# CSM_RH Paper 88 ## Polynomial-Threshold Inertia, Extremal-Family Sieve-Level Conservation, and a Threshold-Type-II Gate for the Prime-Excess Tail **Project:** CSM_RH **Paper:** 88 **Version:** v0.1 **Date:** 2026-09-09 **Canonical root frontier:** F-RH-017-v3 **One-sided frontier:** F-RH-027+ **Entry state:** v1.78 / Paper 87 v0.1 **Status:** POLYNOMIAL-THRESHOLD INERTIA CERTIFIED / EXTREMAL-FAMILY LOCAL SIEVE LEVEL SHOWN NOT TO IMPROVE WITH FAMILY SIZE / INERTIA ENTROPY DOES NOT REMOVE X^ν RESOLUTION CLASS / THRESHOLD-SENSITIVE TYPE-II FRONTIER IDENTIFIED **RH_PROVED:** FALSE **RH_DISPROVED:** FALSE **GLOBAL_RH_CERTIFICATE:** FALSE --- # Abstract Paper 87 showed that controlling the prime-excess tail $$ \mathcal E_\nu^+ = \left\{ x: \Delta_H(x)>HX^{-\nu} \right\} $$ by global factorial moments requires polynomial order $$ X^\nu\log X. $$ It proposed a threshold-native alternative: > use inertia to compress the exceptional set into a maximal separated family and sieve only that family. The present paper audits this strategy. Let $$ H=X^{1-\tau} $$ and $$ \boxed{ T=HX^{-\nu}. } $$ The first result is a polynomial-threshold inertia theorem. For fixed $H$ and $x,y\asymp X$, $$ \Delta_H(x)=\psi(x+H)-\psi(x)-H. $$ Since $$ \Lambda(n)\le\log(4X) $$ on the relevant range, $$ \boxed{ |\Delta_H(y)-\Delta_H(x)| \le 2(|y-x|+1)\log(4X). } $$ Therefore, if $$ \Delta_H(x)>T, $$ then $$ \boxed{ \Delta_H(y)>\frac{T}{2} } $$ whenever $$ \boxed{ |y-x| \le L := \frac{T}{8\log(4X)} } $$ for sufficiently large $X$. Thus $$ \boxed{ L=X^{1-\tau-\nu+o(1)}. } $$ The same statement holds for the deficiency tail. This extends the power-scale content of the classical inertia property to the polynomially shrinking threshold relevant to F-RH-017-v3. Bazzanella and Perelli's published fixed-threshold inertia theorem requires $$ \delta-\delta' \ge \exp(-\sqrt{\log X}), $$ which does not include $$ \delta=X^{-\nu}. $$ The elementary bounded-jump argument above loses only logarithms and is sufficient for the exponent ledger. Now select a maximal $L$ -separated family of excess centers $$ \boxed{ \mathcal C=\{x_1,\ldots,x_J\}. } $$ A maximal-net covering gives $$ \boxed{ |\mathcal E_\nu^+| \ll JL } $$ at fixed-power resolution. Therefore failure of an exceptional bound $$ |\mathcal E_\nu^+| \ll X^{1-c} $$ forces $$ \boxed{ J\gg X^{\tau+\nu-c-o(1)}. } $$ A greedy extraction of an $H$ -separated subfamily gives at least $$ \boxed{ K\gg J\frac{L}{H} = JX^{-\nu-o(1)}. } $$ Hence $$ \boxed{ K\gg X^{\tau-c-o(1)}. } $$ This produces an important regime split. ### Strip-dominated exceptional mass If $$ d<\tau, $$ then F-RH-017-v3 only requires $$ c>d. $$ One may test a violation at some $$ d\tau. $$ A failure of the desired bound need not produce polynomially many $H$ -separated intervals. It may be supported in only one or a bounded number of $H$ -scale clusters. Thus an extremal-family method is structurally better suited to the regime $$ d<\tau. $$ Combining this with the Poisson-resolution window $$ d<\nu<\frac{1-\tau}{2} $$ gives: ## Early-bootstrap family window There exist parameters satisfying $$ \boxed{ d<\tau } $$ and $$ \boxed{ d<\nu<\frac{1-\tau}{2} } $$ if and only if $$ \boxed{ d<\frac13. } $$ Equivalently, $$ \boxed{ \kappa<\frac23. } $$ So a separated-family excess strategy can only live in the clean Poisson large-deviation window as an early-to-middle bootstrap mechanism. The second main result is the family-size cancellation law for local sieve information. Associate to the centers the multiset sequence $$ \boxed{ a_{\mathcal C}(n) = \sum_{j=1}^J \mathbf1_{(x_j,x_j+H]}(n). } $$ Its total mass is $$ \boxed{ A_{\mathcal C} = JH+O(J). } $$ For every modulus $q$, $$ \boxed{ A_{\mathcal C,q} := \sum_{q\mid n} a_{\mathcal C}(n) = \frac{JH}{q} + O(J). } $$ Therefore, for every fixed divisor weight exponent $B$, $$ \boxed{ \sum_{q\le D} \tau(q)^B \left| A_{\mathcal C,q} - \frac{A_{\mathcal C}}{q} \right| \ll_B JD(\log D)^{O_B(1)}. } $$ Relative to the total mass, $$ \boxed{ \frac{\text{sieve remainder to level }D}{A_{\mathcal C}} \ll \frac{D}{H}X^{o(1)}. } $$ The number of exceptional intervals $J$ cancels completely. Consequently, collecting a large family does not by itself increase the deterministic local-divisibility level beyond the length of one interval. At prime-detection scale, classical upper-sieve bounds retain a fixed multiplicative gap. For a single polynomial interval $$ H=X^\theta, \qquad \theta=1-\tau, $$ Brun–Titchmarsh gives $$ \boxed{ \pi(x+H)-\pi(x) \le \left( \frac{2}{\theta}+o(1) \right) \frac{H}{\log X}. } $$ Summing over the multiset family gives exactly the same constant: $$ \boxed{ \sum_j \left( \pi(x_j+H)-\pi(x_j) \right) \le \left( \frac{2}{1-\tau}+o(1) \right) \frac{JH}{\log X}. } $$ Since $$ X^{-\nu}\to0, $$ this cannot contradict the near-mean excess $$ \boxed{ \sum_j \left( \pi(x_j+H)-\pi(x_j) \right) \ge \left( 1+cX^{-\nu} \right) \frac{JH}{\log X}. } $$ The standard dimension-one linear sieve gives the same structural diagnosis. If one uses only the deterministic divisor remainder above, the available level is $$ D\lesssim H. $$ When the upper linear sieve is in the range $$ 1\le s\le3, $$ its function is $$ \boxed{ F(s)=\frac{2e^\gamma}{s}. } $$ At a prime-scale sifting point $$ z\asymp X^{1/2}, $$ the parameter is $$ s=\frac{\log D}{\log z} \lesssim 2(1-\tau). $$ Thus the classical sieve constant remains bounded away from $1$. The family size does not change $s$. The third result concerns entropy. The inertia scale compresses the continuum of possible starts to a grid of size $$ \boxed{ B_{\rm grid} \asymp \frac{X}{L} = X^{\tau+\nu+o(1)}. } $$ But the target exceptional estimate $$ |\mathcal E_\nu^+| \ll X^{1-c} $$ corresponds to an exceptional fraction $$ \boxed{ \frac{J}{B_{\rm grid}} \ll X^{-c+o(1)}. } $$ Thus inertia reduces the number of candidate positions, but not the power of $X$ required in the tail probability. A representative-wise Chernoff or factorial-moment argument still needs logarithmic cost $$ c\log X. $$ At relative deviation $$ \delta=X^{-\nu}, $$ the order remains $$ \boxed{ r \asymp \delta^{-1}\log B_{\rm grid} = X^\nu(\tau+\nu+o(1))\log X. } $$ Hence the polynomial complexity class $X^\nu$ survives inertia compression. The analysis identifies the only place where a separated family can still help: > Type-II or other genuinely prime-specific cancellation across the selected family. To make this precise, define the comparison density $$ \boxed{ \rho_{\mathcal C} = \frac{JH}{X}. } $$ On the dyadic ambient interval define $$ \boxed{ w_{\mathcal C}(n) = a_{\mathcal C}(n) - \rho_{\mathcal C} \mathbf1_{[X,2X]}(n). } $$ At threshold precision $$ \delta A_{\mathcal C} = JHX^{-\nu}, $$ the aggregate Type-I error condition $$ JD\ll\delta JH $$ only reaches $$ \boxed{ D\ll HX^{-\nu}. } $$ Thus the threshold-accurate Type-I exponent is $$ \boxed{ \gamma_{\rm th} = 1-\tau-\nu. } $$ A power-preserving Vaughan / prime-producing-sieve extraction would therefore require a complementary Type-II range strong enough to bridge the missing width $$ \boxed{ 1-\gamma_{\rm th} = \tau+\nu. } $$ Open: ```text F-RH-028+ EXCEPTIONAL_FAMILY_THRESHOLD_TYPE_II ``` Given a maximal separated excess family $\mathcal C$, prove a Type-II estimate for $w_{\mathcal C}$ at relative precision $X^{-\nu}$ over a complementary log-range sufficient to combine with $$ \gamma_{\rm th}=1-\tau-\nu $$ in a Vaughan / Heath–Brown / Ford–Maynard prime-producing identity. A schematic dyadic form is $$ \boxed{ \left| \sum_{\substack{m\sim M\\n\sim N\\mn\asymp X}} \alpha_m\beta_n w_{\mathcal C}(mn) \right| \ll JH X^{-\nu-\eta} } $$ for bounded coefficients in the required balanced ranges. If such a theorem held with the necessary range geometry, then one could obtain $$ \sum_n\Lambda(n)a_{\mathcal C}(n) = JH + O\left( JHX^{-\nu-\eta'} \right), $$ contradicting the defining prime excess of every interval in the family. No such theorem is currently certified. The key point is that it is now the only family-specific place where $J$ could provide new cancellation. Current extremal interval-sieve work provides useful calibration. Banks, Ford and Tao study extremal interval sieves and show that sieve survivor geometry can be highly nontrivial. Jha's 2026 Poisson-tail work successfully combines extremal interval sieve estimates with concentration inequalities, but under a strong Hardy–Littlewood hypothesis and in a regime where the interval mean grows slower than every fixed power of $\log X$. The F-RH regime has $$ \lambda=X^{1-\tau}/\log X, $$ and seeks unconditional polynomial relative precision. Thus those results validate the architecture but do not supply F-RH-028+. No RH theorem is claimed. --- # 1. Elementary polynomial-threshold inertia Let $$ T=HX^{-\nu}. $$ For integers $x,y\in[X,2X]$ and fixed $H$, $$ \begin{aligned} \Delta_H(y)-\Delta_H(x) &= \psi(y+H)-\psi(x+H) \\ &\quad -\left(\psi(y)-\psi(x)\right). \end{aligned} $$ Each $\psi$ increment over an interval containing at most $|y-x|+1$ integers is bounded by $$ (|y-x|+1)\log(4X). $$ Hence: ## Theorem 1.1 — Polynomial-threshold inertia $$ \boxed{ |\Delta_H(y)-\Delta_H(x)| \le 2(|y-x|+1)\log(4X). } $$ If $$ \Delta_H(x)>T $$ and $$ |y-x| \le \frac{T}{8\log(4X)}-1, $$ then $$ \boxed{ \Delta_H(y)>\frac{T}{2}. } $$ The same holds with both signs reversed. Create: ```text B-RH-169 POLYNOMIAL_THRESHOLD_PRIME_INTERVAL_EXCEPTIONS_HAVE_INERTIA_LENGTH_H_X_MINUS_NU_UP_TO_LOGARITHMS CERTIFIED ``` --- # 2. Relation to classical inertia Bazzanella and Perelli prove that for fixed relative thresholds $$ 0<\delta'<\delta $$ with $$ \delta-\delta' \ge e^{-\sqrt{\log X}}, $$ an exceptional point generates an interval of exceptions of length comparable to $$ (\delta-\delta')H. $$ For $$ \delta=X^{-\nu}, $$ the published hypothesis eventually fails. Theorem 1.1 replaces that input in the present polynomial regime, with only a logarithmic loss. --- # 3. Maximal separated family Let $$ L=HX^{-\nu}(\log X)^{-1}. $$ Choose a maximal $L$ -separated subset $$ \mathcal C = \{x_1,\ldots,x_J\} \subset \mathcal E_\nu^+. $$ Maximality implies a covering by $O(L)$ neighborhoods. Therefore: ## Theorem 3.1 — Exceptional mass to center count $$ \boxed{ |\mathcal E_\nu^+| \ll JL. } $$ If $$ |\mathcal E_\nu^+| \ge X^{1-c}, $$ then $$ \boxed{ J\gg X^{\tau+\nu-c-o(1)}. } $$ Create: ```text B-RH-170 A_FAILURE_OF_THE_ONE_SIDED_EXCEPTIONAL_EXPONENT_FORCES_X_TO_TAU_PLUS_NU_MINUS_C_SEPARATED_EXCESS_CENTERS CERTIFIED ``` --- # 4. Extraction of disjoint H-scale intervals Because the centers are $L$ -separated, any interval of length $O(H)$ contains at most $$ O(H/L) = X^{\nu+o(1)} $$ centers. A greedy algorithm therefore selects an $H$ -separated subfamily of size $$ \boxed{ K\gg J\frac{L}{H}. } $$ Hence if the exceptional measure is at least $X^{1-c}$, $$ \boxed{ K\gg X^{\tau-c-o(1)}. } $$ Create: ```text B-RH-171 EXCEPTIONAL_MASS_X_TO_ONE_MINUS_C_FORCES_X_TO_TAU_MINUS_C_DISJOINT_H_SCALE_EXCESS_INTERVALS_WHEN_C_IS_BELOW_TAU CERTIFIED ``` --- # 5. Strip-dominated versus lag-dominated family geometry Recall the root forcing exponent $$ \min(d,\tau). $$ If $d<\tau$, choose $d\tau$ and the disjoint-family lower bound becomes subconstant. Create: ```text O-RH-192 THE_EXTREMAL_DISJOINT_INTERVAL_FAMILY_MECHANISM_IS_INTRINSICALLY_A_STRIP_DOMINATED_D_LESS_THAN_TAU_TOOL CERTIFIED ``` --- # 6. Compatibility with the Poisson window The upper-tail concentration window is $$ d<\nu<\frac{1-\tau}{2}. $$ Together with $d<\tau$, such parameters exist if and only if $$ d<\frac13. $$ Equivalently, $$ \kappa<\frac23. $$ --- # 7. The interval-family multiset Define $$ a_{\mathcal C}(n) = \sum_{j=1}^J \mathbf1_{(x_j,x_j+H]}(n). $$ Then $$ A_{\mathcal C} = JH+O(J). $$ For every modulus $q$, $$ A_{\mathcal C,q} = \sum_j \left( \frac{H}{q}+O(1) \right). $$ Thus: ## Theorem 7.1 — Family divisor law $$ \boxed{ A_{\mathcal C,q} = \frac{JH}{q}+O(J). } $$ Create: ```text B-RH-172 A_MULTISET_OF_J_LENGTH_H_INTERVALS_HAS_DIVISOR_REMAINDER_O_J_PER_MODULUS_INDEPENDENT_OF_THE_FAMILY_SIZE CERTIFIED ``` --- # 8. Aggregate Type-I level For fixed $B$, $$ \sum_{q\le D} \tau(q)^B \left| A_{\mathcal C,q} - \frac{A_{\mathcal C}}q \right| \ll JD(\log D)^{O_B(1)}. $$ Since $$ A_{\mathcal C}\asymp JH, $$ the relative error is $$ \boxed{ \frac{D}{H}X^{o(1)}. } $$ Create: ```text B-RH-173 THE_DETERMINISTIC_TYPE_I_LEVEL_OF_AN_EXTREMAL_INTERVAL_MULTISET_IS_CONTROLLED_BY_H_NOT_BY_JH CERTIFIED ``` --- # 9. Brun–Titchmarsh family bound For $$ H=X^{1-\tau}, $$ Brun–Titchmarsh gives $$ \pi(x_j+H)-\pi(x_j) \le \frac{2H}{\log H}(1+o(1)). $$ Therefore $$ \boxed{ \sum_j \left( \pi(x_j+H)-\pi(x_j) \right) \le \left( \frac{2}{1-\tau}+o(1) \right) \frac{JH}{\log X}. } $$ The defining excess is only $1+X^{-\nu}$, so the classical family upper sieve leaves a constant gap much larger than the target deviation. Create: ```text O-RH-193 SUMMING_BRUN_TITCHMARSH_OVER_AN_EXTREMAL_EXCESS_FAMILY_DOES_NOT_APPROACH_THE_ONE_PLUS_X_MINUS_NU_RESOLUTION CERTIFIED ``` --- # 10. Linear-sieve interpretation The dimension-one upper linear sieve has $$ F(s)=\frac{2e^\gamma}{s} $$ in the initial range. With deterministic level $D\lesssim H$ and prime sifting scale $z\asymp X^{1/2}$, $$ s\lesssim2(1-\tau). $$ The family size $J$ does not appear. This is a method-scope result, not an impossibility theorem for added Type-II information. --- # 11. Inertia grid entropy The inertia scale is $$ L=X^{1-\tau-\nu+o(1)}. $$ Hence $$ \boxed{ B_{\rm grid} = X/L = X^{\tau+\nu+o(1)}. } $$ The target center count is $$ J\ll X^{\tau+\nu-c+o(1)}. $$ Thus the desired exceptional fraction remains $$ X^{-c+o(1)}. $$ --- # 12. Resolution order after inertia compression A near-Poisson representative event with relative deviation $\delta=X^{-\nu}$ still requires order $$ \delta^{-1} \log(B_{\rm grid}/J). $$ At the target fraction, $$ \log(B_{\rm grid}/J)=c\log X+o(\log X). $$ Therefore: ## Theorem 12.1 — Inertia does not change the resolution complexity class $$ \boxed{ r_{\rm grid}\asymp X^\nu\log X. } $$ Create: ```text O-RH-194 INERTIA_COMPRESSES_THE_NUMBER_OF_CANDIDATE_INTERVALS_BUT_DOES_NOT_REMOVE_THE_POLYNOMIAL_X_TO_NU_CONCENTRATION_COMPLEXITY CERTIFIED ``` --- # 13. Threshold-accurate Type-I level Prime detection must be accurate to $$ \delta A_{\mathcal C} = JHX^{-\nu}. $$ The aggregate Type-I error to level $D$ is $$ JDX^{o(1)}. $$ Therefore threshold-small Type I only reaches $$ \boxed{ D\ll HX^{-\nu-o(1)}. } $$ Thus $$ \boxed{ \gamma_{\rm th}=1-\tau-\nu. } $$ Create: ```text B-RH-174 AT_RELATIVE_PRECISION_X_MINUS_NU_THE_EXTREMAL_INTERVAL_SEQUENCE_HAS_DETERMINISTIC_TYPE_I_EXPONENT_ONE_MINUS_TAU_MINUS_NU CERTIFIED ``` --- # 14. Why a Type-II theorem is the only surviving family-specific leverage A Vaughan / Heath–Brown identity can preserve fixed-power Type-I / Type-II errors. But threshold-accurate Type I now reaches only $$ X^{1-\tau-\nu}. $$ The complementary logarithmic gap is $$ \boxed{ \tau+\nu. } $$ Thus any threshold-native prime asymptotic for the selected family must obtain genuinely prime-specific balanced information across a substantial complementary product range. --- # 15. F-RH-028+ — exceptional-family threshold Type II Let $$ \rho_{\mathcal C} = \frac{JH}{X} $$ and $$ w_{\mathcal C}(n) = a_{\mathcal C}(n) - \rho_{\mathcal C}\mathbf1_{[X,2X]}(n). $$ Open: ```text F-RH-028+ EXCEPTIONAL_FAMILY_THRESHOLD_TYPE_II ``` A schematic target is: for a sufficiently broad balanced range with $MN\asymp X$, $$ \boxed{ \left| \sum_{m\sim M} \sum_{n\sim N} \alpha_m\beta_n w_{\mathcal C}(mn) \right| \ll JHX^{-\nu-\eta} } $$ for bounded coefficients, uniformly over every maximal separated excess family $\mathcal C$. If this can be combined with the Type-I exponent $1-\tau-\nu$ in a quantitative prime-producing identity, then $$ \sum_n\Lambda(n)a_{\mathcal C}(n) = JH+O(JHX^{-\nu-\eta'}), $$ contradicting the defining excess. No proof is given. --- # 16. Current extremal-interval literature calibration Bazzanella and Perelli introduced the inertia and decrease properties of the short-interval PNT exceptional set. Banks, Ford and Tao show that extremal interval-sieve survivor geometry is nontrivial and cannot be replaced by naive independent Bernoulli heuristics. Jha's 2026 Poisson-tail work combines extremal interval sieve estimates with concentration inequalities under a strong Hardy–Littlewood hypothesis; its interval mean grows subpolynomially in $\log X$, far below the polynomial mean here. These works validate the architecture but do not supply F-RH-028+. --- # 17. Strategic verdict for F-RH-027+ The inertia / maximal-family attack does achieve two things: 1. it turns exceptional measure into the size of a separated structured family; 2. in the strip-dominated regime it produces polynomially many disjoint prime-rich intervals. But by itself it does not solve the resolution problem. The two naive routes retain the old complexity: ```text LOCAL SIEVE: family size J cancels from the divisor level. REPRESENTATIVE-WISE CONCENTRATION: inertia grid still needs X^nu-order resolution. ``` Thus F-RH-027+ survives only through a genuinely new balanced / Type-II theorem for the exceptional family. --- # 18. State transition Advance candidate state $$ v1.78\to v1.79. $$ Add: ```text B-RH-169 POLYNOMIAL_THRESHOLD_PRIME_INTERVAL_EXCEPTIONS_HAVE_INERTIA_LENGTH_H_X_MINUS_NU_UP_TO_LOGARITHMS B-RH-170 A_FAILURE_OF_THE_ONE_SIDED_EXCEPTIONAL_EXPONENT_FORCES_X_TO_TAU_PLUS_NU_MINUS_C_SEPARATED_EXCESS_CENTERS B-RH-171 EXCEPTIONAL_MASS_X_TO_ONE_MINUS_C_FORCES_X_TO_TAU_MINUS_C_DISJOINT_H_SCALE_EXCESS_INTERVALS_WHEN_C_IS_BELOW_TAU B-RH-172 A_MULTISET_OF_J_LENGTH_H_INTERVALS_HAS_DIVISOR_REMAINDER_O_J_PER_MODULUS_INDEPENDENT_OF_THE_FAMILY_SIZE B-RH-173 THE_DETERMINISTIC_TYPE_I_LEVEL_OF_AN_EXTREMAL_INTERVAL_MULTISET_IS_CONTROLLED_BY_H_NOT_BY_JH B-RH-174 AT_RELATIVE_PRECISION_X_MINUS_NU_THE_EXTREMAL_INTERVAL_SEQUENCE_HAS_DETERMINISTIC_TYPE_I_EXPONENT_ONE_MINUS_TAU_MINUS_NU O-RH-192 THE_EXTREMAL_DISJOINT_INTERVAL_FAMILY_MECHANISM_IS_INTRINSICALLY_A_STRIP_DOMINATED_D_LESS_THAN_TAU_TOOL O-RH-193 SUMMING_BRUN_TITCHMARSH_OVER_AN_EXTREMAL_EXCESS_FAMILY_DOES_NOT_APPROACH_THE_ONE_PLUS_X_MINUS_NU_RESOLUTION O-RH-194 INERTIA_COMPRESSES_THE_NUMBER_OF_CANDIDATE_INTERVALS_BUT_DOES_NOT_REMOVE_THE_POLYNOMIAL_X_TO_NU_CONCENTRATION_COMPLEXITY ``` Open: ```text F-RH-028+ EXCEPTIONAL_FAMILY_THRESHOLD_TYPE_II OPEN ``` F-RH-027+ remains open. No RH certificate is created. --- # 19. Recommended next action Attack F-RH-028+. The next paper should derive an unconditional baseline Type-II estimate for arbitrary separated interval families: $$ a_{\mathcal C}(n) = \sum_j \mathbf1_{x_j