15 A uniform quantum Airy transition for the regular family · Polyhedra, Moduli, and Quantum Geometry

15.1 The exact normalization and the result

Take $b>0$ , $Q=b+b^{-1}$ , and $\alpha=\frac{\pi-\eta}{2\pi b},\quad \eta_{\mathrm E}=\arccos(1/3),\quad h=2\pi b^2,\quad \lambda=h^{1/3}.$ For $\eta$ close to $\eta_{\mathrm E}$ and sufficiently small $b$ , choose an upward vertical contour $C$ satisfying $Q<\operatorname{Re}w<4\alpha$ . This lies in the usual separating interval $3\alpha<\operatorname{Re}w<\min(4\alpha,2Q)$ and puts all three integrand arguments in their common fundamental strip. With the integrand of (14.25), namely $I_b(w,\eta)=S_b(w-3\alpha)^4 S_b(4\alpha-w)^3S_b(2Q-w),$ define $\mathcal R_b(\eta)= \frac{S_b(3\alpha-Q)^2}{S_b(\alpha)^6} \int_C I_b(w,\eta)\,\frac{\mathop{}\!\mathrm dw}{i}.$ (15.1) This is the equal-label symmetric formula (2.27) of [96, arXiv version 3], with the measure convention specified as $\mathop{}\!\mathrm dw/i$ . If that formula’s printed $\mathop{}\!\mathrm dw$ is read literally, its value is $i\mathcal R_b$ . Keeping this constant phase explicit avoids imposing an unstated Barnes-measure convention. The four identical trivalent square roots multiply to the displayed quotient, with no remaining square-root choice. Absolute convergence and independence of the allowed vertical line are proved below.

The functional equation $S_b(x+b^{-1})=2\sin(\pi x/b)S_b(x)$ gives the exact identity $\frac{S_b(3\alpha-Q)^2}{S_b(\alpha)^6} = \frac{1}{4\sin^2((3\eta-\pi)/(2b^2))} \frac{S_b(3\alpha-b)^2}{S_b(\alpha)^6}.$ (15.2) Accordingly define $\widehat{\mathcal R}_b(\eta) = 4\sin^2\!\left(\frac{3\eta-\pi}{2b^2}\right)\mathcal R_b(\eta) = \frac{S_b(3\alpha-b)^2}{S_b(\alpha)^6} \int_C I_b(w,\eta)\,\frac{\mathop{}\!\mathrm dw}{i}.$ (15.3) The right-hand side defines the continuation across the apparent singularities in the product on the left.

Proposition 15.1 (Uniform regular-family transition). For every fixed $Z>0$ , put $\eta=\eta_{\mathrm E} -\frac{h^{2/3}}{6\sqrt2}\,\zeta, \qquad |\zeta|\le Z.$ As $b\downarrow0$ , the integral (15.3) satisfies, uniformly for these real $\zeta$ , $\boxed{\quad \widehat{\mathcal R}_b(\eta)= \frac34\,\frac{h^{1/3}}{b} \left\{ \operatorname{Ai}(\zeta) -3\sqrt2\,h^{1/3}\operatorname{Ai}'(\zeta) +O_Z(h^{2/3}) \right\}. \quad}$ (15.4) The error inside braces is additive. The physical compact hyperbolic side has $\zeta\ge0$ ; negative $\zeta$ describes the analytic continuation across the Euclidean transition.

In particular the leading prefactor is $\frac34(2\pi)^{1/3}b^{-1/3}$ . The correction can equivalently be written by replacing the Airy argument with $\zeta-3\sqrt2\,h^{1/3}$ , at an error $O_Z(h^{2/3})$ . The proof below controls the exact quantum integrand, the whole contour, its orientation, and the first correction.

15.2 A uniform double-sine estimate from its real integral

Let $\Lambda$ be the analytic primitive on $0<\operatorname{Re}z<\pi$ specified by $\Lambda'(z)=-\log(2\sin z),\qquad \Lambda(\pi/2)=0,$ with the logarithm real on $(0,\pi)$ . On that real interval it is the usual Lobachevsky function.

Lemma 15.2 (Strip estimate). For $z$ in a compact subset of $0<\operatorname{Re}z<1$ and real $\kappa$ in a bounded set, $S_b(z/b+\kappa b) = \exp\!\left(-\frac{\Lambda(\pi z)}{\pi b^2}\right) (2\sin\pi z)^{\kappa-1/2} \bigl(1+O(b^2)\bigr).$ (15.5) The logarithmic error and its derivatives are uniform on smaller compact subsets. For $\varepsilon\le\operatorname{Re}z\le1-\varepsilon$ , the real part of the logarithmic error is $O_\varepsilon(b^2)$ uniformly for all $\operatorname{Im}z$ .

Proof. We use $S_b(x)=\Gamma_b(x)/\Gamma_b(Q-x)$ , with $\Gamma_b(Q/2)=1$ . The convergent real-integral normalization is $\log\Gamma_b(x)=\int_0^\infty\frac{\mathop{}\!\mathrm du}{u} \left\{ \frac{e^{-xu}-e^{-Qu/2}}{(1-e^{-bu})(1-e^{-u/b})} -\frac{(Q-2x)^2}{8}e^{-u}-\frac{Q-2x}{2u} \right\},\qquad \operatorname{Re}x>0.$ The first term in braces is $(Q/2-x)/u+(Q/2-x)^2/2+O(u)$ at zero, which fixes both subtractions. In [96, arXiv version 3, (A.1)] the last denominator is printed as $u$ ; the factor $2$ above is necessary for convergence. Subtracting the integrals at $x$ and $Q-x$ therefore gives, for $0<\operatorname{Re}x<Q$ , $\log S_b(x)=\int_0^\infty\frac{\mathop{}\!\mathrm du}{u} \left\{ \frac{\sinh((Q/2-x)u)} {2\sinh(bu/2)\sinh(u/(2b))} -\frac{Q-2x}{u} \right\}.$ This fixes a holomorphic logarithm, real on $(0,Q)$ and zero at $Q/2$ . At the centered argument $x=z/b+b/2$ , substitute $u=2bs$ and separate the term $1/(b^2s)$ from $\operatorname{csch}(b^2s)$ . The result is $\log S_b(z/b+b/2) =-\frac{\Lambda(\pi z)}{\pi b^2}+D_b(z),$ (15.6) where $D_b(z)=\frac12\int_0^\infty \frac{\sinh((1-2z)s)}{s\sinh s} \left(\operatorname{csch}(b^2s)-\frac1{b^2s}\right)\mathop{}\!\mathrm ds.$ (15.7) Here the leading integral is the elementary identity $\frac12\int_0^\infty \left\{\frac{\sinh((1-2z)s)}{\sinh s}-(1-2z)\right\} \frac{\mathop{}\!\mathrm ds}{s^2}=-\frac{\Lambda(\pi z)}{\pi}.$ For example, differentiating twice gives $\pi\cot(\pi z)$ by the partial fraction expansion of the cotangent. Its first derivative at $1/2$ is $\int_0^\infty(s^{-2}-(s\sinh s)^{-1})\mathop{}\!\mathrm ds=\log2$ , as follows from $\operatorname{csch}s=\coth(s/2)-\coth s$ and a change of scale. Its value at $1/2$ is zero. These three facts determine the primitive and its constant. For $u>0$ , $|\operatorname{csch}u-u^{-1}|\le C u$ . Near $s=0$ the numerator $\sinh((1-2z)s)$ is $O(s)$ , and at infinity the strip condition supplies exponential decay. This proves $D_b=O(b^2)$ on compact subsets. Holomorphy and Cauchy’s estimate give the derivative statement.

For the global real-part bound use $\operatorname{Re}\sinh((1-2z)s) =\sinh((1-2\operatorname{Re}z)s) \cos(2\operatorname{Im}z\,s).$ The same integrable bound is then independent of $\operatorname{Im}z$ . Finally replace $z$ in (15.6) by $z+(\kappa-\frac12)b^2$ and Taylor expand the leading primitive. Its second and higher derivatives are bounded on closed substrips, including as $|\operatorname{Im}z|\to\infty$ . This gives (15.5) and the corresponding global modulus estimate. The centered variable is $z=bx-b^2/2$ . Its strip $0<\operatorname{Re}z<1$ is narrower than the fundamental $x$ -strip; no uniform small- $b$ estimate is asserted up to the boundary of that larger strip. ◻

15.3 The smooth quantum phase and amplitude

Use the corrected variable $v=2\pi bw$ , and set $\begin{aligned} \theta_1&=\tfrac12(v-3\pi+3\eta),& \theta_2&=\tfrac12(4\pi-4\eta-v),\\ \theta_3&=\tfrac12(4\pi-v),& s&=\tfrac12(\pi-\eta). \end{aligned}$ (15.8) The three integrand arguments are $\theta_1/(\pi b)$ , $\theta_2/(\pi b)$ , and $\theta_3/(\pi b)+2b$ . The cleared prefactor uses $3s/(\pi b)-b$ and $s/(\pi b)$ . All five scaled arguments are in a closed substrip near the fold. Lemma 15.2 therefore gives $\frac{S_b(3\alpha-b)^2}{S_b(\alpha)^6}I_b(w,\eta) = \mathcal A(v,\eta)\,e^{G(v,\eta)/h} \bigl(1+O(h)\bigr)$ (15.9) on compact subsets, where $\begin{aligned} G(v,\eta) &=-2\{4\Lambda(\theta_1)+3\Lambda(\theta_2) +\Lambda(\theta_3)\} +12\Lambda(s)-4\Lambda(3s),\\ \mathcal A(v,\eta) &=(2\sin\theta_1)^{-2}(2\sin\theta_2)^{-3/2} (2\sin\theta_3)^{3/2} \left(\frac{\sin s}{\sin3s}\right)^3. \end{aligned}$ The powers are continued from their positive real values. For modulus estimates on the full vertical line, the same formula holds with a bounded multiplicative error, uniformly in its imaginary coordinate; this follows from the real-part estimate in the lemma.

At $v_*=4\pi-5\eta_{\mathrm E}$ the exact values are $\begin{array}{c|ccc} &\theta_1&\theta_2&\theta_3\\ \hline \sin\theta_j&1/3&1/\sqrt3&1/(9\sqrt3)\\ \cot\theta_j&2\sqrt2&\sqrt2&-11\sqrt2 \end{array} \qquad \frac{\sin s}{\sin3s}=3.$ Consequently $\mathcal A_*=\frac94,\qquad (\log\mathcal A)_v\big|_*=7\sqrt2.$ (15.11) The phase has the following fold data; its value is justified below, and its derivatives follow from (15.10): $\begin{gathered} G_*=G_v|_*=G_{vv}|_*=G_\eta|_*=0,\\ G_{vvv}|_*=54,\qquad G_{v\eta}|_*=18\sqrt2,\qquad G_{\eta\eta}|_*=36\sqrt2,\\ G_{vvvv}|_*=-648\sqrt2,\qquad G_{vv\eta}|_*=-18. \end{gathered}$ (15.12) The value $G_*=0$ requires fixing the constant, not just differentiating the phase. Use the real Lobachevsky form of [83, Section 1.2, (1.14)–(1.18)], extending the real function by oddness and period $\pi$ . For equal angles its potential and correction are $\begin{split} U_\Lambda(v,\eta) &=\Lambda(v/2)+3\Lambda(v/2+2\eta-\pi) -4\Lambda((v+3\eta-\pi)/2),\\ \Delta_\Lambda(\eta) &=2\Lambda((3\eta-\pi)/2)-6\Lambda((\eta-\pi)/2). \end{split}$ Periodicity and oddness give, term by term, $\begin{split} U_\Lambda&=-4\Lambda(\theta_1)-3\Lambda(\theta_2) -\Lambda(\theta_3),\\ \Delta_\Lambda&=6\Lambda(s)-2\Lambda(3s),\qquad G=2(U_\Lambda+\Delta_\Lambda). \end{split}$ At the two unit-circle saddle roots on the compact hyperbolic side, Theorem 2 of that paper, in this real formulation, says $|U_\Lambda+\Delta_\Lambda|=W$ . Both roots converge to the lift $v_*$ , while the regular tetrahedral volume tends to zero by (14.20). Continuity proves $G_*=0$ . This argument fixes the real constant without selecting unspecified branches of a complex dilogarithm.

15.4 The full contour and its tail estimate

The poles of the first double-sine factor are at $w=3\alpha-mb-n/b$ ; the other two factors have their poles at $w=4\alpha+mb+n/b$ and $w=2Q+mb+n/b$ . Thus the separating interval remains nonempty for $0<\alpha<2Q/3$ . Decreasing the equal real labels from $\alpha=Q/2$ to the compact values preserves this interval. Along this path $\alpha>Q/4$ , so one may choose $\max(3\alpha,Q)<\operatorname{Re}w<4\alpha\le2Q$ and keep all three arguments in the common strip throughout. The integral has no pinch on this path.

First justify moving an arbitrary defining contour to the fold neighborhood at each fixed $b$ . The vertical asymptotics [96, (A.11) and (A.13)] give $|S_b(a+iT)|=e^{\pi(a-Q/2)|T|}(1+o(1)),\qquad |T|\to\infty,$ uniformly for $a$ in a fixed compact real interval. The sum of the coefficients in the eight factors of $I_b$ is $4(w_R-3\alpha-Q/2)+3(4\alpha-w_R-Q/2) +(2Q-w_R-Q/2)=-2Q.$ Consequently $|I_b(w_R+iT,\eta)|=O(e^{-2\pi Q|T|})$ , uniformly between any two fixed separating lines. Horizontal closing segments vanish and no poles are crossed. This proves absolute convergence, contour independence, and agreement with the continued contour in (2.27). The same bound is locally uniform for complex $\alpha$ near the real continuation path, so the integral is holomorphic there. The external factor is meromorphic; (15.2) removes its poles on this path. In a neighborhood of the fold, both remaining prefactor arguments are in $(0,Q)$ , and the cleared expression is holomorphic and real for real $\eta$ .

Only after this fixed- $b$ relocation do we apply the uniform centered-strip estimates near $v_*$ . They apply to the entire vertical line there, including its unbounded imaginary part.

On the vertical line $v=v_*+iy$ at $\eta=\eta_{\mathrm E}$ , put $r=\tanh(y/2)$ . For $y>0$ , direct differentiation gives $-\frac{\mathop{}\!\mathrm d}{\mathop{}\!\mathrm dy}\operatorname{Re}G(v_*+iy,\eta_{\mathrm E}) =\Psi(r),$ (15.13) where $\Psi(r)=4\arctan(2\sqrt2r) +3\arctan(\sqrt2r)-\arctan(11\sqrt2r).$ Here $\Psi(0)=0$ and $\Psi'(r)= \frac{864\sqrt2\,r^2(3+11r^2)} {(1+8r^2)(1+2r^2)(1+242r^2)}>0 \qquad(r>0).$ (15.14) It follows that the fold is the unique maximum of the real part of $G$ on this vertical line. In addition, $\operatorname{Re}G(v_*+iy,\eta_{\mathrm E}) =-27\sqrt2\,y^4+O(y^6),\qquad \Psi(\tanh(y/2))\longrightarrow2\pi \quad(y\to+\infty).$ Complex conjugation gives the corresponding estimates for negative $y$ .

These bounds persist in the form needed below under small changes of $\eta$ and of the real contour coordinate. On bounded annuli in $y$ this follows by continuity. For large positive $y$ , write $a_j=\theta_j(v_R,\eta)\in(0,\pi)$ . The arguments of $2\sin(a_1+iy/2)$ and $2\sin(a_j-iy/2)$ , $j=2,3$ , are respectively $\pi/2-a_1+O(e^{-y})$ and $a_j-\pi/2+O(e^{-y})$ . Since $4a_1+3a_2+a_3=2\pi$ , it follows that $\partial_y\operatorname{Re}G=-2\pi+O(e^{-y})$ , uniformly for nearby real $v_R,\eta$ . Conjugation treats the negative tail. The amplitude in (15.10) is $O(e^{-|y|})$ there. The global modulus bound from Lemma 15.2 transfers these estimates to the exact integrand, so all tails outside a fixed fold neighborhood are exponentially small in $h^{-1}$ . More explicitly, for each sufficiently small fixed $\delta>0$ there are $m_\delta>0$ and $Y_0>\delta$ , uniform in nearby real parameters, such that $\operatorname{Re}G(v_R+iy,\eta) \le -m_\delta-\pi(|y|-Y_0)_+, \qquad |y|\ge\delta.$

15.5 Airy localization and the first correction

Proof of Proposition 15.1. Write $t=\eta_{\mathrm E}-\eta$ and $\lambda=h^{1/3}$ . Choose a fixed $c>0$ and move the contour to $v=v_*+c\lambda+iy.$ For all sufficiently small $b$ this deformation stays within the common pole-free strip. The tail bounds just proved make the horizontal closing segments vanish, so no residues or undetermined cycle coefficients enter.

At this displaced real point, $G_{vv}=54c\lambda+O_Z(\lambda^2)>0$ and $G_{vvvv}=-648\sqrt2+O_Z(\lambda)<0$ . Taylor expansion in $iy$ , with real derivatives at the real point, has real part $-G_{vv}y^2/2+G_{vvvv}y^4/24+O(y^6)$ . It gives constants $c_1,c_2,\delta>0$ such that $\begin{split} &\operatorname{Re}\{G(v_*+c\lambda+iy,\eta) -G(v_*+c\lambda,\eta)\}\\ &\quad \le -c_1\lambda y^2-c_2y^4, \qquad |y|\le\delta. \end{split}$ (15.15) The constants can be taken uniform for $|\zeta|\le Z$ . Outside this neighborhood use the previously established fixed tail gap and linear decay.

Now set $x=v-v_*=\frac{\lambda}{3}Y,\qquad t=\frac{\lambda^2}{6\sqrt2}\zeta.$ The rescaled contour is upward with $\operatorname{Re}Y=3c$ . The derivative table gives, through weighted order four, $\begin{aligned} G(v_*+x,\eta_{\mathrm E}-t) &=9x^3-18\sqrt2\,tx +18\sqrt2\,t^2-27\sqrt2\,x^4+9tx^2 +R_G(x,t),\\ \frac{\mathcal A(v_*+x,\eta_{\mathrm E}-t)}{9/4} &=1+7\sqrt2\,x+O(x^2+|t|). \end{aligned}$ Analyticity gives the explicit local bound $|R_G(x,t)|\le C\bigl(|x|^5+|t||x|^3+t^2|x|+|t|^3\bigr).$ The exact quantum error in (15.9) is $O(h)=O(\lambda^3)$ and hence does not affect the first correction. Consequently, the rescaled integrand is $\frac94\,e^{Y^3/3-\zeta Y} \left\{1+\lambda P(Y,\zeta)+O_Z(\lambda^2)\right\},$ (15.17) where the pointwise remainder is locally uniform and $P(Y,\zeta)= \frac{7\sqrt2}{3}Y-\frac{\sqrt2}{3}Y^4 +\frac{\zeta}{6\sqrt2}Y^2 +\frac{\sqrt2}{4}\zeta^2.$ (15.18)

For completeness, the integrated remainder is also $O_Z(\lambda^2)$ . On $|Y|\le\lambda^{-1/12}$ , analytic Taylor remainders in (15.16), followed by exponentiation, give an error bounded by $C_Z\lambda^2(1+|Y|^8)$ , against the Gaussian $|e^{Y^3/3-\zeta Y}|\le C_Ze^{-3c(\operatorname{Im}Y)^2}$ . Indeed $|R_G|/h\le C_Z\lambda^2(1+|Y|^5)$ , and the square of the order- $\lambda$ phase correction has degree eight. The growing cutoff makes both exponent remainders uniformly small. Equation (15.15) supplies a uniform Gaussian majorant for the exact rescaled integrand, since $G(v_*+c\lambda,\eta)/h=O_Z(1)$ . The part with $\lambda^{-1/12}<|Y|$ inside the fixed fold neighborhood is smaller than every power of $\lambda$ ; the global contour tail is exponentially small in $h^{-1}$ . This justifies integration of the expansion and its first correction.

With the chosen orientation, $\frac1{2\pi i}\int_{\operatorname{Re}Y=3c} e^{Y^3/3-\zeta Y}\,\mathop{}\!\mathrm dY=\operatorname{Ai}(\zeta);$ this follows from [54, (9.5.4)] with its contour oriented from the lower ray to the upper ray. To deform those rays to $\operatorname{Re}Y=3c$ , close at imaginary heights $\pm T$ . On each horizontal connector, with $3c\le x=\operatorname{Re}Y\le T/\sqrt3$ , $\operatorname{Re}(Y^3/3-\zeta Y) \le-\tfrac89xT^2+Zx,$ so the connector integrals vanish. Inserting $Y^k$ yields $(-1)^k\operatorname{Ai}^{(k)}(\zeta)$ . Using $\operatorname{Ai}''(\zeta)=\zeta\operatorname{Ai}(\zeta), \qquad \operatorname{Ai}^{(4)}(\zeta) =2\operatorname{Ai}'(\zeta) +\zeta^2\operatorname{Ai}(\zeta)$ in (15.18), all terms proportional to $\zeta^2\operatorname{Ai}$ cancel. The result is $-3\sqrt2\,\operatorname{Ai}'(\zeta)$ . Finally, $\mathop{}\!\mathrm dw/i=\lambda\,\mathop{}\!\mathrm dY/(6\pi i b)$ . Multiplication by $9/4$ gives the coefficient $3\lambda/(4b)$ in (15.4). ◻

15.6 Numerical evidence and reproducibility status

The values below come from a new evaluation of the exact cleared integral (15.3), implemented in experiments/section15_quantum/. Write $Q_b(\zeta)=\frac{4b}{3h^{1/3}}\widehat{\mathcal R}_b(\eta), \qquad Q_b^{(1)}(\zeta)=\operatorname{Ai}(\zeta) -3\sqrt2\,h^{1/3}\operatorname{Ai}'(\zeta).$ The evaluator uses the exact centered identity (15.6). A finite expansion of $D_b$ in derivatives of the cotangent has an explicit rational remainder bound uniform in a declared centered strip. This approximates the double-sine correction, not the Airy phase. All five logarithmic terms, with weights $4,3,1,2,-6$ , are combined before exponentiation. The logarithms and the Lobachevsky primitive have the branches fixed above.

The outer Gauss–Legendre quadrature uses the upward contour $v=v_*+\frac{\lambda}{3}(C+it),\qquad C>0,$ with measure $\mathop{}\!\mathrm dw/i$ ; thus $C=3c$ in the proof. Each of the seven parameter pairs described below is evaluated at 60 decimal digits with $C=1$ , $|t|\le12$ , then at 90 digits with $C=1$ , $|t|\le14$ , and again at 90 digits with $C=5/4$ and two additional correction terms. The contribution beyond a shorter cutoff is recorded separately. The table rounds the 90-digit, $C=1$ values to six decimals.

$b$	$\zeta$	Computed $Q_b$	Airy $Q_b^{(1)}$
$0.0125$	$0$	$0.459086$	$0.464164$
$0.00625$	$0$	$0.422572$	$0.423779$
$0.003125$	$0$	$0.398047$	$0.398339$
$0.0125$	$1$	$0.214333$	$0.202400$
$0.00625$	$1$	$0.182316$	$0.177567$
$0.003125$	$1$	$0.163805$	$0.161924$

For the three $\zeta=1$ rows, the newly computed values of $(Q_b-Q_b^{(1)})/h^{2/3}$ are approximately $1.208081684$ , $1.211366819$ , and $1.208878763$ . A negative transition parameter is also checked: $Q_{1/160}(-1)\approx0.517966671856531$ . The largest observed 60/90-digit difference in these seven values is less than $4\times10^{-57}$ ; changing the contour and correction order changes them by less than $1.3\times10^{-91}$ . These are comparisons between numerical runs, not guaranteed numbers of correct digits.

Direct quadrature of $D_b$ and of the convergent real $\log S_b$ integral, together with the reflection and shift identities, checks the special-function evaluation independently of the correction expansion. A separate computation integrates the combined Barnes logarithm directly, using QUADPACK in ordinary double precision; its seven normalized values agree with the multiprecision calculation within $2\times10^{-12}$ . The rational bound controls correction-series truncation only. Its propagation uses a numerical absolute-integral estimate; the outer quadrature errors, sign tests, and tail-majorant formulas are evaluated without outward rounding. No total interval enclosure of the quantum integral is claimed.

The supplied draft’s evaluator and raw output remain unrecovered. Its six reported values are preserved in experiments/section15/historical-table.json. The new first value is $0.459086488736078952\ldots$ , replacing the previously printed $0.459087$ by $0.459086$ ; the original rounding procedure is unknown. The other five values agree after rounding. This is new computational evidence with its own provenance, not a recovery of the missing original calculation.

The earlier companion experiments/section15/check.py checks the finite fold derivatives, the rational identity for $\Psi'$ , and the first-correction cancellation in exact arithmetic with $\sqrt2$ adjoined. It also checks the Airy column at 80 and 120 digits against a separately coded ODE series [54, Sections 9.2 and 9.4]; its existing result remains a frozen record of those finite checks. Reproduction instructions for the new evaluator are in experiments/section15_quantum/README.md; its full result is written to artifacts/section15_quantum/result.json. Source identities, independent reviews, and generation records are collected in docs/execution/section15-analytic-review-2026-11-01/README.md. The analytic proof of Proposition 15.1 and this finite numerical evidence remain distinct.

15.7 Interpretation and scope

The correction to the earlier pole argument is substantive. The exact quantum integrand is smooth at the correct fold; the remaining real- $b$ resonances are an explicit scalar factor outside the integral. For example, at $\eta=\eta_{\mathrm E}$ the scalar factor has candidate resonances along $b^2=\frac{3\eta_{\mathrm E}-\pi}{2\pi m}, \qquad m=1,2,\ldots.$ The numerator in (15.3) is nonzero for sufficiently small $b$ , uniformly in each fixed physical window $0\le\zeta\le Z$ : the continuous positive function $\operatorname{Ai}(\zeta)$ has a positive minimum on $[0,Z]$ , and the additive correction and error tend uniformly to zero. Thus zeros of the squared sine give genuine double poles of the original normalization in this window, including all sufficiently large $m$ in the displayed sequence. For negative $\zeta$ , the additive asymptotic remains uniform even at zeros of $\operatorname{Ai}$ ; no uniform noncancellation or relative-error assertion is made there. The cleared quantity has the uniform asymptotic in the proposition even through these resonances.

In geometric terms, $\zeta=\frac{A}{2\sqrt3\,h^{2/3}} +O\!\left(\frac{A^2}{h^{2/3}}\right), \qquad A\asymp b^{4/3}.$ The decaying Airy branch selected by the contour agrees with the decaying volume exponential: its classical saddle exponent is $-2W/h=-W/(\pi b^2)$ . The proposition is uniform for bounded $\zeta$ ; it does not, by itself, justify taking $\zeta\to\infty$ simultaneously with $b\to0$ or keeping a nonzero tetrahedral volume fixed. Those matching limits require additional estimates.

This proves a quantum transition for one specified regular family, beyond the classical cubic calculation. Extension to arbitrary nondegenerate Euclidean tetrahedral shapes, comparison of normalizations used in a global state integral, and gluing into a topological invariant remain separate problems. In particular no new case of the colored-Jones volume conjecture, and no identification of the integrated quantum measure with the cone-metric measure, follows from this proposition alone.