Disposition: PASS for the bounded boundary/identity audit; NO ADMISSION for sign research. The original all-height estimate remains OPEN.
Reviewed remote baseline: d93aaf6b8dbe1fcbebf6fff4f6d1f71d993f4208
(PR75), with audit scope commit edc6563. I read AGENTS, the proposal
template, REVIEW_BRIEF, the complete current derivation, all 14 continuation
demands and their scopes, the frozen-input table, current README/map audit
additions, the current task record, evidence/MISSING, and reviews 06, 07,
08, 10 and 11. The parent owns the refreshed 104-node conclusions review;
this note does not claim an independent replay of every historical proof,
certificate or source theorem. No scan, new candidate proof, new agents,
commit, push or outreach was performed. Both original-recovery groups
remain OPEN.
Wall check: Same open gap. Closest: NS100/101 and PR74/75. What changes: a decaying primitive removes the separate-transform domain problem. No independently estimated signed arithmetic expression changes. There is also a known wall for the narrower inference from the shared positive primitive, envelope and unit tail to the corrected sign: the already recorded NS101 control satisfies those premises and fails that conclusion. This does not close the distinguished original theta route.
With the draft’s complete original objects, define
f(u) = 2*cosh(u/2)-h(u),
Y(r) = integral_R f(u)*exp(i*r*u) du,
p(r) = r^2+1/4,
J(r) = p(r)^2*(Y'(r)^2-Y(r)*Y''(r))
+2*(r^2-1/4)*Y(r)^2.
The domain and boundary work justifies the exact identity
J(r) = 16*(X'(r)^2-X(r)*X''(r)), every real r.
Consequently J(r)>=0 at every real r is precisely the first-Laguerre
target in new variables. It is not an independent sufficient estimate.
The draft gives useful absolute envelopes for f and its pair moments,
but these do not give a signed lower bound for J. Positivity of a spatial
kernel or of its moments does not imply nonnegativity of its cosine
transform. The term 2*(r^2-1/4)*Y^2 must remain; it changes sign at
abs(r)=1/2 and the undivided expression is regular at every real zero.
The tail regularization uses exact original theta data to identify f, but the proposed sign argument discards precisely the distinction those data could provide. For each fixed admissible beta, NS101’s complete reciprocal mixture has the same positive primitive, exponential envelope, unit tail and algebraic identities. Its known analytic negative margin transfers to J without approximation. Constants controlling derivatives may depend on beta; no beta-uniform derivative assertion is needed for this matched counterexample. Its changed arithmetic and lost original Jacobi IVP are concrete mismatches for a genuinely IVP-specific estimate, not evidence that such an estimate has been found.
The following is an admission assessment of the recorded reviews, not an exhaustive claim about all possible methods.
| Method assessed | What survives | Exact missing step / stop |
|---|---|---|
| Original Jacobi clocks (review 06) | The distinguished IVP is absent from NS101; it remains an eligible source of new arithmetic information. | C_lambda(t)=C(lambda*t) has the same positive-definiteness question for every lambda>0 and a singular dilation generator at zero. Starting from the constant kernel at lambda=0 gives no nonsingular positive evolution. No independent inward-cone estimate for every finite matrix is supplied. |
| Complete moment bounds (review 07) | The origin correction is checked; generic positive moments give a specified local interval of first-Laguerre positivity. | No sign-preserving propagation or signed margins on a cover of the entire real line follow. NS101 has the same moment positivity and still fails at its stated moving witness. Complete moments may determine a transform; their positivity alone does not estimate its all-height sign. |
| Euler logarithmic derivative (review 08) | The corrected expression can be written with the exact original prime-power coefficients and gamma/polynomial terms. | The absolutely convergent prime series is justified only for Re(s)>1. Continuation to Re(s)=1/2 is not a termwise limit or a one-sided bound. Residue/correction terms and every zero neighborhood remain unestimated. The older fixed-damping NS47 obstruction has different hypotheses and does not close this pointwise question. |
| Existing curvature/admissibility results (review 11) | The source scope and normalization can be checked; original real-kernel curvature is a distinct available premise. | The primitive has exponential tails and actual transform poles, outside the cited entire-transform admissible class. Real-coordinate curvature and coefficient inequalities do not furnish all-frequency first-Laguerre positivity. No reviewed source transfers the missing sign. |
Thus the suggested mechanisms end at a missing estimate, rather than at a candidate whose assumptions and method have been independently supplied. Requiring a complete invariant cone, an all-height logarithmic derivative lower bound, or positive definiteness of C would merely name that missing estimate unless a method for proving it is also specified.
| Registered group | Why this audit does not meet its demand or thaw condition |
|---|---|
| ZERO-GEOMETRY | The relevant group. No new original-arithmetic sign input absent from the matched NS101 properties has been estimated. The original theta coefficients/IVP are preserved in the definition, but not used in a surviving sign proof. |
| MOBIUS-COST | No estimate of the full projected signed Mobius cost or cross term, pointwise or averaged. |
| MOBIUS-MARGIN | No compatible actual residual numerator or complete grouped lower margin. |
| NB-GAIN | No bound on corrected loads or full projected Gram gain for actual optimized residuals; no nonsummable cofinal gain. The theta identity supplies no such transfer. |
| NB-SAMPLES | No integer approximants or complete signed sample-error bound tending to zero. |
| NB-GROWTH | No internal canonical signed-cancellation or subpolynomial norm-growth estimate. |
| NB-DILATION | No cumulative normalized actual dilation-defect estimate. |
| NB-CONTINUOUS | No continuous-to-integer bridge or original arithmetic zero-defect estimate. |
| WEIL-FLOOR | No complete admissible-form comparison with a common cofinal constant and source/parity constraints. A single first-Laguerre inequality, even if proved, has not been transferred to that floor. |
| KERNEL-API | No full-window arithmetic nullvector exclusion with all shifts, endpoints and poles. A one-dimensional scalar resolvent identity is not API. |
| CCM-LIMIT | No complete relative ground-selection or cofinal entire-transform control. The primitive transform is meromorphic and is not a replacement CCM limit. |
| LEVEL-PACKETS | No arithmetic level mass or independent bounded-energy source-admissible coverage. |
| HEAT-TRANSPORT | No external zero-cluster interaction or support-cost bound. The proposed Jacobi clock is a different evolution. |
| OPERATOR-BRIDGES | A valid elementary scalar Green representation does not construct any registered missing closed realization, evaluator, unshifted-sign or number-field positivity transfer. |
These are distinct unfulfilled demands. Shared RH motivation does not identify them or propagate a freeze closure from one to another. In particular NS74/83 concern NB target/rate domains and are not matched controls for this theta pair. A future NB implication would have to be given separately with those original domains and rate restrictions.
No five-item estimating candidate can honestly be filled from the current audit. The exact expression, original-object hypotheses and all-real-height range are known; the independent arithmetic estimating method is missing. The required specification for a future proposal is:
Before computation, the proposed method must also say what success and failure change and pass the actual matched-control/relevance gate. A new identity, a finite height table, a changed resolvent parameter or a source-coefficient sign comparison alone does not meet this threshold. No additional computation budget follows from this review.
The current README and ZERO-GEOMETRY scope append correctly describe boundary validation, the polynomial correction, and application of an existing control. They do not assert a thaw, new node, original sign estimate or general route closure. The continuation demands remain the same. The board’s audit row should be marked completed only after the parent finishes integrating the requested review and validation.
One presentational correction remains in the draft opening: replace “local self-review, not a fresh independent review” with the actual twelve assigned review scopes once integrated. Do not turn that into a claim of independent replication of every historical result. This same review count and scope should agree across the report, index and record.
Publication is appropriate as one bounded audit PR, with no research row, manuscript version, theorem node or thaw. The result to publish is that the transform problem has been repaired and the original signed target has not been estimated. Stop the primitive-only positivity attempt. The original nonlinear theta route remains unexcluded and requires the specific new arithmetic estimating mechanism above before further sign research. RH, G2 and the cofinal signed-arithmetic lower bound remain open.
Final wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR74/75.