# Reachability of the remaining endpoint domain under nonzero-kernel improvement **Twin-prime infinitude remains OPEN, and the sufficient margin C_2*x+E_dagger(x) >= c_0*x/(log x)^K remains OPEN.** This note establishes no arithmetic estimate. It prices the grouped-moment block budgets of [grouped-divisor-moment.md](grouped-divisor-moment.md) against the domain they have to cover, and identifies the part of that domain the budgets cannot reach whatever happens to the nonzero kernel. A larger controlled exponent area is not a lower bound on twins, and exponent area does not measure remaining signed mass; that caveat applies to every table below. The ceiling here is a limitation of one argument shape. It is not an impossibility theorem, and it does not say that some other argument cannot reach the same region. Companion validator: [reachability-validation.js](reachability-validation.js), artifact [reachability-validation.json](reachability-validation.json). It is closed-form rational arithmetic over the stated inequalities and validates none of the imported analytic estimates. ## 1. Coordinates, budgets and the two edges Notation follows [residual-coverage.md §1](residual-coverage.md) and [grouped-divisor-moment.md §4](grouped-divisor-moment.md): d~x^delta, e~x^nu, a=delta+6/25, b=nu+1/20, with the original domain delta in [6/25,19/25], nu in [1/20,19/20], so a in [12/25,1] and b in [1/10,1]. The right block budgets after [grouped-divisor-moment (14)](grouped-divisor-moment.md) are (1+a)/2 for the zero frequencies, a/2+3b/2 for the nonzero Weil term and a for the complete periods; the left orientation swaps a and b. Introduce the edge distances \[ p=\frac{19}{25}-\delta=1-a,\qquad q=\frac{19}{20}-\nu=1-b,\qquad 0\le p\le\frac{13}{25},\quad 0\le q\le\frac9{10}. \] DERIVED, by substitution (validator section A, 2,438 rational boxes): | orientation | zero | cross, generic | periods | |---|---|---|---| | right | 1 - p/2 | 2 - p/2 - 3q/2 | 1 - p | | left | 1 - q/2 | 2 - q/2 - 3p/2 | 1 - q | Since 1-p/2 >= 1-p for p>=0, **the zero budget dominates the period budget everywhere in the domain**, so the period term never binds where the zero term does not. Let gamma>=0 denote a uniform total saving in the moment exponent relative to the generic nonzero budget 3b on the right and 3a on the left. The cross budgets become 2-p/2-3q/2-gamma/2 and 2-q/2-3p/2-gamma/2. **Dictionary to the local target.** In [grouped-divisor-moment (21)](grouped-divisor-moment.md) the target at (delta,nu)=(8/25,9/20) is X_small <= C*x^(36/25-eta), and 36/25 = 3/2 - 3/50, so **gamma = 3/50 + eta_(21)**. That note's "needs more than 3/50" is gamma>3/50 at that box. CHECKED: first Cauchy gives sqrt(F^2 * X) with F^2 = M = x^(14/25), and sqrt(x^(14/25)*x^(3/2-gamma)) = x^(103/100-gamma/2) = x^(a/2+3b/2-gamma/2), which is x^(1-eta_(21)/2) at gamma = 3/50+eta_(21). Using eta_(21) in place of gamma is a live error mode; it is an active negative control in the validator (`wrongCauchyDictionary`). Fix the required strict margin eta_0>0, meaning all three budgets of an orientation are at most 1-eta_0. DERIVED and checked as an equivalence on 6,156 (box,gamma) pairs in validator section B: \[ \hbox{right usable}\iff p\ge2\eta_0\ \hbox{and}\ p+3q\ge2-\gamma+2\eta_0, \] \[ \hbox{left usable}\iff q\ge2\eta_0\ \hbox{and}\ 3p+q\ge2-\gamma+2\eta_0. \tag{1} \] The factor 2 on eta_0 in the zero condition is exact: (1+a)/2 <= 1-eta_0 is a <= 1-2*eta_0. The period condition p >= eta_0 is implied by it. ## 2. The unreachable set ### 2.1 The corner, for every saving The cross term appears in neither the zero nor the period budget. Therefore, DERIVED from (1) for every gamma, including an unbounded one, the right orientation is unusable when p<2*eta_0 and the left orientation is unusable when q<2*eta_0. The set no nonzero-kernel improvement reaches in either orientation is exactly \[ S_0(\eta_0)=\{p<2\eta_0\}\cap\{q<2\eta_0\} =\{d>x^{19/25-2\eta_0}\}\cap\{e>x^{19/20-2\eta_0}\}, \tag{2} \] equivalently, with k=n/d and t=(n-2)/e the two cofactors carrying the beta weights, \[ Vfloor(x^(3/4)), d^5e^2>floor(x^(49/20)) and (d>floor(x^(151/200)) or de^3>floor(x^(321/200))) ([grouped-divisor-moment (19)](grouped-divisor-moment.md)). On S_0: | condition | exponent on S_0 | threshold | holds | |---|---|---|---| | de | 171/100 - 4*eta_0 | 3/4 | yes | | d^5e^2 | 57/10 - 14*eta_0 | 49/20 | yes | | d | 19/25 - 2*eta_0 | 151/200 | yes for eta_0 < 1/400 | So the endpoint mass of S_0 belongs to E_dagger and not to an already-controlled cut. The validator confirms membership for the corner and for the four named edge points used below. ### 2.3 The edges are structural, not a choice of cutoff a = delta + w <= (1-w) + w = 1 for every admissible cutoff exponent w, because k>V=x^w together with dk<=x forces delta<=1-w; likewise b = nu + v <= 1. Equivalently M <= 2*D_0*V <= 2x and N <= 2*E_0*Z <= 2x: the expanded left divisor can be comparable to n itself and the expanded right divisor comparable to n-2. Raising w toward 1/4, the ceiling from [prime-detection-spec §3 (8)](prime-detection-spec.md) where D=UV must stay under the BV level, or v toward 3/50, the ceiling from [shifted-prime-decomposition §2](shifted-prime-decomposition.md), moves where the edge sits but never removes a=1 or b=1. DERIVED. ### 2.4 What does not cover S_0 | device | status on S_0 | reason | |---|---|---| | trivial branch MN <= x^(1-tau) | does not cover | a+b = 2 - p - q > 2 - 4*eta_0, so MN is far above x^(1-tau) | | untwisted density | covers the density, which is not E_dagger | at fixed e the allowed d form the interval (x^(19/25-2eta_0), D_0], so the excluded-prime Mobius mean of [signed-divisor-grouping §2 (7)](signed-divisor-grouping.md), uniform for U/2<=t<=x, applies. E_dagger is the endpoint sum after that subtraction, [residual-coverage (16)](residual-coverage.md) | | absolute bound on the endpoint terms | does not cover | section 2.6 | | the pre-grouped-moment budgets | do not cover | at a=b=1 the [residual-coverage §3](residual-coverage.md) zero budgets are 1/2+(a+b)/2-v/2 = 59/40 on the right and 1/2+(a+b)/2-w/2 = 69/50 on the left, both above one. The grouped moment's (1+a)/2 = 1 is the smallest available at the edge | ### 2.5 Fixed margins and the proposed shrinking corner The analytic bounds currently require a fixed positive margin. Substituting eta_0=C*loglog x/log x into the displayed exponents alone does not prove an O_H(x/log^H x) estimate: the proofs also carry O_epsilon(x^epsilon) losses, and for fixed epsilon>0, x^epsilon/(log x)^(cC) grows without bound. Nor may epsilon vary with x without controlling its constants. A uniform logarithmic-loss replacement has not been derived. The following is therefore **geometric bookkeeping conditional on a new uniform error bound**, not the analytic size of the proved remainder: - Dyadic steps in delta have size log 2/log x, so S_0 covers 2*eta_0*log x/log 2 = 2C*loglog x/log 2 boxes per side, that is an **O(loglog x) by O(loglog x) block of dyadic boxes at the top corner**, rather than a strip of fixed width. Sample counts at C=3, from the validator: 56 by 56 boxes at x=2^1024 out of 532 d-boxes, and 236 by 236 at x=2^(2^40) out of 5.7e11 d-boxes. The block is a vanishing fraction of the boxes and is never empty. - The block does not disappear. The last dyadic box has p = O(1/log x), hence right zero budget 1 - log 2/(2 log x), giving x^(1-O(1/log x)), which is a constant multiple of x and not O(x/log^H x). For fixed eta_0, (1) still proves failure to obtain that *prescribed* margin on S_0. It does not exclude every smaller positive saving at its interior points. With hypothetical gamma=2 and p=q=eta_0, for example, the zero and cross budgets are 1-eta_0/2 and 1-2eta_0. The exact endpoint p=q=0 retains zero budget 1. See [independent review F3](history/reviews-0906/20-independent-handoff-review.md). ### 2.6 Absolute bounding on S_0, quantified Two counts, both against the target x/log^K x. **Pair count, DERIVED.** The number of (d,e) in S_0 is (D_0-floor(x^(19/25-2eta_0)))*(E_0-floor(x^(19/20-2eta_0))) = (1+o(1))*D_0*E_0. [residual-coverage §4](residual-coverage.md) bounds the total absolute unaggregated endpoint mass on the whole domain by O(x^2 log^C x), using exactly D_0*E_0*V*Z. So S_0 carries 1-o(1) of that crude mass: because exponents are logarithmic, a thin strip in exponents is nearly everything in counts. **Weighted term mass, DERIVED.** Restrict to k=r prime in (x^(6/25), x^(6/25+2eta_0)] and t=r' prime in (x^(1/20), x^(1/20+2eta_0)], with d=n/r and e=(n-2)/r' both squarefree. Fix such a coprime pair r,r'. By CRT the n<=x with r|n and r'|n-2 form one progression modulo r*r' with x/(r r')+O(1) terms. Within it, for each prime P not dividing r r', the condition P^2 | n/r has density 1/P^2 and so does P^2 | (n-2)/r', while for P | r r' the densities are no larger. The union bound over all primes costs at most 2*sum_P 1/P^2 = 2*0.45224... = 0.90449... < 1, leaving density at least 0.09 and hence >> x/(r r') admissible n. Summing, sum_r 1/r = log(1+(25/3)*eta_0)+o(1) and sum_{r'} 1/r' = log(1+40*eta_0)+o(1) by Mertens, and each term carries beta_V(k)*beta_Z(t) = (log r)(log r') >= (3/250)*log^2 x. Hence the sum of absolute values of the terms of R restricted to S_0 is \[ \gg\eta_0^2\,x\log^2x \] with an absolute implied constant. Numerically, at eta_0=1/100 and x=2^1024 the leading count before the squarefree factor is 162.8*x. **One top dyadic box, DERIVED.** Take a single dyadic band on each side, r in (R,2R] with R=x^(6/25) and r' in (Z',2Z'] with Z'=x^(1/20). Mertens gives sum 1/r = log(1+log 2/log R)+o(1) and likewise for r'. The logarithmic weights (log r)(log r') cancel the two log x factors exactly, leaving leading absolute mass (log 2)^2 * x = 0.4805...*x before the squarefree factor. Validator: 0.4801*x at x=2^16384, tending to (log 2)^2 = 0.4805. The qualitative consequence is the same in all three readings: a triangle-inequality treatment of S_0 misses x/log^K x by at least a factor log^K x (by log^(K+2) at fixed eta_0, by (loglog x)^2 log^K x at eta_0 of order loglog x/log x, by log^K x on the single top box). This bounds absolute term mass, both signs together, and says nothing about the signed value. It is the S_0-local analogue of [singleton-fiber-audit.md](singleton-fiber-audit.md)'s global c*x*log x absolute-mass lower bound, without that note's per-sign split: a per-sign bound on S_0 would need the sign of mu along the CRT progression, which is the OPEN two-point correlation itself. ### 2.7 Exactly which limitation this is Three ingredients produce the ceiling, all of them properties of the argument shape and not of the arithmetic. 1. **Arbitrary bounded coefficients.** Equations (2) and (3) of grouped-divisor-moment hold for any |b_u|<=B, and no Mobius sign of b_u is used. 2. **One Cauchy inequality in the other divisor**, costing F = (sum_m |A_left(gm)|^2)^(1/2). That note bounds F by sqrt(M)*log x; this is an upper bound under its general hypotheses. A uniform lower bound would require nondegeneracy of the actual coefficient and support; it is not supplied by those hypotheses. The region arithmetic uses the available upper-budget exponent a/2. 3. **Zero frequencies bounded absolutely.** [grouped-divisor-moment §2 (5)](grouped-divisor-moment.md) applies the triangle inequality to the R=0 class and states that no cancellation is claimed there. That class contains u_1=u_2, h_1=h_2, whose contribution is sum_u sum_h |b_u|^2 |c_h|^2 sum_{m,(m,u)=1} |Phi_{u,h}(m)|^2 >= 0, at most a constant times B^2 C^2 f^2 MN/A. It can vanish for zero coefficients, empty harmonics or coincident endpoints. The upper budget reaches x on A ~ MN/x; that is not a uniform lower bound. Keeping these upper budgets unchanged leaves the right block estimate sqrt(M*x) = x^((1+a)/2), even if the nonzero-frequency budget improves. This is a limitation of the displayed estimates, not a lower bound on the actual sum or moment. A sharper coefficient norm, diagonal estimate or different organization could change it; no necessary cancellation mechanism is proved here. DERIVED, inside this budget calculation only. ## 3. Structure of the edge strips ### 3.1 The d-edge: k is essentially one prime just above V On p<2*eta_0, k lies in (V, x/d] with x/d <= x^(6/25+p), and beta_V(k)!=0 forces a prime power r>V dividing k, so k = r*s with s < x^(1-a) = x^p < x^(2*eta_0). Since k < V^2 for eta_0<3/25, at most one prime base has powers above V dividing k; several powers of that same base can occur. The proper-prime-power class is negligible because sum_{P^j>V,j>=2} P^(-j) << V^(-1/2): split P at sqrt(V), bound each small-base tail by 2/V, and sum 2/P^2 over the larger bases. Counting multiples and divisor multiplicities costs O_epsilon(x^(22/25+epsilon)), which is O_H(x/log^H x) when epsilon<3/25. After discarding this class, r is prime and beta_V(k)=log r. DERIVED; this repairs the false inference P^j>V implies P>sqrt(V) for every j>=2. Write C_{Y,Z}(N) = sum over e|N with e>Y, N/e>Z, e<=E_0 of mu(e)*beta_Z(N/e), the exact right-hand coefficient of R. The d-edge strip contribution to R, and after subtracting its controlled density to E_dagger, is \[ \sum_{s\le x^{2\eta_0}}\ \sum_{r\ \rm prime>V}(\log r) \sum_{d\ \rm in\ strip,\ drs\in J_x}\mu(d)\,C_{Y,Z}(drs-2) \;+\;O(x^{22/25}\log^Cx), \] and aggregating the long variable as Q=d*s, with |lambda_s(Q)| = |mu(Q/s)| <= 1, \[ =\sum_s\sum_{Q,\ s\mid Q}\lambda_s(Q) \sum_{r\ \rm prime,\ Qr\in J_x}(\log r)\,C_{Y,Z}(Qr-2)+O(x^{22/25}\log^Cx), \tag{3} \] with Q of size x^(19/25) up to x^(19/25+2*eta_0) and r of size x^(6/25). ### 3.2 The e-edge: t is essentially one prime just above Z Symmetrically t = r'*s' with r' a prime power above Z and s' < x^(1-b) = x^q < x^(2*eta_0). The same prime-power tail bound with W=Z costs O_epsilon(x^(39/40+epsilon)), which is O_H(x/log^H x) for epsilon<1/40. This includes small bases with powers of exponent at least three; no false lower bound on the prime base is used. Then \[ \sum_{s'}\sum_{E,\ s'\mid E}\lambda'_{s'}(E) \sum_{r'\ \rm prime>Z}(\log r')\,C'_{U,V}(Er'+2),\qquad C'_{U,V}(N)=\sum_{d\mid N,\ d>U,\ N/d>V,\ d\le D_0}\mu(d)\beta_V(N/d), \tag{4} \] with E of size x^(19/20) and r' of size x^(1/20). DERIVED. ### 3.3 The corner: both sides degenerate at the same time On the s=s'=1 subfamily of S_0 the determinant relation becomes d*r - e*r' = 2 with d~x^(19/25), e~x^(19/20), r~x^(6/25), r'~x^(1/20). For fixed coprime (r,r'), the condition r' | d*r-2 fixes d modulo r'; writing d = d_0 + r'*m and e = c_0 + r*m, the inner sum is \[ \sum_{m\sim x^{71/100}}\mu(d_0+r'm)\,\mu(c_0+rm), \tag{5} \] weighted by (log r)(log r') and summed over the two prime bands. The prime-cofactor subfamily is therefore an averaged binary Mobius correlation along two independent linear forms with dilations r~x^(6/25) and r'~x^(1/20) and inner length x^(71/100); the total mass is x^(29/100) parameter pairs times x^(71/100) inner length, that is x. This is algebra. It is not an obstruction claim. The full corner also contains k=rs,t=r's' with s or s' greater than one, including the non-squarefree and gcd-two branches. They have not been discarded at an admissible error rate. The exact full object is (4) of [corner-correlation.md](corner-correlation.md); its nonnegative Möbius-product form (5) applies only to the subfamily above. ### 3.4 Pricing the shapes ASSESSMENT only. No estimate is claimed, and the levels below are what the shapes would require, not what any theorem supplies. | edge | object | nearest known class | level or split required | comment | |---|---|---|---|---| | d-edge, general nu | (3) | Type II bilinear against a shifted divisor-type coefficient, the Deshouillers-Iwaniec, Drappeau, Topacogullari, Pascadi input class | split (19/25, 6/25) = (0.76, 0.24); the short variable exponent 6/25 sits just below 1/4 | U=V=x^(6/25) was chosen as an interior point below 1/4 so that D=UV<=x^(12/25) stays under the BV level ([prime-detection-spec §3 (8)](prime-detection-spec.md)); the split inherits that choice | | d-edge, read as BV | equidistribution of alpha(n)=sum_{n=Qr} lambda(Q) log r on n congruent to 2 modulo e, e up to E_0 | Bombieri-Vinogradov and large-moduli extensions | level 19/20 | far beyond BV at 1/2 and beyond well-factorable extensions. This reading over-demands: the true object carries mu(e) and a further t-sum, so it is bilinear, not an absolute-value average | | d-edge, read as dispersion in r | Cauchy in Q, then pairs (r_1,r_2) with sum_Q C(Qr_1-2) conj C(Qr_2-2) | Linnik dispersion, BFI | completion length Q~x^(19/25) against joint modulus e_1e_2 up to x^(19/10) | concrete failure point: when e_1e_2 exceeds Q there is no complete period to remove, and that is exactly the sub-range e near E_0, the corner | | e-edge, general delta | (4) | same class | split (19/20, 1/20) = (0.95, 0.05) | harsher on the short variable, and v may be raised only to below 3/50 | | corner | (5) | Chowla-type, or averaged Chowla; the Matomaki-Radziwill-Tao averaging is over additive shifts, not multiplicative dilations | log^(K+2) cancellation on average over x^(29/100) dilation pairs | no interface reviewed in [structural-literature-audit.md](structural-literature-audit.md) closes it. The average over (r,r') is a genuine resource whose transfer from additive shifts to dilations is not established in anything read here | The two edges are the same phenomenon at different scales: residual prime x^(6/25) at the d-edge against x^(1/20) at the e-edge, a factor 4.8 in exponent. That asymmetry shows up in the required savings of section 4. ## 4. CONDITIONAL: what a uniform saving would buy **Everything in this section assumes an unproved estimate.** Assume that for every retained box and harmonic band, in both orientations, the nonzero small-common-divisor kernel obeys a one-sided bound improving the moment exponent by a fixed gamma>0 uniformly. The actual open target [grouped-divisor-moment (21)](grouped-divisor-moment.md) is one box, one orientation, gamma = 3/50 + eta_(21). Uniformity across the domain is a much larger assumption and is not implied by (21). Nothing below is a theorem. ### 4.1 Added region and the four leftover pieces From (1), the added region is \[ \{\delta<19/25\ \hbox{and}\ \delta+3\nu<161/100+\gamma\}\ \cup\ \{\nu<19/20\ \hbox{and}\ 3\delta+\nu<123/100+\gamma\}, \] and the leftover decomposes into four overlapping pieces, with S_0 contained in both strips: | piece | definition | binding terms | empties when | |---|---|---|---| | S_0, corner | p<2eta_0, q<2eta_0 | right zero and left zero | never | | S_R, d-edge strip | p<2eta_0, 3p+q<2-gamma+2eta_0 | right zero, left cross | gamma >= 2 | | S_T, e-edge strip | q<2eta_0, p+3q<2-gamma+2eta_0 | left zero, right cross | gamma >= 2 | | B, bulk | p+3q<2-gamma+2eta_0 and 3p+q<2-gamma+2eta_0 | both cross terms | gamma >= 2+2eta_0 | The strip thresholds are DERIVED and exact: S_R minus S_0 requires q>=2*eta_0 and 3p+q<2-gamma+2*eta_0 with p<2*eta_0, whose infimum of 3p+q is 2*eta_0, so the piece is nonempty precisely when gamma<2. The same holds for S_T. **The values 11/10 and 37/25 below are the cheapest single points of the two edges, not clearing thresholds**; clearing either edge apart from the corner costs gamma tending to 2, the same as clearing the bulk. ### 4.2 Required saving at named boxes gamma_req is the minimum, over the orientations whose zero budget is usable, of 2-p-3q on the right and 2-3p-q on the left, equivalently of delta+3nu-161/100 and 3delta+nu-123/100. Exact rationals, validator section D: | (delta,nu) | (p,q) | right | left | gamma_req | note | |---|---|---|---|---|---| | (2/5, 2/5) | (9/25, 11/20) | -1/100 | 37/100 | already controlled | benchmark, right budget 199/200 | | (47/150, 67/150) | (67/150, 151/300) | 13/300 | 47/300 | 13/300 | old frontier meeting point | | (8/25, 9/20) | (11/25, 1/2) | 3/50 | 9/50 | 3/50 | the current local target box | | (1/2, 1/2) | (13/50, 9/20) | 39/100 | 77/100 | 39/100 | | | (6/25, 63/100) | (13/25, 8/25) | 13/25 | 3/25 | 3/25 | old product-supremum point | | (1/2, 7/10) | (13/50, 1/4) | 99/100 | 97/100 | 97/100 | interior, expensive direction | | (19/25, 9/20) | (0, 1/2) | right unusable | 3/2 | 3/2 | on the d-edge, inside W_dagger | | (19/25, 1/2) | (0, 9/20) | right unusable | 31/20 | 31/20 | on the d-edge, inside W_dagger | | (19/25, 1/20) | (0, 9/10) | right unusable | 11/10 | 11/10 | cheapest point of the d-edge | | (6/25, 19/20) | (13/25, 0) | 37/25 | left unusable | 37/25 | cheapest point of the e-edge | | (19/25, 19/20) | (0,0) | unusable | unusable | unbounded | the corner S_0 | Over the whole d-edge gamma_req runs over [11/10, 2), and over the whole e-edge over [37/25, 2). The validator samples 286 edge points; 180 d-edge points exceed 11/10 and 104 e-edge points exceed 37/25, which is the active control against reading either cheapest value as a threshold. ### 4.3 Leftover as gamma grows Exact rational vertices and areas of the bulk B, taken at the limiting region boundaries 19/25 and 123/50 and with eta_0 tending to 0. Domain area 117/250. This is bookkeeping over inequalities, not a measurement. | gamma | bulk B vertices (delta,nu) | area | percent of domain | |---|---|---|---| | 0 | (6/25,63/100) (6/25,19/20) (19/25,19/20) (19/25,17/60) (67/200,17/40) (47/150,67/150) | 35347/120000 | 62.9 | | 3/50 | (6/25,63/100) (6/25,19/20) (19/25,19/20) (19/25,91/300) (101/325,589/1300) | 27863/97500 | 61.1 | | 1/5 | (6/25,71/100) (6/25,19/20) (19/25,19/20) (19/25,7/20) (31/100,1/2) | 651/2500 | 55.6 | | 39/100 | (6/25,9/10) (6/25,19/20) (19/25,19/20) (19/25,31/75) (143/400,219/400) | 25871/120000 | 46.1 | | 1/2 | (13/50,19/20) (19/25,19/20) (19/25,9/20) (77/200,23/40) | 3/16 | 40.1 | | 1 | (32/75,19/20) (19/25,19/20) (19/25,37/60) (51/100,7/10) | 1/12 | 17.8 | | 11/10 | (23/50,19/20) (19/25,19/20) (19/25,13/20) (107/200,29/40) | 27/400 | 14.4 | | 3/2 | (89/150,19/20) (19/25,19/20) (19/25,47/60) (127/200,33/40) | 1/48 | 4.5 | | 37/25 | (44/75,19/20) (19/25,19/20) (19/25,233/300) (63/100,41/50) | 169/7500 | 4.8 | | 7/4 | (203/300,19/20) (19/25,19/20) (19/25,13/15) (279/400,71/80) | 1/192 | 1.1 | | 2 | the single point (19/25,19/20) | 0 | 0 | The row at 37/25 exceeds the row at 3/2 because 37/25<3/2, although that row is printed later. At increasing gamma the sets are nested and their areas are nonincreasing; clipping does not reverse monotonicity. **These areas are taken at the limiting boundaries, not at the concrete cuts.** In (p,q) the concrete cuts of W_dagger read p+q<24/25 and 5p+2q<13/4, while the limiting boundaries read p+q<=19/20 and 5p+2q<=81/25. The genuine W_dagger, with the concrete cuts 3/4, 49/20 and the disjunction (d>x^(151/200) or de^3>x^(321/200)), is a union of two convex pieces with exact area 141979/480000 + 93/80000 = 142537/480000 = 0.296952, that is 63.45 percent of the domain, against 35347/120000 = 0.294558, that is 62.94 percent, at the limiting level. The table above is internally consistent at the limiting level; the genuine figure is the larger one. Two flat readings. First, the entire currently sought saving gamma=3/50, applied uniformly across the whole domain and both orientations, which is far more than (21) asks for, removes about 3 percent of the leftover area, 62.9 to 61.1. Second, in the limiting eta_0=0 geometry a bulk interior persists for gamma<2 by taking small positive p=q<(2-gamma)/4. At a prescribed positive eta_0, the bulk with p,q>=2eta_0 is empty once gamma>=2-6eta_0. Edge strips still prevent full coverage outside S_0 for gamma<2; see §4.5 and round-review-0906 §5. ### 4.4 Exact coupled cuts for the next E With a saving gamma, choose fixed kappa<19/25, lambda<161/100+gamma, rho<19/20, sigma<123/100+gamma and set \[ \mathcal A=\{de\le\lfloor x^{3/4}\rfloor\},\qquad \mathcal B=\{d^5e^2\le\lfloor x^{49/20}\rfloor\}, \] \[ \mathcal C=\{d\le\lfloor x^{\kappa}\rfloor\ \hbox{and}\ de^3\le\lfloor x^{\lambda}\rfloor\},\qquad \mathcal D=\{e\le\lfloor x^{\rho}\rfloor\ \hbox{and}\ d^3e\le\lfloor x^{\sigma}\rfloor\}. \] E_new is then the exact weighted compatible CRT endpoint sum of [residual-coverage (16)](residual-coverage.md) restricted to the complement of the union, namely \[ de>\lfloor x^{3/4}\rfloor,\quad d^5e^2>\lfloor x^{49/20}\rfloor, \] \[ (d>\lfloor x^{\kappa}\rfloor\ \hbox{or}\ de^3>\lfloor x^{\lambda}\rfloor), \quad (e>\lfloor x^{\rho}\rfloor\ \hbox{or}\ d^3e>\lfloor x^{\sigma}\rfloor). \] Separation follows [residual-coverage §4](residual-coverage.md) with up to six truncated Perron integrals: A and B each have one scalar cutoff, C and D each have two. Use real part 1/log x, height x^10 and half-integer-shifted thresholds. In the order A,B,C_vertical, C_monomial,D_vertical,D_monomial the combined twist is d^(-s1-5s2-s3-s4-3s6) e^(-s1-2s2-3s4-s5-s6). All real parts are nonnegative, so the endpoint bounds apply uniformly in the imaginary parts; the cost is O(log^6 x) and the indicator error O(x^(2+max(3/4,49/20,kappa,lambda,rho,sigma)-10) log^C x). The untwisted density is bounded before separation, since at fixed e each of the four sets restricts d to an interval and unions and intersections take maxima and minima of endpoints. A concrete choice at gamma=1/10 is kappa=151/200, lambda=341/200, rho=189/200, sigma=265/200, which puts the (8/25,9/20) box, where de^3 = x^(167/100), inside C. At gamma=3/50 exactly, lambda would have to equal 167/100 and that box sits on the boundary with no fixed slack, which is the note's "equality leaves no fixed slack" restated as a cut. ### 4.5 The threshold is gamma=2, and only gamma=2 CONDITIONAL, derived from (1). At gamma=2 the added region is {p>=2*eta_0} union {q>=2*eta_0}, since the cross conditions reduce to p+3q>=2*eta_0 and 3p+q>=2*eta_0 and are implied there. The leftover is then exactly S_0. For gamma<2 the point p=2eta_0,q=0 fails the right cross inequality and the left zero inequality; the symmetric point does likewise. Thus both edge strips minus S_0 persist, although the bulk with p,q>=2eta_0 is already empty when gamma>=2-6eta_0. The overall threshold remains gamma=2, not a lower-bound theorem for the actual arithmetic remainder. What gamma=2 asks for. At the corner a=b=1, the generic nonzero budget is N^3 = x^3 and the zero part is x, so gamma>=3b-1 = 2 is exactly the statement that the nonzero frequencies contribute no more than the diagonal. HEURISTIC diagnostic: full square-root cancellation across the roughly N*A pairs would leave N^2/A, and at the corner A ~ MN/x = x and N = x, so N^2/A = x and the heuristic best possible gamma is 3-1 = 2. At the corner the heuristic ceiling on gamma therefore coincides with the requirement, with no slack, while the zero budget is simultaneously exactly 1, also with no slack. At the current local target box the same heuristic gives an available 3/2 - 47/50 = 14/25 against a requirement of 3/50, so that box has heuristic room and the corner has none. This is heuristic and assumes square-root cancellation across coupled pairs, which is exactly what is unproved. ## 5. Assessment ### 5.1 Derived 1. At a prescribed fixed margin eta_0, S_0 fails that margin for every gamma in both orientations, because the zero budget is unchanged. This does not exclude smaller margins at its fixed interior points. 2. gamma=2 is the threshold at which the leftover reduces to S_0; both edge strips minus S_0 persist for gamma<2. At fixed eta_0 the bulk is already empty for gamma>=2-6eta_0. 3. The distance between what is currently sought and what would be needed: | goal | gamma needed | multiple of the current 3/50 target | |---|---|---| | the single box (8/25,9/20) | above 3/50 | 1 | | the interior point (1/2,7/10) | 97/100 | 16.2 | | the d-edge point (19/25,9/20) | 3/2 | 25 | | the whole d-edge apart from the corner | tending to 2 | tending to 33.3 | | the whole e-edge apart from the corner | tending to 2 | tending to 33.3 | | leftover reduced to S_0 | 2 | 33.3 | 4. Absolute bounding on S_0 misses x/log^K x by log^(K+2): section 2.6. 5. Neither the trivial MN<=x^(1-tau) branch nor the density term reaches E_dagger on S_0. 6. The a=1 and b=1 edges survive every admissible cutoff choice, w in (0,1/4) and v in (0,3/50). 7. Choosing eta_0 of order loglog x/log x gives the stated geometric box count, but analytic coverage at that margin is unproved (§2.5). 8. Only the s=s'=1 subfamily reduces to (5). The full corner retains small-cofactor and non-squarefree branches. Proper-prime-power classes are negligible by the corrected tail bounds in §§3.1–3.2. ### 5.2 Judgement These are opinions, flagged as such, and are not results. - A succession of local kernel improvements does not exhaust W_dagger. This is items 1 and 2 restated: the programme of moving the frontier lines delta+3nu = 161/100+gamma and 3delta+nu = 123/100+gamma outward has a hard stop at S_0, and reaching even that stop asks for gamma=2 uniformly, 33 times the saving currently priced at one box. Confidence high, conditional only on the three-budget list being correct and complete for this argument, which was re-derived here from (14) but not from (2). - The global consumer therefore needs a different argument on S_0. Inside this framework the thing that must change is the first Cauchy inequality in the left divisor: the moment is diagonal-dominated and the left norm is sharp, so at a=1 the whole loss is that step. That points at [residual-coverage §6 (19)-(20)](residual-coverage.md)'s signed cross-divisor sum, which keeps mu(ge_1)mu(ge_2) before the extra step, rather than at further kernel work. Confidence medium-high. Caveat: that target is written for one sector at (2/5,2/5), its analogue at a=b=1 is not written anywhere in this corpus, and the corner is where both divisors degenerate at once, which section 6 there does not address. - The one-sided or scale-averaged consumer of [endpoint-target-audit.md](endpoint-target-audit.md) is a more likely route for S_0 than any absolute bound, since it needs only a lower bound on E_dagger restricted to S_0. Item 4 says even that needs log^(K+2) of genuine cancellation there, so it is not a free reduction. Confidence medium. - On the pricing in section 3.4, the d-edge is the one with a recognisable estimate shape; the e-edge split (19/20, 1/20) is worse on the short variable and its clearing price is no better. Confidence medium, and this is triage, not a result. - No obstruction theorem for S_0 was found and none is asserted. The reduction to (5) is suggestive of the standard parity difficulty, but an obstruction claim needs its own method, statistics, error tolerance and quantifiers, and (5) carries an average over x^(29/100) dilation pairs, which is the kind of resource that has broken related correlation problems elsewhere. S_0 is OPEN and hard, not closed. ### 5.3 What would falsify this - A block-budget list for this moment shape whose zero-frequency term is below (1+a)/2: an argument that does not pay the left norm at the first Cauchy inequality, or one that extracts cancellation from the R=0 class. Either would invalidate item 1 as stated. **This check has not been run**; the three exponents of (14) and the nonnegativity of the diagonal were verified, but no attempt was made to construct such an argument. - A demonstration that the top dyadic boxes carry o(x/log^K x). Section 2.6's third count addresses exactly the last box and gives leading absolute mass (log 2)^2 * x, so this would have to come from signed cancellation, not from counting. - Any error in the imported budget list, in the dictionary gamma = 3/50+eta_(21), or in the claim that the R=0 class is bounded absolutely in the current derivation. ## 6. Validation, controls and limits [reachability-validation.js](reachability-validation.js) checks, in exact BigInt rationals: the budget identities and the domination of the period budget by the zero budget on 2,438 boxes; the equivalence of the (delta,nu) and (p,q) usability forms on 6,156 (box,gamma) pairs; the unreachability of the corner at fourteen values of gamma up to 1000; the leftover vertices and areas at twelve values of gamma; the concrete-cut W_dagger area as a union of two convex pieces; the required savings at eleven named boxes; the range of gamma_req along both edges over 286 sampled points; the dyadic block size under eta_0 = 3*loglog x/log x; and the two mass constants. Active negative controls, all of which fire: dropping the zero-budget conditions wrongly clears the corner at gamma=9/4; treating 11/10 and 37/25 as clearing thresholds is contradicted by 284 sampled edge points; the limiting boundaries and the concrete cuts give different areas; using eta_(21) in place of gamma = 3/50+eta_(21) fails to clear the target box; using the period budget rather than the zero budget as the binding one wrongly declares 41 edge boxes reachable; and dropping the vertical cut delta<19/25 wrongly clears 12 edge boxes. Limits. Everything here is bookkeeping over inequalities that other notes derived. It revalidates none of them: not the moment bound (2)-(3), not the pre-grouped-moment region of [residual-coverage (11)](residual-coverage.md), not the Pascadi Kloosterman inputs, not the Vaaler majorant argument of [endpoint-pairing §5](endpoint-pairing.md), and not the excluded-prime Mobius mean. A finite rational check establishes no asymptotic rate and no onset. The sufficient twin margin remains OPEN.