"""v296 -- FLATAWAY.A2.CLOSED.01: the EXACT a_2 coefficient in CLOSED FORM, replacing the validated-numeric sum of v295. The infinite mode sum is evaluated analytically: the two operator tails telescope to a geometric series, leaving a finite (4k-1)-term middle. The second-order heat coefficient of the smooth Z4 mode 4k (from v295): W_k(t) = (1/2) sum_{n in Z} Phi(|n|, |n+4k|), Phi(a,b) = [(b-a) t e^{bt} + e^{at} - e^{bt}] e^{-t(a+b)} / (a-b)^2, Phi(a,a)=(t^2/2)e^{-ta}. [E] 1. TAILS TELESCOPE. for n >= 0, Phi(n,n+4k) = e^{-nt} C_k/(16k^2) with the FIXED constant C_k = 4kt + e^{-4kt} - 1 (the n-dependence is the pure geometric factor e^{-nt}); likewise for n <= -4k with D_k = e^{4kt} - 4kt - 1. Summing the two geometric tails and using C_k + D_k e^{-4kt} = 4kt(1 - e^{-4kt}) gives the closed tail W_k^tail(t) = t (1 - e^{-4kt}) / (8k (1 - e^{-t})). [E] 2. MIDDLE IS FINITE. the remaining modes -4k < n < 0 give a finite sum (1/2) sum_{m=1}^{4k-1} Phi(m, 4k-m), which contains the degenerate diagonal Phi(2k,2k) = (t^2/2) e^{-2kt} -- exactly the first-order level-splitting term. [E] 3. CLOSED FORM. W_k(t) = t(1-e^{-4kt})/(8k(1-e^{-t})) + (1/2) sum_{m=1}^{4k-1} Phi(m,4k-m), an exact closed form (geometric tail + finite middle). For k=1: W_1(t) = t (2t e^{t} + e^{3t} + 3 e^{2t} + e^{t} - 1) e^{-3t} / 8. It reproduces the v295 numeric coefficient to machine precision (k=1,2,3: 0.20826, 0.14795, 0.10491 at t=0.5). [E] 4. POSITIVITY MANIFEST. in the closed form the tail is > 0 (1-e^{-4kt} > 0, 1-e^{-t} > 0) and every middle term Phi(m,4k-m) > 0, so W_k(t) > 0 analytically -- the v295 convexity positivity is now visible term by term in the closed form. [E] 5. SMALL-t LEADING ORDER (with an exact cancellation). as t -> 0 the closed form gives W_k(t) = t/2 + O(t^3): the leading coefficient W_k(t)/t -> 1/2 is MODE-INDEPENDENT, and the t^2 term VANISHES because the tail's t^2 piece (1/4 - k) exactly cancels the finite middle's (4k-1)/4 -- so the mode index k enters only at O(t^3). A clean analytic small-t law with a non-trivial cancellation. Status: [E] the closed-form evaluation of the exact a_2 coefficient (geometric tail + finite middle) + manifest positivity + the small-t leading order; reproduces v295 to machine precision. An exact analytic upgrade of v295's validated numeric. Python (sympy closed form + numpy validation). """ import numpy as np import sympy as sp from tfpt_constants import check, summary, reset _t = sp.symbols("t", positive=True) def _Phi_sym(a, b): a, b = sp.Integer(a), sp.Integer(b) if a == b: return (_t ** 2 / 2) * sp.exp(-_t * a) return ((b - a) * _t * sp.exp(b * _t) + sp.exp(a * _t) - sp.exp(b * _t)) \ * sp.exp(-_t * (a + b)) / (a - b) ** 2 def _W_closed(k): tail = _t * (1 - sp.exp(-4 * k * _t)) / (8 * k * (1 - sp.exp(-_t))) middle = sp.Rational(1, 2) * sum(_Phi_sym(m, 4 * k - m) for m in range(1, 4 * k)) return sp.simplify(tail + middle), sp.simplify(tail) def _W_num(k, tt, N=200): def phin(a, b): if a == b: return (tt ** 2 / 2) * np.exp(-tt * a) return ((b - a) * tt * np.exp(b * tt) + np.exp(a * tt) - np.exp(b * tt)) \ * np.exp(-tt * (a + b)) / (a - b) ** 2 return sum(phin(abs(n), abs(n + 4 * k)) for n in range(-N, N + 1)) / 2 def run(): reset() print("v296 FLATAWAY.A2.CLOSED.01: the exact a_2 coefficient in CLOSED FORM (geometric tail + finite middle)") # 1. tails telescope to a geometric closed form k = sp.symbols("k", positive=True, integer=True) Ck = 4 * k * _t + sp.exp(-4 * k * _t) - 1 Dk = sp.exp(4 * k * _t) - 4 * k * _t - 1 combine = sp.simplify(Ck + Dk * sp.exp(-4 * k * _t) - 4 * k * _t * (1 - sp.exp(-4 * k * _t))) check("TAILS TELESCOPE [E]: Phi(n,n+4k)=e^{-nt} C_k/(16k^2) (n>=0) sums geometrically; " "C_k + D_k e^{-4kt} = 4kt(1-e^{-4kt}) (residual %s) so the two tails close to " "W_k^tail = t(1-e^{-4kt})/(8k(1-e^{-t}))" % combine, combine == 0) # 2-3. closed form vs numeric for k=1,2,3 errs = [] for kk in (1, 2, 3): Wc, _ = _W_closed(kk) val = float(Wc.subs(_t, sp.Rational(1, 2))) errs.append(abs(val - _W_num(kk, 0.5))) W1, tail1 = _W_closed(1) check("MIDDLE FINITE + CLOSED FORM [E]: W_k = t(1-e^{-4kt})/(8k(1-e^{-t})) + " "(1/2)sum_{m=1}^{4k-1} Phi(m,4k-m) (incl. the degenerate diagonal " "Phi(2k,2k)=(t^2/2)e^{-2kt}); W_1(t) = %s; reproduces the v295 numeric to " "machine precision (k=1,2,3 abs.err %s)" % (sp.sstr(W1), [f"{e:.1e}" for e in errs]), all(e < 1e-9 for e in errs)) # 4. positivity manifest in the closed form tail_val = float(tail1.subs(_t, sp.Rational(1, 2))) mids = [float(_Phi_sym(m, 4 - m).subs(_t, sp.Rational(1, 2))) for m in (1, 2, 3)] check("POSITIVITY MANIFEST [E]: in the closed form the tail > 0 (1-e^{-4kt}>0, " "1-e^{-t}>0; tail(k=1,t=.5)=%.4f) and every middle term Phi(m,4k-m) > 0 " "(k=1: %s) => W_k(t) > 0 analytically (the v295 convexity, term by term)" % (tail_val, [round(m, 4) for m in mids]), tail_val > 0 and all(m > 0 for m in mids)) # 5. small-t leading order: W_k(t)/t -> 1/2 (mode-independent), then (4k-1)t^2/4 leads = [] subs = [] for kk in (1, 2, 3): Wc, _ = _W_closed(kk) leads.append(sp.nsimplify(sp.limit(Wc / _t, _t, 0))) subs.append(sp.nsimplify(sp.limit((Wc - _t / 2) / _t ** 2, _t, 0))) ok_lead = all(sp.simplify(l - sp.Rational(1, 2)) == 0 for l in leads) ok_sub = all(sp.simplify(s) == 0 for s in subs) # t^2 coefficient vanishes check("SMALL-t LEADING ORDER [E]: W_k(t) = t/2 + O(t^3) as t->0 -- the leading " "coefficient W_k/t -> 1/2 is MODE-INDEPENDENT (k=1,2,3: %s) and the t^2 term " "VANISHES (tail's (1/4-k) cancels the middle's (4k-1)/4; (W_k-t/2)/t^2 -> %s), " "so k enters only at O(t^3)" % ([str(l) for l in leads], [str(s) for s in subs]), ok_lead and ok_sub) return summary("v296 FLATAWAY.A2.CLOSED.01: exact a_2 coefficient in closed form W_k(t)=t(1-e^{-4kt})/(8k(1-e^{-t}))+(1/2)Σ Phi(m,4k-m), manifestly positive, leading W_k~t/2 (t^2 term cancels); reproduces v295 exactly") if __name__ == "__main__": raise SystemExit(1 if run() else 0)