R87/G02 — polynomial locality estimate for the gap bootstrap
Current-agent derivation, 2026-10-24. This unpacks the standard spectral-flow construction of Nachtergaele–Sims–Young, arXiv:2010.15337, equations(4.16)–(4.27), retaining the gap dependence. It is not a claim that this construction is new.

Hypotheses and conventions.
Work on finite d-dimensional tori, d=2 in the application. Metric balls have at most C(1+r)^d sites, uniformly in the torus size. H_s=H_0+sV has an exponentially summable anchored interaction, with a uniform real-time Lieb–Robinson velocity for 0<=s<=1 and J<=1. The reference local energy h_x has fixed radius and bounded norm. V=sum_x V_x, where V_x has exponentially small radius tails bounded by J times fixed constants. Suppose the ground eigenvalue of H_s is simple and separated by at least gamma/2 on the bootstrap interval. All constants below depend on fixed moment orders and interaction geometry, not on gamma, J, volume, or s. Take 0<gamma<=1.

Fix an integer a0>C_*+2d+10. Set
 b=a0+2d+4, M=b+d+4=a0+3d+8, A=2M+1.
The polynomial F-function F_b(r)=(1+r)^(-b) is summable and has a convolution constant depending only on b,d. Constants can increase at each displayed estimate; these finitely many changes do not alter the powers.

1. Filters and ordinary time evolution.
Choose one real even Schwartz function w with integral1 and Fourier support in(-1/8,1/8). A smooth compactly supported Fourier function with value1 at0 supplies such a w; positivity is unnecessary. Let w_gamma(t)=gamma w(gamma t). Define the odd integrated kernel F_gamma(t) with the sign and Fourier normalization giving the inverse-frequency multiplier outside the Fourier support of w_gamma. Explicitly its absolute value is bounded by int_{|t|}^infinity |w_gamma(u)|du. Then for every fixed integer k,
 int |t|^k |w_gamma(t)|dt <=C_k gamma^(-k),
 int |t|^k |F_gamma(t)|dt <=C_k gamma^(-k-1).             (1)
The filter map T_s(A)=int w_gamma(t) tau_t^s(A)dt removes ground/excited matrix elements on the bootstrap interval. The generator D_s=int F_gamma(t) tau_t^s(V)dt, with the corresponding conventional sign, transports the ground projection exactly. No approximation of the spectral projector is used.
For a fixed-radius A at x, normalized partial trace Pi_{x,r} and exponential LR imply
 ||tau_t^s(A)-Pi_{x,r}tau_t^s(A)||
 <= min(C||A||,C||A|| exp(v|t|-mu r))
with harmless polynomial radius factors absorbed by reducing mu. Split |t| at c r, where c is fixed sufficiently small. Equation(1) gives
 ||T_s(A)-Pi_{x,r}T_s(A)||<=C||A|| gamma^(-M)(1+r)^(-M). (2)
The same argument, after an exponentially summable decomposition of V_x, gives the anchored generator tail
 ||D_{s,x}-Pi_{x,r}D_{s,x}||<=CJ gamma^(-M-1)(1+r)^(-M). (3)
This also bounds its norm by CJ/gamma, and hence by the coarser prefactor in(3).

2. The flow velocity is made polynomial, not exponentiated.
Decompose D_{s,x} by differences of successive local partial traces. The radius-r term has norm at most CJ gamma^(-M-1)(1+r)^(-M), including the bounded number of smallest radii. For two sites y,z, anchors whose radius-r ball contains both number at most C(1+r)^d, and d(y,z)<=2r. Consequently
 ||D_s||_{F_b} <= CJ gamma^(-M-1) sum_r(1+r)^(b+d-M)
 <= CJ gamma^(-M-1)=:kappa.                              (4)
Here M=b+d+4, so the summand decays as r^-4. This explicitly pays the anchor multiplicity; it is not a termwise-to-interaction-norm shortcut.
Impose J<=c gamma^(M+1), with c small enough that kappa<=1. The parameter-time LR estimate for the flow alpha_s(A)=U_s^* A U_s then has e^(C kappa)-1<=C kappa. For A supported on X, and X(r) its r-neighborhood,
 ||(alpha_s-id)(A)||<=C kappa |X| ||A||,
 ||(id-Pi_{X(r)})(alpha_s-id)(A)||
 <=C kappa |X| ||A|| (1+r)^(d-b).                       (5)
The first inequality follows directly by integrating alpha_u(i[D_u,A]) and summing interactions touching X. The second follows from the F_b LR commutator estimate and summing its tail outside X(r); the untransformed A is already supported in X. Both retain the factor kappa. The corresponding estimates for alpha_s itself have a bounded constant in place of kappa and include ||A||.

3. Differentiating the energy filter costs a fixed moment.
For fixed-radius A, Duhamel gives
 partial_s tau_t^s(A)=i int_0^t tau_{t-u}^s([V,tau_u^s(A)])du
(up to reversing u, which does not affect the bounds). Summing the commutators with V_x inside a real-time light cone, and using the exponential LR estimate outside it, gives
 ||partial_s tau_t^s(A)||<=CJ ||A|| |t|(1+|t|)^d.         (6)
For completeness, a localization version keeps the factor J. When |t|<=c r, split anchors of V into distance<=r/4 and >r/4 from x. For far anchors the summed commutator norm is CJ times a polynomial in r times exp(v|t|-mu r/8), by LR and the exponential anchored tails. For near anchors, truncate V_x to radius r/8 and tau_u(A) to radius r/2. Their commutator is supported within radius r/2. Localizing its subsequent tau_{t-u} evolution to radius r loses at most another polynomial in r times the same exponential. There are O(r^d) near anchors and at most O(r^d) sites in each intermediate support. Integrating u therefore yields, with a fixed integer K (e.g.2d+3),
 ||(id-Pi_{x,r}) partial_s tau_t^s(A)||
 <=CJ ||A|| (1+r)^K exp(v'|t|-mu' r).                    (7)
Choose c<mu'/(2v'). The short-time integral of(7) is bounded by CJ||A|| times an exponential in r. For |t|>cr, use twice(6) and the M-th moment in(1):
 int_{|t|>cr}|w_gamma(t)||t|(1+|t|)^d dt
 <=C gamma^(-M)(1+r)^(-(M-d-1)).
Thus both the norm and radius tail of (T_s-T_0)(A) are bounded by
 CJ||A|| gamma^(-M)(1+r)^(-(M-d-1)),                    (8)
where the norm statement uses r=0. This derivation avoids subtracting two separately local filters, which would lose the small factor J.

4. Composition and the globally block-diagonal local terms.
Use the exact three-term identity from NSY(4.20):
 Y_x=(alpha_s-id)T_s(h_x)+(T_s-T_0)(h_x)+s alpha_s T_s(V_x).
Decompose T_s(h_x) into successive radius-R shells using(2). Their norms are bounded by C gamma^-M(1+R)^-M, and their support sizes by C(1+R)^d. For R<=r/2 apply(5); for R>r/2 apply its norm bound and sum the tail. The resulting bound for the first term is
 C kappa gamma^-M[(1+r)^(d-b)+(1+r)^(d-M+1)]
 <=CJ gamma^(-2M-1)(1+r)^(-a0).                          (9)
Equation(8) gives the same weaker bound for the second term. Apply the bounded version of(5) to the filtered exponential shells of V_x for the third; it is bounded by CJ gamma^-M(1+r)^-a0. All these estimates also give norm bounds at r=0. Scalar subtraction does not change a localization tail.
As in NSY Proposition4.2, Y_x commutes with the reference ground projector P_0. Since P_0 is rank one, c_x=<Omega_0,Y_x Omega_0> is real and Z_x=Y_x-c_x I satisfies Z_x P_0=0. Moreover
 sum_x Z_x=U_s^*H_sU_s-H_0-E_s I,
 E_s=sum_x c_x=ground energy of H_s,
 ||Z_x||<=CJ gamma^-A,
 ||Z_x-Pi_{x,r}Z_x||<=CJ gamma^-A(1+r)^-a0.              (10)
The sum identity uses T_s(H_s)=H_s and T_0(H_0)=H_0. It does NOT assume T_0(h_x)=h_x term by term.

Conclusion and restrictions.
The previously missing locality estimate C1 in work-03 is supplied with A=2(a0+3d+8)+1, under the additional smallness J<=c gamma^(M+1). The stability argument is allowed to impose that restriction: its purpose is to establish some polynomial perturbation window, not an optimal one. Every use of a filter has a specified fixed moment; every use of the flow has kappa<=1. Thus no exp(poly(gamma^-1)) is hidden in a coefficient. This does not yet give an implementation cost for the filter, the parent, or U_s; those are algorithmic questions separate from this analytic existence argument.
