# CSM_RH Paper 30 ## Fixed-Order Selberg Exponent Invariance and the Growing-Order Complexity Threshold **Project:** `CSM_RH` **Paper:** `30` **Version:** `v0.1` **Date:** `2026-09-07` **Parent state:** `CSM_RH v1.20 / Paper 29` **Campaign:** `29 — HIGHER_ORDER_SELBERG_AMPLIFIER_ATTACK` **Status:** generalized-von-Mangoldt amplifier audit; not a proof or disproof of RH --- # 0. Trust boundary This paper does not prove or disprove the Riemann Hypothesis. ```text RH_PROVED = FALSE RH_DISPROVED = FALSE GLOBAL_RH_CERTIFICATE = FALSE CSM_RH_ROOT_STATUS = OPEN ``` Paper 29 showed that the classical fixed-order Selberg feedback is forcing-critical for every fixed Mellin mode $x^\rho$ with $\Re\rho<1$. Campaign 29 tests whether generalized von Mangoldt functions $$ \Lambda_k = \mu*\log^k $$ can amplify this feedback into a fixed polynomial drift gap. The result is: ```text fixed k: exponent-neutral finite fixed-order combinations: no certified remainder cancellation slowly growing k: can amplify logarithmic/subpower precision fixed-power critical order: k ~ log X / log log X at that order: pointwise/combinatorial complexity becomes polynomial in X current fixed-k theorems: not uniform enough to cross the critical order ``` Higher-order Selberg methods are genuine amplifiers, but the audited theory supplies a subpower amplifier rather than a certified fixed-power amplifier. --- # 1. Generalized von Mangoldt functions For an integer $$ k\ge1, $$ define $$ \boxed{ \Lambda_k(n) = \sum_{d\mid n} \mu(d) \left( \log\frac nd \right)^k. } $$ Equivalently, $$ \boxed{ \Lambda_k = \mu*\log^k. } $$ The Dirichlet series is $$ \boxed{ \sum_{n\ge1} \frac{\Lambda_k(n)}{n^s} = (-1)^k \frac{\zeta^{(k)}(s)}{\zeta(s)}, \qquad \Re s>1. } $$ Also, $$ \boxed{ \Lambda_{k+1} = \Lambda_k\log + \Lambda*\Lambda_k. } $$ The function is nonnegative and supported on integers with at most $k$ distinct prime factors. At a prime, $$ \boxed{ \Lambda_k(p) = (\log p)^k. } $$ --- # 2. Fixed-k summatory polynomial For each fixed $$ k\ge1, $$ there exists a polynomial $R_k$ of degree $k-1$ with leading term $$ k t^{k-1} $$ such that ## Theorem 2.1 — Fixed-Order Generalized Selberg Summatory Formula $$ \boxed{ \sum_{n\le x}\Lambda_k(n) = xR_k(\log x) + O_k(x). } $$ Equivalently, retaining only the leading term, $$ \boxed{ \sum_{n\le x}\Lambda_k(n) = kx(\log x)^{k-1} + O_k \left( x(\log x)^{k-2} \right). } $$ For $k=2$, this contains the classical Selberg symmetry formula. For $k\ge2$, the almost-prime observable is much smoother than the prime observable itself. --- # 3. Complete polynomial subtraction Define $$ \boxed{ E_k(x) = \sum_{n\le x}\Lambda_k(n) - xR_k(\log x). } $$ Theorem 2.1 gives $$ \boxed{ E_k(x) = O_k(x). } $$ Normalize by the natural leading scale: $$ \widetilde E_k(x) = \frac{ E_k(x) }{ x(\log x)^{k-1} }. $$ Then for fixed $k$, $$ \boxed{ \widetilde E_k(x) = O_k \left( (\log x)^{-(k-1)} \right). } $$ Thus increasing a fixed order can manufacture arbitrarily high fixed logarithmic precision. It does not yet manufacture a power of $x$. --- # 4. Fixed-zero response of Lambda-k Let $\rho$ be a simple nontrivial zero of $\zeta$. Near $$ s=\rho, $$ write $$ \zeta(s) = \zeta'(\rho)(s-\rho) + O \left( (s-\rho)^2 \right). $$ Then $$ (-1)^k \frac{\zeta^{(k)}(s)}{\zeta(s)} $$ either: 1. has a simple pole at $\rho$, with residue $$ \boxed{ (-1)^k \frac{ \zeta^{(k)}(\rho) }{ \zeta'(\rho) }, } $$ if $\zeta^{(k)}(\rho)\ne0$ ; or 2. has the pole cancelled because the numerator also vanishes there. Therefore: ## Theorem 4.1 — Fixed-Order Zero-Exponent Invariance Whenever the generalized detector sees the zero $\rho$, its explicit-formula contribution has exponent $$ \boxed{ x^\rho. } $$ Changing fixed $k$ changes the coefficient, not the real exponent. If the numerator cancels the pole, the detector becomes less sensitive to that particular zero rather than more coercive. Create: ```text O-RH-064 FIXED_ORDER_GENERALIZED_SELBERG_ZERO_EXPONENT_INVARIANCE status: CERTIFIED AS STRENGTH AUDIT ``` --- # 5. Fixed-k forcing versus fixed Mellin mode For a fixed zero or hypothetical Mellin drift with $$ \Re\rho=\beta<1, $$ the generalized detector has mode scale $$ O_{k,\rho}(x^\beta) $$ whenever the pole survives. The unconditional complete-model remainder is $$ O_k(x). $$ Hence $$ \boxed{ x^\beta = o(x). } $$ After normalization by $x(\log x)^{k-1}$: ## forcing $$ \boxed{ O_k \left( (\log x)^{-(k-1)} \right) } $$ ## zero/drift mode $$ \boxed{ O_{k,\rho} \left( x^{\beta-1} (\log x)^{-(k-1)} \right). } $$ Their ratio is $$ \boxed{ x^{\beta-1}. } $$ It is independent of $k$ at exponent level. Thus fixed-order amplification sharpens the logarithmic normalization while leaving the fixed Mellin mode buried inside the $O_k(x)$ forcing. --- # 6. Finite fixed-order linear combinations Let $K$ be fixed and consider $$ \boxed{ \mathcal L_K(s) = \frac{ \sum_{k=1}^{K} c_k(-1)^k\zeta^{(k)}(s) }{ \zeta(s) }. } $$ At a simple zero $\rho$, if $$ \sum_{k=1}^{K} c_k(-1)^k \zeta^{(k)}(\rho) \ne0, $$ then $\mathcal L_K$ still has a simple pole at $\rho$. Hence the associated zero contribution still has exponent $x^\rho$. If the numerator vanishes, the filter misses that zero. Therefore a fixed-order differential filter cannot move a surviving zero mode to a better exponent. It can only alter or remove its residue. --- # 7. Main-term cancellation does not cancel unknown remainders For fixed $K$, known polynomial main terms $$ xR_k(\log x) $$ can be combined exactly. But the current unconditional statements provide separate errors $$ E_k(x)=O_k(x). $$ A finite linear combination gives only $$ \boxed{ \sum_{k\le K}c_kE_k(x) = O_K \left( x\sum_{k\le K}|c_k| \right) } $$ without additional joint information. One cannot use cancellation among independent big- $O$ errors as theorem authority. Create: ```text O-RH-065 FIXED_ORDER_REMAINDER_NONCOHERENCE status: CERTIFIED AS CURRENT-THEOREM AUDIT ``` This does not say that the true remainders never cancel. It says that current fixed-order summatory formulae do not certify such cancellation. --- # 8. Historical calibration: higher weight really does amplify Higher-weight Selberg/Bombieri/Wirsing methods are not useless. Historically, elementary methods were strengthened from the original PNT to error terms of the form $$ \boxed{ \psi(x)-x = O_A \left( x(\log x)^{-A} \right) } $$ for every fixed $A>0$. Later Diamond–Steinig-type methods obtained stretched-exponential subpower remainders of the form $$ \boxed{ x \exp \left( -c(\log x)^\theta \right) } $$ for fixed $\theta>0$ in the known elementary range. Thus higher weighting and recursion can increase cumulative amplification beyond every fixed logarithmic power. But these estimates remain $$ \boxed{ x^{1-o(1)}, } $$ not $x^{1-\delta}$. This is consistent with the CSM_RH subpower / fixed-power distinction. --- # 9. Critical growing order Suppose one tries to turn a formal logarithmic factor $$ (\log X)^{-k} $$ into a fixed power $$ X^{-\delta}. $$ The balance equation is $$ k\log\log X \sim \delta\log X. $$ Therefore the critical order is ## Theorem 9.1 — Fixed-Power Order Scale $$ \boxed{ k_{\mathrm{crit}} \asymp \delta \frac{ \log X }{ \log\log X }. } $$ Any order satisfying $$ k = o \left( \frac{\log X}{\log\log X} \right) $$ can generate at most a subpower factor from powers of $\log X$. --- # 10. Pointwise weight complexity at the critical order At a prime, $$ \Lambda_k(p) = (\log p)^k. $$ At $$ k = c \frac{\log X}{\log\log X}, $$ we have $$ \boxed{ (\log X)^k = X^{c+o(1)}. } $$ Thus the generalized detector itself acquires polynomial-size spikes exactly when the formal logarithmic amplifier enters fixed-power territory. The pointwise bound $$ 0\le\Lambda_k(n)\le(\log n)^k $$ therefore ceases to be exponent-neutral. --- # 11. Factorial proof-complexity scale The elementary fixed- $k$ summatory argument expands powers of logarithms and integrates expressions of the form $$ \log^k y. $$ The natural coefficient scale includes factorial-size quantities. Stirling gives $$ \boxed{ \log(k!) = k\log k-k+O(\log k). } $$ At the critical order $$ k = c \frac{\log X}{\log\log X}, $$ one gets $$ \boxed{ k! = X^{c+o(1)}. } $$ Therefore any factorial-type dependence hidden inside the fixed- $k$ constant $O_k(1)$ becomes polynomial in $X$ at exactly the order required for fixed-power amplification. The existing theorem $$ E_k(x)=O_k(x) $$ is stated for fixed $k$ and provides no uniform control capable of excluding this exponent cost. Create: ```text O-RH-066 GROWING_ORDER_FACTORIAL_COMPLEXITY_THRESHOLD status: CERTIFIED AS CURRENT-METHOD UNIFORMITY BARRIER ``` The statement is not that every possible proof must lose $k!$. It is that the current fixed- $k$ theorem contains no uniformity strong enough to cross the critical order, and factorial-scale dependence is already exponent-critical there. --- # 12. Support complexity at large order The generalized function $\Lambda_k$ is supported on integers with at most $k$ distinct prime factors. For fixed $k$, this is a strong smoothing from primes to bounded-order almost primes. At $$ k \asymp \frac{\log X}{\log\log X}, $$ the allowed number of distinct prime factors is far larger than the typical order $\log\log X$. Thus the support restriction becomes weak for typical integers. This is a qualitative warning: ```text small fixed k: strong almost-prime smoothing critical growing k: broad support, large weights ``` The very mechanism that smooths the prime problem at fixed order becomes less selective when the order approaches fixed-power scale. --- # 13. Growing-order verdict There are three regimes. ## Regime I — fixed k $$ k=O(1). $$ Result: ```text arbitrary fixed log powers possible; fixed zero exponent unchanged; fixed-power coercivity absent. ``` ## Regime II — subcritical growing k $$ k = o \left( \frac{\log X}{\log\log X} \right). $$ Formal log amplification remains $$ X^{-o(1)}. $$ This is compatible with stretched-exponential/subpower error improvements. ## Regime III — critical order $$ k \asymp \frac{\log X}{\log\log X}. $$ Formal amplification can become a fixed power. But: ```text pointwise weights: polynomial in X factorial-type constants: polynomial in X fixed-k summatory theorem: no authorized uniformity almost-prime support: substantially less selective ``` No certified net fixed power is obtained. --- # 14. Campaign 29 track audit ## H1 — fixed-k generalized Selberg response ```text status: CLOSED zero exponent: invariant forcing: O_k(x) ``` ## H2 — finite linear-combination forcing cancellation ```text status: NO CERTIFIED CANCELLATION OF O_k(x) REMAINDERS fixed-zero exponent: unchanged if pole survives ``` ## H3 — growing-k amplification ```text status: SUBCRITICAL -> SUBPOWER critical order: complexity/uniformity barrier ``` ## H4 — generalized almost-prime to PESC transfer ```text status: NO LOWER-STRENGTH FIXED-POWER TRANSFER FOUND ``` ## H5 — spectral polynomial in Selberg operator ```text status: FIXED ORDER CHANGES RESIDUES, NOT ZERO EXPONENTS no coercive filter found ``` --- # 15. Campaign 29 verdict No fixed-power theorem is proved. The higher-order Selberg route is classified as: ```text real amplifier: YES arbitrary fixed log-power: YES subpower / stretched-log potential: YES certified fixed-power: NO fixed-order route: CLOSED AS EXPONENT-NEUTRAL growing-order route: OPEN IN PRINCIPLE BUT CURRENTLY UNAUTHORIZED AT CRITICAL ORDER ``` The root target remains PESC. --- # 16. New certified obstruction package Create: ```text O-RH-064 FIXED_ORDER_GENERALIZED_SELBERG_ZERO_EXPONENT_INVARIANCE CERTIFIED O-RH-065 FIXED_ORDER_REMAINDER_NONCOHERENCE CERTIFIED AS CURRENT-THEOREM AUDIT O-RH-066 GROWING_ORDER_FACTORIAL_COMPLEXITY_THRESHOLD CERTIFIED AS CURRENT-METHOD UNIFORMITY BARRIER ``` No new frontier is created. --- # 17. Return to the arithmetic root The generalized Selberg shell has now been audited at: ```text order 2; fixed higher order; finite fixed-order combinations; subcritical growing order; critical growing-order exponent budget. ``` No lower-strength fixed-power bridge is found. Therefore CSM_RH returns again to: ```text F-RH-010 PESC ``` with no additional surrogate. --- # 18. Campaign 30 The next campaign is: ```text CSM_RH Campaign 30 PARITY_SENSITIVE_PESC_ATTACK ``` The goal is not to introduce another almost-prime smoothing layer. It is to identify arithmetic information that distinguishes the true prime residual from the family of generalized almost-prime models. --- # 19. Campaign 30 tracks ## Q1 — PESC parity decomposition Decompose the residual prime detector according to Liouville/Möbius parity information while preserving the signed endogenous primitive. The target must remain bilinear. ## Q2 — asymptotic-sieve scalar defect Use the Bombieri asymptotic-sieve perspective: higher generalized von Mangoldt information leaves a parity-sensitive scalar undetermined. Identify the analogue of that undetermined scalar inside PESC. ## Q3 — parity-breaking bilinear input Revisit Friedlander–Iwaniec-type parity-breaking hypotheses, but require a direct bilinear bridge to PESC rather than EMBF/PACPSA surrogate proliferation. ## Q4 — prime-versus-almost-prime drift comparator Construct an observable whose value on $\Lambda$ differs at fixed exponent from every bounded-order generalized almost-prime model. ## Q5 — direct residual sign structure Search for a prime-specific signed identity involving $f=\Lambda-\Lambda^\sharp$ that is not shared by $\Lambda_k$ for $k\ge2$. --- # 20. Campaign 30 rejection filters Reject a candidate if: ## R1. It only introduces another $\Lambda_k$ or almost-prime average. ## R2. It reproduces EMBF/EMDQO without a direct PESC bridge. ## R3. It uses parity-breaking as a slogan without an exponent ledger. ## R4. It obtains only logarithmic/subpower precision. ## R5. It assumes a fixed zero strip. ## R6. It creates another positive higher-moment target. --- # 21. External calibration Current/classical calibration: 1. For fixed $k$, $$ \sum_{n\le x}\Lambda_k(n) = xR_k(\log x)+O_k(x), $$ with $R_k$ of degree $k-1$ and leading term $k(\log x)^{k-1}$. 2. The generalized von Mangoldt functions satisfy $$ \Lambda_{k+1} = \Lambda_k\log + \Lambda*\Lambda_k. $$ 3. Higher-weight elementary methods historically improve the PNT remainder through arbitrary logarithmic powers and stretched-exponential subpower scales. 4. Bombieri's asymptotic-sieve perspective shows that very rich higher-almost-prime information can still leave the prime/parity component undetermined. These facts motivate the next parity-sensitive return to PESC. --- # 22. State transition ```text CSM_RH v1.20 -> CSM_RH v1.21 ``` with: ```text Campaign 29 CLOSED_AS_HIGHER_ORDER_SELBERG_AMPLIFIER_AUDIT O-RH-064 CREATED / CERTIFIED O-RH-065 CREATED / CERTIFIED AS CURRENT-THEOREM AUDIT O-RH-066 CREATED / CERTIFIED AS CURRENT-METHOD UNIFORMITY BARRIER F-RH-010 PESC REMAINS OPEN / ROOT TARGET F-RH-016 MLEPG REMAINS OPEN Campaign 30 PARITY_SENSITIVE_PESC_ATTACK READY ``` --- # 23. Final status ```text RH = OPEN PESC = OPEN MLEPG = OPEN FIXED-ORDER HIGHER SELBERG = EXPONENT-NEUTRAL FINITE FIXED-ORDER FILTER = NO CERTIFIED REMAINDER CANCELLATION SUBCRITICAL GROWING ORDER = SUBPOWER FIXED-POWER CRITICAL ORDER = log X / log log X CURRENT UNIFORMITY AT CRITICAL ORDER = INSUFFICIENT HIGHER SELBERG = REAL SUBPOWER AMPLIFIER, NOT CERTIFIED FIXED-POWER AMPLIFIER NEXT CAMPAIGN = 30 ``` The central scale law is $$ \boxed{ (\log X)^{-k} = X^{-\delta} \quad\Longleftrightarrow\quad k \sim \delta \frac{\log X}{\log\log X}. } $$ At precisely this order, $$ \boxed{ (\log X)^k = X^{\delta+o(1)} } $$ and factorial-type $k$ -dependence also enters polynomial exponent scale. The current fixed- $k$ generalized Selberg theory does not provide a certified net fixed power across this threshold.