{
  "target": "local-downstream-fourier-bounds",
  "statement_status": "established",
  "original_failed_claim": "original-periodic-smoothing",
  "original_graph_preserved": true,
  "origin": "independent_model_assessment_not_formal_certificate",
  "edges": [
    {
      "from": "binary-block-coordinates",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "block-expansion-definition",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "block-projection-definition",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "elementary-analysis",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "periodic-step-definition",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "walsh-orthonormality",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "walsh-product-definition",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "periodic-step-definition",
      "to": "R1-shifted-kernel-polynomial",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "elementary-analysis",
      "to": "R1-shifted-kernel-polynomial",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R1-shifted-kernel-polynomial",
      "to": "R2-uniform-dyadic-grid-error",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "periodic-step-definition",
      "to": "R2-uniform-dyadic-grid-error",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "elementary-analysis",
      "to": "R2-uniform-dyadic-grid-error",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R1-shifted-kernel-polynomial",
      "to": "R3-products-and-finite-expansion",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R2-uniform-dyadic-grid-error",
      "to": "R3-products-and-finite-expansion",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "walsh-product-definition",
      "to": "R3-products-and-finite-expansion",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "elementary-analysis",
      "to": "R3-products-and-finite-expansion",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "block-projection-definition",
      "to": "R4-correct-block-coefficient-norms",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "block-expansion-definition",
      "to": "R4-correct-block-coefficient-norms",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "binary-block-coordinates",
      "to": "R4-correct-block-coefficient-norms",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "walsh-product-definition",
      "to": "R4-correct-block-coefficient-norms",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "walsh-orthonormality",
      "to": "R4-correct-block-coefficient-norms",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "elementary-analysis",
      "to": "R4-correct-block-coefficient-norms",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R3-products-and-finite-expansion",
      "to": "R5-projected-approximation-error",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R4-correct-block-coefficient-norms",
      "to": "R5-projected-approximation-error",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "elementary-analysis",
      "to": "R5-projected-approximation-error",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R1-shifted-kernel-polynomial",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R2-uniform-dyadic-grid-error",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R3-products-and-finite-expansion",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R4-correct-block-coefficient-norms",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    },
    {
      "from": "R5-projected-approximation-error",
      "to": "local-downstream-fourier-bounds",
      "type": "replacement_proof_dependency"
    }
  ],
  "replacement_lemmas": [
    {
      "id": "R1-shifted-kernel-polynomial",
      "statement": "For every integer ell>=1, set delta=2^(-ell-2) and M=2^(2ell+2). There is an explicit real 1-periodic trigonometric polynomial p_ell with |p_ell|<=1, Fourier support |b|<M<=2^(4ell), and sum_b |p_hat_ell(b)|<=C0 ell for an absolute C0>=2. Its coefficients are p_hat_ell(b)=integral_0^1 p_ell(t) exp(-2 pi i b t) dt. Let D_jump={0,1/2}, with distance taken on the circle. Whenever distance(t+delta,D_jump)>=a>0, with a<=1/2, one has |p_ell(t)-h(t+delta)|<=1/(Ma).",
      "proof": "Write e(t)=exp(2 pi i t) and define K_M(s)=M^(-1)|sum_{r=0}^{M-1} e(rs)|^2. Expanding the square gives K_M(s)=sum_{|b|<M}(1-|b|/M)e(bs). Consequently K_M is real, nonnegative, 1-periodic, and has integral 1 over any interval of length 1; the integral assertion follows by integrating each integer exponential.\n\nLet ||s||_T denote distance to the nearest integer. The geometric-series identity gives, for nonintegral s, K_M(s)=sin^2(pi M s)/(M sin^2(pi s)). The supplied sine inequalities therefore give K_M(s)<=1/(4M||s||_T^2). Hence, for 0<a<=1/2, integral_{a<=|s|<=1/2} K_M(s) ds <= 2 integral_a^(1/2) (4Ms^2)^(-1) ds <=1/(2Ma).\n\nDefine p_ell(t)=integral_{-1/2}^{1/2} K_M(s)h(t+delta-s) ds. Nonnegativity and unit mass give real-valuedness and |p_ell(t)|<=1. Expanding the finite kernel sum and changing variables in each integral gives p_ell(t)=sum_{|b|<M} c_b e(bt), where c_b=(1-|b|/M)e(b delta)h_hat(b) and h_hat(b)=integral_0^1 h(v)e(-bv)dv. Thus p_ell is a 1-periodic trigonometric polynomial. Integrating this finite expansion against e(-bt) verifies that c_b are precisely the required Fourier coefficients, with coefficients zero outside |b|<M.\n\nDirect integration of the two constant pieces of h gives h_hat(0)=0 and, for b!=0, h_hat(b)=(1-(-1)^b)/(pi i b). Therefore sum_b |c_b| <= (4/pi)sum_{b=1}^{M-1}1/b. Since M=2^(2ell+2), grouping the positive integers into dyadic intervals shows sum_{b=1}^{M-1}1/b <=2ell+2<=4ell. Any fixed C0>=max(2,16/pi) suffices. Also 2ell+2<=4ell, proving the asserted cutoff.\n\nFinally, put v=t+delta. If distance(v,D_jump)>=a, then h(v-s)=h(v) whenever |s|<a; possible equality endpoints do not affect the integral. Since the difference is always at most 2, |p_ell(t)-h(v)| <=2 integral_{a<=|s|<=1/2}K_M(s)ds <=1/(Ma).",
      "depends_on": [
        "periodic-step-definition",
        "elementary-analysis"
      ]
    },
    {
      "id": "R2-uniform-dyadic-grid-error",
      "statement": "For p_ell from R1 and every power of two Q>=2, Q^(-1)sum_{k=0}^{Q-1}|h(k/Q)-p_ell(k/Q)|^2 <=17\u00b72^(-ell). This includes the prescribed right-sided values of h at 0 and 1/2.",
      "proof": "Use delta and M from R1, so M delta=2^ell. First suppose Q<=(2delta)^(-1). Both jumps are grid points. A grid point t not at a jump has distance at least 1/Q>=2delta from the jump set. Consequently h(t+delta)=h(t) and distance(t+delta,D_jump)>=delta. At either jump itself, the prescribed value of h is its value immediately to the right, so again h(t+delta)=h(t), and the shifted point has distance delta from the jump set. R1 therefore gives |p_ell(t)-h(t)|<=1/(M delta)=2^(-ell) at every grid point. In particular, |p_ell(0)-1| and |p_ell(1/2)+1| are at most 2^(-ell). The mean squared error is at most 2^(-2ell).\n\nNow suppose Q>(2delta)^(-1). Declare a grid point bad when its circular distance from D_jump is at most 2delta. Each of the two corresponding circular arcs has length 4delta. An arc of length L<1 contains at most LQ+2 grid points: after splitting once if it wraps around 0, multiply coordinates by Q and count integers in each ordinary interval. Thus the bad proportion is at most 8delta+4/Q<=16delta. At every other point t, shifting by delta crosses no jump and leaves distance at least delta from the jump set. Hence h(t+delta)=h(t), and R1 again bounds the error by 2^(-ell). At bad points the error is at most 2, because |p_ell|<=1 and |h|=1. The mean squared error is therefore at most 4(16delta)+2^(-2ell) <=17\u00b72^(-ell).",
      "depends_on": [
        "R1-shifted-kernel-polynomial",
        "periodic-step-definition",
        "elementary-analysis"
      ]
    },
    {
      "id": "R3-products-and-finite-expansion",
      "statement": "For every integer m>=1 and A subset {0,...,m-1}, define p_A(y)=product_{j in A}p_ell(y/2^(j+1)) on 0<=y<2^m. Then p_A is real, |p_A|<=1, and 2^(-m/2)||w_A-p_A||_2 <=sqrt(17)|A|2^(-ell/2), using counting measure. Writing c_b=p_hat_ell(b), p_A has the finite expansion sum_{(b_j): |b_j|<M} (product_{j in A}c_{b_j})e(y sum_{j in A}b_j/2^(j+1)). Its coefficient-product l1 sum is at most (C0 ell)^|A|, hence at most (C0 ell)^K0 whenever |A|<K0 and K0>=1. Empty products are 1.",
      "proof": "Fix j<m and put Q=2^(j+1). Since Q divides 2^m, the fractional parts of y/Q, as y runs through 0,...,2^m-1, visit each Q-grid point equally often. R2 therefore gives 2^(-m)sum_y |h(y/Q)-p_ell(y/Q)|^2 <=17\u00b72^(-ell).\n\nOrder the elements of A. The difference between the two products is a telescoping sum, each term consisting of one difference h(y/2^(j+1))-p_ell(y/2^(j+1)) multiplied by other h and p_ell factors. Those other factors have absolute value at most 1. The L2 triangle inequality thus gives the asserted normalized product error. For A empty both products are exactly 1, and the error is zero. Reality and the sup norm bound follow directly from the same properties of every factor, and hold for all real y as well.\n\nSubstituting the finite expansion from R1 into each factor and multiplying proves the displayed expansion. It is also an integer-frequency expansion for the period 2^m: its term indexed by (b_j) has integer harmonic index k=sum_{j in A}2^(m-j-1)b_j and exponential e(ky/2^m). On the sampled cyclic group, indices may be reduced modulo 2^m. Combining coincident indices cannot increase the l1 norm, by the triangle inequality.\n\nFinite-sum factorization gives sum_{(b_j)}|product_{j in A}c_{b_j}|=(sum_{|b|<M}|c_b|)^|A| <=(C0 ell)^|A|. The empty tuple contributes 1. Since C0 ell>=2, increasing the exponent from |A| to K0 preserves the required upper bound.",
      "depends_on": [
        "R1-shifted-kernel-polynomial",
        "R2-uniform-dyadic-grid-error",
        "walsh-product-definition",
        "elementary-analysis"
      ]
    },
    {
      "id": "R4-correct-block-coefficient-norms",
      "statement": "Let z=(u,w) denote the n-m outside bits, so there are 2^(n-m) values of z and 2^m values of y. For the supplied projection onto |A intersection J|<K0, every allowed block coefficient satisfies g_A(z)=2^(-m)sum_{y=0}^{2^m-1}f(z,y)w_A(y). Consequently |g_A(z)|<=1, sum_{|A|<K0}|g_A(z)|^2<=1, ||g_A||_{2,z}<=2^((n-m)/2), and sum_{|A|<K0}||g_A||_{2,z}^2=2^(-m)||g||_2^2<=2^(-m)||f||_2^2<=2^(n-m). In particular, ||g||_2<=||f||_2<=sqrt(N), with all displayed norms using counting measure.",
      "proof": "The binary block coordinates give a bijection between x and (z,y), with y in 0,...,2^m-1. By the definition of h, h(x/2^(j+1)) is the binary sign determined by bit j. Thus a global Walsh character factors as v_T(z)w_A(y), where T indexes outside bits and A indexes the block bits relabeled as 0,...,m-1.\n\nUse the supplied orthonormal Walsh basis to expand f in these tensor-product characters. Its global coefficients are normalized counting inner products, namely 2^(-n)sum_x f(x)w_S(x). The projection retains every outside index T precisely for those block indices A with |A|<K0. For such A, orthogonality in y gives g_A(z)=sum_T f_hat(T union A_J)v_T(z)=2^(-m)sum_y f(z,y)w_A(y), where A_J is the corresponding set of global block positions. This also verifies the coefficient appearing in the supplied block expansion.\n\nThe averaging formula and |f|<=1 immediately give |g_A(z)|<=1. For each fixed z, expand f(z,\u00b7) in the complete block Walsh basis. Expanding the finite squared norm and applying orthogonality yields sum_{all A}|2^(-m)sum_y f(z,y)w_A(y)|^2=2^(-m)sum_y|f(z,y)|^2<=1. Restricting this sum proves the pointwise selected-coefficient bound.\n\nLikewise, block orthogonality gives ||g||_2^2=2^m sum_{allowed A}||g_A||_{2,z}^2. Summing the preceding coefficient identity over z proves that this is at most ||f||_2^2<=N. Finally, |g_A(z)|<=1 and the outside domain size 2^(n-m) give the individual coefficient L2 bound. If f is real, its displayed partial coefficients are real.",
      "depends_on": [
        "block-projection-definition",
        "block-expansion-definition",
        "binary-block-coordinates",
        "walsh-product-definition",
        "walsh-orthonormality",
        "elementary-analysis"
      ]
    },
    {
      "id": "R5-projected-approximation-error",
      "statement": "For n sufficiently large, integer 1<=m<n/100, and real K0>=1, define tilde g(z,y)=sum_{A subset {0,...,m-1}, |A|<K0}g_A(z)p_A(y). Then ||g-tilde g||_2 <= C n^K0 2^(-ell/2)sqrt(N), with an absolute C. If f is real, tilde g is real.",
      "proof": "Put d_A(y)=w_A(y)-p_A(y). At each (z,y), finite Cauchy-Schwarz and R4 give |sum_A g_A(z)d_A(y)|^2 <= (sum_A|g_A(z)|^2)(sum_A|d_A(y)|^2) <=sum_A|d_A(y)|^2, with sums over allowed A. Summing over z and y, and applying R3, yields ||g-tilde g||_2^2 <=2^(n-m)sum_A||d_A||_{2,y}^2 <=17N\u00b72^(-ell)sum_A|A|^2.\n\nLet r be the largest integer with 0<=r<=m and r<K0. If r=0, every d_A is zero. Otherwise, the binomial theorem gives k<=2^k, hence k^2<=4^k. Also binom(m,k)<=m^k, directly from its product formula. Therefore sum_A|A|^2=sum_{k=1}^r k^2 binom(m,k) <=sum_{k=1}^r(4m)^k <=(4/3)(4m)^r. Since 4m<n and r<K0, this is at most (4/3)n^K0. The preceding error bound consequently gives even the stronger estimate C n^(K0/2)2^(-ell/2)sqrt(N), which implies the requested bound. All constants are independent of m, n, K0, and ell. Reality when f is real follows from R4 and the real-valued p_A.",
      "depends_on": [
        "R3-products-and-finite-expansion",
        "R4-correct-block-coefficient-norms",
        "elementary-analysis"
      ]
    }
  ]
}