# Agent 3 — smooth modular completion and the missing sign margin Reviewed mathematical baseline: `5434d96f0034f842b0229f1d69dbbf3fb7fea270` (merged PR73). Working-tree HEAD when read: `971c8bd02c834cf31ab98ea2a50060289fbece40` (the coordinator's audit-scope record). The coordinator refreshed the remote and reviewed the complete 104-node register, checkpoints, freezes and missing-evidence ledger. This lane read the current agent instructions, proposal gate, ZERO-GEOMETRY scope, NS100/101 arguments, PR72's source/transfer notes, PR73's current report, and the September 29 Jacobi-product admission report. This is a paper applicability/error-budget audit, not a replay of historical proofs or certificates. No numerical or symbolic scan was run. **Wall check: Same open gap.** Closest results: NS101, PR72/73 and the September 29 product audit. What changes: a particular smooth reciprocal completion permits an explicit, uniform-in-height error comparison; it has no reflected-prefix cusp. What does not change: no independently proved lower bound for the complete first Laguerre expression is supplied. The completion below is a supporting comparison, not an admitted sign method or a thaw of ZERO-GEOMETRY. ## Exact question and dependency Can a smooth completion of the original theta coefficients both evade PR73 and transfer a lower first-Laguerre bound to the original function? Write ```text Theta(x) = sum_(n in Z) exp(-pi*n^2*x), h(u) = exp(u/2)*Theta(exp(2u)), D = (d^2/du^2 - 1/4)/4, phi = D h, X(r) = integral_R phi(u)*exp(i*r*u) du, L1[X] = X'^2-X*X''. ``` The exact dependency will be ```text L1[X_M](r) >= B_M(r) for every real r, with the specified complete error B_M (or a pointwise choice M=M(r)) => L1[X](r)>=0 for every real r => positive definiteness of the complete associated kernel C. ``` This is only the first-Laguerre target, not the all-order criterion or RH. The lower-reference inequality on the first line is absent. The work below makes the error and the missing lower margin explicit; it does not rename that margin an estimate. ## 1. A specified completion, including the weight derivatives Fix `a=pi/4` (a completion parameter, unrelated to the Jacobi state a), and define for integers `M>=1` ```text H_M(u) = exp(u/2)*(1+2*sum_(1<=n<=M) exp(-pi*n^2*exp(2u))), w(u) = 1/(1+exp(-2*a*sinh(2u))), h_M(u) = w(u)*H_M(u)+w(-u)*H_M(-u), phi_M(u) = D h_M(u), X_M(r) = integral_R phi_M(u)*exp(i*r*u) du. ``` All derivatives of w are included in `D h_M`. This is not reflection of a half-line differential kernel and not the finite Jacobi-product object from September 29. Because `w(u)+w(-u)=1`, h_M is even and smooth. It is holomorphic in a neighborhood of the closed strip `|Im u|<=pi/8`, as verified below. Thus every odd boundary derivative vanishes. There is no distributional atom at zero and no algebraic cusp tail of PR73's kind. In the limited sense of reciprocity, it is a modularly compatible completion: `Theta_M(x)=x^(-1/4)*h_M((log x)/2)` satisfies `Theta_M(x)=x^(-1/2)*Theta_M(1/x)`. It is **not** claimed to satisfy every theta modular law, to be a one-lattice Gaussian series globally, or to solve the original normalized nonlinear Jacobi initial-value problem. The complete original remainder is retained exactly: ```text T_M(u) = h(u)-H_M(u) = 2*exp(u/2)*sum_(n>M) exp(-pi*n^2*exp(2u)), Delta_M(u) = h(u)-h_M(u) = w(u)*T_M(u)+w(-u)*T_M(-u), phi-phi_M = D Delta_M. (1) ``` Here the second equality uses the original theta reciprocity `h(-u)=h(u)`. Both modular tails in (1) matter. In particular, the negative-u evaluation of a finite original sum cannot be silently replaced by the even original kernel. The original coefficients and n-squared frequencies enter the estimates of T_M below; the completion operation alone is generic and contains no arithmetic sign information. At both real infinities h_M has leading homogeneous mode `exp(|u|/2)`; its corrections and their differentiated versions have double-exponential decay. Therefore phi_M is a smooth, real, even, rapidly decreasing function. The endpoint normalization is retained, rather than assumed: ```text integral_R exp(u/2)*phi_M(u) du = (1/4)*[exp(u/2)*(h_M'(u)-h_M(u)/2)]_(-infinity)^(infinity) = (1/4)*(0-(-1)) = 1/4. (2) ``` The negative endpoint in (2) is essential. Evenness supplies the same normalization at Laplace parameter -1/2. These facts do not assign a sign to phi_M or to L1[X_M]. ## 2. A necessary warning: this completion has a negative far tail For each fixed M and real `u->+infinity`, set `y=exp(2u)`. Direct expansion of the finite sum and of w gives ```text H_M(u) = exp(u/2)*(1+O_M(exp(-pi*y))), H_M(-u) = (2*M+1)*exp(-u/2)*(1+O_M(1/y)), w(-u) = exp(-a*y)*(1+O(1/y)), h_M(u)-exp(u/2) = -exp(u/2)*exp(-a*y)*(1+O_M(y^(-1/2))). (3) ``` The remainders come from differentiable exponential/power expansions, so applying D preserves the leading term and yields ```text phi_M(u) ~ -a^2*exp(9*u/2)*exp(-a*exp(2u)) < 0. (4) ``` The original phi is positive on the real line. Its exact remainder in (1) cancels this slower, negative artificial tail (`a=0, C_j = [8/(1-exp(-beta))] * integral_0^infinity (u+v)^j*J(u) du, j=0,1,2. ``` These are explicit finite positive constants, independent of r and M. They are defined by convergent elementary integrals, not fitted or numerically evaluated constants. Here is the denominator check for the weight. For `z=u+i*eta`, `u>=0`, `|eta|<=v`, put `q=exp(-2*a*sinh(2*z))` and ```text u_* = (1/2)*asinh(log(2)/(2*a*cos(2*v))). ``` For `u>=u_*`, `|q|<=1/2`, so `|1+q|>=1/2`. For `0<=u<=u_*`, ```text |arg q| <= sqrt(pi^2/8 + log(2)^2) < pi/2 ``` using the continuously specified exponent argument, hence `Re q>=0` and `|1+q|>=1`. Consequently, throughout that half-strip, ```text |w(z)|<=2, |w(-z)|<=2*exp(-2*a*sinh(2*u)*cos(2*eta)). (5) ``` This proves the required absence of poles and provides an explicit bound, rather than treating smoothness on the real axis as a complex-strip bound. Let `y=exp(2u)>=1`, `b=pi*cos(2*eta)>=B` and `d_eta=a*cos(2*eta)>=d`. From (1) and (5), ```text |Delta_M(u+i*eta)| <= 4*y^(1/4)*sum_(n>M) exp(-b*n^2*y) + 4*exp(a)*y^(-1/4)*sum_(n>M) exp(-d_eta*y-b*n^2/y). (6) ``` For the first sum, `n^2*y>=(n^2+y)/2` and `n^2>=n` give a factor `exp(-beta*n)*exp(-(B/2)*y)`. For the second, split its exponent in halves and use ```text (d_eta*y+b*n^2/y)/2 >= sqrt(d_eta*b)*n >= beta*n. ``` In the remaining half discard only the additional nonnegative `b*n^2/(2*y)`. Summing the geometric majorant gives ```text |Delta_M(u+i*eta)| <= [4/(1-exp(-beta))]*exp(-beta*(M+1))*J(u). (7) ``` Evenness supplies the same bound with `|u|` on the other half-strip. Thus there are no omitted real tails or internal endpoint pieces. Let `Dhat_M(r)=integral_R Delta_M(u)*exp(i*r*u) du`. Contour shifting to `Im u=sign(r)*v`, justified by (7) on the vertical boundaries, yields ```text |Dhat_M^(j)(r)| <= C_j*exp(-v*|r|-beta*(M+1)), j=0,1,2, r real. (8) ``` The factor `(u+v)^j` above bounds `|u+i*v|^j`. At r=0 the same inequality follows by shifting either way or by continuity. This is a uniform all-height estimate for a specified approximation error. It is not an all-height lower bound. ## 4. Transfer to X, X', X'' and L1 Because Delta_M and its derivatives decay at both ends, integrating D in (1) produces no boundary term: ```text E_M(r) := X(r)-X_M(r) = -(r^2+1/4)*Dhat_M(r)/4. (9) ``` Put `p=r^2+1/4`, `q_M(r)=exp(-v*|r|-beta*(M+1))`, and ```text S_0(r) = p*C_0/4, S_1(r) = (2*|r|*C_0+p*C_1)/4, S_2(r) = (2*C_0+4*|r|*C_1+p*C_2)/4, epsilon_j(r,M) = S_j(r)*q_M(r). ``` Differentiating (9), rather than an upper envelope, proves ```text |X^(j)(r)-X_M^(j)(r)| <= epsilon_j(r,M), j=0,1,2, (10) ``` for **every real r and every integer M>=1**. Every r derivative is at fixed M. A later selection M(r) is pointwise selection among these inequalities; one must not differentiate the discontinuous composite `X_(ceil(r))(r)` and call those derivatives `X_M'` or `X_M''`. Expansion of the complete quadratic expression gives the explicit conditional transfer ```text |L1[X](r)-L1[X_M](r)| <= B_M(r), B_M = 2*|X_M'|*epsilon_1 + epsilon_1^2 + |X_M|*epsilon_2 + |X_M''|*epsilon_0 + epsilon_0*epsilon_2. (11) L1[X_M](r) >= B_M(r) => L1[X](r)>=0. (12) ``` The cross terms in (11) are indispensable. Small transform error alone is not a sign-transfer argument. One can remove unknown transform values even from the upper error bound. For `j=0,1,2` define the explicit original-theta majorants ```text A_j = 2*integral_0^infinity (u+v)^j * sum_(n>=1) [2*pi^2*n^4*exp(9*u/2) +3*pi*n^2*exp(5*u/2)] *exp(-B*n^2*exp(2*u)) du. ``` These constants converge, and shifting the original smooth theta kernel in the same strip proves `|X^(j)(r)|<=A_j*exp(-v*|r|)`. Expanding the difference in terms of X and E_M instead of X_M gives the alternative fully specified upper error ```text Btilde_M(r) = [2*A_1*S_1 + A_0*S_2 + A_2*S_0] *exp(-2*v*|r|-beta*(M+1)) + [S_1^2+S_0*S_2] *exp(-2*v*|r|-2*beta*(M+1)), (13) |L1[X]-L1[X_M]| <= Btilde_M. ``` There is no lower-sign estimate hidden in the definition of A_j. In particular, choosing `M(r)>=ceil(|r|)+M_0`, `M_0>=1`, gives ```text Btilde_(M(r))(r) = O((1+r^2)*exp(-(pi/4+beta)*|r|-beta*(M_0+1))) + O((1+r^2)^2*exp(-(pi/4+2*beta)*|r|-2*beta*(M_0+1))), (14) ``` with constants already specified in (13). Both rates are faster than `exp(-pi*|r|/2)`, since `beta>pi/4`. This illustrates how a growing, smooth completion could have adequate *absolute error rate*. It proves neither a useful lower scale nor nonnegativity of the reference expression. For example, if an independent argument supplied ```text [L1[X_M]](r) evaluated at M=M(r) >= m*(1+|r|)^(-k)*exp(-pi*|r|/2), every |r|>=R, ``` for explicit `m>0`, finite k and R, (13) would give an explicit eventual transfer threshold; the remaining compact interval would still require its own complete proof. No such premise is proved here. A strict margin also requires care around possible vanishing of L1: an arbitrary positive absolute allowance cannot transfer a zero margin. Even an all-height nonnegative-reference theorem by itself would not establish (12). ## 5. Why the original nonlinear trajectory does not finish (12) For the actual theta kernel the clock is `L=2u`, and the distinguished Jacobi state satisfies ```text a'=(U+a^2-1)/2, U'=2*U*(a+chi), chi'=a*chi-U, a(0)=chi(0)=0, U(0)=Gamma(1/4)^8/(64*pi^4). ``` The retained original remainder makes the complete phi exactly this kernel. It therefore retains the original odd-derivative cancellation, the nonlinear trajectory, and both factors of C. The finite h_M has only the declared reciprocal symmetry and endpoint normalization; (4) already shows that the source's positive-kernel real-orbit hypotheses do not transfer to it. Replacing (12) by real monotonicity of the original state would again drop the signed oscillatory comparison. The source [Planat–Sole, arXiv:2608.19160v1](https://arxiv.org/html/2608.19160v1), Sections 5 and 7, distinguishes the original Jacobi orbit and gives real-coordinate curvature/Turán conclusions. Its quoted theorem is not an all-frequency lower bound for the Fourier first-Laguerre expression. No external finite certificates were replayed here. The classical associated-kernel/Laguerre setting is also explicit in [Csordas, arXiv:1309.0055](https://arxiv.org/abs/1309.0055); a new completion does not create a new criterion. The analysis above uses original arithmetic to control a complete approximation error. It does **not** use the nonlinear arithmetic to estimate the missing lower-reference sign. This distinction is precisely why the output remains a supporting comparison rather than a candidate for admission. The coordinator's separately reviewed Bessel reference is a different object. Nothing above estimates `X-B_a` or its first two derivatives relative to a Bessel Laguerre margin. A positive first-Laguerre formula for that reference cannot be substituted for the missing lower bound for X_M in (12). This lane supplies no all-height original-theta-to-Bessel comparison. ## 6. Matched controls and overlap - **PR73:** its hypothesis is an even-reflected fixed finite half-line theta sum with a nonzero odd boundary derivative. This h_M is smooth even before D is applied, so that cusp theorem does not apply. This is an explicit hypothesis change, not a larger M in the rejected family. - **September 29 finite Jacobi product:** that completion has a retained atom at zero and a different differential density. Its negative radial-sign diagnostic does not decide the smooth family above, and a radial sign away from the imaginary axis is not the same as first Laguerre at x=0. Its normalization warning is addressed by (2), not by dropping an endpoint. - **NS101 reciprocal mixture:** it shares smooth reciprocal completion and general transform calculus but changes the exact Gaussian lattice and fails the original normalized Jacobi ODE (PR72). The general error-transfer lemma would apply whenever analogous error hypotheses were proved. Such a lemma is permitted to be generic; the missing sign premise is separate. The mixture's off-axis zeros do **not** by themselves prove failure of its first-Laguerre sign, and none is claimed. - **NS100:** its original complete negative slice excludes per-slice positivity, which is never assumed in (11). Its log-concave control lacks the original theta data and cannot be imported as a first-Laguerre counterexample merely from its radial-sign failure. - **Davenport–Heilbronn:** its conductor, coefficients and gamma/theta law differ from the original integer-square theta input used in (1), (6) and the Jacobi IVP. It is not a matched counterexample to those original-data premises. The generic perturbation inequality is valid for it too if its error hypotheses hold; it never supplies its sign. This is an analytic admission/scope assessment. No control scan, sampled pass or new numerical sign claim is recorded. ## Disposition and bounded next budget **No admitted sign method emerges.** We have an explicit cusp-free reciprocal approximation with all-height error bounds (10) and (13), complete endpoint normalization, and a warning that its real differential kernel has a negative artificial tail. Constructing and accurately approximating the complete original object is therefore separable from estimating its sign. The missing lower bound in (12) is still the NS100/101 full first-Laguerre comparison. Success of a future independent lower-reference estimate meeting (12), with all heights and any zero margins addressed, would prove the stated first-Laguerre target only. Failure for this smooth reference family would stop that comparison; it would not close smooth completions or the exact original nonlinear evolution generally. There is currently no reason to fund a height/M scan. A bounded next step, only if this comparison is needed by another specified estimating method, is a one-hour independent paper check of (5)–(13) and of the proposed method's actual lower-margin hypothesis. No automatic search of adjacent weight parameters is proposed. **Final wall check: Same open gap.** The supporting approximation rate is explicit; the complete arithmetic sign remains unestimated. No research row, manuscript change, computation, commit, push or outreach. RH, G2 and the cofinal signed-arithmetic lower bound remain open.