Exact identities and proved implications with explicit hypotheses. No numerical experiment or new certified numerical bound is claimed. G2 and RH remain open. The v1.43 enclosure is unchanged.
At a fixed window, define the minorant loss
L_a[f] = integral (beta_a - m_a) |Fourier(f)|^2, where
m_a = -D_a + sum w_j 1_Gj <= beta_a. For each source-admissible exact
pencil vector, the weighted concentration test is exactly
L_a[v_k]/q_a^+[v_k] <= nu_k + eta_a ||v_k||^2/q_a^+[v_k].
At eta_a=0 the relative loss allowance is nu_k. With allowed negative error, the eta term must be retained. Small nu_k alone does not refute the weaker bounded-error goal or force relative accuracy nu_k.
The diagnostic uses the uncompressed 49-dimensional even head (N=48).
It does not establish source orthogonality. In exact pencil coordinates,
write R for the loss Gram, G for the ordinary Gram, and b_k=<v_k,u_a>.
The entire admissible head requires diag(nu)-R+eta_a G >= 0 on ker b.
Diagonal tests alone miss cross terms. Projecting raw vectors changes their
ratios. If the first twelve exact values were below 1e-8, their span would
contain at least eleven source-admissible dimensions with ratio below 1e-8;
the reported threshold remains diagnostic, not a verified hypothesis.
These are single-window head statements. The live sufficient criterion still requires the complete source complement on a cofinal family with uniform errors: eta_a -> 0 in v1.31, or uniformly bounded errors in both parities with the residual/coupling hypotheses of v1.36. No such estimate, minorant construction or transfer from the finite head is supplied. This is a quantifier finding, not a failure of the criterion or the object. The task stops here as requested.
pencil_concentration.tex: the exact proposition and proof integrated in the manuscript.scope_review.md: source-admissibility, cross-term and quantifier checks.provenance.json: hashes of the existing diagnostic; no diagnostic was rerun.build_report.json: actual full-manuscript build and source hash.research-report-2026-09-21-v2.html: current styled user-facing report (NS-24).research-report-2026-09-21-v1.html: preserved report before task-ID reconciliation.The diagnostic citation is evidence/diag_true_symbol/results.md, section 7,
with pencil.py and pencil_output.txt. Its lambda=4 N=48 figures (twelve
nu_k below 1e-8, nu_0 about 2.9e-65) are numerical illustrations only. The
positive/negative forms in the proposition are exact complete Fourier
integrals, not their floating-quadrature approximations.
No new window, tail metric, numerical run or zero-enclosure refinement. The two NS-1 evidence gaps remain OPEN. This translation does not recover the missing v1.31 scripts or certify the old concentration diagnostics.
Verify and build from the repository root:
python3 tools/verify_manifest.py v144
./manuscript/build.sh