# CSM_RH Paper 03 ## Principal Zeta-Packet Bottleneck, Positive-Gate No-Bypass, and Campaign 02 Closure **Project:** `CSM_RH` **Paper:** `03` **Version:** `v0.1` **Date:** `2026-09-05` **Parent state:** `CSM_RH v0.3 / Paper 02` **Status:** structural reduction / gate-strength theorem; not a proof or disproof of RH **中文標題：** CSM_RH 論文 03：主 Zeta 封包瓶頸、正閘門不可繞道與 Campaign 02 閉合 **作者：** Neo.K **機構：** EveMissLab／一言諾科技有限公司 **Language:** English --- # 0. Trust boundary This paper does not prove or disprove the Riemann Hypothesis. Canonical root state: ```text RH_PROVED = FALSE RH_DISPROVED = FALSE GLOBAL_RH_CERTIFICATE = FALSE CSM_RH_ROOT_STATUS = OPEN ``` The purpose of this paper is narrower: 1. isolate the $q=1$ principal zeta packet inside the positive EMAE gate; 2. prove an exact equivalence between its fixed-power bound and a fixed zeta zero strip; 3. show that conductor weights, character-family averaging, zero-density estimates, and Deuring–Heilbronn mechanisms cannot by themselves bypass this $q=1$ necessary condition; 4. close `CSM_RH Campaign 02` as a reduction campaign rather than a proof campaign; 5. define the next gate-minimality search. No live GLM-5.3-Flash run is claimed. --- # 1. Starting state Paper 02 proved that the smooth $q=1$ zeta-zero packet has exact exponential type. Let $$ \Theta_\zeta = \sup_{\rho} \Re\rho $$ over nontrivial zeta zeros, and let $$ \Delta_\zeta = \Theta_\zeta-\frac12. $$ By functional-equation symmetry, $$ \Theta_\zeta = \frac12+\Delta_\zeta. $$ Paper 02 established, for the Hilbert packet $F(T)$, $$ \inf \left\{ a: \|F(T)\| = O(e^{aT}) \right\} = \Theta_\zeta. $$ For the associated positive $q=1$ pole-zero energy, $$ \mathcal E_\zeta(e^T;U) = e^T \|F(T)\|^2, $$ the exact type is $$ \boxed{ \tau_\zeta = 1+2\Theta_\zeta = 2+2\Delta_\zeta. } $$ This paper studies what that exact identity does to the full arithmetic EMAE program. --- # 2. The $q=1$ packet is an unavoidable positive summand The v3.18 character major-arc energy is $$ \mathfrak Z_2(x;Q,U) = \sum_{\substack{ q\le Q\\ q\ {\rm squarefree} }} \frac{ |\mu(q)|^2 }{ \phi(q)^3 } \sum_{\chi\bmod q} |\tau(\chi)|^2 \int_{|\epsilon|\le U/x} |W_{x,\epsilon}(1)|^2 |Z_\chi(x,\epsilon)|^2 d\epsilon. $$ At $$ q=1, $$ all arithmetic weights are equal to one. The unique character is the trivial character, and the corresponding primitive $L$ -function is the Riemann zeta function. Therefore the $q=1$ term is exactly $$ \boxed{ \mathcal E_\zeta(x;U) = \int_{|\epsilon|\le U/x} |W_{x,\epsilon}(1)|^2 |Z_\zeta(x,\epsilon)|^2 d\epsilon. } $$ Since every term in $\mathfrak Z_2$ is nonnegative, $$ \boxed{ \mathfrak Z_2(x;Q,U) \ge \mathcal E_\zeta(x;U). } $$ This inequality is independent of every nonprincipal character estimate. --- # 3. Principal Zeta-Packet Fixed Power For fixed $$ 0\le\eta\le1, $$ define: ## ZPPF $(\eta)$ For every $$ \varepsilon>0, $$ $$ \boxed{ \mathcal E_\zeta(x;U) \ll_{\varepsilon,w,U} x^{3-\eta+\varepsilon}. } $$ Equivalently, $$ \mathcal E_\zeta(x;U) \ll x^{3-\eta+o(1)}. $$ The acronym is: ```text ZPPF PRINCIPAL ZETA-PACKET FIXED POWER ``` --- # 4. Exact ZPPF / zero-strip equivalence ## Theorem 4.1 — Principal Zeta-Packet Equivalence For fixed $$ 0\le\eta\le1, $$ the following are equivalent: $$ \boxed{ \operatorname{ZPPF}(\eta) } $$ and $$ \boxed{ \Theta_\zeta \le 1-\frac{\eta}{2}. } $$ Equivalently, $$ \boxed{ \Delta_\zeta \le \frac12-\frac{\eta}{2}. } $$ ### Proof: ZPPF implies the strip ZPPF $(\eta)$ gives $$ \tau_\zeta \le 3-\eta. $$ Paper 02 gives $$ \tau_\zeta = 1+2\Theta_\zeta. $$ Therefore $$ 1+2\Theta_\zeta \le 3-\eta, $$ so $$ \Theta_\zeta \le 1-\frac{\eta}{2}. $$ ### Proof: the strip implies ZPPF Paper 02 also gives the direct upper bound $$ \|F(T)\| \le e^{\Theta_\zeta T} \sum_\rho \|v_\rho\|. $$ The coefficient sum is finite. Hence $$ \mathcal E_\zeta(e^T;U) \ll e^{(1+2\Theta_\zeta)T}. $$ If $$ \Theta_\zeta \le 1-\frac{\eta}{2}, $$ then $$ 1+2\Theta_\zeta \le 3-\eta. $$ Therefore $$ \mathcal E_\zeta(x;U) \ll x^{3-\eta}. $$ Thus ZPPF $(\eta)$ holds. $\square$ --- # 5. RH endpoint Set $$ \eta=1. $$ Theorem 4.1 gives $$ \operatorname{ZPPF}(1) \Longleftrightarrow \Theta_\zeta \le \frac12. $$ Functional-equation symmetry gives $$ \Theta_\zeta \ge \frac12. $$ Therefore $$ \boxed{ \operatorname{ZPPF}(1) \Longleftrightarrow RH. } $$ The $q=1$ positive packet is therefore an exact RH diagnostic at the endpoint. --- # 6. EMAE necessity theorem Recall $$ \operatorname{EMAE}(\eta): \qquad \mathfrak Z_2(x;Q,U) + \mathfrak Z_4(x;Q,U) \ll x^{3-\eta+o(1)}. $$ Because $$ \mathfrak Z_4\ge0 $$ and $$ \mathfrak Z_2 \ge \mathcal E_\zeta, $$ we obtain: ## Theorem 6.1 — EMAE necessarily contains a zeta strip For every fixed $$ \eta>0, $$ $$ \boxed{ \operatorname{EMAE}(\eta) \Longrightarrow \operatorname{ZPPF}(\eta) \Longrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. } $$ Thus a proof of positive EMAE at any fixed power already proves a new fixed zero strip for zeta before any nonprincipal character issue is considered. This is a necessary-condition theorem. It is stronger than the earlier single-zero heuristic because the packet interference question was closed in Paper 02. --- # 7. Positive-Gate No-Bypass theorem The previous theorem extends beyond the exact EMAE definition. Let $$ \mathcal G(x)\ge0 $$ be any proposed positive sufficient gate. Assume that for some nonnegative weight $c(x)$, $$ \mathcal G(x) \ge c(x) \mathcal E_\zeta(x;U). $$ ## Theorem 7.1 — Subpolynomial $q=1$ retention Suppose $$ c(x) = x^{-o(1)}. $$ If $$ \mathcal G(x) \ll x^{3-\eta+o(1)} $$ for a fixed $$ \eta>0, $$ then $$ \boxed{ \Theta_\zeta \le 1-\frac{\eta}{2}. } $$ ### Proof Since $$ c(x)^{-1} = x^{o(1)}, $$ we have $$ \mathcal E_\zeta(x;U) \le c(x)^{-1}\mathcal G(x) \ll x^{3-\eta+o(1)}. $$ Apply Theorem 4.1. $\square$ --- # 8. Fixed-power weight-loss law More generally, suppose $$ c(x) \ge x^{-\lambda+o(1)} $$ for fixed $$ \lambda\ge0. $$ Then $$ \mathcal G(x) \ll x^{3-\eta+o(1)} $$ implies $$ \mathcal E_\zeta(x;U) \ll x^{3-(\eta-\lambda)+o(1)}. $$ Hence, if $$ \eta>\lambda, $$ then $$ \boxed{ \Theta_\zeta \le 1-\frac{\eta-\lambda}{2}. } $$ Consequently, a positive gate can avoid any fixed-strip consequence only by losing at least the full fixed power in its $q=1$ retention. But such a loss creates a separate target-fidelity question: > does the weakened gate still control the actual $q=1$ prime-pair mismatch? That question cannot be assumed. --- # 9. Why conductor weighting cannot solve the full EMAE gate For $$ q>1, $$ v3.18 identified useful weights such as $$ q^{-2+o(1)} $$ in the pole-zero channel. These weights can genuinely suppress high-conductor characters. However, at $$ q=1, $$ the conductor weight is exactly one. Therefore no conductor-weight argument can reduce the zeta exponent $$ 1+2\Theta_\zeta. $$ Conductor weighting remains useful for the nonprincipal remainder after the $q=1$ problem has been separately handled. It is not a bypass of the principal bottleneck. --- # 10. Why height weights cannot change fixed exponential type For each fixed zeta zero $$ \rho=\beta+i\gamma, $$ the smooth Mellin coefficient $$ v_\rho $$ may be very small for large $$ |\gamma|. $$ But Paper 02 proved $$ v_\rho\neq0 $$ for every zero and $$ \tau_F=\Theta_\zeta. $$ Therefore any $x$ -independent nonzero coefficient suppression in zero height changes constants but not the exponential type. In particular, rapid vertical decay is sufficient for convergence, but it does not turn $$ x^\beta $$ into $$ x^{\beta-\delta} $$ for a fixed zero. This closes the following false inference: ```text very small high-zero coefficient -> fixed-power suppression of every near-extremal zero ``` The implication is invalid at the asymptotic exponent level. --- # 11. Current prime exponential-sum input at the $q=1$ core Maynard–Pandey–Radziwiłł (2026) prove that if $$ \alpha=\frac aq+\epsilon $$ and $$ B=\max(q,qN|\epsilon|), $$ then $$ \left| \sum_{n1-\frac{\eta}{2} $$ for a prescribed fixed $$ \eta>0. $$ By Theorem 4.1, one such zero is enough to rule out ZPPF $(\eta)$. Therefore: ```text ZERO POPULATION CONTROL does not imply ZPPF FIXED POWER. ``` This is now a direct consequence of the exact packet-type theorem. --- # 13. Current zero-free regions are not fixed strips Bellotti–Trudgian–Yang (2026) prove an explicit zeta zero-free region of the form $$ \zeta(\sigma+it)\neq0 $$ when $$ t\ge3 $$ and $$ \sigma \ge 1-\frac{1}{4.896\log t}. $$ This is valuable explicit progress. However the width $$ \frac{1}{4.896\log t} $$ tends to zero with height. It does not imply a fixed $$ \delta>0 $$ such that $$ \Theta_\zeta \le1-\delta. $$ Therefore it does not close ZPPF $(\eta)$ for any fixed $$ \eta>0. $$ --- # 14. Deuring–Heilbronn does not bypass $q=1$ Modern explicit Deuring–Heilbronn results give zero repulsion for Dirichlet $L$ -functions under the existence of a Landau–Siegel zero. This mechanism is relevant to exceptional real characters. But the necessary principal subgate in Theorem 6.1 is the Riemann zeta packet at $$ q=1. $$ Therefore Deuring–Heilbronn may improve the nonprincipal character remainder but cannot, by itself, establish the $q=1$ ZPPF gate. This is a scope obstruction, not a criticism of the theorem. --- # 15. Correction to the v3.18 weighted-extreme-zero optimism v3.18 correctly observed that the full character major-arc energy contains: - conductor weights; - arc widths; - height localization; - Mellin decay; - Gauss-sum weights. It therefore suggested that a weighted theorem might permit some zeros closer to one without requiring a uniform Dirichlet fixed strip. Paper 03 refines that statement. For nonprincipal characters, this opening remains valid. For the complete positive EMAE gate, however, the $q=1$ zeta packet has: ```text conductor weight = 1 character averaging = none Gauss-sum discount = none fixed nonzero Mellin coefficient per zero ``` Therefore: $$ \boxed{ \text{weighted character flexibility} \neq \text{principal zeta flexibility}. } $$ Any full positive EMAE theorem still contains a fixed zeta zero-strip theorem as a necessary subresult. --- # 16. Campaign 02 candidate audit ## Candidate C02-A — Density + conductor + height boxes ```text status: REJECTED_AS_FULL_EMAE_BYPASS reason: useful for q>1 boxes; cannot discharge q=1 ZPPF ``` ## Candidate C02-B — Nonexceptional-modulus prime exponential-sum improvement ```text status: REJECTED_AS_PRINCIPAL_CORE_BYPASS reason: q=1 core has B=O(1); N/B^(1/2) term retains N-scale ``` ## Candidate C02-C — Deuring–Heilbronn exceptional-zero repulsion ```text status: PARTIAL / NONPRINCIPAL ONLY reason: does not solve q=1 zeta packet ``` ## Candidate C02-D — Positive conductor / height reweighting ```text status: REJECTED_IF_q1_WEIGHT_IS_x^(-o(1)) certificate: Theorem 7.1 ``` ## Candidate C02-E — Remove / subtract the principal zeta packet ```text status: REJECTED_WITHOUT_TARGET_FIDELITY_BRIDGE reason: deleting the blocker from the sufficient gate is not closure unless the resulting gate still controls the original prime-pair target ``` ## Candidate C02-F — Direct zeta fixed-strip arithmetic theorem ```text status: VALID TARGET strength: BREAKTHROUGH_LEVEL comment: not a simplification; exactly the required new mathematics ``` ## Candidate C02-G — Signed principal mismatch / non-positive gate ```text status: SURVIVOR reason: Theorem 7.1 blocks positive gates retaining q=1, but does not rule out a genuinely signed target-faithful cancellation mechanism. proof status: OPEN ``` --- # 17. Campaign 02 closure verdict Campaign 02 asked: > find the weakest genuinely arithmetic theorem that yields EMAE $(\eta)$ for some fixed $\eta>0$. The result is: ```text DIRECT FULL EMAE BY CURRENT CHARACTER-FAMILY TECHNOLOGY = NOT CLOSED PRINCIPAL NECESSARY SUBGATE = EXACTLY IDENTIFIED q=1 POSITIVE GATE STRENGTH = EXACTLY FIXED ZERO STRIP CURRENT DENSITY / CONDUCTOR / HEIGHT / REPULSION INPUTS = INSUFFICIENT TO BYPASS q=1 SIGNED TARGET-FAITHFUL ALTERNATIVE = SURVIVOR / OPEN ``` Therefore Campaign 02 is closed as a reduction and no-go audit. It is not closed as a proof of EMAE. --- # 18. New canonical frontier object Create: ```text F-RH-005 PRINCIPAL_ZETA_PACKET_FIXED_POWER abbrev: ZPPF status: OPEN type: EXACT_EQUIVALENCE_FRONTIER ``` For fixed $\eta$: $$ \boxed{ F\text{-}RH\text{-}005(\eta) \Longleftrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. } $$ This frontier is intentionally marked as an exact-equivalence frontier. It is not advertised as an easier reformulation. --- # 19. New obstruction Create: ```text O-RH-006 POSITIVE_GATE_PRINCIPAL_NO_BYPASS status: CERTIFIED ``` Statement: > Any positive fixed-power sufficient gate which retains the $q=1$ principal zeta packet with only subpolynomial loss already contains a fixed zeta zero-strip theorem. This obstruction applies to: ```text positive character-family energies positive conductor reweightings positive height reweightings positive large-sieve envelopes positive zero-density envelopes ``` whenever they retain the principal packet at $x^{-o(1)}$ strength. --- # 20. New survivor Create: ```text S-RH-008 SIGNED_TARGET_FAITHFUL_PRINCIPAL_GATE status: OPEN / SURVIVOR ``` The survivor asks whether the original prime-pair target admits a sufficient gate that: 1. preserves the actual signed structure; 2. does not dominate $\mathcal E_\zeta$ as a positive summand; 3. still controls the target pair-spectrum deviation; 4. has a weaker fixed-power strength profile than ZPPF; 5. does not subtract zeta zeros by assumption; 6. does not insert RH into the model. This is not known to exist. It is the only structural bypass left by the present positive-gate audit. --- # 21. Campaign 03 — Principal Gate Minimality Audit The next worker campaign is: ```text CSM_RH Campaign 03 PRINCIPAL_GATE_MINIMALITY ``` Primary question: > Is the current positive $q=1$ energy gate stronger than necessary for the actual prime-pair target, or is every target-faithful fixed-power principal gate still forced to have zero-strip strength? This is a gate-design question before it is a new estimate question. --- # 22. Campaign 03 candidate families Workers may generate candidates from: ```text A SIGNED PRINCIPAL MISMATCH B LINEARIZED POLE-RESIDUAL OBSERVABLE C POLARIZED CROSS-CORRELATION D SCALE-FREQUENCY SIGNED TRANSFORM E DIRECT PAIR-SPECTRUM DIFFERENCE WITHOUT POSITIVE DOMINATION F CERTIFIED CANCELLATION BETWEEN PRINCIPAL SUBTERMS ``` These are search classes, not theorem claims. --- # 23. Campaign 03 rejection tests Every candidate must answer: ## T1. Target fidelity Does the proposed gate still imply the original prime-pair / pair-spectrum target? ## T2. Principal retention Has the zeta packet merely been deleted, renamed, or assumed small? ## T3. Hidden positivity Does the proof eventually dominate a positive object containing $$ x^{-o(1)} \mathcal E_\zeta? $$ If yes, `O-RH-006` applies. ## T4. Hidden fixed strip Does any intermediate estimate already imply $$ \Theta_\zeta \le1-\delta $$ for fixed $\delta>0$? If yes, classify it honestly as breakthrough-strength. ## T5. Signed cancellation certificate If cancellation is used, is it structurally forced or only heuristic? ## T6. Quantifier audit Does the argument work uniformly as $$ x\to\infty $$ for all fixed zero witnesses? ## T7. Supremum audit Does it avoid assuming that $\Theta_\zeta$ is attained? --- # 24. Worker architecture `CSM_RH` remains the protocol. GLM-5.3-Flash may be used as a current high-volume worker provider. Canonical roles: ```text Designer Builder Verifier ``` The provider remains replaceable. Worker agreement is not theorem authority. Campaign artifacts must be promoted only after: ```text strength audit target-fidelity audit counterexample attempt quantifier audit independent frontier review ``` --- # 25. External evidence snapshot As of 2026-09-05: ## Maynard–Pandey–Radziwiłł `Exponential sums over primes`, arXiv:2608.14777, submitted 2026-08-14. Main current pointwise estimate: $$ |S_N(\alpha)| \le N^{o(1)} \left( \frac{N}{B^{1/2}} + N^{19/24} \right). $$ The authors explicitly interpret the first term morally as reflecting possible exceptional zeros close to one. ## Guth–Maynard `New large value estimates for Dirichlet polynomials`, Annals of Mathematics 203 (2026), 623–675. Zero-density consequence: $$ N(\sigma,T) \le T^{30(1-\sigma)/13+o(1)}. $$ ## Chen–Gupta–Li `Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet L-Functions`, arXiv:2507.08296v2, revised 2026-07-27. Character-family zero-density estimate: $$ \sum_{\chi\bmod q} N(\sigma,T,\chi) \ll_\varepsilon (qT)^{7(1-\sigma)/3+\varepsilon}. $$ ## Bellotti–Trudgian–Yang `Zero-free regions inspired by work of Heath-Brown`, arXiv:2603.21490. Explicit region: $$ \sigma \ge 1-\frac{1}{4.896\log t}, \qquad t\ge3. $$ ## Benli–Goel–Twiss–Zaman `Explicit Deuring-Heilbronn phenomenon for Dirichlet L-functions`, arXiv:2410.06082v3, revised 2026-01-08. Scope: ```text Landau-Siegel-zero conditional repulsion for Dirichlet L-functions modulo q ``` These results are external inputs or consistency checks only. None is promoted here into a fixed zeta zero strip. --- # 26. State transition The canonical transition is: ```text CSM_RH v0.3 -> CSM_RH v0.4 ``` with: ```text Campaign 02: CLOSED_AS_REDUCTION_AUDIT F-RH-003: EMAE OPEN but now has certified necessary subgate F-RH-005 F-RH-005: PRINCIPAL_ZETA_PACKET_FIXED_POWER CREATED OPEN / EXACT_EQUIVALENCE_FRONTIER O-RH-006: POSITIVE_GATE_PRINCIPAL_NO_BYPASS CREATED / CERTIFIED S-RH-008: SIGNED_TARGET_FAITHFUL_PRINCIPAL_GATE CREATED / OPEN next: Campaign 03 PRINCIPAL_GATE_MINIMALITY ``` --- # 27. Final status ```text RH = OPEN MAJOR-ARC PACKET GLOBALIZATION = CLOSED_AT_EXPONENTIAL_TYPE FULL POSITIVE EMAE = OPEN q=1 PRINCIPAL POSITIVE FIXED POWER = EXACTLY EQUIVALENT TO A FIXED ZETA ZERO STRIP CURRENT CHARACTER-FAMILY ADVANCES = DO NOT BYPASS q=1 POSITIVE-GATE BYPASS = CLOSED / NO SIGNED TARGET-FAITHFUL BYPASS = OPEN / SURVIVOR NEXT CAMPAIGN = PRINCIPAL_GATE_MINIMALITY ``` The central result is: $$ \boxed{ \operatorname{ZPPF}(\eta) \Longleftrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. } $$ and therefore: $$ \boxed{ \operatorname{EMAE}(\eta) \Longrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. } $$ The remaining design choice is now explicit: > either prove genuinely new fixed-strip mathematics, or abandon the current positive-dominating principal gate and prove that a weaker signed target-faithful gate is sufficient. That is the canonical output of Campaign 02.