The interval computations below certify strict positivity of the complete canonical semilocal Weil operator at the exact window lambda=3, in both reflection sectors, using the proved operator/core, matrix, and tail identities in the working manuscript. All Fourier modes are included: the infinite complement is bounded analytically rather than discarded.
This is a computer-assisted fixed-window result. It does not establish G2 uniformly as the window grows, ground-state parity or simplicity, the required endpoint/graph-normalized asymptotics, or RH. The interval arithmetic is a proof enclosure, whereas the earlier floating-point spectral plots were numerical evidence only.
Here N is the head cutoff, M the finite support cutoff of the lifted columns, J the last explicitly computed output row, and r the number of terms in the remote geometric expansion. Both runs use 768-bit Arb arithmetic, 128 terms in the special-function assembly series, exact lambda=3, and tau=1/1000000 in the remote Young inequality.
| Reflection sector | N | M | J | r | Head dimension | Positive LDL pivots |
|---|---|---|---|---|---|---|
| Even | 256 | 512 | 4096 | 80 | 257 | 257 |
| Odd | 512 | 1024 | 4096 | 100 | 512 | 512 |
Both use diagonal inverse weights, not a common inverse coercivity constant. The minimum tail constants are respectively greater than 2.0690317004 and 1.1907506685. The geometric remainder coefficients delta^2 are enclosed near 1.6557866048e-148 and 3.2874200718e-124.
The reported minimum LDL pivots are approximately 9.1749869395e-9 and 9.6802634105e-8. These are not lower eigenvalue bounds. Positivity follows because every pivot is strictly positive; the magnitude of a pivot does not by itself quantify the spectral gap.
| Canonical operator and domain. On the trigonometric basis the exact off-diagonal entries are W_nm=(b_n-b_m)/(n-m), with bounded real odd b, and the separately defined physical diagonal satisfies d_n=log( | n | /L)+O(1). Thus W=diag(d)+[M_b,H], where the discrete Hilbert transform H is bounded. The operator is self-adjoint on the log-squared weighted sequence domain, finite sequences are an operator core, and its quadratic-form domain has one logarithmic weight. Agreement on the established trigonometric form core identifies this operator with the canonical closed Weil form. Reflection yields an orthogonal even/odd decomposition. |
Exact matrix assembly. assembly.py uses the physical logarithmic diagonal, the pole term, every nonzero prime-power shift at lambda=3, and certified digamma/trigamma formulas with explicit geometric tail radii. The shift at m=9 is identically zero. All 8,321 even/odd matrix entries through cutoff 64 overlap the independent integration certificate; see assembly_independent_overlap.json. No prolate-source or ground-state identification is assumed in this sign certificate.
Positive diagonal lower bound on the entire tail. The cosh-weighted prime Schur estimate, archimedean Hilbert-commutator estimate, and pole estimate prove T=Q_N W Q_N >= D=diag(g(n))>0 for every omitted Fourier index. With the notation of the sharp-tail proof,
g(n)=Gamma_N+log(n/(N+1))+E(t_{N+1})-E(t_n).
The common commutator and pole losses are retained at cutoff N. Hence g is increasing, and inverse order gives T^{-1}<=D^{-1}. This is the step that makes the final bounds substantially sharper than the old scalar Gamma_N^{-1} budget.
Exact finite lift and residual. The numerical intermediate inverse solve is used only to choose a finite lift Z, then frozen to exact dyadic midpoints. The code computes
K_Z=F-B^*Z-Z^*B+Z^*TZ,
R=B-TZ,
with every finite intermediate residual retained. The exact Schur complement obeys
F-B^*T^{-1}B = K_Z-R^*T^{-1}R
>= K_Z-R^*D^{-1}R.
| All infinite residual rows are included. Rows N<n<=J are evaluated directly with the correct diagonal. For | n | >J, the identity (n-m)^{-1}=sum_{j<r}m^j n^{-j-1}+(m/n)^r/(n-m) gives exact moment rows. Their Gram is bounded by the Hurwitz-zeta moment matrix plus an explicit geometric-remainder Gram. The full signed-index parity factors, including sqrt(2) and the two tails, are retained. The monotonicity of g bounds the weighted remote contribution by U_remote/g(J+1). |
Finite interval sign check and closed-form conclusion. Arb interval LDL verifies strict positivity of
L_cert=K_Z-R_near^*D_near^{-1}R_near
-U_remote/g(J+1).
Therefore the exact Schur complement is positive. The tail is already positive. A bounded invertible block-triangular congruence then proves strict positivity of the full operator in that reflection sector. Applying this in both sectors covers the whole canonical fixed-window form. The congruence argument gives a positive operator lower bound in principle; no numerical value for that lower bound is asserted here.
Run certify_infinite_schur.py with:
--N 256 --M 512 --J 4096 --r 80 --bits 768 --parity even --diagonal
--N 512 --M 1024 --J 4096 --r 100 --bits 768 --parity odd --diagonal
Report SHA-256 values at this checkpoint:
infinite_schur_N256_M512_J4096_r80_even_b768_diagonal.json
7bfbda6db0664eb4667b2567cb05f69098e3d9031dce876ad2c550c6dc868e8a
infinite_schur_N512_M1024_J4096_r100_odd_b768_diagonal.json
a52654a03664dd7dee81ce9857f34ed042dc08305b6781f63ab1b31be3ad70d5
The exact finite-lift witnesses are saved as compressed JSON. Their uncompressed SHA-256 values are:
schur_witness_N256_M512_even_b768.json.gz
993c5b325c1cc7dce64c1dbdab9384dd54dea6a480cce8739cf623e9e8bce43c
schur_witness_N512_M1024_odd_b768.json.gz
4054507110d6a806c3dfa926c716c1692de4c9b7dc7801263d50b92cbd8dfdea
The sequence caches preserve interval radii. Arbitrary recomputation may produce different enclosing intervals while proving the same inequalities; the hashes identify these particular checkpoint reports and witnesses. Independent higher-precision replay is recorded separately when completed.
The odd scalar-Gamma budget at N=256 failed even before the remote tail was added. The full odd scalar-Gamma budget at N=512/M=1024/J=2048 also failed, and the smaller-head odd diagonal budget N=256/M=512/J=4096 failed. Those were negative directions of conservative lower bounds, not negative directions of the actual Weil operator. Their records remain in the bundle. They explain why directional moments, diagonal inverse weights, and the larger odd lift were needed; none is evidence against the established final positivity statement.
At this fixed window, run shifted certificates for W-aI, for example with a=1e-36. Positivity of the odd shifted sector together with a lower even Schur matrix having exactly one negative direction, and the already certified even trial Rayleigh quotient below 5.32e-38, would prove the whole-window ground state is simple and even with a quantitative separation. Such a shift requires subtracting a consistently from every physical diagonal and tail lower bound and verifying the complete indefinite LDL inertia, not stopping at the first negative pivot.
After that fixed-window ordering check, the unresolved G2 task remains obtaining estimates that hold along the required growing-window family and interact with the physical endpoint, graph defect, and required normalization. No finite list of positive window certificates replaces that uniform proof.