# The determinant corollary: a legal transfer that adds no region **Twin-prime infinitude and the sufficient margin C_2*x+E_dagger(x)>=c_0*x/log^K x remain OPEN. Nothing here changes the controlled region, W_dagger, E_dagger or the required saving in [grouped-divisor-moment.md (21)](grouped-divisor-moment.md).** This note prices one literature interface and returns a negative. A larger controlled exponent area would not be a lower bound on twins; that is said once and applies to every region statement below. The one thing this note does change is *which* obstruction is binding. The wave-2026-09-06 literature scout located Bettin-Chandee Corollary 1 as an asymptotic for our exact determinant equation with our exact Mobius coefficients, failing on one hypothesis: our two cofactor weights are prime-power detectors, not smooth. **That hypothesis failure is removable.** The identity beta_W(k)=log k-sum_{r|k,r<=W}Lambda(r) moves the detector onto the arbitrary coefficient side, where the corollary imposes no condition at all. What actually stops the interface is the size of its error term. ## 1. The transfer and the four pieces ### 1.1 Two constraints are redundant Throughout, x=2^j, J_x=(x/2,x] intersected with the integers, and the objects are those of [RESEARCH-HANDOFF.md §3](RESEARCH-HANDOFF.md): U=V=floor(x^(6/25)), Y=Z=floor(x^(1/20)), w=6/25, v=1/20, and \[ R(x)=\sum_{\substack{d>U,\ k>V,\ e>Y,\ t>Z\\ dk-et=2,\ dk\in J_x}} \mu(d)\mu(e)\beta_V(k)\beta_Z(t),\qquad \beta_W(k)=\sum_{r\mid k,\ r>W}\Lambda(r). \] Since beta_W(k)=0 for k<=W, the constraints k>V and t>Z may be deleted from the summation without changing R(x). This is the step that makes the transfer possible: after the split below, a retained k>V would be a sharp condition coupling the smooth variable to the coefficient index, and no dyadic decomposition of the coefficient would separate it. Deleting it first leaves d>U and e>Y as the only sharp divisor conditions, and those sit on the arbitrary coefficients, where they are free. ### 1.2 The exact four-piece decomposition Use the identity sum_{r|k}Lambda(r)=log k, i.e. beta_W(k)=log k-sum_{r|k,\ r<=W}Lambda(r), on both cofactors, and the exact bijection (d,r) -> l=dr of [endpoint-fourier.md §2](endpoint-fourier.md). Write n_2 for the left coefficient index, m_1 for the left smooth variable, n_1 for the right coefficient index, m_2 for the right smooth variable, so that the equation is m_1*n_2-m_2*n_1=2 with Delta=2. Then, exactly, \[ R(x)=T_{11}-T_{10}-T_{01}+T_{00}, \tag{1} \] with all four sums taken over m_1*n_2-m_2*n_1=2, m_1*n_2 in J_x, and | piece | n_2 coefficient | n_1 coefficient | f(m_1) | g(m_2) | |---|---|---|---|---| | T_11 | mu(d)1_{d>U}, n_2=d | mu(e)1_{e>Y}, n_1=e | log m_1 | log m_2 | | T_10 | B_V(n_2)=sum_{n_2=dr,\ d>U,\ r<=V}mu(d)Lambda(r) | mu(e)1_{e>Y} | 1 | log m_2 | | T_01 | mu(d)1_{d>U} | A_Z(n_1)=sum_{n_1=er',\ e>Y,\ r'<=Z}mu(e)Lambda(r') | log m_1 | 1 | | T_00 | B_V(n_2) | A_Z(n_1) | 1 | 1 | Both aggregated coefficients are arbitrary complex numbers as far as the import is concerned, and are pointwise bounded: |B_V(n_2)|<=sum_{r|n_2}Lambda(r)=log n_2, |A_Z(n_1)|<=log n_1. Hence \[ \|\alpha\|\,\|\beta\|\ll\sqrt{N_1N_2}\,\log^2x. \tag{2} \] The support is sparser than that bound uses -- n_2=dr with r a prime power in a band has about N_2/log x admissible values, each of size about log x -- so the sharper norm is sqrt(N_2 log x). That is a logarithmic gain only; there is no power saving available from the sparsity, and none is claimed. Identity (1) and the redundancy of k>V, t>Z are verified exactly on five small parameter sets in [determinant-corollary-validation.js](determinant-corollary-validation.js), with the negative control that re-imposing m_1>V, m_2>Z inside the four pieces destroys the identity. ### 1.3 Supports Decompose d~x^delta, e~x^nu dyadically as in [residual-coverage.md §1](residual-coverage.md), and the moved prime powers into bands r~x^rho, r'~x^zeta with rho in [0,w], zeta in [0,v] (rho=0 or zeta=0 meaning the piece with no moved prime power). Then \[ N_2\asymp x^{\delta+\rho},\quad N_1\asymp x^{\nu+\zeta},\quad M_1\asymp x^{1-\delta-\rho},\quad M_2\asymp x^{1-\nu-\zeta}, \tag{3} \] because m_1*n_2=dk lies in J_x and m_2*n_1=dk-2. Write A=delta+rho and B=nu+zeta for the two coefficient exponents. The worst band, rho=w and zeta=v, gives exactly the expanded exponents a=delta+6/25 and b=nu+1/20 of the existing notes. Two bookkeeping points. The product n_2=dr over dyadic d and dyadic r has ratio four, so it splits into O(1) blocks of ratio at most two; a sharp restriction of an arbitrary coefficient is free. The smooth variables need a partition of unity into intervals [M/2,M]; that is exact, costs O(log x) pieces of which O(1) are nontrivial, and has eta of order one. ### 1.4 No g in {1,2} branch, and the main term Corollary 1 acts on the equation itself, so the compatibility bookkeeping of [signed-divisor-grouping.md (15)-(18)](signed-divisor-grouping.md) does not arise. Its main term already carries the condition (n_1,n_2)|Delta, which at Delta=2 is exactly our g=(a,b) in {1,2}; pairs with (n_1,n_2) not dividing 2 contribute nothing to T because the equation has no solution, which is the same exactness as "incompatible congruences contribute zero". The CRT origin n_0, the inverse residue and the two gcd branches are never formed. This is a genuine simplification of the bookkeeping relative to our own route; it is not a saving. The corollary's main term is \[ \sum_{\substack{n_1,n_2\\ (n_1,n_2)\mid2}} \frac{(n_1,n_2)}{n_1n_2}\,\alpha_{n_1}\beta_{n_2} \int_{\mathbb R} f\!\left(\frac{y+2}{n_2}\right) g\!\left(\frac{y}{n_1}\right)dy. \tag{4} \] Its density factor (n_1,n_2)/(n_1n_2)=g/(ab) and its compatibility condition are **identical** to the density term M_> of [signed-divisor-grouping.md §5](signed-divisor-grouping.md); the modulus in both is the lcm [n_1,n_2]=ab/g. The two differ only in that (4) integrates the smooth weights we inserted while M_> sums the sharp counted weights over integers of J_x, and that difference is bounded by the same smoothing error priced in §2. At fixed e, r, r' the n_2 sum in (4) is a coprimality-restricted sum of mu(d)/d against a weight of bounded variation, which is precisely the shape handled in [endpoint-fourier.md §3](endpoint-fourier.md) by the uniform excluded-prime Mobius mean of [signed-divisor-grouping.md §2 (7)-(8)](signed-divisor-grouping.md); partial summation absorbs the smooth weight. So (4) is O_H(x/log^H x) for every fixed H, by reuse, and the corollary would control the endpoint remainder on the pieces where its error is below x^(1-eta'). **The main terms match; there is no mismatch to report.** This reuse is DERIVED by citation, not rewritten line by line here; nothing in the verdict depends on it, because the error term fails first. ## 2. Meeting the smoothness hypothesis: cost eta=x^kappa The remaining sharp condition is dk in J_x, which couples m_1 to n_2. Separate it by truncated Perron rather than by blocking the coefficient, which would multiply the error by the number of blocks. **Order of operations.** Deleting k>V and t>Z in §1.1 leaves the four-variable sum with no a priori upper bound on m_1 or m_2 except the one dk in J_x itself supplies. So fix the dyadic boxes first: the sharp restrictions d~x^delta, e~x^nu, r~x^rho, r'~x^zeta go onto the arbitrary coefficients (free), the partitions of unity in m_1 and m_2 are inserted as smooth factors summing to one over all positive integers, and only then is dk in J_x separated, inside that fixed box. The partitions are exact over the whole range, so no mass is lost to a box outside the O(1) that dk in J_x makes nontrivial; on those boxes M_1N_2 and M_2N_1 are both within a bounded factor of x, which is what makes R=O(1) below. Separating first and blocking afterwards would be circular, since the box bound on m_1 is exactly what the separated condition supplies. Fix kappa>0, put c=1/log x, T_P=x^kappa, and use, for n=m_1n_2 inside such a box, \[ 1_{n\le x}=\frac1{2\pi i}\int_{c-iT_P}^{c+iT_P}\left(\frac xn\right)^s\frac{ds}s +O\!\left(\left(\frac xn\right)^c\min\!\left(1, \frac1{T_P|\log(x/n)|}\right)\right). \tag{5} \] Two such integrals give 1_{n in J_x}. The total error from (5) over all terms of R(x) is O(x^(1-kappa) log^C x): the terms with |n-x|>x/T_P contribute at most T_P^{-1} sum_n |a_n| x/|n-x| and those with |n-x|<=x/T_P at most sum_{|n-x|<=x/T_P}|a_n|, where |a_n|<=d(n)d(n-2)log^2x. The same bound holds by smoothing the cutoff at relative scale x^(-kappa) instead; both routes give the same eta. Inside the integral, (x/n)^s=x^s*n_2^(-s)*m_1^(-s). The factor x^s has modulus x^c=O(1); the factor n_2^(-s) has modulus at most one and joins the arbitrary coefficient; the factor m_1^(-s) joins f. With a fixed bump phi supported in [1/2,1], \[ f(m_1)=\phi(m_1/M_1)\,(m_1/M_1)^{-s},\qquad f^{(j)}\ll_j(1+T_P)^jM_1^{-j}, \tag{6} \] and |f|=O(1). So eta is of order x^kappa>1, as the corollary requires. The log-weights are normalised the same way: log m_1=log M_1+log(m_1/M_1), the constant log M_1<=0 and some eta>1. Then T equals \[ \sum_{\substack{n_1\in\mathcal N_1,n_2\in\mathcal N_2\\(n_1,n_2)\mid\Delta}} \frac{(n_1,n_2)}{n_1n_2}\alpha_{n_1}\beta_{n_2} \int_{\mathbb R}f\!\left(\frac{y+\Delta}{n_2}\right)g\!\left(\frac y{n_1}\right)dy +O\!\left((\eta R)^{3/2}\|\alpha\|\|\beta\| (N_1N_2)^{7/20}(N_1+N_2)^{1/4+\epsilon}(M_1M_2)^{\epsilon}\right), \] R=M_1N_2/(M_2N_1)+M_2N_1/(M_1N_2), norms being L2. **Hypotheses, checked against §§1-2.** Delta=2 is nonzero, which is all the corollary asks; the proof (its §9) additionally observes that one may assume |Delta|<=4(M_1N_2+M_2N_1), which holds with room. Supports are dyadic after the O(1) resplit of n_2=dr. alpha and beta are arbitrary complex, so d>U, e>Y, r<=V, r'<=Z, the Perron twists d^(-s), the sparse prime-power support, the prime 2, repeated prime powers and every low sector are admissible without further hypotheses. f and g satisfy (6) with eta=x^kappa>1. R=O(1). No coprimality between n_1 and n_2 is required. **No hypothesis fails.** The corollary is uniform in nothing that we need to be uniform in beyond what is displayed; the sum over d|Delta inside its proof has two terms at Delta=2. **Not checked, assumed:** the corollary's own proof, and Theorem 1 on which it rests. Both were read as statements, not verified. **Pricing.** With (2) and (3), one application costs \[ x^{\,3\kappa/2+\frac{17}{20}(A+B)+\frac14\max(A,B)+\epsilon}, \tag{7} \] and O(log^C x) applications plus the Perron error O(x^(1-kappa)log^C x) do not change the exponent. Since kappa>0 is fixed but arbitrarily small, the sufficient condition, with a fixed margin eta'>0, is \[ \boxed{\ \tfrac{17}{20}(A+B)+\tfrac14\max(A,B)<1 \iff 22\max(A,B)+17\min(A,B)<20.\ } \tag{8} \] The equivalence is verified on a rational grid in the validator. ## 4. The region, in exact rationals Because 22max+17min>=17(A+B)+(5/2)(A+B)=(39/2)(A+B), (8) forces A+B<40/39. Requiring **every** piece and **every** band to pass means taking the worst band A=a=delta+6/25, B=b=nu+1/20, so \[ \delta+\nu<\tfrac{40}{39}-\tfrac{6}{25}-\tfrac1{20} =\tfrac{2869}{3900}=0.735641\ldots<\tfrac{19}{25}. \tag{9} \] The all-pieces region is therefore **strictly inside** the first of the three already controlled regions, delta+nu<19/25. Its added area is exactly zero. | object | condition on (delta,nu) | vertices | area | added area | |---|---|---|---|---| | all four pieces, all bands | 2200*delta+1700*nu<1387 **and** 1700*delta+2200*nu<1482 | (6/25,1/20), (651/1100,1/20), (266/975,361/780), (6/25,537/1100) | 342383/4290000 = 0.079810 | **0** | | log-log piece only (rho=zeta=0) | 22*delta+17*nu<20 **and** 17*delta+22*nu<20 | (6/25,1/20), (19/25,1/20), (19/25,82/425), (20/39,20/39), (6/25,199/275) | 167659/729300 = 0.229890 | 17013209/280280000 = 0.060701 | Domain area 117/250; the log-log added area is 12.97 percent of it. Its vertices are (3263/4900,771/2450), (20/39,20/39), (6/25,199/275), (6/25,63/100), (47/150,67/150), (67/200,17/40), and in the reachability report's coordinates p=19/25-delta, q=19/20-nu (an area-preserving affine map) (461/4900,3113/4900), (241/975,341/780), (13/25,249/1100), (13/25,8/25), (67/150,151/300), (17/40,21/40). Named points, with the error exponent of (7) at kappa=epsilon=0: | point | a, b | all pieces | log-log piece | currently controlled | |---|---|---|---|---| | benchmark (2/5,2/5) | 16/25, 9/20 | 2173/2000=1.0865, fails | 39/50, passes | yes | | next target (8/25,9/20) | 14/25, 1/2 | 1041/1000=1.041, fails by 41/1000 | 767/1000, passes | no | | (1/2,1/2) | 37/50, 11/20 | 2563/2000=1.2815, fails | 39/40, passes | no | | corner (19/25,19/20) | 1, 1 | 39/20=1.95, fails | 1691/1000, fails | no | | diagonal delta=nu | -- | passes iff delta<1387/3900=0.35564 | passes iff delta<20/39=0.51282 | -- | **What the log-log row does and does not mean.** T_11 on a single box is a divisor-correlation object, and Corollary 1 does give it an asymptotic on a region 12.97 percent of the domain larger than the current one. Summed over *all* boxes with d>U, e>Y, T_11 reconstructs a truncated form of the twin sum itself, so this is not a statement that the easy part got easier: it is a statement about individual boxes, on three-quarters of which the other three pieces are uncontrolled. It changes nothing about the region controlled for R, which is governed by the worst piece. Note also that the diagonal constraint a+b<40/39 is the same constraint that [endpoint-fourier.md](endpoint-fourier.md)'s use of Bettin-Chandee **Theorem 1** already forces (recorded in the wave review of residual-coverage). That is not a coincidence: Corollary 1 is proved from Theorem 1 and inherits its (N_1N_2)^(7/20)(N_1+N_2)^(1/4) bracket. ## 5. Splitting by the size of the moved prime power adds nothing The remaining question is whether the small-r pieces can go to Corollary 1 while the large-r pieces go to the existing dispersion estimates. **What the existing notes give for a fixed band, derived here.** The owning notes state their bounds for the full expanded box, not per band. The per-band version is an immediate re-application, not a new estimate: the band coefficient \[ A_1^{(s),\rho}(\ell)=-\sum_{r\mid\ell,\ r\sim x^{\rho},\ 2\le r\le W} \mu(\ell/r)(\ell/r)^{-s}1_I(\ell/r)\Lambda(r) \] is pointwise bounded by sum_{r|l}Lambda(r)=log l, since it is a subsum of nonnegative Lambda weights against coefficients of modulus at most one, and its support is l<=1 and arbitrary bounded coefficients, so it applies verbatim with M=x^(delta+rho), N=x^(nu+zeta), giving right-orientation budgets \[ \tfrac12(1+\delta+\rho),\qquad \tfrac12(\delta+\rho)+\tfrac32(\nu+\zeta),\qquad \delta+\rho, \tag{10} \] and the left orientation swaps them. Likewise [residual-coverage.md (9)](residual-coverage.md) per band replaces W by x^zeta and N by x^(delta+rho). Summing O(log^2 x) bands with a uniform margin is free. At the top band rho=w, zeta=v, (10) reproduces exactly the recorded conditions delta<19/25 and delta+3nu<161/100 -- verified on a rational grid in the validator. **Why no split helps.** Every per-band budget above, and the Corollary 1 exponent (7), is nondecreasing in both rho and zeta (checked on a grid). So each estimate's per-band admissible set is downward closed in (rho,zeta). A split covers a point (delta,nu) exactly when every band is admissible for some estimate; by downward closedness that happens exactly when the **top** band rho=w, zeta=v is admissible for some estimate. The set of such (delta,nu) is the union of the current region with the all-pieces Corollary 1 region, and by (9) the latter is contained in the former. **The split region equals the current region.** Verified on a rational grid. Two things this does not close. It does not exclude a decomposition of beta other than log-minus-Lambda, and it does not exclude an estimate that uses the band structure jointly rather than band by band. It closes only the specific "route small r here, large r there" proposal. For the record, at the next local target (8/25,9/20) Corollary 1 does cover rho<223/1100 when zeta=1/20, and zeta<3/1700 when rho=6/25; the top band misses by 41/1000 in the exponent. Covering a sub-band does not reduce the saving needed on the binding band, so this does not lower the >3/50 requirement of [grouped-divisor-moment.md (21)](grouped-divisor-moment.md). ## 6. Effect on the global consumer, and the corner Because the controlled region is unchanged, the cuts (17)-(19) of [grouped-divisor-moment.md](grouped-divisor-moment.md) are unchanged: **W_dagger does not shrink**, E_dagger is the same exact weighted compatible CRT endpoint sum, and (20) reads exactly as before. The uniform product threshold is still every fixed exponent below 19/25, and the witness (8/25,11/25) of grouped-divisor-moment §5 is still outside the all-pieces region of §4 (22*(14/25)+17*(49/100)=413/20>20). Its log-log piece alone is inside the log-log region, which is exactly the point of the previous paragraph: one controlled piece out of four leaves the witness uncontrolled. The corner (delta,nu)=(19/25,19/20) fails, and it fails for a reason no choice of smoothing repairs. At delta=19/25 the cofactor k=n/d has size comparable to V, and beta_V(k)!=0 forces the prime-power part of k to exceed V. In the log-minus-Lambda decomposition that means the whole mass sits in the top band rho=w, so A=a=1 and (7) gives at least 17/20*(1+B)+1/4>=11/10>1. In the direct decomposition k=r*k' with r>V the same thing happens with N_2=dr>x^(19/25+6/25) and m_1=k' short, so again A tends to 1. The literature scout's phrasing -- that Corollary 1 fails "on smoothness" -- is therefore right at the corner for a sharper reason than smoothness: at the corner the weight cannot be moved to either side without driving one coefficient exponent to 1, and the error term then exceeds x. This is the same a=1, b=1 edge that the wave-2026-09-06 reachability report (a session working file, not part of this repository) identifies in its Q1 as unreachable by nonzero-kernel improvement; it is reached here by a different route. ## 7. Was Corollary 1 already applied here? No. The distinction matters because Theorem 1 of the same paper is used repeatedly in this repository. | where | what was imported | to which object | |---|---|---| | [endpoint-fourier.md §5](endpoint-fourier.md), [endpoint-pairing.md](endpoint-pairing.md), [coefficient-structure.md](coefficient-structure.md), [dispersion-range.md](dispersion-range.md), [prime-power-dispersion.md](prime-power-dispersion.md) | Bettin-Chandee **Theorem 1** and Remark 1 | the post-Vaaler trilinear Kloosterman fraction with the endpoint phase | | [small-divisor-kernel.md §5A](small-divisor-kernel.md) | Bettin-Chandee **Theorem 1** and Remark 1; DFI 1997 bound (1.1) as quoted by them | the post-reciprocity kernel (7) of that note; priced at block exponent 129/125 and 1267/1200, adding no region | | [structural-literature-audit.md §3](structural-literature-audit.md) | §3B is Blomer-Pascadi and Pascadi; §3F is Friedlander-Iwaniec's asymptotic sieve | neither mentions Bettin-Chandee at all | | the wave-2026-09-06 literature scout report, section F (a session working file, not part of this repository) | Bettin-Chandee **Corollary 1**, read at source | our determinant equation; recorded the smoothness hypothesis as the blocker, did not attempt the transfer | **New here:** the observation that the smoothness hypothesis is removable by the log-minus-Lambda identity once k>V and t>Z are deleted, the resulting exact four-piece decomposition (1), the pricing (7)-(9), and the split argument of §5. All of it is negative. **Duke-Friedlander-Iwaniec, primary sources.** Both attempts to reach the primary text failed today, and that is a statement about the fetcher: *Bilinear forms with Kloosterman fractions*, Invent. Math. 128 (1997) 23-43 (Springer, DOI 10.1007/s002220050135) returns the paywall page, and no open copy of *Representations by the determinant and mean values of L-functions* (1995) was located. Their statements are taken from Bettin-Chandee's own text, which was read at the primary source: DFI 1997's bound is B_a(M,N)<<||alpha||||beta||(a+MN)^(3/8)(M+N)^(11/48+epsilon) (their (1.1)), and DFI 1995 obtained the determinant corollary with the error (eta R)^(19/8)||alpha||||beta||(N_1N_2)^(3/8)(N_1+N_2)^(11/48+epsilon)(M_1M_2)^epsilon. Priced the same way, the DFI 1995 version needs (7/8)(A+B)+(11/48)max(A,B)<1: on the diagonal that is 48/95=0.505263 against Bettin-Chandee's 20/39=0.512821, and its all-pieces region is contained in the Bettin-Chandee one. The containment is not pointwise domination of the two brackets -- 7/8>17/20 but 11/48<1/4 -- so it needs its line. With S=A+B and M=max(A,B), \[ \Bigl(\tfrac78S+\tfrac{11}{48}M\Bigr)-\Bigl(\tfrac{17}{20}S+\tfrac14M\Bigr) =\tfrac1{40}S-\tfrac1{48}M \ge\Bigl(\tfrac1{40}-\tfrac1{48}\Bigr)S=\tfrac S{240}>0, \tag{11} \] using M<=S. So the DFI 1995 exponent exceeds the Bettin-Chandee exponent at every (A,B), its admissible set is contained in (8)'s, and it likewise adds nothing. Reading the DFI originals would not change the verdict. ## 8. Uncontrolled sectors, and what would change the verdict Everything in E_dagger remains uncontrolled; §§4-5 add no rectangle to [grouped-divisor-moment.md (19)](grouped-divisor-moment.md). Within this note, two dependencies are cited rather than rewritten: the identification of (4) with M_> reuses [endpoint-fourier.md §3](endpoint-fourier.md) and the excluded-prime Mobius mean, and the per-band budgets (10) are a re-application of grouped-divisor-moment (1),(3) rather than a statement in that note. Neither affects the verdict, which is driven by the error term. The verdict would change if any of the following were established, and none is: * an error term for the determinant equation of the shape ||alpha||*||beta||*(N_1N_2)^gamma*(N_1+N_2)^kappa with a smaller exponent. At the next target (8/25,9/20), where a=14/25 and b=1/2, the requirement is (1/2+gamma)(a+b)+kappa*a<1. Holding kappa=1/4, that needs gamma<33/106=0.31132 against Bettin-Chandee's 7/20; holding gamma=7/20, it needs kappa<99/560=0.17679 against 1/4. Those are the same order as the improvements [small-divisor-kernel.md §5A](small-divisor-kernel.md) computes for Theorem 1 on its own object at the same box (gamma<9/28, kappa<27/140), which is what one expects from two consumers of the same bracket. At the benchmark (2/5,2/5), already controlled by other means, the deficit in (7) is 173/2000; * a version averaged over Delta (Bettin-Chandee point at one, citing Bettin-Chandee-Radziwill), which our fixed Delta=2 cannot use; * a decomposition of beta_V other than log-minus-Lambda that keeps both coefficient exponents below the boundary of (8) at the top band; * an estimate that exploits the band structure jointly rather than band by band. A failed upper bound closes its own scope. Nothing above shows that the determinant formulation is a dead end, that Mobius cancellation is necessary, or that any other method is obstructed. ## 9. Validation [determinant-corollary-validation.js](determinant-corollary-validation.js) runs in 0.2 s and carries its output under `research/qc/embed.js` custody. It checks, in exact BigInt rationals, the equivalence of the two forms of (8), the half-plane form of both regions, the containment (9), the added areas and their vertices, the named points, the band thresholds at (8/25,9/20), the monotonicity of every per-band budget, that the top band of (10) reproduces delta<19/25 and delta+3nu<161/100, that the current union is unchanged by the top-band Corollary 1 condition, and the DFI 1995 comparison. In finite enumeration it checks beta_W(k)=log k-sum_{r|k,r<=W}Lambda(r), that beta_W vanishes below W, identity (1) on five parameter sets, and |B_V(n_2)|<=log n_2. Four negative controls fire: swapping max and min in the error term, using (delta,nu) in place of (a,b), re-imposing m_1>V and m_2>Z inside the four pieces, and dropping the norm factor from (7). These certify exponent bookkeeping and finite identities. They cannot establish an asymptotic saving, an effective onset, or anything about the sign or size of E_dagger.