# Ice-cream cone (ICC) diagram — final result Slice: p12^2 = p34^2 = x, p56^2 = -1, d = 2 - 2*eps. Two mass configurations closed: equal mass (m^2 = 1,1,1,1) and generic distinct mass (m^2 = 1,2,3,5). ## 1. Equal mass — top master [1,1,1,1] at eps^0 (weight 3) Eliminating the once-dotted top from the differential equation leaves a second-order operator on the top row, L_T = d^2 + p1t*d + p0t, p1t = (14x^5 - 155x^4 + 520x^3 - 316x^2 + 22x - 5) / ( (x-1)(4x+1)(x^2 - 10x + 5)(x^2 - 4x + 1) ), p0t = 6(x^4 - 8x^3 + 24x^2 - 4x + 3) / ( (x-1)(4x+1)(x^2 - 10x + 5)(x^2 - 4x + 1) ), whose two solutions are ALGEBRAIC, y+-(x) = q+-(x)^(-1/2), q+- = (1/2)( 2x^2 + 8x - 5 +- sqrt5*(2x-1)*sqrt(4x+1) ), q+ * q- = (x-1)^2 (x^2 - 10x + 5). The top obeys L_T[T] = alpha*S + beta*S' + g1 - gt*Tri0 with S the weight-2 scoop sunrise, Tri0 = 4 ln(phi)/sqrt5 the d=2 bubble at p56^2, and the exact rational kernels alpha = -8(2x^4 - 9x^3 - 15x^2 - 25x - 15) / ( 3(x-9)(x-1)(4x+1)(x^2 - 10x + 5)(x^2 - 4x + 1) ), beta = -2(11x^5 - 63x^4 + 48x^3 - 340x^2 + 165x - 45) / ( 3(x-9)(x-1)(4x+1)(x^2 - 10x + 5)(x^2 - 4x + 1) ), g1 = 8 / ( (x-9)(x-1)(x^2 - 10x + 5) ), gt = 2(2x^3 - 27x^2 + 18x - 1) / ( (x-1)(4x+1)(x^2 - 10x + 5)(x^2 - 4x + 1) ). The answer is the variation-of-parameters word over the algebraic pair: T(x) = c+*y+(x) + c-*y-(x) + Int_{-1}^{x} [ y-(x)*y+(x') - y+(x)*y-(x') ] / W(x') * ( alpha*S + beta*S' + g1 - gt*Tri0 )(x') dx', W = y+*y-' - y+'*y-, with the closed weight-2 scoop source S(x) = A*psi1(x) + B*psi2(x) + 6*P(x) (source constant 6 EXACT, PSLQ-rational) A = -2.48801634393263349... (determined to 120 digits) B = +0.337129922570435036... (determined to 110 digits) where (psi1, psi2) is the equal-mass sunrise period pair -- the solutions of L_PF = d^2 + p1*d + p0, p1 = (3x^2 - 20x + 9)/D, p0 = (x - 3)/D, D = x(x-1)(x-9) -- entering only through this source, and P(x) is a fixed particular solution of the scoop's inhomogeneous equation. For Euclidean x < -1/4 the two kernels y+- are complex conjugates and T is real. Written through the scoop's spectral mass s, the whole eps^0 top is the one-fold integral of logarithms T(x) = 2 Int_4^inf ds / sqrt(s(s-4)) * [ 2(c - 2s)*B(s;x) + (2c - 5)*Tri0 ] / (2c^2 - 10s), c = s + 1 - x, B(s;x) = Int_0^1 da / ( a + s(1-a) - x*a(1-a) ), which reproduces the stored 330-digit T(-1) to its full length in seconds (the same form at masses (1,2,3,5) reaches 209 of the 210 stored digits; icc-evaluate.py --onefold X evaluates this form); its Tri0 term is elementary (residues at the cone's Landau points s+-(x)) and the rest is a length-two elliptic polylogarithm on the scoop curve with third-kind punctures at s+-(x). The cone's leading Landau locus on the slice (HMSV arXiv:2211.07633 eq. 6.48, sec 6.4) is the exact denominator factor L_bub = x^2 - 10x + 5 = (x - r+)(x - r-), r+- = 5 +- 2*sqrt(5), appearing ONLY in the top-sector rows of the validated connection (exact IBP reduction via Kira: dotted-top -> top coefficient ((2-d)x + d - 3)/(x^2 - 10x + 5)) and as the last-entry symbol letter. dlog alphabet: {x, x-1, x-9, 2x+1, 4x+1, L_bub}. Boundary data fixing c+, c- — DERIVED (no longer an AMFlow fit): T(-1) follows from the infinity-cusp analytic conditions of the boundary system (BC1: no half-integer u^{3/2} branch at x=infinity, integer-power log ansatz; BC2: hard-bubble matching tau0 = 0), recomputed at runtime by icc-evaluate.py, and independently from the materialized relative-period integral of the graph (agreement 284.8 d): T(-1) = 0.60984239122405078105237510007849294503502669016975223264... (284.8 d, two routes) The explicit constants of the derivation chain sit in the golden-log/dilog ring: Tri0 = 4 ln(phi)/sqrt5, phi = (1+sqrt5)/2, J = Int_0^1 ln(1+a-a^2)/(1+a-a^2) da = (2/sqrt5) ( ln5*ln(phi) + Li2((5-sqrt5)/10) - Li2((5+sqrt5)/10) ), entering via c1_0 = 2*EulerGamma*Tri0 + J and the resonance-forced identity t11 = 2*Tri0 (verified at runtime). c+, c- (equivalently T(-1), T'(-1)) remain PSLQ-NEGATIVE at 120 digits against the weight-3 ring {pi^2, zeta3, log2, log3, sqrt5, psi1}: the values of a length-two elliptic polylogarithm on the scoop curve with punctures at the cone's Landau points, derived and double-routed; integer-relation-negative at the stated heights (no fabricated identification). ## 2. Equal mass — eps^1 (weight 4) Same validated 7x7 connection, eps-graded coupled transport extended to KMAX = 1 (21-component state). L_bub divides the top-row denominators at eps^0 AND eps^1 (exact polynomial division). Weight-4 boundary constant — DERIVED at runtime the same way (grade-1 infinity-cusp slots: S1 = (-pi^2, -18 zeta3) Mellin residues, T1 Lambda^0 = -2 zeta2 Tri0 two-region bracket, F50 = 0 by decay, F60 unique series), confirmed at 133.5 d by the materialized eps^1 relative-period integral: T^(1)(-1) = -0.67751873381094124976928955837140216282816254544618352556173467483778159541427717163716516552497941281862415293... PSLQ-negative at 110 digits against the focused weight-4 ring. ## 3. Generic distinct mass m^2 = (1,2,3,5) — eps^0 and eps^1 At generic mass the top sector is again rank two with algebraic (Liouvillian) solutions (Lairez–Vanhove; Doran–Harder–Pichon-Pharabod–Vanhove), the (1,3,5) scoop and the (2,3,5) sub-sunrise entering through the source; our 19-master elimination there is third-order and we have not reduced it to the second-order form, so the generic result is delivered as the transported connection with its derived boundary data. The scoop curve is Q_sc = x^4 - 36x^3 + 302x^2 - 564x + 121 = prod_{+-} ( x - (+-1 +- sqrt3 +- sqrt5)^2 ). Explicit rational data (19x19 validated connection): A_15_15 = (-270x^4 + 2706x^3 - 8658x^2 + 10676x - 4004) / ( (4x+1)*(x^4 - 40x^3 + 348x^2 - 880x + 196) ) cS = A_15_5 = (-192x^9 + 8550x^8 - 118439x^7 + 691024x^6 - 1762968x^5 + 1499645x^4 + 880795x^3 - 1000411x^2 - 61114x + 58410) / ( (2x+1)(4x+1)*(x^4 - 40x^3 + 348x^2 - 880x + 196)*(x^4 - 36x^3 + 302x^2 - 564x + 121) ) with the cone's own quartic L_bub = x^4 - 40x^3 + 348x^2 - 880x + 196 (irreducible over Q) in the denominators of the top-sector rows {15,16,17,18} only. Boundary data, partially runtime-derived: 21 of the 46 stored grade<=0 seed literals are now computed at runtime by icc-evaluate.py (tadpole^2 masters: exact Gamma(eps)^2 P^-eps expansions; triangles: exact W = ln(1+sqrt2)/sqrt2 plus live quadrature; sunrises S/S14: live materialized spectral integrals). The top boundary value is RETAINED interim (AMFlow eps-Laurent at x = -1, ~110 digits) — the analytic boundary route has derived it (tau0_gen = ln(15)*W with the tau1 = 2W identity) but gates end-to-end at 86.5 d, below the stored string, and a deeper materialized relative-period value is still to come; both routes are wired as runtime cross-checks: T(-1) = 0.13489705723149886247058591911842832927628482575287... The (1,3,5) curve is non-modular: no certified constant ring exists, so c+, c- are reported numerically (nothing fabricated). ## 4. What the deliverable is (DE-transport closure) There is no compact closed form for the boundary constants; the deliverable is a runtime boundary derivation + validated connection + transport: - Equal-mass 7x7 connection matrix, validated by two independent builds (byte-identical output). - Generic-mass 19x19 connection matrix, validated the same way. - Equal-mass graded Taylor transport (series order 60 at 130-digit working precision); eps^1 transport (order 50 at 75-digit working precision). - Generic-mass transport (order 45 at 90-digit working precision; 76-component eps-graded state, matching icc-evaluate.py: 19 masters x 4 eps-orders). - Boundary data: equal mass — ALL x=-1 seeds (eps^-2..eps^1) DERIVED at runtime from infinity-cusp analytic conditions (icc_blib.py; no AMFlow input), with the superseded oracle strings kept only as a held-out de-fit check; generic mass — 21/46 seeds runtime-computed, the rest retained interim eps-graded AMFlow evaluations (~110 d). - Secondary route: materialized relative-period integrals of the graph confirm the derived T(-1) to 284.8 d and T^(1)(-1) to 133.5 d (icc-derived.json; cross-check only, never input). - Certified sunrise period seed computed to 769 digits. - Scoop closure verified to 62 digits, double-confirmed against the exact Bessel-moment representation. - Oracle engine: an independent C++ port of AMFlow (arXiv:2201.11669), run as a separate process at held-out points never used in the construction. ## 5. Gate summary (held-out verification) | Configuration | Order | Held-out CV | Status | |--------------------------|------------------|----------------------------------------------|--------| | Equal mass, m^2=1 | eps^0 (wt 3) | 98.5 d at x=-2; 93.0-95.2 d at x=-3,-4,-6,-8 | closed | | Equal mass, m^2=1 | eps^1 (wt 4) | 75.0 d at x=-2; 70 d non-circular | closed | | Generic, m^2=1,2,3,5 | eps^0 (wt 3) | 85.1 d (x=-2), 88.8 d (x=-3); 88.9 d vs independent verification run | closed | | Generic, m^2=1,2,3,5 | eps^1 (wt 4) | 68.6 d at x=-2 | closed | | Scoop sub-block (eq.m.) | eps^0 | 62 d, double-confirmed vs Bessel-exact | closed | Live re-verification (icc-evaluate.py, 2026-07-04 back-port run, boundary constants DERIVED at runtime): 134.9 d / 134.8 d (equal mass eps^0/eps^1, dps-limited) and 108.1 d / 108.6 d (generic, capped by the 110 d retained literals) at held-out x=-2; equal-mass derived seeds vs superseded stored strings 156.6 d (held-out de-fit); T(-1) vs materialized relative-period integral 158.2 d (of 284.8 d available), T^(1)(-1) 133.5 d; scoop Bessel cross-check 35.0 d; total wall 503 s.