCDC-17-T09 $$ \boxed{ s_r \le 3(1-t)+o(1). } $$ for： $$ 1/3\le t\le1/2. $$ so： $$ \boxed{ 3/2 \le s_r \le 2 } $$ at the best possible rowwise scale。 --- # 23. Best Standard Linear Upper on Each Row linear sieve： $$ \boxed{ F(s) = \frac{ 2e^\gamma }{s} } $$ for： $$ 11. } $$ Therefore： ## Provider No-Go TCDC-17-N01 within the rank-one benchmark normalization： $$ \boxed{ \text{best-case standard fixed-row linear semiprime upper} + \text{ideal mixed upper} } $$ still cannot satisfy the separate bad-mass simplex gate。 --- # 28. Minimum Coupling Discount for This Stack T18： $$ \boxed{ \chi_{\mathrm{req}} \approx 0.078526978880. } $$ even with： - perfect neutral lower： $$ \alpha=1; $$ - ideal mixed upper： $$ \kappa_{PS}=1. $$ So coupled transport must recover at least about： $$ \boxed{ 7.85\% } $$ of the neutral benchmark mass just to compensate for the rowwise semiprime-upper architecture。 --- # 29. Equivalent Impossible Mixed Multiplier if no coupling： $$ \chi=0, $$ and semiprime upper remains： $$ \kappa_{SP}^{row}, $$ then mixed multiplier would need： ## Corollary TCDC-17-C05 $$ \boxed{ \kappa_{PS} < \frac{ 1-\kappa_{SP}^{row}s_* }{ p_*s_*} } $$ numerically： $$ \boxed{ \kappa_{PS} < 0.675224805224. } $$ This lies below： $$ 1, $$ the rank-one ideal benchmark。 So within this benchmark， no physically reasonable standalone mixed upper can rescue the rowwise semiprime provider。 --- # 30. Conclusion of the Rowwise Audit the main problem is now clear： $$ \boxed{ \text{semiprime marginal cannot be handled by independent fixed-row standard linear uppers}. } $$ This conclusion is stronger than： > current theorem constants happen to be loose。 Even granting each row its full interval-scale distribution， the shell-averaged standard multiplier remains too large for the bad-mass simplex。 --- # 31. Aggregate Level-Lift as the Escape Route fixed row： $$ r $$ offers only one prime variable： $$ p. $$ aggregate over： $$ r,p $$ creates a prime-product convolution： $$ \boxed{ rp } $$ and divisibility condition： $$ d\mid2m-rp. $$ This permits mean-value distribution unavailable to a single row。 Therefore： # Semiprime Marginal Escape Route is： $$ \boxed{ \text{$r$-average} + \text{convolution distribution} + \text{upper sieve}. } $$ --- # 32. Effective Sieve-Parameter Profile let aggregate technology provide an effective profile： $$ \boxed{ \sigma(t) } $$ for shell： $$ r=X^t. $$ Then linear-upper functional： ## Definition TCDC-17-D05 $$ \boxed{ \mathcal K_{SP}[\sigma] = \frac1{\log2} \int_{1/3}^{1/2} F(\sigma(t)) \frac{ dt }{ t(1-t) }. } $$ Target： $$ \boxed{ \mathcal K_{SP}[\sigma] < \kappa_{SP}^{\mathrm{target}}. } $$ --- # 33. Target if Mixed Upper Is Ideal from C03： $$ \boxed{ \kappa_{SP}^{\mathrm{target}} = 1.852078931739 } $$ when： $$ \alpha=1, \quad \kappa_{PS}=1. $$ so aggregate distribution must reduce： $$ \boxed{ 2.043896\ldots \to <1.852079\ldots } $$ at minimum。 --- # 34. Constant Effective Parameter Heuristic if： $$ \sigma(t)\equiv s_{\mathrm{eff}} $$ and： $$ 1 1.923322367603. } $$ --- # 35. Fixed Rows Reach This Only Near the Cube-Root Edge best fixed row： $$ s_r=3(1-t). $$ condition： $$ 3(1-t)>s_{\mathrm{eff}}^{req} $$ gives： ## Diagnostic TCDC-17-D07 $$ \boxed{ t < 0.358892544132. } $$ But semiprime shell extends to： $$ 1/2. $$ So only a narrow least-factor zone： $$ \boxed{ 1/3 \le t < 0.3589\ldots } $$ has enough rowwise sieve parameter even to meet the mixed-ideal target pointwise。 The majority of the logarithmic shell requires aggregate level lift。 --- # 36. Provider-Lift Functional define gain relative to rowwise ceiling： $$ \boxed{ G_{\mathrm{lift}} = \kappa_{SP}^{row} - \mathcal K_{SP}[\sigma]. } $$ minimum needed： $$ \boxed{ G_{\mathrm{lift}} > \kappa_{SP}^{row} - \kappa_{SP}^{target}. } $$ numerically： $$ \boxed{ G_{\mathrm{lift}} > 0.191817461887. } $$ if mixed upper ideal and no coupling。 --- # 37. Coupling Reduces Required Level Lift with coupling discount： $$ \chi, $$ simplex gate： $$ \kappa_{SP}s_* + \kappa_{PS}p_*s_* - \chi < \alpha. $$ so semiprime target becomes： ## Theorem TCDC-17-T11 $$ \boxed{ \kappa_{SP} < \frac{ \alpha - \kappa_{PS}p_*s_* + \chi }{ s_*}. } $$ Thus every coupled discount unit： $$ \chi $$ relaxes semiprime multiplier by： $$ \boxed{ \chi/s_*. } $$ --- # 38. Neutral Lower Loss Tightens the Target same T11： every neutral loss： $$ 1-\alpha $$ tightens allowable semiprime multiplier by： $$ \boxed{ (1-\alpha)/s_*. } $$ So aggregate semiprime upper and neutral lower cannot be optimized independently。 --- # 39. Modern Large-Moduli Technology Role Maynard's well-factorable large-moduli theorem reaches moduli： $$ \boxed{ x^{3/5-\varepsilon} } $$ for suitably weighted prime distribution。 Uniform-residue-class results also exceed： $$ x^{1/2} $$ for moduli with conveniently sized divisors。 These results do not automatically prove the exact TCDC semiprime marginal upper， but they supply precisely the kind of： # Aggregate Level Lift that fixed rows cannot access。 --- # 40. Required Theorem Shape TCDC semiprime marginal needs an estimate schematically： $$ \boxed{ \sum_{r\in\mathcal R} \sum_{\substack{ p\in\mathbb P\\ rp\le X }} \mathbf1_{ \operatorname{RootRough}_z(2m-rp) } } $$ from above， with： - $r$ prime； - $p$ prime； - $r$ spanning： $$ X^{1/3}\text{ to }X^{1/2}; $$ - growing Goldbach residue： $$ 2m; $$ - enough weighted mean distribution across $r$。 This is not the same as a collection of independent fixed-row Brun–Titchmarsh bounds。 --- # 41. Shell Partition partition： $$ \left[ 1/3,1/2 \right] = I_1\cup\cdots\cup I_J. $$ For each shell： $$ I_j, $$ derive： $$ \boxed{ \kappa_{SP,j} } $$ using the best legal provider。 Then total： ## Theorem TCDC-17-T12 $$ \boxed{ \kappa_{SP} = \frac{ \sum_j w_j\kappa_{SP,j} }{ \sum_jw_j } } $$ where： $$ \boxed{ w_j = \int_{I_j} \frac{ dt }{ t(1-t) }. } $$ This makes semiprime provider optimization a weighted shell-allocation problem。 --- # 42. Provider Portfolio possible shell choices： ### Near cube root fixed-row upper may already be adequate。 ### Middle shell use aggregate convolution / well-factorable estimates。 ### Near square root short rows are poor individually； use： - switching； - dual factorization； - transport LP； - exact center locks。 So the optimal semiprime marginal theorem need not use one provider globally。 --- # 43. TCDC Resource Allocation Returns TCDC-09 used LP shadow prices across rows。 TCDC-17 uses： $$ \boxed{ w(t) = 1/[t(1-t)] } $$ as asymptotic bad-mass value density。 So analytic effort should target shells with high product： $$ \boxed{ \text{shell mass} \times \text{multiplier excess} \times \text{available level lift}. } $$ This is the continuum analogue of proof-resource allocation。 --- # 44. Coupled Bad-Mass LP separate simplex： $$ U_{SP}+U_{PS}. $$ But the same semiprime row contributes simultaneously to： - marginal mass； - possible mixed mass； - internal $SS$ flow。 Therefore a coupled transport polytope can directly maximize： ## Definition TCDC-17-D08 $$ \boxed{ B_{\max} = \sup \left( n_{SP}+M_{PS} \right). } $$ subject to all TCDC failure constraints。 Then： $$ \boxed{ M_{PP} \ge L_S-B_{\max}. } $$ --- # 45. Coupled Discount Is Not Abstract TCDC-12 / 13 provide explicit mechanisms： - prime-sink saturation； - mirror receiver competition； - center locks； - network doubling； - mixed-capacity deficiencies。 All of these can lower： $$ B_{\max} $$ below： $$ U_{SP}+U_{PS}. $$ So the required： $$ \chi_{\mathrm{req}} $$ in T18 has concrete transport sources。 --- # 46. Finite Exact Simplex Audit using canonical cube-root RootRough states， selected exact cases： | $m$ | $S_m$ | $a=n_{SP}/S$ | $b=M_{PS}/S$ | bad sum $a+b$ | good $M_{PP}/S$ | |---:|---:|---:|---:|---:|---:| | 19 | 7 | 0.285714 | 0.285714 | 0.571429 | 0.428571 | | 96 | 38 | 0.236842 | 0.184211 | 0.421053 | 0.578947 | | 255 | 98 | 0.204082 | 0.142857 | 0.346939 | 0.653061 | | 496 | 70 | 0.357143 | 0.271429 | 0.628571 | 0.371429 | | 783 | 184 | 0.255435 | 0.190217 | 0.445652 | 0.554348 | | 870 | 234 | 0.205128 | 0.145299 | 0.350427 | 0.649573 | | 969 | 202 | 0.242574 | 0.113861 | 0.356436 | 0.643564 | | 1000 | 136 | 0.257353 | 0.198529 | 0.455882 | 0.544118 | | 4801 | 369 | 0.330623 | 0.254743 | 0.585366 | 0.414634 | These are exact finite counts for the stated boundary construction。 --- # 47. Finite Scan to $m\le5000$ computational scan： $$ 3\le m\le5000. $$ observed： ## Computational Diagnostic TCDC-17-D09 maximum bad-mass fraction： $$ \boxed{ \max(a_m+b_m) = 0.628571\ldots } $$ at： $$ \boxed{ m=496. } $$ equivalently minimum observed ordered $PP$ density： $$ \boxed{ \min \frac{M_{PP}}{S_m} = 0.371428\ldots } $$ at the same case。 benchmark rank-one $PP$ density： $$ \boxed{ p_*^2 \approx 0.348827388387. } $$ So finite sample lies even farther from failure face than the rank-one benchmark。 This is computational evidence only。 --- # 48. Finite Semiprime Marginal same scan observed maximum： $$ \boxed{ a_m = n_{SP}/S_m } $$ approximately： $$ \boxed{ 0.357143 } $$ in the scanned range， below benchmark： $$ s_* \approx 0.409383890850. $$ Again this is finite evidence， not an asymptotic theorem。 --- # 49. Why This Does Not Solve the Constant Problem finite bias may be caused by： - small center factors； - endpoint geometry； - small cube-root cutoff； - arithmetic singular series； - finite-size parity bias。 So no extrapolation to： $$ m\to\infty $$ is legal。 The finite scan is useful only to test whether the simplex framework captures actual data coherently。 --- # 50. Bad-Mass Simplex vs Parity Plane TCDC-15 parity coordinates： $$ (\Lambda,\mathcal C). $$ TCDC-17 bad coordinates： $$ (a,b). $$ The transformation is exact： ## Theorem TCDC-17-T13 $$ \boxed{ a = \frac{ 1+\Lambda }{2}, } $$ $$ \boxed{ b = \frac{ 1-\mathcal C }{4}. } $$ inverse： $$ \boxed{ \Lambda = 2a-1, } $$ $$ \boxed{ \mathcal C = 1-4b. } $$ So the bad-mass simplex and parity plane are the same geometry in different coordinates。 --- # 51. Failure Face Correspondence bad-mass failure： $$ a+b=1. $$ substitute T13： $$ \frac{ 1+\Lambda }{2} + \frac{ 1-\mathcal C }{4} = 1. $$ equivalent： $$ \boxed{ \mathcal C = 2\Lambda-1. } $$ exactly TCDC-15 failure line。 So： # Capacity Coordinates and： # Parity Coordinates are affine-dual descriptions of the same failure surface。 --- # 52. Why Bad-Mass Coordinates Are Better for Positive Sieve Providers in parity coordinates， need： - sign cancellation； - Liouville means。 in bad-mass coordinates， need： - positive neutral lower； - positive bad upper。 Thus T13 justifies using different coordinates for different tools。 TCDC does not insist on one representation。 --- # 53. Main Research Decision after TCDC-17 standard fixed-row semiprime upper is now ruled out as the sole separate provider architecture at benchmark level。 Therefore next effort should prioritize： 1. aggregate $r$-averaged semiprime marginal upper； 2. shell-specific convolution level lift； 3. coupled bad-mass discount； 4. only secondarily improve standalone mixed upper。 This is a sharper priority ordering than TCDC-16。 --- # 54. Formal Theorem Queue ```yaml TCDC_17: T01: exact_matrix_in_bad_mass_coordinates T02: bad_mass_simplex T03: Goldbach_failure_face T04: bad_mass_distance_equals_PP_density T05: benchmark_distance_equals_p_star_squared T06: separate_bad_mass_gate T07: provider_loss_budget T08: coupled_simplex_gate T09: fixed_row_sieve_parameter_ceiling T10: optimistic_fixed_row_semiprime_multiplier T11: coupling_relaxes_semiprime_multiplier T12: shell_weighted_provider_portfolio T13: affine_equivalence_bad_mass_and_parity_coordinates ``` --- # 55. Provider No-Go Queue ```yaml TCDC_17_NoGo: N01: best_case_fixed_row_linear_semiprime_upper plus_ideal_mixed_upper: exceeds_simplex_boundary row_semiprime_multiplier: 2.043896393626 semiprime_bad_cost: 0.836738258118 with_ideal_mixed_bad_cost: 1.078526978880 minimum_coupling_discount: 0.078526978880 mixed_multiplier_needed_without_coupling: 0.675224805224 ``` --- # 56. Barrier Queue ```yaml TCDC_17_Barriers: B01: aggregate_semiprime_marginal_upper_not_proved B02: effective_shell_level_profile_unknown B03: growing_Goldbach_residue_uniformity B04: neutral_lower_efficiency_not_jointly_matched B05: mixed_upper_still_has_constant_loss B06: coupled_bad_mass_discount_not_uniformly_quantified B07: no_positive_simplex_gap ``` --- # 57. Acceptance Gate TCDC-17 formal core前需： 1. bad-mass coordinates exact； 2. feasible triangle derived from matrix positivity； 3. failure edge exact； 4. benchmark point explicitly heuristic / transferred benchmark； 5. provider normalization separated from actual theorem； 6. loss-budget algebra checked； 7. neutral efficiency threshold stated benchmark-conditionally； 8. fixed-row distribution ceiling correctly scoped； 9. linear-sieve $F(s)$ valid range checked； 10. shell integral evaluated； 11. rowwise no-go labeled provider-architecture benchmark； 12. aggregate level-lift only target, not claimed theorem； 13. finite scan labeled computational； 14. no Goldbach proof claim。 --- # 58. Verdict 本輪可以正式給： $$ \boxed{ \mathrm{TCDC\mbox{-}17\ BAD\mbox{-}MASS\ SIMPLEX\ PASS}. } $$ 最重要 exact geometry： $$ \boxed{ \frac{ M_{PP} }{ S_m } = 1-a_m-b_m, } $$ with： $$ \boxed{ a_m = n_{SP}/S_m, \qquad b_m = M_{PS}/S_m. } $$ feasible： $$ \boxed{ 0\le b\le a, \qquad a+b\le1. } $$ failure face： $$ \boxed{ a+b=1. } $$ rank-one Buchstab benchmark point： $$ \boxed{ (a_*,b_*) = (s_*,p_*s_*), } $$ has failure-face slack： $$ \boxed{ p_*^2 \approx 0.348827388387. } $$ Therefore all provider losses share one budget： $$ \boxed{ (1-\alpha) + s_*(\kappa_{SP}-1) + p_*s_*(\kappa_{PS}-1) < p_*^2+\chi. } $$ The key analytic no-go： even with the **best possible row-scale distribution** for every fixed semiprime row， standard linear upper sieve gives： $$ \boxed{ \kappa_{SP}^{row} = \frac{ 2e^\gamma }{ 3\log2 } \left( \frac12+\log2 \right) \approx 2.043896393626. } $$ Hence semiprime upper alone costs： $$ \boxed{ 0.836738258118 } $$ of the neutral benchmark。 Even granting ideal mixed upper： $$ p_*s_* \approx 0.241788720763, $$ total bad mass becomes： $$ \boxed{ 1.078526978880 >1. } $$ So： $$ \boxed{ \text{independent fixed-row standard semiprime uppers cannot close the simplex}. } $$ At least one of the following is mandatory： 1. aggregate convolution level lift； 2. switching； 3. coupled bad-mass discount； 4. a genuinely stronger nonstandard semiprime upper。 With ideal mixed upper and perfect neutral lower， the fixed-row stack already requires coupled recovery： $$ \boxed{ \chi> 0.078526978880. } $$ or roughly： $$ \boxed{ 7.85\% } $$ of the normalized neutral mass。 Thus the next bottleneck is no longer vague： # Lift the Semiprime Marginal Upper off the Fixed-Row Ceiling 目前： $$ \boxed{ \mathrm{NO\ UNIFORM\ SIMPLEX\ GAP\ YET}. } $$ 因此： $$ \boxed{ \mathrm{GOLDBACH\ STILL\ OPEN}. } $$ --- # 59. 下一輪建議 # TCDC-18 ## Aggregate Semiprime-Marginal Convolution / Level-Lift Closure 下一輪停止 finite-row upper。 直接攻： $$ \boxed{ n_{SP} = \sum_{z 3(1-t) $$ from averaging in： $$ r. $$ ### Part C — Upper Functional evaluate： $$ \boxed{ \mathcal K_{SP}[\sigma] = \frac1{\log2} \int F(\sigma(t)) \frac{dt}{t(1-t)}. } $$ target： $$ \boxed{ \mathcal K_{SP} < 1.852079\ldots } $$ if mixed upper remains ideal benchmark。 ### Part D — Technology Match audit： - Maynard well-factorable； - Maynard uniform residue； - Yang convolution-BV； - switching / complementary divisor； shell by shell。 ### Part E — Coupling Trade if only： $$ \mathcal K_{SP} = 1.9 $$ is reachable， compute exact： $$ \chi_{\mathrm{req}} $$ rather than abandoning the route。 This will be the first round to measure precisely how much **aggregate distribution level** is needed to move the semiprime bad-mass point inside the Goldbach simplex。 --- # 附錄 A：Bad-Mass Simplex ```yaml coordinates: a: n_SP/S b: M_PS/S matrix: PP: 1-a-b PS: b SS: a-b feasible: 0 <= b <= a a+b <= 1 failure: a+b == 1 ``` --- # 附錄 B：Benchmark Point ```yaml p_star: 0.590616109150 s_star: 0.409383890850 a_star: s_star b_star: p_star*s_star failure_distance: p_star^2: 0.348827388387 ``` --- # 附錄 C：Provider Budget ```yaml neutral_lower: alpha semiprime_upper: kappa_SP*s_star mixed_upper: kappa_PS*p_star*s_star coupling_discount: chi closure: kappa_SP*s_star + kappa_PS*p_star*s_star - chi < alpha loss_form: (1-alpha) + s_star*(kappa_SP-1) + p_star*s_star*(kappa_PS-1) < p_star^2+chi ``` --- # 附錄 D：Fixed-Row No-Go ```yaml least_factor: r: X^t t_range: 1/3_to_1/2 row_prime_scale: X^(1-t) max_row_distribution: D_r: X^(1-t) best_s: 3*(1-t) linear_upper: F: 2*exp(gamma)/s weighted_average_multiplier: 2.043896393626 bad_cost: 0.836738258118 plus_ideal_mixed: 1.078526978880 required_coupling: 0.078526978880 ``` --- # 附錄 E：Finite Simplex Audit | $m$ | $S_m$ | $a=n_{SP}/S$ | $b=M_{PS}/S$ | bad sum | $PP/S$ | |---:|---:|---:|---:|---:|---:| | 19 | 7 | 0.285714 | 0.285714 | 0.571429 | 0.428571 | | 96 | 38 | 0.236842 | 0.184211 | 0.421053 | 0.578947 | | 255 | 98 | 0.204082 | 0.142857 | 0.346939 | 0.653061 | | 496 | 70 | 0.357143 | 0.271429 | 0.628571 | 0.371429 | | 783 | 184 | 0.255435 | 0.190217 | 0.445652 | 0.554348 | | 870 | 234 | 0.205128 | 0.145299 | 0.350427 | 0.649573 | | 969 | 202 | 0.242574 | 0.113861 | 0.356436 | 0.643564 | | 1000 | 136 | 0.257353 | 0.198529 | 0.455882 | 0.544118 | | 4801 | 369 | 0.330623 | 0.254743 | 0.585366 | 0.414634 | scan $3\le m\le5000$: - maximum observed bad sum: $0.628571\ldots$ at $m=496$; - minimum observed $PP/S$: $0.371428\ldots$ at $m=496$; - all statements computational only. --- # 附錄 F：Commit Decision ```yaml research_round: TCDC-17 decision: bad_mass_simplex: proved provider_loss_budget: proved neutral_efficiency_threshold: derived fixed_row_semiprime_linear_upper: audited fixed_row_provider_stack: no_go_at_benchmark_level aggregate_level_lift: required coupled_discount_target: quantified uniform_simplex_gap: not_proved Goldbach_proved: false next: TCDC-18_AggregateSemiprimeMarginal_Convolution_LevelLift_Closure ``` --- # References / External Benchmarks 1. Terence Tao, **254A, Supplement 5: The linear sieve and Chen's theorem**, 2015. Relevant fact: for the dimension-$1$ linear sieve, $$ F(s)=\frac{2e^\gamma}{s} $$ for $1