21 September 2026. The new shifted computations have passed at the unequal parity thresholds specified in the final section, with exact-witness replays at higher precision. The abstract common-threshold argument is retained below to make the inertia logic explicit. The existing v1.16 result already supplies strict positivity of the complete fixed-window form at lambda=3.
Let W denote the canonical self-adjoint Weil operator at lambda=3, with its established logarithmic operator/form domains and compact resolvent. It commutes with reflection and splits into its even and odd sectors. Fix an exact rational threshold a>0. In a chosen sector decompose
[ W-aI=\begin{pmatrix}F-aI&B^*\B&T-aI\end{pmatrix},\qquad T_a=T-aI. ]
The off-diagonal sequence b_n is unchanged. Every actual diagonal d_n and every inverse-majorizing weight g_n must be replaced by d_n-a and g_n-a respectively. In particular the common-cut loss constants are retained; shifting does not authorize a new distant-cut denominator for the whole inverse.
For the same exact frozen finite-support witness Z,
[ K_{Z,a}=K_{Z,0}-a(I+Z^*Z),\qquad R_a=B-T_aZ=R_0+aZ. ]
The second identity changes all intermediate residual rows where Z is supported. Outside that support the residual is unchanged. The remote moment matrix, including its complete leading Gram and geometric remainder, is unchanged because G=I-Z and all remote rows see only unchanged off-diagonal entries. The new finite lower matrix is
[ L_a=K_{Z,a}-\sum_{N<n\le J}\frac{R_{a,n}^*R_{a,n}}{g_n-a} -\frac{U_J}{g_{J+1}-a}. ]
Here n runs over normalized positive-index parity rows, and U_J is evaluated in the full signed-index embedding. If g_{N+1}>a, the existing inverse-order proof gives the exact form-order inequality
[ S_a:=F-aI-B^*T_a^{-1}B\succeq L_a. ]
Subtracting aI only from the final unshifted lower matrix would be invalid: it misses the aZ residual and the change of inverse weights.
Assume the even lower matrix L_a has one strictly negative eigenvalue and all its other eigenvalues strictly positive; assume the odd lower matrix is positive definite. A complete interval LDL factorization whose every pivot excludes zero and whose sign count is exactly one negative certifies the stated even inertia. Pivot sizes do not certify eigenvalue sizes. A routine that stops after the first negative pivot is not an inertia certificate.
For the even matrix, finite-dimensional min–max and S_a>=L_a show that its second eigenvalue is strictly positive. Thus S_a has at most one nonpositive eigenvalue. This stronger conclusion is needed: merely bounding the number of strictly negative eigenvalues would not by itself exclude a zero eigenvalue at a.
Let u be the existing finite-supported, normalized, even rational candidate, embedded in the complete space. Its certified Rayleigh value is
[ \alpha=\langle Wu,u\rangle<r:=5.32\,10^{-38}. ]
The finite compression evaluates this full quadratic form exactly because u has no omitted input coefficients. There is no finite-to-infinite approximation in this Rayleigh statement. If r<a, then the even shifted form is negative on u. Since T_a is positive, the square-completion identity
[ q_a[x+y]=\langle S_ax,x\rangle+ |T_a^{1/2}(y+T_a^{-1}Bx)|^2 ]
forces S_a to have a strictly negative direction. Therefore its first eigenvalue is strictly negative and its second is strictly positive. In particular S_a is invertible, has precisely one negative eigenvalue, and has no zero eigenvalue. The odd Schur matrix is positive definite by its lower matrix.
The square-completion map is boundedly invertible on the Hilbert direct sum and preserves the form domain, since T_a^{-1}B maps the finite head into the operator domain of T_a. It identifies the negative index of q_a with that of S_a. Also
[ \ker(W-aI)\cong\ker S_a, \qquad x\longmapsto x-T_a^{-1}Bx. ]
Consequently the complete even operator has exactly one eigenvalue below a, counted with multiplicity, and no eigenvalue at a; the odd operator has none at or below a. Compact resolvent excludes continuous spectrum or finite accumulation at the threshold. Writing the complete eigenvalues in increasing order as mu_0<=mu_1<=…, the conclusion is
[ 0<\mu_0<r<a<\mu_1. ]
The initial strict positivity is the previously proved v1.16 result. The new shifted argument supplies simplicity and evenness of the actual complete ground state and a gap mu_1-mu_0>a-r. For a=10^-36 this gives a gap greater than 9.468 x 10^-37; for a=10^-34 it gives a gap greater than 9.99468 x 10^-35. These numerical statements remain conditional until the corresponding shifted inertia certificates pass.
The same argument can alternatively avoid invoking compact resolvent for the gap: the positive part of the finite Schur matrix and the positive entire tail have a positive common lower bound; the bounded square-completion map transports this to a positive bound on a closed codimension-one subspace. Min–max then gives a strictly positive second shifted spectral value. Compact resolvent is already established here and is the simpler formulation.
The two parity thresholds need not be equal. It is enough that the even lower Schur matrix has one negative pivot at a_even, the odd lower matrix is positive at a_odd, and both thresholds exceed the even Rayleigh upper bound r. Then the ground is simple and even; the full second eigenvalue exceeds min(a_even,a_odd). For an exactly even trial vector, every overlap estimate below may nevertheless use a_even: its spectral expansion has no odd component. This preserves the stronger even-sector angle bound when the odd certificate requires a smaller threshold.
Let xi_0 be a unit complete ground eigenvector and P_0 its rank-one orthogonal projection. Prior positivity and the new bound mu_1>a yield, for every unit vector v in the form domain with Rayleigh value beta,
[ \beta\ge a|(I-P_0)v|^2. ]
For the exact rational source candidate u this gives
[ \sin\angle(u,\xi_0)<\sqrt{r/a}. ]
Thus the overlap conclusion concerns the actual infinite-dimensional ground state, not merely its finite compression. The bounds are sin(angle)<0.230652 at a=10^-36, or <0.0230652 at a=10^-34. Without the previous nonnegativity, the formula would require an independent lower spectral bound ell and become sin^2(angle)<(alpha-ell)/(a-ell); a possibly negative unknown ground would not permit the displayed r/a bound.
There is a modest improvement from the already certified finite inverse solve. In the N=64 compression let
[ A=C-\alpha I,\quad z=A^{-1}r_{\rm fin},\quad q=|z|, \quad E=\langle z,r_{\rm fin}\rangle. ]
The vector v=(u-z)/sqrt(1+q^2) is finite-supported and has the exact full-form Rayleigh value
[ \beta=\alpha-\frac{E}{1+q^2}. ]
| This follows from z perpendicular to u, <Wu,z>=E, and <Wz,z>=E+alpha q^2. Existing enclosures give alpha<5.312382 x 10^-38, E>1.668982 x 10^-38, and q<0.000216. Elementary rational arithmetic therefore gives beta<3.644 x 10^-38. Also | P_u-P_v | =q/sqrt(1+q^2)<0.000216. The triangle inequality for rank-one orthogonal projections gives |
[ \sin\angle(u,\xi_0) <0.000216+\sqrt{3.644\,10^{-38}/a}. ]
Conservative decimal bounds are 0.19112 at a=10^-36 and 0.01931 at a=10^-34. This does not assume that the finite ground is the complete ground, nor that either is the exact PSWF source.
certify_ground_overlap_constants.py checks these scalar inequalities against the existing interval certificate. At 256 bits it certifies the conservative trial Rayleigh upper value 3.643400077868021 x 10^-38 < 3.644 x 10^-38 and the two sine upper values 0.191108639983841 and 0.019305263998385. The scalar output is ground_overlap_scalar_certificate.json. It does not supply the separate shifted inertia hypotheses.
| For another normalized vector v_true with a rigorously bounded projector distance | P_true-P_u | <=delta, add delta to either actual-ground angle bound. The existing certified approximation of the exact N=64 source projection may be used this way if its normalization error is retained. The full unprojected source additionally needs its omitted ordinary-norm tail. Point evaluation is unbounded in ordinary L2, so none of these angle bounds establishes endpoint recovery or endpoint relative error. |
| A genuine full-operator residual bound epsilon= | (W-beta)v | supplies the additional standard estimate sin(angle(v,xi_0))<=epsilon/(a-beta), provided beta<a. Only the residual of the complete operator is admissible; a truncated residual cannot replace it. This optional estimate has not been evaluated here. |
The parent’s 768-bit run at a=10^-34 certified exactly one negative even lower-matrix pivot (index 24), but the odd lower matrix also had one negative pivot (index 85). The latter does not pass the needed odd positivity gate. It is not a proof of an actual odd eigenvalue below 10^-34. The same exact even witness passed its 896-bit replay with the identical sign count and witness hash. The proposed resolution is to retain the strong even threshold and use a smaller odd threshold above the trial Rayleigh value; the overlap observation above then retains its strong even threshold.
The same exact frozen witnesses were replayed successfully at 896 bits:
| Sector | Threshold | Dimension | Negative pivots | Positive pivots |
|---|---|---|---|---|
| Even | 10^-34 | 257 | 1 (index 24) | 256 |
| Odd | 10^-36 | 512 | 0 | 512 |
Every pivot excludes zero. The 768-bit and 896-bit witness hashes match exactly in each sector:
993c5b325c1cc7dce64c1dbdab9384dd54dea6a480cce8739cf623e9e8bce43c.4054507110d6a806c3dfa926c716c1692de4c9b7dc7801263d50b92cbd8dfdea.The shifted script updates the true diagonal before every block or residual-row construction, subtracts the exact rational shift from the full diagonal lower function, retains all frozen-solve residuals, and correctly continues interval LDL through negative pivots. The remote moment bound is unchanged for the justified disjoint-support reason. No blocking issue was found in the proof or implementation. The replay shares the same Arb arithmetic and code; it is not a separate software implementation or external review.
Combining the accepted inertia counts, previous complete positivity, and the new finite-supported trial proves that the complete lambda=3 ground is simple and even, with
[ 0<\mu_0<3.644\,10^{-38},\qquad \mu_1>10^{-36}. ]
Thus its full spectral gap exceeds 9.6356 x 10^-37. Within the even sector all non-ground eigenvalues exceed 10^-34. The exact rational candidate and the previously certified normalized exact N=64 source projection each have sine angle less than 0.01931 to this complete ground direction.
These conclusions are confined to lambda=3. No ground-state endpoint estimate, unprojected-source comparison, uniform growing-window result, G2 closure, or RH proof is inferred. The earlier odd test at 10^-34 remains an unsuccessful sufficient lower-bound test and is not evidence of a negative or unusually low odd eigenvalue. No further sign experiments were run after the successful replay.