工程紀錄 · 第五弧線 v3.16 · 2026-09-03 · ACTUAL_PRIME_DEVIATION · ENDPOINT_PAIR_VARIANCE_GATE · LOG_UNIFORMITY_BARRIER · RH_CLAIM_FALSE # Four-Point Prime Deviation：Actual $\Lambda$ Barrier、Endpoint Pair Variance Gate 與 Log-Uniformity Limitation **RH-FourPointPrimeDeviation v3.16** 本節點承接： - `RH-AxisFree-MixedRamanujanTail v3.15` - `RH-MeanSquare-ArithmeticGap v3.5` - `RH-CenteredShift-FourierGap v3.6` v3.15 closed the entire deterministic refined-singular-series contribution： $$ \boxed{ \mathfrak M_{4,N} \ll_\varepsilon N^{9/2+\varepsilon}. } $$ Thus： $$ \boxed{ \eta_M=\frac12. } $$ The remaining quartic object is： $$ \boxed{ \mathfrak Q_N = \mathfrak M_{4,N} + \mathfrak E_{4,N}, } $$ where： $$ \boxed{ \mathfrak E_{4,N} } $$ is the **actual-prime deviation** from the correctly centered four-point Hardy–Littlewood / refined-singular-series model。 v3.16 does not prove a fixed power saving for： $$ \mathfrak E_{4,N}. $$ Instead it completes three reductions： 1. derives the exact local actual-prime deviation field； 2. identifies an endpoint prime-pair variance theorem that is sufficient for the entire RH progress gate； 3. audits current 2026 prime-correlation / higher-uniformity technology and finds no unconditional fixed： $$ N^{-\eta} $$ gain。 The new canonical status is： ```text DETERMINISTIC MODEL CLOSED AT eta_M = 1/2 ACTUAL PRIME DEVIATION OPEN CURRENT GENERAL HIGHER UNIFORMITY LOG-POWER / o(1) SAVING FIXED POWER PRIME VARIANCE OPEN ``` No RH claim is made。 **RH_CLAIM = False.** --- # 0. Claim register ```text RH_PROVED = FALSE RH_DISPROVED = FALSE DETERMINISTIC_MODEL_ETA_M = 1/2 LOCAL_PRIME_DEVIATION_IDENTITY = CLOSED ENDPOINT_PAIR_RESIDUAL_IDENTITY = CLOSED EPV_GATE = CLOSED_AS_SUFFICIENT_REDUCTION GRH_PAIR_VARIANCE_CALIBRATION = EXTERNAL_CONDITIONAL GRH_EPV_ETA = 1/2_UP_TO_LOGS HIGHER_UNIFORMITY_PSEUDORANDOM_SAVING = LOG_POWER HIGHER_UNIFORMITY_FIXED_POWER_ETA = NOT_OBTAINED UNCONDITIONAL_EPV_ETA_POSITIVE = OPEN UNCONDITIONAL_FPD_ETA_POSITIVE = OPEN COMPLETE_QUARTIC_ETA_Q_POSITIVE = OPEN MEANSQUARE_KAPPA_I_POSITIVE = OPEN GLOBAL_RH_CERTIFICATE = FALSE ``` --- # 1. Centered prime sequence Define： $$ \boxed{ a_n = \Lambda(n)-1. } $$ For： $$ h\ge1, $$ write： $$ \boxed{ \mu_h = \mathfrak S(h)-1. } $$ Define the pointwise centered pair fluctuation： $$ \boxed{ P_h(r) = a_r a_{r+h} - \mu_h. } $$ --- # 2. Refined four-point main term For： $$ h,d\ge1, \qquad h\ne d, $$ define the parallelogram： $$ \boxed{ \mathcal H_{h,d} = \{0,h,d,h+d\}. } $$ Let： $$ \boxed{ \mathfrak S_0(\mathcal H_{h,d}) } $$ be the refined / centered Hardy–Littlewood four-point singular-series main term。 Define the pointwise four-point prime-pattern error： $$ \boxed{ T_{h,d}(r) = a_r a_{r+h} a_{r+d} a_{r+h+d} - \mathfrak S_0( 0,h,d,h+d ). } $$ --- # 3. Deterministic covariance model v3.6 defines： $$ \boxed{ \mathfrak K_4(h,d) = \mathfrak S_0( 0,h,d,h+d ) - \mu_h^2. } $$ The actual pair-pair product is： $$ P_h(r)P_h(r+d). $$ --- # 4. Exact actual-prime deviation field Expand： $$ P_h(r)P_h(r+d). $$ Then： $$ \begin{aligned} &P_h(r)P_h(r+d) - \mathfrak K_4(h,d) \\ &= a_ra_{r+h}a_{r+d}a_{r+h+d} - \mathfrak S_0 \\ &\quad - \mu_h \left[ P_h(r) + P_h(r+d) \right]. \end{aligned} $$ Therefore： ## Theorem 4.1 · Prime Deviation Identity $$ \boxed{ P_h(r)P_h(r+d) - \mathfrak K_4(h,d) = T_{h,d}(r) - \mu_h \left[ P_h(r) + P_h(r+d) \right]. } $$ This splits the actual-prime deviation into： ```text GENUINE FOUR-POINT PRIME PATTERN ERROR + PAIR-ERROR CORRECTION. ``` --- # 5. Exact weighted FPD object Recall the v3.6 four-distinct coordinates： $$ r, \quad r+h, \quad r+d, \quad r+d+h. $$ Define： $$ \boxed{ W_N(r,h,d) = 2 w_N(r+h) w_N(r+d+h). } $$ The valid region is： $$ r,h,d\ge1, $$ $$ h\ne d, $$ $$ r+h+d\le2N-1. $$ Then： ## Theorem 5.1 $$ \boxed{ \begin{aligned} \mathfrak E_{4,N} &= \sum_{r,h,d} W_N(r,h,d) T_{h,d}(r) \\ &\quad - \sum_{r,h,d} W_N(r,h,d) \mu_h \left[ P_h(r) + P_h(r+d) \right]. \end{aligned} } $$ Thus the remaining v3.15 obstruction is genuinely prime-specific。 --- # 6. Endpoint pair residual For： $$ 1\le h0 $$ such that： $$ \boxed{ \mathcal P_N \ll N^{4-\eta+o(1)}. } $$ Then Theorem 9.1 gives： $$ \boxed{ \mathcal V_N \ll N^{5-\eta+o(1)}. } $$ --- # 11. EPV immediately gives PNT mean-square progress v3.5–v3.6 show that： $$ \mathcal V_N \ll N^{5-\eta} $$ implies： $$ \boxed{ I(N) = \int_N^{2N} |\psi(x)-x|^2dx \ll N^{3-\eta/2+o(1)}. } $$ Therefore： $$ \boxed{ \Theta \le 1-\frac{\eta}{4}. } $$ Thus： ## Theorem 11.1 · EPV Zero-Strip Gate Any unconditional： $$ \boxed{ \operatorname{EPV}(\eta), \qquad \eta>0, } $$ produces a genuine fixed zero strip。 This is why no known routine prime-correlation estimate can close EPV。 --- # 12. Relation to the deterministic closure v3.15 has： $$ \boxed{ \mathfrak M_{4,N} \ll N^{9/2+\varepsilon}. } $$ If one proves the more targeted： ## FPD$(\eta)$ $$ \boxed{ |\mathfrak E_{4,N}| \ll N^{5-\eta+o(1)}, } $$ then： $$ \boxed{ |\mathfrak Q_N| \ll N^{ 5-\min(\eta,1/2)+o(1) }. } $$ Hence： $$ \boxed{ I(N) \ll N^{ 3-\frac12 \min(\eta,1/2) +o(1) }. } $$ EPV is a stronger sufficient prime gate；FPD is the minimal post-v3.15 gate。 --- # 13. Chou–Haag–Huryn–Ledoan variance For one endpoint： $$ N, $$ define： $$ \boxed{ E_{\rm pair}(N) = \sum_{ 1\le|h|\le N } \left[ \psi_2(N,h) - \mathfrak S(h)(N-|h|) \right]^2. } $$ Chou–Haag–Huryn–Ledoan emphasize that obtaining upper bounds for this variance is extremely hard。 Their conjectural scale is： $$ \boxed{ E_{\rm pair}(N) \asymp N^2(\log N)^2. } $$ They prove： $$ \boxed{ E_{\rm pair}(N) = \Omega( N^{1+2\Theta-\varepsilon} ) } $$ for every： $$ \varepsilon>0. $$ Thus a fixed-power upper bound on this standard pair variance already has zero-strip consequences。 --- # 14. GRH calibration The same paper records that, assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions： $$ \boxed{ E_{\rm pair}(N) \ll N^{5/2} (\log N)^C } $$ for some constant： $$ C. $$ Under GRH, RH also gives： $$ \boxed{ E(x) \ll x^{1/2} (\log x)^2. } $$ Use Theorem 7.1。 Then： $$ \begin{aligned} \sum_{h0. $$ They also obtain asymptotically small short-interval Gowers norms and Hardy–Littlewood patterns with a short average over one variable。 --- # 18. What higher uniformity gives here The parallelogram pattern： $$ r, \quad r+h, \quad r+d, \quad r+h+d $$ is a finite-complexity linear system and is naturally controlled by Gowers-uniformity methods after the structured approximation： $$ \Lambda^\sharp $$ is removed。 Therefore these results strongly support control of the **pseudorandom component** of the four-point deviation。 However their published quantitative scale is： $$ \boxed{ (\log N)^{-A} } $$ or： $$ o(1), $$ not： $$ N^{-\eta}. $$ After restoring the： $$ N^5 $$ quartic weight scale, such bounds remain： $$ \boxed{ N^{5-o(1)} } $$ at fixed-exponent resolution。 Thus： ```text HIGHER UNIFORMITY = major method progress FIXED ZERO-STRIP ETA = still zero ``` --- # 19. Why the parallelogram $U^2$ structure does not automatically solve the gate The unweighted identity： $$ \sum_{r,h,d} f(r) \overline{f(r+h)} \overline{f(r+d)} f(r+h+d) $$ is a $U^2$ / fourth Fourier-moment object。 But for primes, the relevant Fourier square： $$ |S(\alpha)|^2 $$ contains the full prime-pair spectrum。 Chou–Haag–Huryn–Ledoan write the pair-error variance exactly as an： $$ L^2 $$ distance between： $$ |S(\alpha)|^2 $$ and a singular-series / major-arc model。 Thus the special parallelogram Fourier structure **is exactly where the hard prime-pair variance already lives**。 It does not bypass that barrier。 --- # 20. Major-arc approximation barrier For the Selberg-type approximation： $$ \Lambda_R, $$ Chou et al. record： $$ \boxed{ \int_0^1 | S(\alpha)-S_R(\alpha) |^2d\alpha = N\log\left( \frac NR \right) + O(N). } $$ A related Ramanujan major-arc approximation： $$ \lambda_R(n) = \sum_{q\le R} \frac{\mu(q)}{\phi(q)} c_q(-n) $$ has controlled second moment only in the classical regime： $$ R\le N^{1/2}, $$ up to the standard error terms。 This is another expression of the familiar： $$ \sqrt N $$ major-arc / large-sieve threshold。 --- # 21. Why current minor-arc technology is not enough by itself Classical and modern exponential-sum estimates can give strong cancellation away from structured major arcs。 But the fixed-power EPV problem also requires quantitative control of the structured prime-pair approximation itself across endpoints and shifts。 That structured component is sensitive to Dirichlet $L$-function zeros and cannot be removed merely by a generic minor-arc supremum bound。 This is consistent with the GRH dependence of the best quoted pair-variance upper bound。 --- # 22. Actual-prime problem split The remaining prime-specific research problem should be separated into： ## A. Four-point pseudorandom error Control： $$ \boxed{ T_{h,d}(r) } $$ after subtracting the refined four-point model。 Higher uniformity is the main current tool。 ## B. Pair correction Control： $$ \boxed{ \mu_h [ P_h(r)+P_h(r+d) ]. } $$ Endpoint prime-pair variance is the natural current interface。 A proof of FPD must preserve cancellation between these pieces when appropriate。 --- # 23. Strong sufficient endpoint gate The most concrete next theorem is： ## Integrated Endpoint Pair Variance $\operatorname{IEPV}(\eta)$ $$ \boxed{ \sum_{j=N}^{2N-1} \sum_{h=1}^{j-1} \left| r_j(h) - E(j) - E(j-h) + E(h) \right|^2 \ll N^{4-\eta+o(1)}. } $$ This is exactly EPV written in standard arithmetic objects。 Any： $$ \eta>0 $$ would close the first zero-strip gate。 --- # 24. More targeted four-point gate A weaker theorem than EPV may still suffice： ## Four-Point Prime Deviation $\operatorname{FPD}(\eta)$ $$ \boxed{ \left| \sum_{r,h,d} W_N(r,h,d) \left\{ T_{h,d}(r) - \mu_h[ P_h(r)+P_h(r+d) ] \right\} \right| \ll N^{5-\eta+o(1)}. } $$ This preserves the exact signed three-parameter cancellation and is therefore the canonical **minimal** post-v3.15 target。 --- # 25. Current literature strength audit ### Published / established - deterministic singular-series geometry： closed in AMRAL v3.15 at： $$ N^{9/2+\varepsilon}; $$ - higher uniformity of： $$ \Lambda-\Lambda^\sharp $$ with arbitrary log-power saving； - Hardy–Littlewood patterns with averaged variables； - conditional GRH prime-pair variance： $$ N^{5/2}\operatorname{polylog}N. $$ ### Not currently available - unconditional： $$ E_{\rm pair}(N) \ll N^{3-\eta} $$ for any fixed： $$ \eta>0; $$ - unconditional EPV$(\eta)$； - unconditional FPD$(\eta)$。 Therefore： $$ \boxed{ \eta_{\rm prime}=0 } $$ remains the rigorous state of the actual-prime branch。 --- # 26. Strategic interpretation The fifth arc has now separated three fundamentally different layers： ```text LOCAL CONGRUENCE MODEL solved at fixed power PSEUDORANDOM PRIME PART strong log-power control STRUCTURED PRIME ERROR fixed-power barrier open ``` So future agents should not continue improving the deterministic singular-series model unless a prime-specific proof requires it。 The main target is now an actual-prime variance / structured approximation theorem。 --- # 27. False-progress gates Reject： ### F1 · Log-power pseudorandom saving $$ N^5(\log N)^{-A} $$ still has fixed exponent： $$ 5. $$ ### F2 · GRH pair-variance bound used inside an RH proof Circular / conditional calibration only。 ### F3 · Minor-arc fixed power alone Must also control structured major-arc pair error。 ### F4 · Deterministic $\eta_M=1/2$ promoted to prime $\eta_Q$ Actual-prime deviation is additive and unresolved。 ### F5 · Generic four-linear Hardy–Littlewood asymptotic An： $$ o(1) $$ relative error is not automatically： $$ N^{-\eta}. $$ --- # 28. Autonomous candidate schema Every prime-side candidate should record： ```text target EPV / FPD / four-point pseudorandom / pair correction structured approximant Lambda^sharp / Lambda_R / lambda_R / other quantitative saving log^(-A) / o(1) / N^(-eta) endpoint averaging yes / no shift averaging yes / no major-arc error controlled yes / no exceptional-character / zero sensitivity yes / no translation to eta_Q translation to kappa_I ``` Only a proved： $$ N^{-\eta} $$ gain counts as fixed zero-strip progress。 --- # 29. Suggested v3.17 direction Recommended： `RH-PrimeStructuredApproximationGap v3.17` Do not return to deterministic geometry。 Tasks： 1. choose a canonical prime approximant： $$ \lambda_R $$ or： $$ \Lambda^\sharp; $$ 2. decompose： $$ P_h(r) $$ and： $$ T_{h,d}(r) $$ into structured / pseudorandom pieces； 3. use higher uniformity to remove every term containing at least one pseudorandom factor at logarithmic strength； 4. identify the all-structured residual exactly； 5. express that residual in Dirichlet-character / major-arc data； 6. determine which character-zero estimate would imply EPV$(\eta)$； 7. compare this with Chou et al.'s GRH： $$ N^{5/2} $$ variance； 8. isolate the weakest non-GRH character theorem sufficient for any： $$ \eta>0. $$ This is now the canonical actual-prime continuation。 --- # 30. GAP ledger ## CLOSED / REDUCED ### G1. Local FPD identity ```text CLOSED ``` ### G2. Endpoint pair interface ```text CLOSED ``` ### G3. EPV sufficient gate ```text CLOSED_AS_REDUCTION ``` ### G4. Deterministic model ```text eta_M = 1/2 ``` ### G5. Higher-uniformity method class ```text LOG-POWER ONLY AT FIXED-EXPONENT AUDIT ``` --- ## OPEN ### G6. EPV$(\eta>0)$ ```text OPEN ``` ### G7. FPD$(\eta>0)$ ```text OPEN ``` ### G8. Complete quartic： $$ \eta_Q>0 $$ ```text OPEN ``` ### G9. PNT mean-square： $$ \kappa_I>0 $$ ```text OPEN ``` ### G10. RH ```text OPEN ``` --- # 31. Trust boundary ```text RH_PROVED = FALSE RH_DISPROVED = FALSE DETERMINISTIC eta_M = 1/2 ACTUAL PRIME eta = 0 PROVED GRH eta=1/2 = CONDITIONAL CALIBRATION ONLY HIGHER UNIFORMITY LOG SAVING != FIXED POWER SAVING NO NEW ZERO STRIP IN V3.16 GLOBAL_RH_CERTIFICATE = FALSE ``` Forbidden： $$ \text{GRH pair variance} \Longrightarrow \text{unconditional EPV}. $$ Forbidden： $$ \log^{-A}N \Longrightarrow N^{-\eta}. $$ Forbidden： $$ \mathfrak M_{4,N} \ll N^{9/2} \Longrightarrow \mathfrak Q_N \ll N^{9/2}. $$ --- # 32. One-line status > v3.16 begins the actual-prime phase after the deterministic refined-singular-series model was closed at $N^{9/2+\varepsilon}$. Writing $a_n=\Lambda(n)-1$, the pointwise pair fluctuation is $P_h(r)=a_ra_{r+h}-(\mathfrak S(h)-1)$ and the genuine four-point error is $T_{h,d}(r)=a_ra_{r+h}a_{r+d}a_{r+h+d}-\mathfrak S_0(0,h,d,h+d)$. The exact identity $P_h(r)P_h(r+d)-\mathfrak K_4(h,d)=T_{h,d}(r)-(\mathfrak S(h)-1)[P_h(r)+P_h(r+d)]$ splits the remaining prime deviation into a genuine four-linear pattern error plus pair-error corrections. At endpoint level, $Q_j(h)=r_j(h)-E(j)-E(j-h)+E(h)$ and the total centered shift variance satisfies $\mathcal V_N\le N\sum_{j\asymp N}\sum_{h