# CSM_RH Paper 40 ## Averaged Type-II Large Values, the Orthogonality Floor, and the Single-Resonance Obstruction **Project:** `CSM_RH` **Paper:** `40` **Version:** `v0.1` **Date:** `2026-09-07` **Parent state:** `CSM_RH v1.30 / Paper 39` **Campaign:** `39 — AVERAGED_TYPEII_POLYNOMIAL_W_ATTACK` **Status:** averaged large-value replacement 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 39 proved that the current pointwise route to polynomial $W$ is fixed-strip locked. Campaign 39 asks whether modern Dirichlet-polynomial large-value estimates can replace the pointwise hypothesis in the Type-II amplifier. The result is negative at the generic bounded-coefficient level. The Guth–Maynard theorem improves the nontrivial large-value terms, but retains the classical orthogonality term $$ N^2V^{-2}. $$ After normalization to the 1-line, this becomes a $P^{-2}$ level-set floor. That floor yields at best logarithmic control after layer-cake integration and cannot generate a vanishing fixed power. More decisively, the divisor-bounded coefficient class of Lemma 3.5 contains explicit phase-aligned examples with a single constant-size resonant frequency. Such one-frequency resonance already forces an $O(1)$ contribution to the product mean square, whereas polynomial $W$ requires that mean square to tend to zero like a fixed power. Therefore generic large-value technology cannot replace the arithmetic pointwise/nonresonance input. A genuinely arithmetic simultaneous-resonance exclusion for the actual Heath–Brown factors remains open. No live GLM-5.3-Flash run is claimed. --- # 1. Type-II integrated target Recall $$ A(s) = \sum_{m\sim M} a(m)m^{-s}, $$ and $$ B(s) = \sum_{N/30, $$ the right side is $$ X^{-3w/10+o(1)}. $$ Thus the desired integrated product must itself vanish at a fixed polynomial rate. --- # 2. Product polynomial normalization Write $$ F(s)=A(s)B(s). $$ After partitioning its support into $O(1)$ dyadic blocks, one obtains a length- $\asymp X$ polynomial $$ F(1+it) = \frac1X D_F(t) \cdot X^{o(1)}, $$ where $$ D_F(t) = \sum_{\ell\asymp X} c_\ell \ell^{-it} $$ and $$ |c_\ell| \le d_2(\ell)^{O(1)} = X^{o(1)}. $$ After division by an $X^{o(1)}$ factor, Guth–Maynard's bounded-coefficient large-value theorem applies at exponent resolution. --- # 3. Guth–Maynard large-value theorem For a length- $L$ Dirichlet polynomial $$ D(t) = \sum_{LP \} \,dP. $$ If the available general level-set estimate has the form $$ \operatorname{meas} \{ |F|>P \} \ll X^{o(1)}P^{-2}, $$ then its contribution to the layer cake is $$ \boxed{ X^{o(1)} \int \frac{dP}{P}. } $$ This is logarithmic / subpower. It is not $$ X^{-cw} $$ for any fixed $c,w>0$. Create: ```text O-RH-096 GENERIC_LARGE_VALUE_ORTHOGONALITY_FLOOR status: CERTIFIED ``` Statement: > A generic Dirichlet-polynomial large-value theorem whose leading normalized term is $P^{-2}$ cannot by itself imply the polynomially vanishing Type-II product mean square required for polynomial $W$. --- # 6. Sharpness of the first large-value term Guth–Maynard discuss general bounded-coefficient constructions with $$ \gg L^{2-2\sigma} $$ one-separated points at amplitude $$ L^\sigma. $$ This is precisely the scale represented by $$ L^2V^{-2}. $$ Thus the orthogonality term is not merely an artifact of their proof. It reflects a real generic bounded-coefficient phenomenon. Therefore replacing the first term by a fixed-power improvement is impossible without exploiting additional arithmetic structure. --- # 7. Explicit one-frequency resonance There is an even simpler obstruction tailored directly to Lemma 3.5. Fix $$ t_0 $$ in the relevant integration interval and choose $$ \boxed{ a(m)=m^{it_0}, \qquad b(n)=n^{it_0}. } $$ These coefficients satisfy $$ |a(m)|=|b(n)|=1. $$ Then $$ A(1+it) = \sum_{m\sim M} \frac{ m^{-i(t-t_0)} }{m}, $$ and $$ B(1+it) = \sum_{N/3 polynomial-W integrated Type-II saving ``` can hold. The pointwise smallness in Lemma 3.5 is not merely a convenient sufficient condition. It excludes resonances which are legal in the ambient coefficient class. --- # 11. Weak pointwise logarithmic control is still insufficient Suppose one supplements large-value theory by an arbitrary fixed logarithmic pointwise estimate $$ \boxed{ |A(1+it)|,\, |B(1+it)| \le \log^{-K}X } $$ for fixed $K$. A single peak at the maximal allowed scale contributes only a negative power of $\log X$ to the product mean square. But $$ \log^{-C}X = X^{-o(1)} \gg X^{-cw}. $$ Thus a finite or subpower number of subpower-sized resonant peaks is still incompatible with the required fixed-power integrated bound. A fixed-power conclusion requires polynomial control of the total resonance mass, not merely arbitrary fixed log-power peak suppression. --- # 12. Large-value count quantization Large-value theorems count one-separated points. Such a count is an integer. A bound which says $$ R\le X^{o(1)} $$ does not distinguish between: ```text no resonance; one resonance; a subpower number of resonances. ``` But the polynomially vanishing Type-II target is sensitive even to one sufficiently broad subpower-amplitude resonance. Create: ```text O-RH-098 LARGE_VALUE_COUNT_QUANTIZATION_BARRIER status: CERTIFIED AS FIXED-POWER TRANSFERENCE AUDIT ``` This is another formulation of why improving the number of large values is not enough when the target integral itself must tend to zero polynomially. --- # 13. Applying Guth–Maynard separately to the two factors One might attempt to control simultaneous large values through $$ \min \{ R_A(U),R_B(V) \}. $$ For the normalized factor $A$, the first Guth–Maynard term gives $$ \boxed{ R_A(U) \ll X^{o(1)}U^{-2} +\cdots } $$ and likewise $$ \boxed{ R_B(V) \ll X^{o(1)}V^{-2} +\cdots. } $$ The same orthogonality floor remains. Moreover the phase-aligned example makes the two resonances occur at the same $t_0$. Thus independent large-value estimates do not solve simultaneous resonance. --- # 14. Applying the theorem to the product Treating $$ F=AB $$ as one Dirichlet polynomial improves bookkeeping but not the exponent class. The first term becomes $$ P^{-2} $$ and the one-frequency example still survives. Therefore neither: ```text factorwise large-value control ``` nor ```text product large-value control ``` gives the required polynomial mean-square saving generically. --- # 15. Why Guth–Maynard still matters This negative conclusion does not diminish the Guth–Maynard theorem. Their improvement in the critical regime near amplitude $$ V=L^{3/4} $$ is strong enough to improve zero-density estimates and prime short-interval ranges. It is simply a different quantitative task. The current CSM_RH Type-II target asks for an integrated quantity which tends to zero by a fixed power. That requires arithmetic cancellation beyond the generic bounded-coefficient orthogonality floor. --- # 16. Actual Heath–Brown factors are not generic The obstruction above applies to the ambient divisor-bounded coefficient class. The Type-II factors arising from the actual decomposition of $\Lambda$ and $\mu$ have additional structure. Heath–Brown's identity writes the relevant components as convolutions of factors which are: ```text 1 on a dyadic block; log n on a dyadic block; mu(n) on a short dyadic block. ``` In particular, Möbius-bearing factors occur on scales at most a small fixed power of $X$ in the chosen identity. Therefore the generic resonance example does not prove that the actual arithmetic factors possess such resonances. It only proves that general large-value theory cannot rule them out. --- # 17. New surviving arithmetic question The remaining route is: > prove that the actual Möbius-bearing Heath–Brown factors and their complementary factors cannot be simultaneously resonant on a set with enough spectral mass to defeat polynomial $W$. This is stronger than generic large-value theory and weaker in form than demanding pointwise fixed-power Mertens for the full Möbius polynomial. Whether it is genuinely lower-strength is open. No theorem is claimed here. --- # 18. Campaign 39 track audit ## AV1 — large-value replacement of pointwise input ```text status: FAILS GENERICALLY reason: P^(-2) orthogonality floor ``` ## AV2 — Guth–Maynard large-value technology ```text status: IMPROVES CRITICAL LARGE-VALUE TERMS does not remove: orthogonality floor ``` ## AV3 — low-frequency excision ```text status: DOES NOT FIX GENERIC RESONANCE resonance can be placed at any legal t0 > W ``` ## AV4 — product large-value geometry ```text status: GENERIC PRODUCT THEOREM STILL HAS P^(-2) FLOOR single simultaneous resonance: legal ``` ## AV5 — polynomial-W integrated bridge ```text status: NOT OBTAINED GENERICALLY requires: arithmetic resonance exclusion ``` --- # 19. Campaign 39 verdict No polynomial- $W$ integrated theorem is proved. The campaign closes the purely generic averaged escape: ```text pointwise fixed-power input: too strong / fixed-strip locked generic averaged large values: too weak / resonance floor remaining possibility: arithmetic simultaneous-resonance exclusion ``` The component bridge B-RH-014 remains valid. --- # 20. New certified package Create: ```text B-RH-015 GUTH_MAYNARD_PRODUCT_LEVEL_SET_NORMALIZATION CERTIFIED O-RH-096 GENERIC_LARGE_VALUE_ORTHOGONALITY_FLOOR CERTIFIED O-RH-097 SINGLE_RESONANT_FREQUENCY_OBSTRUCTS_GENERIC_POLYNOMIAL_W_MEAN_SQUARE CERTIFIED O-RH-098 LARGE_VALUE_COUNT_QUANTIZATION_BARRIER CERTIFIED ``` No new canonical frontier is created. --- # 21. Canonical root status ```text F-RH-010 PESC OPEN / ROOT TARGET F-RH-016 MLEPG OPEN / DIRECT THEOREM CANDIDATE B-RH-014 POLYNOMIAL_W_TYPEII_FIXED_POWER_AMPLIFIER CERTIFIED COMPONENT BRIDGE generic large-value replacement: CLOSED ``` --- # 22. Campaign 40 The next campaign is: ```text CSM_RH Campaign 40 HEATH_BROWN_ARITHMETIC_RESONANCE_EXCLUSION ``` This is a genuinely arithmetic campaign. No generic bounded-coefficient theorem is admissible as the main step. --- # 23. Campaign 40 tracks ## HB1 — Möbius carrier audit For every Type-II component arising from the chosen Heath–Brown identity for $\Lambda-\Lambda^\sharp$, identify which grouped factor contains a Möbius-bearing short variable. Determine whether at least one factor always retains explicit Möbius structure after grouping. ## HB2 — translated-frequency Möbius large values The large- $T$ major-arc application twists the factors by $n^{iT}$. Study the actual frequency window seen by each Möbius factor and whether its short length relative to $|T|$ permits fixed-power averaged cancellation without a global Mertens theorem. ## HB3 — simultaneous resonance exclusion Define the set where both Type-II factors are larger than polynomial thresholds. Seek a bound on the spectral measure of the simultaneous-large set stronger than the generic $P^{-2}$ floor. ## HB4 — short Möbius factor versus complementary factor Exploit that Möbius factors in the fixed Heath–Brown identity have length at most a small power of $X$. Test whether large values of such a factor force a complementary factor into a regime controlled by mean-value or large-value estimates. ## HB5 — integrated polynomial- $W$ admission The only accepted output is the exact integrated estimate needed by Lemma 3.5 with $$ W=X^w $$ for some fixed $w>0$. If obtained, invoke B-RH-014. --- # 24. Campaign 40 rejection filters Reject a candidate if: ## R1. It treats the actual factors as arbitrary divisor-bounded coefficients. ## R2. It assumes pointwise fixed-power Mertens. ## R3. It assumes a fixed zero-free strip. ## R4. It only reduces the number of resonant frequencies to $X^{o(1)}$. ## R5. It obtains only logarithmic total resonance mass. ## R6. It ignores an allowed Heath–Brown component with no certified Möbius carrier. --- # 25. External calibration The current audit uses: 1. Matomäki–Radziwiłł–Shao–Tao–Teräväinen, *Higher uniformity of arithmetic functions in short intervals II*, Lemmas 3.4 and 3.5. The Type-II product mean square is proved using a Baker–Harman–Pintz parallelogram lemma plus pointwise smallness of one factor. 2. Guth–Maynard, *New large value estimates for Dirichlet polynomials*, Theorem 1.1: $$ R \le T^{o(1)} \left( N^2V^{-2} + N^{18/5}V^{-4} + TN^{12/5}V^{-4} \right). $$ The first term is the generic orthogonality floor. 3. The higher-uniformity paper's Heath–Brown decomposition expresses $\Lambda$ components using dyadic factors of type $1$, $\log$, and short $\mu$. The next campaign uses the third item as arithmetic structure rather than discarding it into the generic divisor-bounded class. --- # 26. State transition ```text CSM_RH v1.30 -> CSM_RH v1.31 ``` with: ```text Campaign 39 CLOSED_AS_GENERIC_AVERAGED_TYPEII_LARGE_VALUE_AUDIT B-RH-015 GUTH_MAYNARD_PRODUCT_LEVEL_SET_NORMALIZATION CREATED / CERTIFIED O-RH-096 GENERIC_LARGE_VALUE_ORTHOGONALITY_FLOOR CREATED / CERTIFIED O-RH-097 SINGLE_RESONANT_FREQUENCY_OBSTRUCTS_GENERIC_POLYNOMIAL_W_MEAN_SQUARE CREATED / CERTIFIED O-RH-098 LARGE_VALUE_COUNT_QUANTIZATION_BARRIER CREATED / CERTIFIED F-RH-010 PESC REMAINS OPEN F-RH-016 MLEPG REMAINS OPEN Campaign 40 HEATH_BROWN_ARITHMETIC_RESONANCE_EXCLUSION READY ``` --- # 27. Final status ```text RH = OPEN PESC = OPEN MLEPG = OPEN POLYNOMIAL-W TYPE-II AMPLIFIER = VALID POINTWISE POLYNOMIAL-W INPUT = FIXED-STRIP LOCKED GENERIC AVERAGED LARGE-VALUE REPLACEMENT = CLOSED GUTH-MAYNARD IMPROVEMENT = REAL BUT RETAINS ORTHOGONALITY FLOOR SINGLE GENERIC RESONANCE = ENOUGH TO KILL VANISHING PRODUCT MEAN SQUARE ARITHMETIC HEATH-BROWN RESONANCE EXCLUSION = OPEN NEXT CAMPAIGN = 40 ``` The decisive generic obstruction is $$ \boxed{ R_F(P) \ll X^{o(1)} \left( P^{-2} +\cdots \right), } $$ together with the legal phase-aligned example $$ \boxed{ a(m)=m^{it_0}, \qquad b(n)=n^{it_0}, } $$ for which $$ \boxed{ \int |A(1+it)B(1+it)|^2dt \gg1. } $$ The breakthrough route, if it exists, must use the arithmetic identity of the actual prime factors rather than generic large-value geometry.