# Exact Section 15 quantum-integral companion This independent companion evaluates the **cleared Barnes/double-sine integral** of Section 15. It does not replace the integrand by an Airy approximation. The historical values in `../section15/historical-table.json` are read-only comparison inputs; their rounding convention is unknown. The supported environment used for development is Python 3.12.11 with mpmath 1.3.0. No installation, optional acceleration, or python-flint is required. Run from any directory, with a new output path whose parent exists: ```sh /tmp/polyhedra-baseline-20261031/bin/python /Volumes/Lexar/lambda/devel/polyhedra/consolidated/experiments/section15_quantum/check.py --profile full --output /absolute/new/result.json ``` `smoke` runs the independent checks at 30 decimal digits and the first historical row at 40 digits. `full` checks the double sine at 40 and 60 digits, then evaluates all six historical rows and one negative-zeta continuation point (`b=1/160, zeta=-1`) under three settings: 60 digits, `C=1,T=12`; 90 digits, `C=1,T=14`; and 90 digits, `C=5/4,T=14` with two extra D terms. The contribution beyond the shorter cutoff (10 for `T=12`, 12 for `T=14`) is separately recorded, so the cutoff comparison does not subtract rounded totals. The command refuses to overwrite an output and does not write a successful result until all declared checks pass. Any change to HEAD or a declared scientific source, including either chapter, between the initial and final snapshots fails the run. Final manuscript edits must occur outside a frozen computation run. The module can also be imported without launching computation, including by its package name. ## Integrand, contour, and normalization Let `h=2*pi*b^2`, `lambda=h^(1/3)`, `eta_E=acos(1/3)`, and `eta=eta_E-lambda^2*zeta/(6*sqrt(2))`. Set `alpha=(pi-eta)/(2*pi*b)` and `Q_Barnes=b+1/b`. The exact cleared integrand is ```text F(w) = S_b(w-3*alpha)^4 S_b(4*alpha-w)^3 S_b(2*Q_Barnes-w) S_b(3*alpha-b)^2 / S_b(alpha)^6. ``` The upward contour is `v=2*pi*b*w=4*pi-5*eta_E+lambda*(C+i*t)/3`, `C>0`. With the explicitly specified measure `dw/i`, the normalized value is `Q_b=2/(9*pi) integral_R F(t) dt = 4/(9*pi) integral_0^infinity Re F(t) dt`. The five centered arguments `x=z/b+b/2` are ```text z1=(v-3*pi+3*eta)/(2*pi)-b^2/2 weight 4 z2=(4*pi-4*eta-v)/(2*pi)-b^2/2 weight 3 z3=(4*pi-v)/(2*pi)+3*b^2/2 weight 1 z4=3*(pi-eta)/(2*pi)-3*b^2/2 weight 2 z5=(pi-eta)/(2*pi)-b^2/2 weight -6. ``` Every real part is checked against `[epsilon,1-epsilon]`, `epsilon=1/100`. This centered strip is narrower than the fundamental strip of the original double-sine argument. All log factors are combined before exponentiation. On `00. ``` Let `P_0(c)=c`, `P_(n+1)(c)=-(1+c^2)*P_n'(c)`. The finite correction is ```text D_K(z) = sum_(k=1)^K a_k*b^(4*k-2)*pi^(2*k-1) *P_(2*k-2)(cot(pi*z))/2^(2*k). ``` In particular its first term is `-pi*b^2*cot(pi*z)/24`. For `a=Re z` in the strip, the absolute remainder is at most ```text b^(4*K+2)*zeta(2*K+2)/pi^(2*K+2) * (2*K)!/2^(2*K+1) * [HurwitzZeta(2*K+1,a)+HurwitzZeta(2*K+1,1-a)]. ``` This follows by bounding the complex sinh by `cosh((1-2*a)*s)` and integrating the positive geometric expansion of `1/sinh(s)`. The code uses the coarser **exact rational** bound obtained from `pi>3`, `zeta(2*K+2)<2`, and `HurwitzZeta(p,a)<=epsilon^(-p)+epsilon^(1-p)/(p-1)`. It multiplies by the sum of absolute weights, 16, for the full logarithm and chooses K until the bound is below `10^(-working_dps-5)`. The bound is global in `Im z`. It controls series truncation only; strip membership and floating-point evaluation are not certified by this rational calculation. Independent checks compare `D_K` with direct real-integral quadrature at all five arguments, compare full `log S_b` with its direct integral at two complex arguments, and check reflection, the `b` shift equation, the Lobachevsky derivative, the original-to-centered coordinate mapping, conjugation, and addition of two D terms. The nearest-strip complex argument is also checked with tanh-sinh quadrature on a different panel layout. For `L>=1`, the direct D integral's tail has the separate exact rational majorant `b^2/(6*epsilon) * 2^(-floor(2*epsilon*L))`. This follows by first bounding it by `b^2*exp(-2*epsilon*L)/(12*epsilon*(1-exp(-2*L)))`, then using `e>2`. The recorded cross-method check uses `epsilon=1/50,L=8192`; this bound covers only the tail, conditional on the numerically checked strip hypothesis. The direct quadrature still integrates over `[0,infinity)`, including this tail. The bound is a separate component diagnostic, not an omitted-tail error. ## Outer tails and evidence boundaries For the three centered real angles `Theta_j=pi*Re(z_j)`, put `c_j=cot(Theta_j)` and `r=tanh(y/2)`. The negative derivative of the real leading phase along the vertical line is `Psi(r)=4*atan(c_1*r)+3*atan(c_2*r)+atan(c_3*r)`. Its derivative has denominator `product_j(1+c_j^2*r^2)`; the code records and checks positivity of each coefficient of the numerator in `r^2`. For positive coefficients, `Psi` increases. Also `|Re D_b(z)|<=pi*b^2*|cot(pi*Re(z))|/24` follows from `|csch(u)-1/u|<=u/6`. At `y=lambda*T/3`, these facts give the normalized two-sided tail majorant formula ```text 4/(9*pi) * 3*h/(lambda*Psi(tanh(lambda*T/6))) * exp(Re(leading_log(T)) +pi*b^2/24*sum_j |weight_j|*|cot(pi*Re(z_j))|). ``` Its numerical evaluation and numerator-sign checks use ordinary mpmath arithmetic **without outward rounding**. They are diagnostic estimates. The propagated D error uses a numerical absolute-integral estimate, and mpmath's quadrature errors are internal heuristic estimates. No combined interval enclosure or rigorous achieved-digit claim is made. The output records parameters as strings, precision, contour and panels, all residuals, exact rational D bounds, separate numerical error components, cutoff/precision/contour comparisons, source hashes at start and finish, commit and dirty status, runtime versions, timing, and historical residuals. The Airy expression is calculated only after quadrature for comparison. Agreement here is finite numerical evidence, separate from the global asymptotic theorem and its mathematical review. During development, a changed Gauss-Legendre panel layout at 60 digits underestimated its actual direct-D quadrature error: a residual around `2.3e-46` accompanied an internal estimate around `1e-66`. The original refined layout and the alternate tanh-sinh method agreed with the bounded series to approximately `1e-63`; both methods at 80 digits agreed to approximately `1e-84`. The outer execution records preserve this failed diagnostic. These observations motivate the cross-method check and the explicitly uncertified error status. Focused checks (no full toolkit suite): ```sh /tmp/polyhedra-baseline-20261031/bin/python -m pytest /Volumes/Lexar/lambda/devel/polyhedra/consolidated/experiments/section15_quantum/test_evaluate.py -q /opt/anaconda3/bin/ruff check /Volumes/Lexar/lambda/devel/polyhedra/consolidated/experiments/section15_quantum ```