Threshold banana — K3 rung

A three-loop integral tuned so its heaviest line sits exactly at the production threshold of the other three: the K3 surface underneath responds with a sign-flipping square-root branch — to our knowledge the first analysis to expand around this point — and the coefficient of that branch is closed here exactly, $c_{3/2} = -\sqrt{3}/(36\,\pi)$.

The integral

The three-loop banana is four propagators stretched between two vertices,

$$ I[\nu_1,\nu_2,\nu_3,\nu_4] \;=\; \int d^d l_1\, d^d l_2\, d^d l_3\; \frac{1}{\bigl(l_1^2 - M_1\bigr)^{\nu_1}\bigl(l_2^2 - M_2\bigr)^{\nu_2}\bigl(l_3^2 - M_3\bigr)^{\nu_3}\bigl((l_1+l_2+l_3-p)^2 - M_4\bigr)^{\nu_4}}\,, \qquad d = 4-2\varepsilon, $$

up to an overall normalization convention. It is a two-point function: a single external momentum $p$ flows in one side and out the other, so the only kinematic variable is $t = p^2$, and the four internal masses-squared are set to $(M_1, M_2, M_3, M_4) = (1,1,1,9)$ in units of the light mass. Its maximal cutthe integral with all four propagators put on shell — it isolates the underlying geometry is the periodthe integral of the surface's holomorphic 2-form over a closed cycle: the basic special function the geometry attaches to $t$ of a K3 surfacea complex two-dimensional Calabi–Yau manifold — the next step up from the elliptic curve that controls the two-loop sunrise, the next rung of the Calabi–Yau ladder above the elliptic sunrise. The object closed on this page is that holomorphic period $\varpi_0(t)$ — a solution of the exact fourth-order Picard–Fuchs operatorthe fourth-order differential equation in $t$ that the period satisfies over $\mathbb{Q}$ — near the threshold. The closure is period-level, exact in $t$; the dimensional regulator never enters the closed-form statement.

With generic masses that period is well understood: every singular point in the kinematic plane has unipotent local monodromythe linear map on the solution space induced by carrying the integration contour once around a singular point and back, meaning that walking the contour once around the singularity shifts the answer by at most a logarithm. The published modular closures of the three-equal-mass family (Duhr–Maggio, arXiv:2511.19245) are built on that assumption. Tune the masses-squared to $(1,1,1,9)$ and the assumption fails. This is a threshold coalescencethe kinematic tuning $\sqrt{M_{L+1}}=\sum_i\sqrt{M_i}$, where the heavy line sits exactly on the normal threshold of the others and that threshold collides with $p^2=0$: since $3 = 1+1+1$, the heavy line sits exactly on the normal threshold $\sqrt{M_4} = \sqrt{M_1}+\sqrt{M_2}+\sqrt{M_3}$, and the Picard–Fuchs operator picks up a half-integer indicial roota local exponent in $\tfrac12+\mathbb{Z}$; it forces a $\sqrt{t}$ branch in the solution and an eigenvalue $-1$ in the local monodromy at $t = p^2 \to 0$, where $\rho$ is the local exponent:

$$ 18432\,(\rho-1)^3\,(2\rho-3) \;\Longrightarrow\; \{\,1,1,1,\tfrac{3}{2}\,\}. $$

The lone $\tfrac{3}{2}$ exponent gives a monodromy eigenvalue of $e^{3\pi i} = -1$. Walk once around the threshold and one of the four period solutions comes back with its sign flipped — an order-two element, genuinely non-unipotenta monodromy with an eigenvalue $\neq 1$; finite-order rather than the pure log-shift that "unipotent" guarantees, the way one circuit of a Möbius band brings you back flipped. These period-level statements (the exponent, the monodromy spectrum) are exact; the finer geometric label of the degenerate K3 fibre at this point is strongly supported but not yet verified.

The full corner master $I[1,1,1,1]$ — each propagator raised to the first power — was checked separately in dimensional regularization at Euclidean points and is analytic at the threshold: the square-root branch is a property of the cut geometry, visible on the maximal cut.

Why it matters

Order-two reflection monodromy is in principle latent in every Feynman K3 — it can be read off the half-integer powers in the published equal-mass Wronskians (arXiv:2108.05310; arXiv:2109.15251; arXiv:1406.2664), whose operators carry indicial exponents $\{0,\tfrac12,1\}$ at their own thresholds. But every prior analysis — the all-loop period papers arXiv:2008.10574 and arXiv:1912.06201, the modular three-equal-mass closure of arXiv:2511.19245 — works near the maximal-unipotent (MUM) point or on the mass-equality locus, where the expansions engage only the unipotent structure; this threshold corner of the function landscape is not emphasized in the modular, MUM-centred reviews, and to our knowledge no Feynman K3 had been expanded around a threshold point of the mixed type described below or had a connection coefficient closed onto a fractional branch. That is what happens here. And the $(1,1,1,9)$ threshold is structurally new even against the equal-mass operators: the fractional exponent arrives in a mixed local structure — $\{1,1,1,\tfrac32\}$, a rank-three unipotent block with the $-1$ eigenvector on top — rather than the clean $\{0,\tfrac12,1\}$, so circling this point does more than shift the period basis by logarithms, breaking the pattern the MUM-centric Calabi–Yau period analyses are built on.

It also has teeth for the function space. The $t^{3/2}$ branch matches no product basis of holomorphic K3 periods — every attempted fit fails at evaluation points withheld from the fit — so the threshold contributes a genuinely new transcendental here (conjecturally at every loop order), one that differential-equation transport can reach and a symbol-alphabet ansatz cannot. This is the K3 lift of the Type III threshold sunrise found at two loops (Sec. 2.1 of the geometry paper), and the second rung of a ladder that BootLoops has now traced through the CY₃ and CY₄ bananas: the threshold monodromy order runs $4,2,4,2,\ldots$ up the tower — order 4 when $L-1$ is odd, order 2 when even — and never drops back to one. Exact indicial data pins that pattern rung-by-rung through CY$_4$ (sunrise, K3, CY$_3$, and CY$_4$ on its two-mass slice); that it persists for every loop order is a structural prediction from the $\mathrm{Sym}^{L-1}$-of-sunrise monodromy, not a rung-by-rung certification — and a naive reading of the pattern already misfired once, forecasting a unipotent collapse at CY$_4$ that the exact indicial data refuted. So: established by exact indicial data through CY$_4$, and structurally expected above.

What was hard

The structure had to be certified before anything could be extracted. The fourth-order Picard–Fuchs operator was built exactly over $\mathbb{Q}$ from the known all-loop period series (Bönisch–Fischbach–Klemm–Nega–Safari, arXiv:2008.10574) and annihilates exactly dozens of series coefficients beyond those used to build it. The non-unipotency was then established by three independent routes: the exact indicial polynomial, an exact rational Frobenius recursion for the $t^{3/2}$ branch, and a blind numerical monodromy computation that returns the eigenvalue spectrum $\{1,1,1,-1\}$ using none of the indicial data. A control at the generic mass point $(1,1,4,9)$, away from the coalescence, returns all monodromy eigenvalues $+1$ — the $-1$ appears only at the threshold tuning. Run after the fact as a Feynman family, the integration-by-parts reduction of the $(1,1,1,9)$ family (seven masters, five in the top sector) gives a maximal-cut operator at $d=2$ that equals this Picard–Fuchs operator exactly, up to a unit shift of the Euler operator; the threshold coefficient recomputed from that operator shares its first 47 digits with the recorded value at the two-precision floor (77 at the higher precision alone), and the physical master is finite at $d=2$, its auxiliary-mass-flow evaluations at two kinematic points agreeing between the two precision goals to 71 digits on every coefficient. The physical master's half-integer threshold coefficient at $d=2$ is bounded by $10^{-32}$ with the log tower absent, so to this precision the fractional monodromy is confined to the maximal cut; the four Frobenius coefficients of its $\varepsilon^{0}$ layer, read in the Frobenius basis of the reduction's operator at $z=-1/p^2=0$ under a stated particular-solution convention, are $a_0 = 19.2329104\ldots$, $a_1 = 0$ (bounded by $10^{-31}$), $a_2 = -13.1833474\ldots$ and $a_3 = -38.5332202\ldots$, the three non-vanishing ones established to 31 digits.

The coefficient itself was the wall. It matches no combination of known constants by direct numerical fitting, so it has to be carried in along the differential equation from a distant anchor. The first attempt — transport the period to the threshold and solve a $4\times 4$ linear system against the local Frobenius basis — stalled at a few significant figures: the square-root admixture is roughly $10^4$ times smaller than the dominant unipotent period, and the linear solve was fishing the answer out of subtraction noise.

The fix is algebraic. The threshold monodromy matrix $M_0$ has a rank-three unipotent block (nilpotent of index three) and a one-dimensional $-1$ eigenspace. The cube $(M_0-\mathbb{1})^3$ therefore kills the entire log tower and acts as $-8$ on the square-root branch, so a single polynomial in $M_0$ serves as the exact spectral projectorthe matrix polynomial in $M_0$ that maps onto one chosen eigenspace and annihilates every other generalised eigenspace onto the non-unipotent branch — the $10^4$ suppression is removed by construction rather than by cancellation. The trick generalises to any non-unipotent monodromy extraction (build the spectral idempotent from the minimal polynomial of $M_0$ instead of solving the mixed linear system) and has since been folded into BootLoops' Calabi–Yau transport library as the default extractor.

The result

Near the threshold the holomorphic period splits over the local Frobenius basis as

$$ \varpi_0(t) \;=\; A_0\,\Phi^{(0)}(t) + A_1\,\Phi^{(1)}(t) + A_2\,\Phi^{(2)}(t) \;+\; c_{3/2}\,\Phi_{3/2}(t), $$

where $\Phi^{(0)}, \Phi^{(1)}, \Phi^{(2)}$ are the three unipotent Frobenius solutions at $t=0$ — power series carrying zero, one and two logarithms — and $\Phi_{3/2}$ is the normalized square-root branch, an exact Frobenius series whose recursion closes over $\mathbb{Q}$:

$$ \Phi_{3/2}(t) \;=\; t^{3/2}\Bigl(1 + \tfrac{19}{216}\,t + \tfrac{181}{17280}\,t^2 + \cdots\Bigr). $$

The one function-level number closed is the connection coefficient $c_{3/2}$ — how much of the square-root branch sits inside the physical holomorphic period; the unipotent coefficients $A_0, A_1, A_2$ are carried numerically and are not closed here.

The connection coefficient closes exactly:

$$ c_{3/2} \;=\; -\frac{\sqrt{3}}{36\,\pi} \;=\; -0.0153146915\ldots\,, \qquad 432\,\pi^2\,c_{3/2}^{\,2} \;=\; 1\,, $$

extracted by the exact spectral projector onto the $-1$ eigenspace of the threshold monodromy $M_0$ — the exact $4\times 4$ linear map the four-dimensional solution space undergoes on one circuit of the threshold $t=0$:

$$ P_{-1} \;=\; -\tfrac{1}{8}\,(M_0 - \mathbb{1})^3\,, \qquad P_{-1}\!\cdot\!\varpi_0 \;=\; c_{3/2}\,\Phi_{3/2}\,. $$

An integer-relation search identifies this closed form at height 36, and the identification is verified to 195 digits. The constant also follows analytically from the position-space Bessel representation of the period: written with a Hankel kernel, the non-oscillatory tail of $J_0(y)^3J_0(3y)$ begins $-\tfrac{\sqrt{3}}{18\pi^2}\,y^{-3}$, where $1/\sqrt{3}=(M_1M_2M_3M_4)^{-1/4}$ and the $1/\pi^2$ comes from the four Bessel asymptotics, and the Mellin transform of $H_0^{(1)}$ turns that term into $-\tfrac{\sqrt{3}}{36\pi}\,t^{3/2}$ exactly. One may also read $36 = (\sum_i \sqrt{M_i})^2$, the normal threshold; at this mass point the two readings coincide. The Chowla–Selberg $\Gamma(a/15)$ period ring of the level-15 K3 motive was tested and comes back empty — $c_{3/2}$ lives in a simpler ring than the bulk K3 periods do. The sign is branch-convention dependent (the order-two monodromy negates $\Phi_{3/2}$), so $\lvert c_{3/2}\rvert = \sqrt{3}/(36\pi)$ is the convention-independent statement. The singular points of the operator sit at $t = p^2 \in \{0,\,4,\,16,\,36,\,11\pm\sqrt{105},\,\infty\}$ — exactly the threshold loci $(\sum_i \pm\sqrt{M_i})^2$ for $\sqrt{M_i} = \{1,1,1,3\}$.

The complete statement and the certified value live in threshold-banana-expression.md; threshold-banana-evaluate.py evaluates the closed form instantly by default, running two fail-closed self-checks on every invocation; --full (or --digits N) prints the certified record string of $c_{3/2}$ to its 195 certified digits with its provenance, the closed form evaluated beside it; --derive rebuilds the operator, the series, the transport and the projector from scratch at runtime — byte-identical to the default — and checks the closed form against an independent evaluation. --family runs the same integral as a Feynman family after the fact: the maximal-cut operator of the integration-by-parts reduction of the $(1,1,1,9)$ family is checked against this Picard–Fuchs operator three ways by exact rational arithmetic at runtime, the connection coefficient is recomputed from that operator (--c32; 47 digits against the recorded value at the two-precision floor), and the physical master's two independent evaluations and its threshold bound are read from the vendored record (--oracle, vendor_row31_family/). The bundle now carries a MANIFEST.sha256 and a CHANGES.md. Two further modules travel in the bundle: threshold_hankel_tail.py derives the threshold connection coefficients of the K3, CY₃ and CY₄ bananas analytically — from the non-oscillatory tail of the Bessel-kernel representation, every step exact over the rationals — and checks them against the recorded closed forms, digit strings and integer relations (--check, under a second); direct_linear_extract.py recovers the K3 and CY₃ coefficients by one direct linear solve against the exact Picard–Fuchs operator in 1400-bit ball arithmetic (--selftest, about a minute). Both read only the files beside them (calpha_rings.py and fixtures/). The Frobenius coefficients of the physical master's $\varepsilon^0$ layer in the operator's $z = 0$ Frobenius basis — for this $(1,1,1,9)$ banana and for the CY₃ $(1,1,1,1,16)$ banana, to 31 significant digits, together with the particular-solution convention they are read in — are served as frobenius_coefficients_rows31_32_20260909T042148Z.json and listed in the manifest.

The closed form is a period-level statement, so the independent evaluations behind it are two transport routes of the same exact rational Picard–Fuchs operator, one in rigorous ball arithmetic (Arb) and one in mpmath; the physical master, evaluated separately by the auxiliary-mass-flow method at two kinematic points, enters only through the family checks above. The closed form carries a rigorous ball-arithmetic certificate to 195 digits, and the independent mpmath route agrees with it to 195 digits; the family-level checks stand at 47 digits for the coefficient recomputed from the reduction's operator (at the two-precision floor; 77 at the higher precision alone), 71 digits for the physical master and 31 digits for its Frobenius coefficients.

References

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