Riemann Hypothesis

Source-reported subject: Current arithmetic route. An exact endpoint or theta subproblem remains open; retain the stated sparse/all-scale, same-function and same-cutoff quantifiers.

Source-reported subject: Current energy route. An exact endpoint or theta subproblem remains open; retain the stated sparse/all-scale, same-function and same-cutoff quantifiers.

Source-reported subject: Missing-reference and phase-gate status. The current source states that reference existence and upper bounds remain missing. T32-01 restricts PFX, while NC and CR remain separate. The seven topic files are cited for full proofs; those attachments have not been inspected here. This is a current route-status record, not a newly proposed existence theorem or attachment verification.

Source-reported subject: Current reference route. An exact endpoint or theta subproblem remains open; retain the stated sparse/all-scale, same-function and same-cutoff quantifiers.

Source-reported subject: Current theta route. An exact endpoint or theta subproblem remains open; retain the stated sparse/all-scale, same-function and same-cutoff quantifiers.

Source-reported subject: Directed nodal graph and endpoint pairing. Leading route: use the source-reported proper analytic nodal graph, strict phase flow, singular normal forms, and classified ends to determine global incidence and prove component-wise phase avoidance.

Source-reported subject: Directed nodal graph and endpoint pairing. Retained Revision 6 modulus-one U=0 boundary-graph route with increasing phase V. Its source-scoped local results and open pairing obligations remain historical mathematics. The current packet instead uses phase-zero Theta=0 fixed witnesses with increasing U; neither chart is silently substituted for the other.

Source-reported subject: Fourier-decay and scalar-growth closures. Current actual Liouville sine, exponential and triangular-source lower-bound routes remain open. The sine condition is required for every sufficiently large k; positivity of selected characters or comparison models does not supply actual Liouville signs.

Source-reported subject: Current fixed-function route. An exact endpoint or theta subproblem remains open; retain the stated sparse/all-scale, same-function and same-cutoff quantifiers.

Source-reported subject: Endpoint phases and rectangle flow balance. Active companion route: determine portal and right-edge phase signs and combine them with multiplicity-safe rectangle accounting to forbid opposite-sign pairing.

Source-reported subject: Endpoint phases and rectangle flow balance. Retained Revision 6 modulus-one U=0 boundary-graph route with increasing phase V. Its source-scoped local results and open pairing obligations remain historical mathematics. The current packet instead uses phase-zero Theta=0 fixed witnesses with increasing U; neither chart is silently substituted for the other.

Source-reported subject: Current signed-kernels route. An exact endpoint or theta subproblem remains open; retain the stated sparse/all-scale, same-function and same-cutoff quantifiers.

Source-reported subject: Uniform compact-profile rectangle. ## 15. Compact profile gate: exact target and stopping rule

The missing finite-domain inequality is

with removable values at t=0. The front expansion is

A uniform finite-width remainder has not been certified. A pointwise Taylor expansion does not fill a rectangle.

The retained quadrature contract allows two overlapping boxes [0,1/2] and [9/20,8/5] in t, with the full a strip. It requires outward arithmetic, positive denominator enclosures, all theta-atom tails and derivative tails, removable singularities, and an explicit seam. The production/covariance formula is preferable to finite-differencing kappa. The analytic large-t theorem already handles t>=8/5; do not recertify the infinite tail merely to postpone the compact interior.

The observed minimum D≈4.78752 near (a,t)≈(1/2,.9059) and covariance diagnostic≈.07815 are not interval certificates. They motivate a finite target but do not satisfy it. data/V25_QUADRATURE_CONTRACT.json, the existing diagnostic scripts, and the complete topic proof specify the integrands and regularizations. A legitimate completion needs a new certificate plus a checker that fails when any domain, tail, seam or denominator bound is removed. No such certificate is present.

Completing this gate plus the inherited tail gives full-profile varrho'<0. The dependency is then full-profile + PFX -> NC, followed by NC + CR -> exclusion of the hypothetical witness. Compact profile alone is not a full RH route, and the dependency checker rejects that promotion.

T32-01 is an additional stopping rule: finishing this physical compact certificate does not cure the exponentially large prefix-to-terminal ratio at hypothetical high stationary contacts. An independent high-contact exclusion or signed transform argument is needed before PFX can be used there.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Test an exponentially resolving or theta-density closure. Either quantify a height-growing Pick/Padé hierarchy below the completed-xi scale or prove a theta-density inequality excluding the exact forced moments.

Source-reported subject: Test an exponentially resolving or theta-density closure. Either quantify a height-growing Pick/Padé hierarchy below the completed-xi scale or prove a theta-density inequality excluding the exact forced moments.

Source-reported subject: Open fixed-function work with exact source conditions. B32-TARGET fixed-function construction; B32 decisive fixed-function work order

Source-reported subject: Open energy work with exact source conditions. Q30-TARGET; Q30-06 same-cutoff reference target; Q30/Q32 decisive energy work order

Source-reported subject: Open overview work with exact source conditions. opening missing-reference and theta-gate status; alternative endpoint availability

Source-reported subject: Critical stationary-contact exclusion. ### 10.4 Critical contacts and the remaining assembly distinction

At a critical contact, $\mathcal Q(w)=r>0$, $\mathcal H'(w)=0$, and $L(-z)+rL(z)$ has a zero of multiplicity at least two at $w$. Its associated normalized positive two-sided density has characteristic nulls

The exact scalar inequality $y^2\ge(\pi^2/2)(1-\cos y)-2y\sin y$ gives $\operatorname{Var}X\ge\pi^2/(2\gamma^2)$, strictly for the continuous theta law. This bound decays with height and does not close CR. A separate positive-convolution moment obstruction is $\gamma^2\mathbb E X^2>12$; its exact three-point ceiling is below 12.04, so optimizing only that constant cannot solve the high-height problem. These are different probability laws and different moments.

The order-$d$ local normal form $\mathcal H-\mathcal H(w_0)=\zeta^d$ supplies $2d$ rays, including a smaller-$a$ ray. That does not force a selected source-to-sink path to use a compatible pair of rays. R21-11.2 is a finite resolved-graph dichotomy with explicit path-class and leaf hypotheses. The unresolved global replacement must prove compatibility under splicing, attainment or stable limiting selection, and every boundary/infinite-height continuation. The fixed-witness theorem above does not supply family-wise compactness. A direct proof of CR would bypass those additional obligations.

Source-reported subject: Noncritical stationary-fold exclusion. ## 13. Positive curvature pencil and the bad-fold interface

Detailed proofs: active_proof_curvature_profile.txt (R21-7 and inherited pencil IDs). For x+y=t, d=x-y, z=ad, define $\eta_c(u)=\mathfrak q(u)-2cu^2\ge0$ for 0<=c<=C_th, and

It is strictly positive for t>0. Put $\mathcal G_a=t^2k_a+a^{-1}\partial_a k_a$, with continuous a=0 extension. Exact integration by parts gives

At a stationary negative-axis contact K=K_gamma=0, $\widehat{\mathcal F}_{0,a}=aK_a$, $\widehat{\mathcal F}_{a^2,a}=a^2K_{\gamma\gamma}$. For noncritical K_a!=0, favorable NC orientation is equivalent to nonvanishing of the affine transform for every c in the closed interval [0,a²]. A flat K_gammagamma=0 is a bad endpoint, not excluded by strict-minimum language. Critical K_a=0 is the separate CR gate.

Source-reported subject: Open reference work with exact source conditions. Q32-07 residual-energy warning; R32 centered-energy missing estimate; R32 decisive centered-reference work order

Source-reported subject: Open arithmetic work with exact source conditions. A28-TARGET all-scale exceptional-count endpoint; A28 sparse count and good-window endpoint; A28 decisive arithmetic work order

Source-reported subject: S30 actual exponential lower bound. Prove, for the actual Liouville coefficients, the source-defined B_exp(x)=Σ λ(n)n^−2(1−e^(−nx)) lower bound B_exp(x)≥−C_η x^(3/2−η) eventually for every η>0. The source reports this family as equivalent to RH and does not obtain the signed-coefficient cancellation merely from a nonoscillatory kernel.

Source-reported subject: Open signed-kernels work with exact source conditions. O28-10 triangular-source endpoint; L30 logarithmic and subexponential all-k endpoints; S30 actual exponential endpoint; L28 selected-character matching and sign bridge; L30/S30/O28 decisive signed-transform work order

Source-reported subject: A28 all-scale exceptional-count bound. Establish the all-scale bound E_h(X)=O_ε(X^(1/2+ε)) for every ε>0 in the source-defined actual exceptional-count problem. The source gives this as a sufficient RH interface and does not establish it. The stronger polylogarithmic count is a separate, unproved target.

Source-reported subject: Open theta work with exact source conditions. flat stationary-fold and critical alternatives; compatible higher-order local-ray selection gap; support-disc noniteration boundary; NC and CR closed-pencil targets; PFX and T32-01 high-contact stopping rule; COMPACT-PROFILE finite rectangle and full-theta closure; Theta decisive NC/CR work order

Source-reported subject: Selected-character matching and sign together. For every sufficiently large k, produce the selected prime character required by L28-05 with both its Liouville-prefix matching and its signed finite-sum lower bound at that same scale. The characters may change with k. The source does not provide one fixed character with arbitrarily long Liouville prefixes, and it excludes universal positivity.

Source-reported subject: Separate arithmetic, energy and theta endpoints remain open. The source adds sparse energy, same-function reference, arithmetic and signed-transform endpoints. The newer phase-zero fixed-witness route is distinct from the retained modulus-one boundary graph.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Boundary geometry and portal law. The packet reports the boundary circle, canonical phase, Wronskian formulas, boundary jets, portal law, regular-edge phase monotonicity, and multiplicity-safe rectangle accounting.

Source-reported subject: Boundary geometry and portal law. The packet reports the boundary circle, canonical phase, Wronskian formulas, boundary jets, portal law, regular-edge phase monotonicity, and multiplicity-safe rectangle accounting.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Directed nodal graph frontier. Revision 6 adds singular geometry, endpoint exclusions, properness and end classification, compact-component exclusion, infinity/right-edge safety, and recasts the root gate as global directed endpoint pairing.

Source-reported subject: Directed nodal graph frontier. Revision 6 adds singular geometry, endpoint exclusions, properness and end classification, compact-component exclusion, infinity/right-edge safety, and recasts the root gate as global directed endpoint pairing.

Source-reported subject: NR-16 — Finite scopes. C103, C163, D26-BAND and profile diagnostics concern different domains; their finite coverage cannot be merged.. NR-16 — Finite scopes. C103, C163, D26-BAND and profile diagnostics concern different domains; their finite coverage cannot be merged.

Source-reported subject: NR-14 — Local rays. An available left ray need not splice to the fixed witness. Admissibility, endpoints and attained extrema remain separate.. NR-14 — Local rays. An available left ray need not splice to the fixed witness. Admissibility, endpoints and attained extrema remain separate.

Source-reported subject: K26 zero-replacement and prescribed-fold constructions have different retained hypotheses; E26 loses exact arithmetic/functional equation; no counterexample to actual zeta.. ## 17. Same-kernel and Euler-data stress tests

Full constructions are in active_proof_contact_geometry.txt, with K26/E26 stable IDs. These are precise scope tests, not counterexamples to the actual zeta function.

K26-01–03: common-kernel zero replacement. Choose simple critical-line pairs at T±delta and write

Then Q-P=4delta²(T²-w²) and $\widetilde X=XQ/P$ replaces those pairs with an off-axis quartet while preserving the full prescribed zero strip. Resolvents realize this through a single common positive even analytic superexponentially decreasing kernel, preserving the listed finite open curvature/shape bounds for sufficiently late choices. The supply of suitable simple pairs uses the cited positive-proportion theorem; that external analytic premise was not independently reproved in the first audit.

K26-04–05: prescribed regular wrong fold. A separate six-jet common-kernel perturbation fixes a0=1/4, high w_tau and epsilon=e^-tau, prescribing the quotient perturbation T(w)=epsilon,T'=i*epsilon,T''=i*epsilon. The resulting actual contact for that perturbed kernel has K_a<0, K_gammagamma>0, radial second derivative 1, and pencil root c=1/20. This construction does not assert that every zero stays in the original strip. Never combine its fold conclusion with the strip-preservation claim from the other family without a new simultaneous construction.

K26-06: lost exact arithmetic. The first deformation gives $\widetilde\zeta(\sigma)=1-4\delta^2/\sigma^2+O(\sigma^{-3})$, inconsistent with the exponentially small right-half-plane tail of the exact ordinary Dirichlet series. Generic positive-kernel and finite-jet conditions thus do not replace exact zeta arithmetic.

E26-01: positive Euler data are also insufficient. The model $Z_{\rm mod}(s)=\zeta(s)/[\zeta(s+1/4+i/4)\zeta(s+1/4-i/4)]$ with the specified small-prime corrections through 13 has positive ordinary and logarithmic-derivative coefficients, the leading PNT scale, and zeros at 3/4±i/4. It loses the exact coefficients and completed functional equation. The O28 inverse perturbations in §5 preserve a different list (integer forcing, jumps, moments) but lose the exact between-jump ODE. These lists cannot be merged into a stronger countermodel than was actually built.

A reopened generic route must name an exact additional hypothesis excluding the relevant retained model. Replacing 'positive kernel' by 'very regular positive kernel' or positive coefficients by 'Euler-positive' does not do so.

Source-reported subject: S30-01–05 actual exponential endpoint remains open; positive squarefree average, squared inverse and smoothing obstructions do not refute the actual Liouville lower bound.. ### 6.2 Exact real-kernel identities and sign obstructions — S30-01–05

Let $B_{\exp}(x)=\sum\lambda(n)n^{-2}(1-e^{-nx})$. Its Mellin factor is $\Gamma(s)/(1-s)$ times the same zeta quotient. The family $B_{\exp}(x)\ge-C_\eta x^{3/2-\eta}$ eventually for every eta>0 is equivalent to RH. A nonoscillatory kernel still requires signed-coefficient cancellation.

For bounded continuous phi and $H_\phi=\sum\lambda(n)n^{-2}\phi(nx)$, define $D_wf=\sum w(d)d^{-2}f(dx)$. Then

If w(1)=1 and w*lambda is coefficientwise nonnegative, then $w=\mu^2*(w*\lambda)\ge\mu^2$, also on a finite positive prefix. The critical absolute norm is therefore at least $12\sqrt M/\pi^2-O(\log M)$. The positive average cancels the detecting poles, not the inverse sign problem.

For the actual coefficients,

A positive increasing source $(1-e^{-x})+5(1-e^{-x})^2$ also has a negative inverse below -1/100 there; the same certificate has $B_{\exp}(\log(4/3))<9/100$. CK056 stores the exact rational finite sums and tails once. These examples do not show that B_exp or the actual sine series is negative.

For bounded continuous nonzero phi>=0 satisfying $\phi(x)=O(x^q)$, q>3/2,

The negative real residue and Landau positivity prove the statement. More positive smoothing can create unavoidable negative values. Subtracting a zero linear term is not the same operation as replacing the original kernel by its square.

For a finite-support normalized filter, $(w*\lambda)(p)=-1$ beyond support. At x=1/X the absolute prime contribution in (X,2X] is of order $1/(X\log X)$, larger than $X^{-3/2+o(1)}$. An absolute far-tail estimate needs support at least $X^{3/2-o(1)}$, hence positive-prefix critical norm at least $X^{3/4-o(1)}$. This limits absolute estimation; it does not determine the complete signed sum.

Source-reported subject: NR-11 — Endpoint jets. All finite algebraic odd-sine endpoint jets vanish. More fixed-order integration by parts does not recover the sign.. NR-11 — Endpoint jets. All finite algebraic odd-sine endpoint jets vanish. More fixed-order integration by parts does not recover the sign.

Source-reported subject: NR-20 — Positive Euler data. E26 has positive ordinary/prime-power data and PNT leading behavior but off-line zeros; exact arithmetic is missing.. NR-20 — Positive Euler data. E26 has positive ordinary/prime-power data and PNT leading behavior but off-line zeros; exact arithmetic is missing.

Source-reported subject: NR-36 — Two different costs. Critical finite W has coefficient Z; the coarse prime-local product has e^gamma_E Z. Keep finite cutoffs in asymptotics.. NR-36 — Two different costs. Critical finite W has coefficient Z; the coarse prime-local product has e^gamma_E Z. Keep finite cutoffs in asymptotics.

Source-reported subject: NR-42 — Threshold example. The ternary reference has nonnegative sums but cost~(2sqrt3/pi)sqrt X. Big-O cost does not imply the little-o terminal condition.. NR-42 — Threshold example. The ternary reference has nonnegative sums but cost~(2sqrt3/pi)sqrt X. Big-O cost does not imply the little-o terminal condition.

Source-reported subject: B32-05 finite seven-frequency constant obstruction >1.0057; CK062/067 are source-reported and share interval backend.. ### 2.8 A finite spectral obstruction and the exact threshold example

Let $\rho_0=1/2+i\gamma_0$ be the simple zero certified near $\gamma_0=14.13472514173469379046$, and put

CK062 proves nonvanishing of $\zeta(1/2+ij\gamma_0)$ for $j=2,\ldots,6$, and $c_0>d_0$. Under an eventual lower barrier $u_g\ge-C$ and weak cost, a seven-frequency positivity matrix gives

Thus every such weak-cost reference has $\liminf S_g(x)/\sqrt x\le-(c_0+d_0)$. The matrix is tridiagonal because the five harmonics are nonzeros. This does not assume rational independence of ordinates or an infinite residue expansion.

Independent finite recheck: CK067 uses a separate explicit-derivative Euler–Maclaurin implementation (192 terms, 18 Bernoulli corrections and a radius-$10^{-12}$ root disk). It certifies $1.0057490559<c_0+d_0<1.0057490569$, inside the repaired CK062 enclosure. Both use the same directed-interval library, so this is implementation independence, not backend independence. Exact rational endpoints govern, and the stable conclusion remains >1.0057. Neither program is an all-height zero check.

Source-reported subject: NR-25 — Universal characters. Genuine characters violate universal positivity even with any fixed matching prefix. No negative actual Liouville value is inferred.. NR-25 — Universal characters. Genuine characters violate universal positivity even with any fixed matching prefix. No negative actual Liouville value is inferred.

Source-reported subject: NR-37 — Moving alpha. Use the finite uniform energy sandwich, not fixed-alpha sample asymptotics with alpha_X substituted.. NR-37 — Moving alpha. Use the finite uniform energy sandwich, not fixed-alpha sample asymptotics with alpha_X substituted.

Source-reported subject: NR-35 — Sparse independence. Logarithmic cost does not bound the hard deterministic mean. Expected energy is comparable to it; exceptional correlations are not excluded.. NR-35 — Sparse independence. Logarithmic cost does not bound the hard deterministic mean. Expected energy is comparable to it; exceptional correlations are not excluded.

Source-reported subject: NR-33 — Fair/block-first references. Low fair energy comes with high cost; fixing a late block forces many positive signs under the stated conditioning.. NR-33 — Fair/block-first references. Low fair energy comes with high cost; fixing a late block forces many positive signs under the stated conditioning.

Source-reported subject: NR-26 — Finite square samples. T_100>0 does not prove eventual control. Matching characters supplies approximation, not a nonnegative sign.. NR-26 — Finite square samples. T_100>0 does not prove eventual control. Matching characters supplies approximation, not a nonnegative sign.

Source-reported subject: NR-18 — Common-kernel stress. K26 full-strip replacement and prescribed wrong-fold constructions are different families, not one combined countermodel.. NR-18 — Common-kernel stress. K26 full-strip replacement and prescribed wrong-fold constructions are different families, not one combined countermodel.

Source-reported subject: NR-02 — Stieltjes order. A5–A6 retains positive measures, decreasing tail ratios, both nulls and negative contact, but the wrong J3 sign.. NR-02 — Stieltjes order. A5–A6 retains positive measures, decreasing tail ratios, both nulls and negative contact, but the wrong J3 sign.

Source-reported subject: NR-21 — Finite sampling. No exception through two million is not tail control. The summable half N^-3/2 sampling loss must be retained.. NR-21 — Finite sampling. No exception through two million is not tail control. The summable half N^-3/2 sampling loss must be retained.

Source-reported subject: NR-28 — Fourier overreach. B=0 fixed-log Fourier bounds force simplicity and derivative-residue bounds. Use L30-03 for the RH-equivalent target.. NR-28 — Fourier overreach. B=0 fixed-log Fourier bounds force simplicity and derivative-residue bounds. Use L30-03 for the RH-equivalent target.

Source-reported subject: B32-06 ternary reference has bounded energy and big-O sqrt-cost; little-o cannot be replaced by big-O. Composite feedback and finite tests are limited.. For the threshold example, define $g_3(3^a m)=\chi_3(m)$ for $3\nmid m$, where $\chi_3$ is the nonprincipal real character modulo 3. Then

This disproves replacing little-o by big-O in the cost-improvement conclusion, not RH itself. A bounded-energy, one-sided reference is not enough without the specified cost.

CK064 independently replays exact energies at $10^6$, the feedback reference and its cost, the conditional mean at $10^5$, and the digit identity through $10^5$. The feedback rule changes a prime only if the previous sum is below $-\sqrt p$; it already fails the literal barrier at the composite integer 32. Its energy is about 5.85805 at $10^6$, but its cost is about 17.1570, giving a transferred upper bound over 1700 rather than the actual energy about 6.20034. These finite facts neither prove nor disprove all possible adaptive constructions.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Positive Stieltjes shifts growing proportionally with height. The source reports oscillation along every positive-slope ray.

Source-reported subject: Positive Stieltjes shifts growing proportionally with height. The source reports oscillation along every positive-slope ray.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Abstract Stieltjes positivity without theta-density structure. The source reports a sharp positive two-atomic realization of the generic moment barrier.

Source-reported subject: Abstract Stieltjes positivity without theta-density structure. The source reports a sharp positive two-atomic realization of the generic moment barrier.

Source-reported subject: NR-43 — Prefix overshoot. T32-01 forces high-contact overshoot regardless of curvature orientation. PFX would already exclude those contacts; existence is not asserted.. NR-43 — Prefix overshoot. T32-01 forces high-contact overshoot regardless of curvature orientation. PFX would already exclude those contacts; existence is not asserted.

Source-reported subject: NR-41 — Fixed barrier. The eventual -sqrt x barrier is impossible in B32 weak-cost references. Prime feedback alone does not control composite descendants.. NR-41 — Fixed barrier. The eventual -sqrt x barrier is impossible in B32 weak-cost references. Prime feedback alone does not control composite descendants.

Source-reported subject: NR-34 — Fixed bias limit. Fixed bias has a polynomial mean-energy term. Its nonuniform near-Liouville remainder cannot be discarded at the endpoint.. NR-34 — Fixed bias limit. Fixed bias has a polynomial mean-energy term. Its nonuniform near-Liouville remainder cannot be discarded at the endpoint.

Source-reported subject: O28-04–08 smooth/diagonal countermodels retain enumerated moments, jumps and decay but lose exact between-jump ODE; hypothetical off-line growth stays conditional.. O28-04–05, resonance limitation. For a smooth cutoff chi=0 below1 and chi=1 above2,

locally uniformly with parameter derivatives. A hypothetical off-line zero allows a power-growing oscillation whose image is smaller than every power; this is not an exhibited off-line zero or an exactly homogeneous solution. An actual critical-line zero also gives an unconditional loglog-growing oscillation with image tending to zero. A compact moment correction and the stated baseline retain negative forcing and its limit, defeating a bounded inverse rule. This alone does not defeat every subpower rule.

O28-06–08, arbitrary small negative sources. The exact inverse for h zero below1 is

For smooth h supported in (Y,2Y), the unconditional Mertens estimate gives $\mathcal Rh(x)\ll_{A,Y}\sqrt x(\log x)^{-A}\mathcal V_Y(h)$, where $\mathcal V_Y(h)=\frac12\int|h|y^{-3/2}+\int|h'|y^{-1/2}$. The moment is absolutely convergent and zero, so $\mathcal D_{\rm ar}\mathcal Rh=h$.

Negative narrow bumps at P/n for $\mu(n)=1$, avoiding integers and all other sample points, produce

A locally finite diagonal choice makes the source h nonpositive, arbitrarily small in sup norm, zero near every integer, and $O_K(x^{-K})$ for every fixed K. The perturbed solution g=F_0+u agrees initially, has the same derivative jumps and moment, and satisfies

Thus exact forcing near integers, pointwise asymptotics to every order and a strict negative margin still do not give a generic subpower inverse theorem. Width choices are made after a variation-controlled tail cutoff; pointwise source decay is not derivative decay.

Source-reported subject: NR-01 — Generic positivity. Positive analytic/log-concave kernels admit off-axis zeros and wrong folds; an additional exact assumption is necessary.. NR-01 — Generic positivity. Positive analytic/log-concave kernels admit off-axis zeros and wrong folds; an additional exact assumption is necessary.

Source-reported subject: NR-07 — Canceled channel. J4 already cancels the full sin(gamma r) channel. Bounding pieces first loses the relevant cancellation.. NR-07 — Canceled channel. J4 already cancels the full sin(gamma r) channel. Bounding pieces first loses the relevant cancellation.

Source-reported subject: NR-30 — Extra smoothing. A nonnegative kernel vanishing faster than x^1.5 has negative real residue. Subtracting a zero linear term is not squaring the kernel.. NR-30 — Extra smoothing. A nonnegative kernel vanishing faster than x^1.5 has negative real residue. Subtracting a zero linear term is not squaring the kernel.

Source-reported subject: NR-15 — Moment ceiling. The stationary-moment method has an exact constant ceiling near12; improving it cannot cover unbounded height.. NR-15 — Moment ceiling. The stationary-moment method has an exact constant ceiling near12; improving it cannot cover unbounded height.

Source-reported subject: R30-01/02/03 independent prime signs; exact squarefree moment and fixed-bias/coherence failures only for the stated classes.. For independent prime signs, write $m_p=\mathbb E g(p)$, $v_p=1-m_p^2$, and $a(n)=\mathbb E g(n)$. The mean is multiplicative but generally not completely multiplicative: its even prime-power values are 1, its odd values are $m_p$. If $a_d$ sets odd means to zero at primes dividing squarefree $d$, then

It follows that $\mathcal E_a\le\mathbb E\mathcal E_g\le R_m(X)\mathcal E_a$, where

The local inequality is the square $(\sqrt p\,|m_p|-1)^2\ge0$. Fair signs have square-indicator mean, $\mathbb E\mathcal E_g(X)=3(\log X)^2/\pi^2+O(\log X)$, and almost surely a polylogarithmic prefix-maximum $Q_g^*$ bound; their comparison cost is not subpower. The order-log mean-energy factor is genuine (R30-03).

Q30-07–09 record precise failed shortcuts: coherent high-prime translations have norm of order $\sqrt X/\log X$; a prime-13 change lowers $Q(13)$ by exactly $8/13$; block-first fair conditioning forces about half the block positive to preserve low mean energy; and fixed negative bias has a polynomial-size Selberg–Delange mean. None proves a universal impossibility for carefully coordinated references. Each counterexample's retained and lost hypotheses are recorded in §18 and the proof topic.

Source-reported subject: NR-03 — Fixed ballot sign. Actual theta germs near heights 82.9 and 84.7 force both proposed J signs; the exact live-ballot check excludes the gate.. NR-03 — Fixed ballot sign. Actual theta germs near heights 82.9 and 84.7 force both proposed J signs; the exact live-ballot check excludes the gate.

Source-reported subject: NR-39 — Auxiliary cancellation. Quarter-power control is for c, not lambda or g. Affine subtraction and decaying source retain hypothetical off-line poles.. NR-39 — Auxiliary cancellation. Quarter-power control is for c, not lambda or g. Affine subtraction and decaying source retain hypothetical off-line poles.

Source-reported subject: L28-03–06 finite altered g and CRT-selected characters only; reported external Conrey v1 counterexample needs separate review, no fixed-character or Liouville-sign inference.. ### 6.3 Genuine-character obstruction and its precise scope — L28-03–06

Take g completely multiplicative, +1 on primes 5000<p<10000 and -1 otherwise. Independent fixed-point and Fraction engines certify at N=10^6, x=10^-4:

The absolute 1/N tails imply a negative infinite g series and a positive actual Liouville series. For each odd p<=N, prescribe $(q/p)=g(p)(-1)^{(p-1)/2}$ and q=3 mod8. Reciprocity, CRT and Dirichlet give infinitely many prime conductors q>N realizing the entire finite sign pattern; mod8 supplies the sign at2. Thus these are genuine prime quadratic characters, not relabelled arbitrary sequences. No explicit conductor was computed.

This contradicts the universal positivity statement of Conrey's arXiv:2404.19647v1, Conjecture1, as stated there, but not his conditional implication or the actual Liouville inequality. No novelty-priority assertion is made. Finite exact logs and the two engines are retained; optimized Python cannot silently skip assertions.

Arbitrary fixed-prefix agreement does not restore universality. The unconditional Mertens bound and the zero harmonic sum give $\mathscr L(x)\ll_A x/(\log(1/x))^A$. On [2/3,5/6], a first-term and tail estimate gives $\mathscr L(y)\le-c_0$, $c_0=1+\sqrt3/2-\pi^2/6>0$. For small x, flip only primes $2/(3x)<p<5/(6x)$. Then g_x agrees with lambda through 2/(3x), but the positive convolution identity yields

Longer finite CRT matching and an absolute tail transfer the negative value to genuine characters at each scale. Characters change with x; no one fixed character has arbitrarily long Liouville prefixes.

The remaining selected-family possibility L28-05 is: for every sufficiently large k find one prime character q_k=3 mod8 matching through k^3 and with $\mathscr S_{\chi_{q_k}}(k^{-2})\ge0$. It would give T_k>=-k^-3, hence RH and simplicity. CRT supplies matching, not the required sign. This optional gate must control scale, match length and sign together and has not been proved.

Source-reported subject: NR-12 — Nearest-zero model. Support-pole models may violate stationarity or the sign. Actual nulls and background terms must be retained.. NR-12 — Nearest-zero model. Support-pole models may violate stationarity or the sign. Actual nulls and background terms must be retained.

Source-reported subject: NR-22 — Count quantifiers. All-window blocks give exponent1 without attainment. Positive lower density requires attainment; o(X) is not enough generally.. NR-22 — Count quantifiers. All-window blocks give exponent1 without attainment. Positive lower density requires attainment; o(X) is not enough generally.

Source-reported subject: NR-13 — Boundary collar. Multiplicity-safe local collars depend on height and do not provide a uniform all-height collar.. NR-13 — Boundary collar. Multiplicity-safe local collars depend on height and do not provide a uniform all-height collar.

Source-reported subject: NR-31 — Energy minimizer. A single prime13 flip decreases Q(13) by8/13. Liouville is not the minimizer; this alone says nothing about every mean comparison.. NR-31 — Energy minimizer. A single prime13 flip decreases Q(13) by8/13. Liouville is not the minimizer; this alone says nothing about every mean comparison.

Source-reported subject: NR-08 — Moving divisor scale. J6 fixed-scale TP2 does not persist under changing physical scale or removal of incomplete causal tails.. NR-08 — Moving divisor scale. J6 fixed-scale TP2 does not persist under changing physical scale or removal of incomplete causal tails.

Source-reported subject: NR-19 — Cubic recursion. The cubic increment is not monotone; fixed ratio slack is already PNT-scale. Absolute inversion loses sqrt x.. NR-19 — Cubic recursion. The cubic increment is not monotone; fixed ratio slack is already PNT-scale. Absolute inversion loses sqrt x.

Source-reported subject: NR-24 — Negative source. O28 defeats generic inverse bounds while losing exact between-jump evolution. Pointwise source decay is not derivative decay.. NR-24 — Negative source. O28 defeats generic inverse bounds while losing exact between-jump evolution. Pointwise source decay is not derivative decay.

Source-reported subject: NR-29 — Positive averages. The squarefree average cancels zeta poles but has a negative actual squared-exponential inverse. Positive-source inversion is not sign-preserving.. NR-29 — Positive averages. The squarefree average cancels zeta poles but has a negative actual squared-exponential inverse. Positive-source inversion is not sign-preserving.

Source-reported subject: NR-17 — Denominator sign. D26 locates supporting zeros, not new stationary contacts. D26-POLY defeats orientation from that sign alone.. NR-17 — Denominator sign. D26 locates supporting zeros, not new stationary contacts. D26-POLY defeats orientation from that sign alone.

Source-reported subject: NR-04 — Old envelope. At a=.01 the old ratio t²k/(k−tk′) turns near t=.5–.72. It is not the current varrho profile.. NR-04 — Old envelope. At a=.01 the old ratio t²k/(k−tk′) turns near t=.5–.72. It is not the current varrho profile.

Source-reported subject: NR-40 — Covariance-only repair. R32-05 conditional means produce a same-sign x/log²x drift despite arbitrary large-prime correlations. Other means are outside scope.. NR-40 — Covariance-only repair. R32-05 conditional means produce a same-sign x/log²x drift despite arbitrary large-prime correlations. Other means are outside scope.

Source-reported subject: NR-32 — Generic operator norm. Coherent large-prime translations have squared norm of order X/log²X. A smaller norm for actual inputs needs arithmetic proof.. NR-32 — Generic operator norm. Coherent large-prime translations have squared norm of order X/log²X. A smaller norm for actual inputs needs arithmetic proof.

Source-reported subject: NR-23 — Spectral filters. A zero-dependent filter proves oscillation, not an arithmetic count upper bound. Omitting its resonance repair kills selected modes.. NR-23 — Spectral filters. A zero-dependent filter proves oscillation, not an arithmetic count upper bound. Omitting its resonance repair kills selected modes.

Source-reported subject: NR-38 — Sparse multiplicity. Q32-06 now permits sparse logarithmic energy cutoffs. It does not permit sparse Fourier sampling.. NR-38 — Sparse multiplicity. Q32-06 now permits sparse logarithmic energy cutoffs. It does not permit sparse Fourier sampling.

Source-reported subject: NR-05 — Unweighted prefix. The z cot z=y prefix fails at y=4/5. Central-range estimates do not extend through y=1 or all y<1.. NR-05 — Unweighted prefix. The z cot z=y prefix fails at y=4/5. Central-range estimates do not extend through y=1 or all y<1.

Source-reported subject: NR-06 — Positive polynomial. Positive physical amplitudes and pencils can have opposite leading sine signs. Positivity is not P6.. NR-06 — Positive polynomial. Positive physical amplitudes and pencils can have opposite leading sine signs. Positivity is not P6.

Source-reported subject: R32-05 fixed theta in (1/2,1), exact small-prime sigma-field and conditional means; extra independence needed for asymptotic equality, no uniform q to -1.. R32-05 addresses a correlated class. Fix $1/2<\theta<1$, $y=X^\theta$, $Y=(\log X)^2$. Change an arbitrary random subset $S$ of primes at most $Y$, keep all other primes at most $y$ negative, and allow arbitrary dependence among primes above $y$, subject to conditional mean zero given $S$. With $R_S=\prod_{p\in S}(p+1)/(p-1)$, the exact conditional mean satisfies uniformly for $X/(\log X)^6\le x\le X$,

Conditional Jensen gives

Conditional independence upgrades this to an asymptotic. A fixed common conditional mean $q>-1$ multiplies the lower coefficient by $(1+q)^2$. No uniform $q\to-1$ assertion is made. Only the small-prime cost is automatically subpower; the full reference need not have low cost. The obstruction is to the exact stated marginals and range, not arbitrary prime-dependent means.

Source-reported subject: NR-44 — Outward intervals. Count16 decimal endpoints rounded inward. CK062 repairs them; independent CK067 encloses the same constant more tightly. Old residual tables remain diagnostic.. NR-44 — Outward intervals. Count16 decimal endpoints rounded inward. CK062 repairs them; independent CK067 encloses the same constant more tightly. Old residual tables remain diagnostic.

Source-reported subject: NR-09 — Componentwise completion. Low Jacobi products defeat completion positivity and constant repair; the wedge defect need not decrease.. NR-09 — Componentwise completion. Low Jacobi products defeat completion positivity and constant repair; the wedge defect need not decrease.

Source-reported subject: NR-45 — Fixed-function gate. B32 requires one fixed g, all-scale lower bounds and a sparse weak-cost sequence. Different g at each cutoff needs the R30 same-pair route.. NR-45 — Fixed-function gate. B32 requires one fixed g, all-scale lower bounds and a sparse weak-cost sequence. Different g at each cutoff needs the R30 same-pair route.

Source-reported subject: NR-27 — Audit scope. Both current audits have stated analytic and executable scopes, not formal verification. Hashes and finite checks do not prove infinite gates; optimized assertions are forbidden.. NR-27 — Audit scope. Both current audits have stated analytic and executable scopes, not formal verification. Hashes and finite checks do not prove infinite gates; optimized assertions are forbidden.

Source-reported subject: NR-10 — Unmatched cones. Hyperbola-Abel and inner-score positivity need a proved intertwiner matching all normalizations and nulls.. NR-10 — Unmatched cones. Hyperbola-Abel and inner-score positivity need a proved intertwiner matching all normalizations and nulls.

Source-reported subject: Root-locus moment chart handles X=0 without dividing by X → Exact CR variance and convolution curvature quantities differ. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Reflection-quotient phase gate. In the target half-strip, RH is equivalent to the implication U(w)=0 implies V(w) is nonzero for the principal logarithm h=U+iV of the reflection quotient.

Historical subject: Riemann Hypothesis → Reflection-quotient phase gate. The source reports the completed-zeta zero condition and the reflection-quotient phase gate as exact reformulations of the same problem.

Historical subject: Global endpoint-incidence and same-sign pairing gate; Zero-phase singular exclusion → Reflection-quotient phase gate. A component-wise phase-avoidance theorem, including singular vertices, would establish the exact quotient gate.

Source-reported subject: A26-01–05 exact prime-power jumps, continuous F, downward slope jumps, Mellin/Landau lower bounds → A26-BARRIER/SLACK requires stated a0+alpha>0, b0>=0 and eventual branch condition or fixed quadratic slack/PNT absorption.. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: C163 phase-contact band 0<gamma<=25/4 differs from C103 off-axis-zero band |gamma|<=82.40 and stationary band [64,81.90] → D_theta chart valid at X=0. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Q32-01/02 positive convolution detector and multiplicity filtering → Q32-07 finite-mode subtraction gives decaying source, not bounded residual input.. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: O28-01–03 exact F0 extension, moment and strict negative forcing → O28-10 one fixed triangular Mobius source. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Q30-CONV/06 exact real cutoffs X/d → R30-04–07 critical 1/(2 sqrt p) reference. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: A26-12–15 clipped slope, nonnegative jumps and essential half N^-3/2 sampling loss → A28-01–03 whole spectral extension differs from zero extension. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: A26-BARRIER/SLACK requires stated a0+alpha>0, b0>=0 and eventual branch condition or fixed quadratic slack/PNT absorption. → A26-12–15 clipped slope, nonnegative jumps and essential half N^-3/2 sampling loss. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Closed pencil c in [0,a^2], NC for noncritical Ka!=0 and independent CR → Full profile plus PFX only implies NC, still requiring CR. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Q32-01/02 positive convolution detector and multiplicity filtering → Q32-05 finite critical ordinates all of exact multiplicity m. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Q32-01/02 positive convolution detector and multiplicity filtering → Q32-03 logarithmic lower baseline, not an asymptotic expansion.. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: A28-05–07 positive compact filters and finite odd-harmonic resonance repair for selected zeros → A28-08–13 every-late-window exceptional blocks without attained edge. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.