# CSM_RH Paper 38 ## Polynomial Shrinking Thresholds, Explicit-Formula Height Tax, and the Right-Edge Subpower Barrier **Project:** `CSM_RH` **Paper:** `38` **Version:** `v0.1` **Date:** `2026-09-07` **Parent state:** `CSM_RH v1.28 / Paper 37` **Campaign:** `37 — SHRINKING_THRESHOLD_TAIL_ATTACK` **Status:** explicit dependence audit of the Gafni–Tao exceptional-set architecture; 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 37 certified that a polynomial shrinking-threshold exceptional-set theorem would imply MLEPG and hence a fixed zero-free strip. Campaign 37 asks whether the current Gafni–Tao exceptional-set proof can already be pushed to such a threshold by exposing all parameter dependence. The answer is negative for the audited architecture. There are three independent reasons: 1. the theorem is proved with $\delta$ and $J$ fixed before $X\to\infty$ ; 2. a polynomial threshold forces the explicit-formula height to increase by a fixed power of $X$ ; 3. the right-edge zero packet is controlled only at stretched-log / subpower strength. The third obstruction is decisive. No new fixed-power theorem is admitted. --- # 1. Shrinking-threshold admission target Let $$ H=X^\theta $$ with fixed $$ 0<\theta<1. $$ Define $$ U_H(x) = \psi(x+H)-\psi(x)-H. $$ The Paper-37 admission target is: $$ \boxed{ \left| \left\{ x\in[X,2X]: |U_H(x)|>HX^{-\eta} \right\} \right| \ll X^{1-c} } $$ for some fixed $$ \eta>0, \qquad c>0. $$ By Paper 37 this implies $$ \mathcal S_\Lambda(X,H) \ll XH^2X^{-\min(2\eta,c)+o(1)}, $$ and therefore MLEPG. --- # 2. Fixed-parameter structure of Gafni–Tao The Gafni–Tao proof begins by fixing $$ 0<\delta<1. $$ It then chooses a natural number $J$ sufficiently large depending on $$ \delta,\theta,\varepsilon, $$ and proves its exceptional-set estimate as $$ X\to\infty $$ with $$ J,\delta,\theta,\varepsilon $$ held fixed. Therefore the published theorem does not by itself authorize the substitution $$ \boxed{ \delta=X^{-\eta}. } $$ Create: ```text O-RH-088 GAFNI_TAO_FIXED_PARAMETER_NONUNIFORMITY status: CERTIFIED AS THEOREM-SCOPE AUDIT ``` This is a theorem-scope statement, not a claim that the proof cannot be uniformized by new work. --- # 3. Spatial localization cost To replace the varying interval length $x^\theta$ by the multiplicatively normalized length $x/\tau$, the proof localizes $x$ to intervals of the form $$ [X,(1+\delta/J)X], $$ where $$ \tau=X^{1-\theta}. $$ Covering $[X,2X]$ requires $$ \boxed{ O(J/\delta) } $$ such intervals. For fixed $\delta,J$ this is a harmless constant. If $$ \delta=X^{-\eta} $$ and $J$ grows polynomially, this becomes an exponent-level cost. --- # 4. Explicit-formula height tax The truncated explicit formula used in the proof has error $$ \boxed{ O \left( \frac{ X(\log X)^2 }{ T } \right). } $$ To resolve the shrinking threshold $$ \delta X^\theta, $$ it is necessary that $$ \frac{ X(\log X)^2 }{ T } \ll \delta X^\theta. $$ Therefore: ## Theorem 4.1 — Polynomial Threshold Truncation Law $$ \boxed{ T \gtrsim \delta^{-1} X^{1-\theta} (\log X)^2. } $$ In particular, if $$ \delta=X^{-\eta}, $$ then $$ \boxed{ T \ge X^{1-\theta+\eta+o(1)}. } $$ Create: ```text B-RH-013 SHRINKING_THRESHOLD_EXPLICIT_FORMULA_HEIGHT_LAW status: CERTIFIED ``` The original proof chooses $$ T = J(\log X)^2X^{1-\theta}. $$ Thus, within that parameterization, polynomial shrinking forces at least $$ \boxed{ J\gtrsim X^\eta. } $$ --- # 5. Zero-density exponent tax Suppose generously that one has the uniform zero-density bound $$ N(\sigma,T) \le T^{A_0(1-\sigma)+o(1)} $$ with $$ A_0=\frac{30}{13}. $$ Under the polynomial truncation height $$ T=X^{1-\theta+\eta+o(1)}, $$ the density contribution becomes $$ \boxed{ X^{ A_0(1-\theta+\eta)(1-\sigma) +o(1) }. } $$ Thus the zero-density part itself pays a fixed exponent tax $$ \boxed{ A_0\eta(1-\sigma). } $$ This is not yet the decisive obstruction, because for $\sigma$ bounded away from $1$ and $\theta$ above the almost-all threshold by a fixed margin, sufficiently small $\eta$ can still leave exponent room. The fatal region is $\sigma\to1$. --- # 6. Right-edge lemma Gafni–Tao isolate a fixed strip $$ I\subset[1-\eta_0,1]. $$ Using the Vinogradov–Korobov zero-free region and a near-one zero-density estimate, they prove $$ \boxed{ \sup_{X\le x\le2X} |S_I(x)| \ll_\theta \exp \left[ -c_\theta \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right] \frac{X}{\tau}. } $$ Since $$ \frac{X}{\tau}=X^\theta, $$ the available relative suppression is $$ \boxed{ \Delta_{\mathrm{edge}}(X) = \exp \left[ -c_\theta \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right]. } $$ This is $$ X^{-o(1)}. $$ --- # 7. Polynomial threshold comparison For any fixed $$ \eta>0, $$ the desired relative threshold is $$ \delta_X=X^{-\eta}. $$ But $$ \boxed{ \Delta_{\mathrm{edge}}(X) \gg X^{-\eta} } $$ as $X\to\infty$. Indeed, $$ \eta\log X \gg \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} }. $$ Therefore the right-edge estimate is asymptotically much larger than the desired polynomial threshold. Create: ```text O-RH-089 GAFNI_TAO_RIGHT_EDGE_SUBPOWER_FLOOR status: CERTIFIED AS CURRENT-PROOF BARRIER ``` The existing Lemma 2.1 cannot dispose of the right-edge zero packet at threshold $X^{-\eta}$. --- # 8. This failure occurs before Markov The right-edge treatment is an $L^\infty$ disposal step. It occurs before the proof invokes the $L^2$ and $L^4$ Markov bounds for the remaining zero strips. Therefore improving bookkeeping in the later Markov step does not repair this failure. The proof already lacks sufficient amplitude suppression on the near-one zero packet. --- # 9. Why the second moment cannot repair the edge Away from the right edge, the second-moment exponent has the form $$ (1-\theta) (1-\sigma) A(\sigma) + 2\theta + 2\sigma - 2. $$ After the shrinking-threshold height tax, the natural generous replacement is $$ \boxed{ (1-\theta+\eta) (1-\sigma) A(\sigma) + 2\theta + 2\sigma - 2. } $$ Markov at threshold $X^{\theta-\eta}$ introduces an additional cost $$ X^{2\eta}. $$ Even ignoring localization and binning costs, the exceptional-set exponent tends to $$ \boxed{ 1+2\eta } $$ as $$ \sigma\to1. $$ The zero-density term is nonnegative and vanishes with $1-\sigma$. Hence no uniform power saving can emerge from the second moment arbitrarily close to $\sigma=1$. --- # 10. Fourth and higher moments have the same edge geometry The fourth-moment exponent has the form $$ (1-\theta) (1-\sigma) A^\ast(\sigma) + 4\theta + 4\sigma - 4. $$ At polynomial threshold $X^{\theta-\eta}$, Markov contributes $X^{4\eta}$. As $$ \sigma\to1, $$ the resulting exceptional-set exponent tends to $$ \boxed{ 1+4\eta. } $$ The same phenomenon persists for a hypothetical $2k$ -th moment of the standard zero packet. The baseline exponent is $$ 2k\sigma-2k = -2k(1-\sigma), $$ while the shrinking threshold costs $$ X^{2k\eta}. $$ If $$ 1-\sigma<\eta, $$ then even the best possible contribution obtained by discarding the nonnegative zero-density term cannot compensate the threshold cost. Create: ```text O-RH-090 MOMENT_ORDER_CANNOT_CURE_POLYNOMIAL_THRESHOLD_AT_SHRINKING_EDGE status: CERTIFIED AS STANDARD_MARKOV-PACKET STRENGTH AUDIT ``` This concerns the standard moment-plus-Markov zero-packet architecture. It is not a universal impossibility theorem for every high-moment method. --- # 11. The zero-free-region scale The Vinogradov–Korobov region only removes zeros in a shrinking neighborhood $$ 1-\sigma \asymp (\log X)^{-2/3} (\log\log X)^{-1/3} $$ at the present height scale, up to constants and the harmless effect of subpower height changes. This width tends to zero. For every fixed $$ \eta>0, $$ eventually $$ 1-\sigma<\eta $$ throughout part of the remaining admissible edge layer. Thus the moment-order audit in Section 10 is consistent with the explicit $L^\infty$ barrier of Section 7. --- # 12. Maximum natural shrinking class of the current edge input The right-edge bound is compatible with thresholds no smaller than a stretched-log class such as $$ \boxed{ \delta_X \gtrsim \exp \left[ -C \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right]. } $$ For such a subpower threshold, $$ \delta_X^{-1}=X^{o(1)}, $$ so the necessary truncation height can remain $$ T=X^{1-\theta+o(1)}. $$ This explains structurally why the existing architecture naturally lives in a subpower-precision class. This section is an architecture calibration, not a new uniform theorem, because the published proof keeps $\delta$ fixed. --- # 13. Interior zero-density region is not the primary failure To localize the obstruction, suppose hypothetically that all zeros with $$ \sigma>1-\eta_0 $$ were absent for one fixed $$ \eta_0>0. $$ Then the remaining $\sigma$ -range is bounded away from $1$. With Guth–Maynard $$ A_0=\frac{30}{13}, $$ and $$ \theta>\frac{2}{15} $$ by a fixed margin, sufficiently small shrinking exponent $\eta$ leaves room in the second-moment exponent ledger. For example, at the most basic $\sigma=1/2$ density point, ignoring localization losses, the condition is $$ \boxed{ \frac{A_0}{2} (1-\theta+\eta) + 2\eta < 1. } $$ This has a positive solution $\eta$ whenever $$ \theta>\frac{2}{15}. $$ Thus the critical issue is not merely the $30/13$ density coefficient. It is the absence of a fixed right-edge gap. --- # 14. Current-proof J coupling adds further costs In the published proof the same $J$ controls: 1. spatial localization; 2. zero-strip subdivision; 3. explicit-formula truncation height. If one formally sets $$ J=X^j, \qquad \delta=X^{-\eta}, $$ then: ## truncation requirement $$ j\ge\eta; $$ ## number of spatial intervals $$ J/\delta = X^{j+\eta}; $$ ## number of zero strips $$ J = X^j; $$ ## pigeonhole threshold per zero strip $$ \frac{\delta}{J} = X^{-(\eta+j)}. $$ These costs were constants in the fixed-parameter proof. They become exponent-level losses when $J$ and $\delta^{-1}$ grow polynomially. Even without the right-edge failure, the proof would require a new uniform re-optimization. --- # 15. Generous current-architecture L2 ledger If one keeps the original $J$ coupling and chooses the minimal polynomial scale $$ j=\eta, $$ then a generous global $A_0$ second-moment calculation produces the schematic exceptional exponent $$ \boxed{ \Xi_2(\sigma) = 2\sigma-1 + A_0 (1-\theta+\eta) (1-\sigma) + 7\eta } $$ after accounting for: 1. the threshold $\delta/J$ ; 2. the $J$ zero strips; 3. the $J/\delta$ spatial blocks. For $\sigma=1/2$, $$ \boxed{ \Xi_2(1/2) = \frac{A_0}{2} (1-\theta+\eta) + 7\eta. } $$ This can remain below $1$ for very small $\eta$ when $\theta$ is above $2/15$ by a fixed margin. But $$ \boxed{ \lim_{\sigma\to1} \Xi_2(\sigma) = 1+7\eta>1. } $$ Again the right edge is fatal. The precise coefficient $7$ is architecture-specific and is not claimed to be optimal. --- # 16. Campaign 37 track audit ## ST1 — fixed-delta dependence ```text status: NONUNIFORM IN PUBLISHED THEOREM delta, J: fixed before X -> infinity ``` ## ST2 — explicit-formula truncation ```text status: POLYNOMIAL HEIGHT TAX delta = X^(-eta) forces T >= X^(1-theta+eta+o(1)) ``` ## ST3 — Guth–Maynard density insertion ```text status: INTERIOR REGION CAN RETAIN EXPONENT ROOM FOR SMALL eta global A0: 30/13 decisive failure: not interior density ``` ## ST4 — right-edge zero packet ```text status: FATAL FOR CURRENT PROOF available suppression: X^(-o(1)) required suppression: X^(-eta) ``` ## ST5 — higher moments ```text status: STANDARD MARKOV MOMENTS DO NOT CURE SHRINKING EDGE reason: threshold penalty survives as sigma -> 1 ``` --- # 17. Campaign 37 verdict No polynomial shrinking-threshold theorem is obtained. The Paper-37 admission target remains valid and open. The current Gafni–Tao architecture is classified as: ```text fixed threshold: valid power exceptional sets: valid subpower shrinking threshold: structurally compatible with edge scale, but not a published uniform theorem polynomial shrinking threshold: not supported first decisive obstruction: right-edge X^(-o(1)) amplitude floor ``` No theorem passes the Campaign-36 breakthrough gate. --- # 18. New certified package Create: ```text B-RH-013 SHRINKING_THRESHOLD_EXPLICIT_FORMULA_HEIGHT_LAW CERTIFIED O-RH-088 GAFNI_TAO_FIXED_PARAMETER_NONUNIFORMITY CERTIFIED AS THEOREM-SCOPE AUDIT O-RH-089 GAFNI_TAO_RIGHT_EDGE_SUBPOWER_FLOOR CERTIFIED AS CURRENT-PROOF BARRIER O-RH-090 MOMENT_ORDER_CANNOT_CURE_POLYNOMIAL_THRESHOLD_AT_SHRINKING_EDGE CERTIFIED AS STANDARD MARKOV-PACKET STRENGTH AUDIT ``` No new frontier is created. --- # 19. Canonical root status ```text F-RH-010 PESC OPEN / ROOT TARGET F-RH-016 MLEPG OPEN / DIRECT THEOREM CANDIDATE B-RH-012 SHRINKING_THRESHOLD_EXCEPTIONAL_SET_TO_MLEPG CERTIFIED BRIDGE polynomial shrinking-threshold theorem: OPEN / NOT PROVED ``` --- # 20. Campaign 38 The zero-density exceptional-set route is now localized to the same right-edge subpower barrier already seen in other explicit-formula arguments. The next campaign therefore switches to a genuinely different arithmetic source: ```text CSM_RH Campaign 38 GROWING_ACCURACY_LAMBDA_RESIDUAL_ATTACK ``` The target is the 2026 higher-uniformity theorem for $$ f=\Lambda-\Lambda^\sharp. $$ The question is: > can its arbitrary fixed log-power accuracy be uniformized to an accuracy parameter growing with $X$ strongly enough to produce $HX^{-\eta}$? --- # 21. Campaign 38 tracks ## GA1 — accuracy parameter scaling The current theorem gives $$ H\log^{-A}X $$ for every fixed $A$. Set formally $$ A \asymp \eta \frac{\log X}{\log\log X} $$ and audit every proof constant. ## GA2 — W-parameter transition The 2026 proof uses a prime major-arc parameter of the form $$ W_\Lambda=\log^{C A}X. $$ At growing $A$, compute the exact point at which $$ W_\Lambda $$ becomes polynomial in $X$. ## GA3 — Vinogradov–Korobov dependency Track where the prime Dirichlet-polynomial input uses the shrinking zero-free region. Determine whether that step alone prevents polynomial $W_\Lambda$. ## GA4 — exceptional-set parameter uniformity The almost-all theorem also gives a logarithmic exceptional set. Audit whether making $A$ grow causes the exceptional-set constant or auxiliary complexity to become exponent-level. ## GA5 — direct bridge to Paper 37 target If a uniform theorem of the form $$ |U_H(x)| \ll HX^{-\eta} $$ outside $$ O(X^{1-c}) $$ intervals emerges, immediately invoke B-RH-012. No new observable is allowed. --- # 22. Campaign 38 rejection filters Reject a candidate if: ## R1. $A$ is treated as growing while a theorem assumes it fixed. ## R2. A hidden $O_A(1)$ constant is ignored. ## R3. $W_\Lambda$ becomes polynomial without re-proving the prime Dirichlet-polynomial estimate. ## R4. The proof imports a fixed zero-free strip. ## R5. The final exceptional set remains only logarithmically small. ## R6. The output is $X^{-o(1)}$ rather than $X^{-\eta}$. --- # 23. External calibration The audited source is Gafni–Tao, *On the number of exceptional intervals to the prime number theorem in short intervals*, published in Essential Number Theory 5 (2026), 221–241. The proof: 1. fixes $\delta$ and $J$ before letting $X\to\infty$ ; 2. uses the truncation height $$ T=J(\log X)^2X^{1-\theta}; $$ 3. disposes of the right-edge zero packet with a Vinogradov–Korobov stretched-log bound; 4. controls the remaining zero strips by $L^2$ and $L^4$ moments and Markov's inequality. The Guth–Maynard zero-density theorem gives the global bound $$ A(\sigma)\le\frac{30}{13}, $$ which yields the all-interval threshold $17/30$ and almost-all threshold $2/15$. Neither theorem currently provides the polynomial shrinking-threshold statement required by B-RH-012. --- # 24. State transition ```text CSM_RH v1.28 -> CSM_RH v1.29 ``` with: ```text Campaign 37 CLOSED_AS_SHRINKING_THRESHOLD_GAFNI_TAO_DEPENDENCE_AUDIT B-RH-013 SHRINKING_THRESHOLD_EXPLICIT_FORMULA_HEIGHT_LAW CREATED / CERTIFIED O-RH-088 GAFNI_TAO_FIXED_PARAMETER_NONUNIFORMITY CREATED / CERTIFIED O-RH-089 GAFNI_TAO_RIGHT_EDGE_SUBPOWER_FLOOR CREATED / CERTIFIED O-RH-090 MOMENT_ORDER_CANNOT_CURE_POLYNOMIAL_THRESHOLD_AT_SHRINKING_EDGE CREATED / CERTIFIED F-RH-010 PESC REMAINS OPEN F-RH-016 MLEPG REMAINS OPEN Campaign 38 GROWING_ACCURACY_LAMBDA_RESIDUAL_ATTACK READY ``` --- # 25. Final status ```text RH = OPEN PESC = OPEN MLEPG = OPEN POLYNOMIAL SHRINKING-THRESHOLD EXCEPTIONAL SET = NOT PROVED GAFNI-TAO FIXED-DELTA THEOREM = NOT UNIFORM IN delta = X^(-eta) EXPLICIT-FORMULA HEIGHT = PAYS +eta EXPONENT RIGHT-EDGE ZERO PACKET = X^(-o(1)) FLOOR STANDARD L2/L4/HIGHER-MOMENT MARKOV = CANNOT CURE SHRINKING EDGE INTERIOR GUTH-MAYNARD DENSITY = NOT THE PRIMARY BARRIER NEXT CAMPAIGN = 38 ``` The decisive comparison is $$ \boxed{ \exp \left[ -c \frac{ (\log X)^{1/3} }{ (\log\log X)^{1/3} } \right] \gg X^{-\eta} } $$ for every fixed $\eta>0$. The present exceptional-set proof reaches the right edge at subpower precision. The breakthrough gate requires polynomial precision.