Exact identities and proved implications, with numerical illustrations
kept separately under evidence/diag_beta_lattice_identity/. G2 and RH
remain open. No new window, head, Weil-tail metric or zero enclosure.
prop:v146-beta-explicit gives the unconditional shifted Perron formula
for the exact beta_a. Its archimedean/log-derivative/lower-limit block
cancels by the functional equation, without RH. The remaining terms are
the complete zero sum, trivial zeros and the strict-cutoff endpoint;
a finite zero prefix retains an explicit Perron remainder.
The saved cutoffs 9,16,36,64 are integers. The endpoint term is +Lambda(x)/sqrt(x) cos(xi log x); dropping it changes the answer. The lattice phase gives alternation, but its cosine and sine envelopes vary. Equal crests and exact quarter-integer zeros are not identities. The zero-frequency amplitude is a weighted arithmetic remainder, not psi(x)-x. The naive zero sum omits the endpoint and trivial terms, shifted denominators and finite-sum remainder, and confuses amplitude with crest-minus-trough.
All forty old low-frequency locations and fourteen deep-trough/centroid locations at lambda=4,8 were reproduced against the unchanged original symbol. The analytic sum uses symmetric height/Perron ordering. The finite numerical evaluation uses Gaussian Perron residues, epsilon=.025, and exact desmoothing to the original sharp cutoff. Sufficient empirical counts of positive zeros (plus conjugates) are 83/83/80/80 at the ten low samples for lambda=3/4/6/8, and 530/609 for the selected deep samples at lambda=4/8. All corresponding residuals are below 1.5e-7 at both precisions. These counts are sample- and convention-specific, not sharp-sum counts or interval certificates. The reference run uses 1200 positive zeros, checks 1000-to-1200 stability, and checks the finite zero count. The three-decimal legacy output is checked by rounding; 1e-6 agreement is against recomputation. Full float NS-22 trough depths agree within 7e-13. Two binding centroids lie where beta is positive, so centroids are not silently relabelled as trough locations.
This is the explicit formula restated, including at the deep troughs. It supplies no new positivity or cofinal zero-spacing/density theorem.
For a finite step minorant m, eta(m) is the negative spectral floor of P F* m F P, not the norm of F*(beta-m)F. The correct packet bounds are
eta(m) >= [(b+B)c_{a,u}(E) - B]_+
when m<=-b on E and m<=B globally, and
eta(m) >= [c delta - Q]_+
when beta-m>=delta on E and an admissible unit packet has E-mass>=c and original Weil energy<=Q. Positive spill/original energy cannot be omitted. A two-value countermodel shows that depth growth and a fixed concentration fraction alone do not imply divergent eta; it is not a counterexample for the arithmetic beta.
For carrier aomega>=10pi, a cosine packet has mass at least 7/10 on a full-cell-width mirrored band and 11/25 on a half-cell-width band. If its source overlap is <=1/20, the projected bounds are 3/5 and 3/8. The fixed-window overlap cost is explicit; the known source convergence makes it vanish cofinally. Odd sine packets obey the same uncompressed bounds and have no constraint from the even source. A single half-cell’s even concentration norm is <0.509, so the diagnostic 0.6-0.8 for the whole negative set cannot be used for an isolated half-cell.
For K distinct values, a proved conditional bound is
eta(m) >= [c D_a/(2K) - Q]_+.
It assumes a source-admissible bounded-energy packet whose pushforward
Fourier-mass distribution under beta dominates (c/D_a) dt throughout
[-D_a,0]. Uniform c>0 and Q
The route therefore remains blocked, not closed. The weighted concentration proposition is still a valid sufficient criterion. This is a gap in the obstruction proof, not a failure of the object.
Actual build: 232 pages; 0 undefined references; 0 duplicate references; 0 overfull boxes. The new Python diagnostics report 0 errors and 0 warnings.
beta_lattice_identity.tex: exact identity and initial numerical discussion.beta_lattice_deep_checks.tex: deep samples, convention and zero counts.step_minorant_scope.tex: explicit packet bounds, conditional level theorem,
countermodel and missing arithmetic hypotheses.scope_review.md: internal algebra, scope and quantifier audit.provenance.json: hashes of new diagnostics and unchanged source inputs.build_report.json: actual final build counts and source hash.research-report-2026-09-21-v1.html: user-facing report.The two NS-1 evidence groups remain OPEN; this work does not recover their
original scripts. Verify with python3 tools/verify_manifest.py v146 and
build with ./manuscript/build.sh from the repository root.