{
  "coverage": "mathematical-material-only",
  "equations": [
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:36-36",
      "line_end": 36,
      "line_start": 36,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "14bdb5565ba3798cf9971db88f8a6a05181c04b4554cec282908b431a866cdca",
      "text": "    ell=F(M)-F(S)>0.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:64-65",
      "line_end": 65,
      "line_start": 64,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "96336ce51ad939e663aa6ae70020560393cfa3ac5fb8d1e56c4dd77c2b7c0bfb",
      "text": "    nu_bar(ell) ~ C ell^(-1/3),                 ell down to 0,\n    E N_bar((0,t])/V_X ~ (3/2) C t^(2/3),       t down to 0.        (1)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:78-81",
      "line_end": 81,
      "line_start": 78,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "96c36b3913251532b200ef4ccc306cc104de89d468fe1e427e0c96d3aa5255cb",
      "text": "    O_(p,q)=(F(p),grad F(p),F(q),grad F(q)),\n    Delta_(p,q)=|det Hess F(p) det Hess F(q)|,\n    tau_(p,q)=1{Hess F(p)<0} 1{Hess F(q) has index d-1},\n    W_(p,q)=Delta_(p,q) tau_(p,q).                           (2)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:91-94",
      "line_end": 94,
      "line_start": 91,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "1ddcdb23e55dde4e216c194505c6a9aca2cbadbaff8d16cbe92d98184f4dfae2",
      "text": "    E sum_(p!=q,grad F(p)=grad F(q)=0) w(p,q,F)\n      =integral_(p!=q) p_(grad F(p),grad F(q))(0)\n                           E[Delta_(p,q) w | both gradients=0]\n                           dvol(p) dvol(q).                (3)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:100-102",
      "line_end": 102,
      "line_start": 100,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "a3b3a5e2e73ef5fa32f83f7a80b08376dc005329afef759285299757d20a0341",
      "text": "    K_sel(p,q,b,s)=p_O(b,0,s,0) E_Q[W_(p,q) sigma(p,q,F)],\n    nu_bar(ell)=V_X^(-1) integral_(p!=q) integral_R\n                       K_sel(p,q,b,b-ell) db dvol(p) dvol(q).   (4)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:123-123",
      "line_end": 123,
      "line_start": 123,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "fc59fa26a7f0bf4ea4aa72e4418117f1ca012141ff1ed89d6269512a09c8029a",
      "text": "    min_(closure B_i) f > lambda for each ball in the chain.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:129-130",
      "line_end": 130,
      "line_start": 129,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "440e97281fec90eb96ee8361db9905a4904f0cfdac66edb836da5c5f553b1858",
      "text": "    D_f(p)=sup{lambda in Q: lambda<f(p),\n         exists z in D with f(z)>f(p) connected to p in {f>lambda}}, (5)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:140-140",
      "line_end": 140,
      "line_start": 140,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "37d81641db6b3ed8c9a4f25da286a82067f679cdd68cb194f08ba01fd32e70c2",
      "text": "    D_f(p)=f(q)<f(p);\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:175-175",
      "line_end": 175,
      "line_start": 175,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "d863ddd34922bd121b351a5c379d22393bdd8ee24010ed07c47f3ec09a0b578e",
      "text": "    M=exp_x(-r u/2), S=exp_x(r u/2).\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:181-181",
      "line_end": 181,
      "line_start": 181,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "f8ea374207247bc48d971eb33da0fd7f8948610f22b8664f74cd1c4236a0bddd",
      "text": "    (h_M,h_S,g_uM,g_uS,(g_iM,g_iS)_(i=1,...,d-1)).\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:185-189",
      "line_end": 189,
      "line_start": 185,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "89fad618b01d7180fb22e8dca36254517e9967ec6acd9362532446c154b301ab",
      "text": "    (h_M+h_S)/2,\n    (h_S-h_M)/r,\n    (g_uS-g_uM)/r,\n    (6/r^2)[g_uM+g_uS-2(h_S-h_M)/r],\n    ((g_iM+g_iS)/2,(g_iS-g_iM)/r)_(i=1,...,d-1).           (6)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:193-193",
      "line_end": 193,
      "line_start": 193,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "b32e048523f61fda15fae865445dd011bf1c74a38327fbf2adbfaa7201adbc36",
      "text": "    v_r=(b-k r^3/2,-k r^2,0,12k,(0,0)_(i=1,...,d-1)).     (7)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:197-200",
      "line_end": 200,
      "line_start": 197,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "5d23671b1f78d638199a93c186d27676d7a6b3fd9f03f489c250ad4b420e9427",
      "text": "    [  1/2       1/2         0        0    ]\n    [ -1/r       1/r         0        0    ]\n    [   0         0        -1/r      1/r   ]\n    [ 12/r^3   -12/r^3      6/r^2    6/r^2 ].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:206-207",
      "line_end": 207,
      "line_start": 206,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "7b23021ae3f9bb3449d29a14f64ab2a445da7d89168d5a2477a6c5916b6c83b5",
      "text": "    |det(O -> U_r)|=12 r^(-(d+3)),\n    p_O(b,0,b-k r^3,0)=12 r^(-(d+3)) pi_r(v_r).            (8)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:215-216",
      "line_end": 216,
      "line_start": 215,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "3da423efc94d33fa5078b2ccc398fe516250e3adc04c5c11ff0f273dd20bf655",
      "text": "    U_0=(F,F_u,F_uu,F_uuu,(F_vi,F_uvi)_(i=1,...,d-1)),\n    v_0=(b,0,0,12k,(0,0)_(i=1,...,d-1)).                 (9)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:236-237",
      "line_end": 237,
      "line_start": 236,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "8c51a12327a8a47a5873e97b2de337a93662b05fb2af39e053a63d1be4b086b7",
      "text": "    H_i=[ r alpha_i   r beta_i^T ]\n        [ r beta_i        A_i   ].                       (10)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:244-245",
      "line_end": 245,
      "line_start": 244,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "d0469d9143feea2b2b78c80795e4a0aace6f310cc208070ea969fc722e8a86ec",
      "text": "    alpha_M -> -6k, alpha_S -> 6k,\n    beta_M,beta_S bounded, A_M,A_S -> A_0.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:253-253",
      "line_end": 253,
      "line_start": 253,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "827660c5815e583fbc2b5d4d5425157d62121603b0a60e596212af3db0b8721a",
      "text": "    det H_i/r=alpha_i det A_i-r beta_i^T adj(A_i) beta_i.   (11)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:263-263",
      "line_end": 263,
      "line_start": 263,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "ef4ac4cf405f71283c311f4b8563317ee79a3a9f6ce3d7100e7cf6890909ee8b",
      "text": "    V_r:=W_r/r^2 -> V_0:=36k^2 (det A_0)^2 1{A_0<0}.     (12)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:356-356",
      "line_end": 356,
      "line_start": 356,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "49cdbb048798c3dc08325c3a1fa8df4ac2c54e9871e21ef5a0a4268e55dfec18",
      "text": "    Psi(x,h)=(exp_x(-h/2),exp_x(h/2))\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:364-364",
      "line_end": 364,
      "line_start": 364,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "746831f8045cc028d30f79d6aa6c31a27ca15b3e590a4320abf7eaf75b08e0d2",
      "text": "    (a,b) -> (a-b/2,a+b/2),\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:371-372",
      "line_end": 372,
      "line_start": 371,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "b7abcfa13c8cdb05e42b9ec202d14fd735f73d52c32bdce0e25ef384b1e8525c",
      "text": "    dvol(M) dvol(S)\n       =J(x,r u) dvol(x) r^(d-1) dr d sigma_x(u).          (13)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:383-383",
      "line_end": 383,
      "line_start": 383,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "99284f9becb38af43180a679059d0f2f21d48e30c6249bafe8143f40269c58e9",
      "text": "    B_r(b,k,x,u)=12 pi_r(v_r) E_Qr[V_r sigma].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:392-395",
      "line_end": 395,
      "line_start": 392,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "fc3dd6ec0e426953fbda8ed278650d8654dd6fa7062de740107508ad003c62ec",
      "text": "    spatial polar volume         r^(d-1) dr,\n    height change s=b-k r^3       r^3 db dk,\n    original observation density 12 r^(-(d+3)) pi_r(v_r),\n    selected determinant weight  r^2 E_Qr[V_r sigma].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:400-400",
      "line_end": 400,
      "line_start": 400,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "24d2b5ba4fedf67cb92e99654219eed050a864d2ebcc256eef9e39cf2c453b80",
      "text": "    (d-1)+3-(d+3)+2=1.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:409-414",
      "line_end": 414,
      "line_start": 409,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "e149cda777474fe0cb62496945987abd266f46c64dfc5bf12d5a1f2fe811a139",
      "text": "    ell^(1/3) nu_near(ell)\n      =(1/(3 V_X)) integral_(SX,R,(0,infinity))\n          1{k>=ell/r_0^3} k^(-2/3)\n          J(x,(ell/k)^(1/3)u)\n          B_((ell/k)^(1/3))(b,k,x,u)\n          dvol(x) d sigma_x(u) db dk.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:425-427",
      "line_end": 427,
      "line_start": 425,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "3f3da91457480b8153b70dfdb13abd226e18e1616e1426aeecf9a59ea7e0bca3",
      "text": "    C=(4/V_X) integral_(SX,R,(0,infinity))\n                    k^(-2/3) pi_0(v_0) z_0(b,k,x,u)\n                    dvol(x) d sigma_x(u) db dk.             (14)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:456-457",
      "line_end": 457,
      "line_start": 456,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "313067933b5b820f803d7f87d2749dc78297a9ffac11619582e20584b9485680",
      "text": "    odd:  G=grad F (dimension d), T=F_uuu (one scalar);\n    even: F, V=Hess F(u,.) (dimension d), A=Hess F|_(u-perp). (15)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:469-470",
      "line_end": 470,
      "line_start": 469,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "af8f2dd62e4a2f73d27cd6eb7bf8bef2dbd8f0a07a2548cc671d9c3f5df60cf9",
      "text": "    tau_(x,u)^2=Var(T|G=0)>0,\n    D_(x,u)=E[(det A)^2 1{A<0} | V=0].                   (16)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:475-476",
      "line_end": 476,
      "line_start": 475,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "aff0fd1f16c08fb3f5750537f25e458a34859effdc777a9ca9163dadf88301ce",
      "text": "    pi_0(v_0)=p_(F,V)(b,0) p_G(0) phi_tau(12k),\n    z_0=36k^2 E[(det A)^2 1{A<0} | F=b,V=0].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:481-483",
      "line_end": 483,
      "line_start": 481,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "422065f9723f0a971af704214075f65641ae1e8dd3d3af036fe466bf503e2cf7",
      "text": "    integral_R p_(F,V)(b,0)\n        E[(det A)^2 1{A<0} | F=b,V=0] db\n      =p_V(0) D_(x,u).                                    (17)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:490-492",
      "line_end": 492,
      "line_start": 490,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "9de9189be21cd9ab881fbdc050238eb47cb35d752b5afa30312794c5064457ed",
      "text": "    144 integral_0^infinity k^(4/3) phi_tau(12k) dk\n      =12^(-1/3) integral_0^infinity t^(4/3) phi_tau(t) dt\n      =Gamma(7/6) tau^(4/3)/(24^(1/3) sqrt(pi)).            (18)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:498-501",
      "line_end": 501,
      "line_start": 498,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "378b606e2f08aeb1f7967f71685b1f06977ccc1a39ea1aa5c50a5465913ce337",
      "text": "    C=Gamma(7/6)/(24^(1/3) sqrt(pi) V_X)\n       *integral_SX p_G(0) p_V(0) tau_(x,u)^(4/3)\n                                             D_(x,u)\n                    dvol(x) d sigma_x(u).                  (19)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:545-547",
      "line_end": 547,
      "line_start": 545,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "7705c6501a5fca0a754db98232b6a96b0b20533854778e7d65966a1437422e1d",
      "text": "    xi_r=m_r+P_r Z,\n    m_r=L_r^T (L_r L_r^T)^(-1) v_r,\n    P_r=I-L_r^T (L_r L_r^T)^(-1) L_r,                     (20)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:564-565",
      "line_end": 565,
      "line_start": 564,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "b3a65afb9a2cd7397f152707ac217074b42427c3091f1d64a01d72e39df1d302",
      "text": "    |det H_i|/r<=C(1+||F||C3)^d,\n    W_r/r^2<=C(1+||F||C3)^(2d).                          (21)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:574-575",
      "line_end": 575,
      "line_start": 574,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "4a6e1dbb96211588c9716b88cc1b08038d87f8e3685c11c3b288f7ad505f7da2",
      "text": "    12 pi_r(v_r) E_Qr[V_r]\n      <=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)].                (22)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:614-615",
      "line_end": 615,
      "line_start": 614,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "1b03e4499184f59bbc69f91197987730f5ea153c4ff8c4d5657b3fbdbe8eb839",
      "text": "    integral_R p_O(b,0,b-ell,0) E_Q[W] db <=C,\n                                          0<ell<=1.       (23)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:663-663",
      "line_end": 663,
      "line_start": 663,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "af6f4ebbb02b3daf07519e7b1371ec64465b3dfa3d1a8428348d57a52f57af3a",
      "text": "    R=S/sqrt(E S^2), F=R H, E S^2=alpha/(alpha+2).\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:672-674",
      "line_end": 674,
      "line_start": 672,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "e0044907b9bd2ac914e708f3adffec553d39cbdd13f7d102b7a25e73e762ee4a",
      "text": "    E N_F((0,t])\n      >=E N_H([a,b]) P(R<=t/b)\n      =c t^alpha, c>0.                                   (24)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:701-702",
      "line_end": 702,
      "line_start": 701,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "862d38aab3f507807ba9e43637ccac4d789390cf9e6df783c614fc2d65dbc674",
      "text": "    s=gamma+m-a+w,\n    r^s B_r dr db dk d(other parameters).\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:706-706",
      "line_end": 706,
      "line_start": 706,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "4a10e3337fe06ce0b1f5172df4df52622c710398910965b831dd893ad1c5178c",
      "text": "    e=(s+1)/m-1=(gamma+1-a+w)/m,\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:710-710",
      "line_end": 710,
      "line_start": 710,
      "path": "experiments/universality/fold_structural_universality/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "47e43a9edc4e1d6a0f94e18bc28496d5f1ffd87cc36abaebdaaa6364bd08a603",
      "text": "    (1/m) integral k^(-(s+1)/m) B_0 db dk d(other parameters). (25)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:27-28",
      "line_end": 28,
      "line_start": 27,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "1eb8a9e1501ed77070323e44383a290f528cf626e7dc1cd43655ab9796d1d60b",
      "text": "    K_24(z)=sum_(j in Z^d) exp(-|z+24j|^2/2)\n             /sum_(j in Z^d) exp(-|24j|^2/2), d=2 or 3.     (1)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:35-36",
      "line_end": 36,
      "line_start": 35,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "9ffa9cde7d2a3308707c92a7aa11d07af0541b74cd6f8c33f99a7dd9299729bf",
      "text": "    K_L(z)=sum_(j in Z^d) exp(-|z+Lj|^2/2)/S_(d,L),\n    S_(d,L)=sum_(j in Z^d) exp(-|Lj|^2/2).                 (2)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:46-47",
      "line_end": 47,
      "line_start": 46,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "36940c390000f5ac920f53d120753265ec47a416e615f813dc45de4c86380200",
      "text": "    q_n=exp(-omega^2 |n|^2/2)/Z_(d,L),\n    Z_(d,L)=sum_(j in Z^d) exp(-omega^2 |j|^2/2), n in Z^d. (3)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:60-61",
      "line_end": 61,
      "line_start": 60,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "951483ec419a87fc721af18b7431ab4faac11f0fe4e2e521c4097ed8479ede15",
      "text": "    nu_bar(ell) ~ c_(d,L) ell^(-1/3),\n    E N_bar((0,t])/L^d ~ (3/2)c_(d,L) t^(2/3),             (4)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:91-93",
      "line_end": 93,
      "line_start": 91,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "d1871f81cb314a86ab3c7297dcddcc1db9a01912b2690036225354430c02a8fb",
      "text": "    a_n=(1/(L^d S_(d,L))) integral_Rd\n                   exp(-|z|^2/2) exp(-i omega n.z) dz\n        =(2pi)^(d/2)/(L^d S_(d,L)) exp(-omega^2 |n|^2/2).\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:107-108",
      "line_end": 108,
      "line_start": 107,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "ebac1b40bf522e88710ea716fe8cd3231e497af78fb72603cb3146738e987230",
      "text": "    F(z)=sqrt(q_0) C_0+sum_pairs sqrt(2q_n)\n                   [C_n cos(omega n.z)+S_n sin(omega n.z)]. (5)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:112-112",
      "line_end": 112,
      "line_start": 112,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "2128f471adb90388553ff476bee5507eb19fd4332a1826d57b841deac942b395",
      "text": "    A_q:=sqrt(q_0)+sum_pairs sqrt(2q_n)(1+omega|n|)^q<infinity.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:122-122",
      "line_end": 122,
      "line_start": 122,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "343b3689959a209dda7d184ac6ff631cf1182d94dbb16eecc9cd0aead5394896",
      "text": "    E ||F||Cq^p<infinity for each integer q>=0 and finite p>0. (6)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:135-135",
      "line_end": 135,
      "line_start": 135,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "125b39093b6c904e585d238648c4dc41df75ce7af5cd029e7aa33f6c0c33c8c2",
      "text": "    theta_a(z)=sum_(j=1,...,d) [1-cos(omega(z_j-a_j))].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:150-151",
      "line_end": 151,
      "line_start": 150,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "772504a6b6a50b571b49230ed4c9d2e09abb4b43646d2821af9897f66d57421e",
      "text": "    h_p,0=B_p,\n    h_p,j=B_p sin(omega(z_j-p_j))/omega, j=1,...,d.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:164-164",
      "line_end": 164,
      "line_start": 164,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "49ab4f82844d210220ce5ce6019a318c4407b7b9674170eb46bc45866f9e3a11",
      "text": "    B_p=theta_x^2 theta_q^2, B_q=theta_x^2 theta_p^2.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:206-206",
      "line_end": 206,
      "line_start": 206,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "0685f78c72e580e97b81b992c03d6e47a1ed91c75269bd170d7ccc6e8b2533e8",
      "text": "    F_r(z)=F(z)+c_r(z) Sigma_r^(-1)(v_r-U_r F).             (7)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:219-220",
      "line_end": 220,
      "line_start": 219,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "f32f811b7132d01c2313bfed81a937c23d5b4c891d96dc4dbe23053bf619aef4",
      "text": "    U_r F -> U_0 F in every finite Lp and almost surely,\n    sup_(x,R,0<r<=1) |U_r F|<=C ||F||C3,\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:232-233",
      "line_end": 233,
      "line_start": 232,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "6897123b1cdea3ed6e4f26c4326ff2a990dc68dc7f2662a79f451dd51c79b79b",
      "text": "    F_0=F+c_0 Sigma_0^-1(v_0-U_0 F),\n    v_0=(b,0,0,12k,0,...,0).                               (8)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:247-247",
      "line_end": 247,
      "line_start": 247,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "608635032c2841e28adef5806ca46f599f8e25423b037d42bea7bf1c5dd2fb31",
      "text": "    ||F_r||Cq<=||F||Cq+C||F||C3+C(|b|+k).                 (9)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:263-265",
      "line_end": 265,
      "line_start": 263,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "414777ebfb710f368e5877f9d25ac90849e6baa384defbe28ad92a3d673da6f2",
      "text": "    12 pi_r(v_r) E_Qr[W_r/r^2]\n      <=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)],\n    integral_Rx(0,infinity) k^(-2/3) RHS db dk<infinity.    (10)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:295-295",
      "line_end": 295,
      "line_start": 295,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "b037af6593c60d088625358817b407a831488f572649a5ca4fd7cc172b0f6e71",
      "text": "    a(p,eta)=-B(p)^(-1)[grad T(p)+C(p)eta].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:317-319",
      "line_end": 319,
      "line_start": 317,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "1f278150e56ea0423a83ce3ec806f656e21a1be9acc29d7e6f2511ad34e6d89a",
      "text": "    E sum_(p!=q,grad F(p)=grad F(q)=0) w(p,q,F)\n      =integral_(p!=q) p_(grad F(p),grad F(q))(0)\n          E[|det H_p det H_q| w | both gradients=0] dp dq. (11)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:368-369",
      "line_end": 369,
      "line_start": 368,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "98605b759509fe4fc52d23097a10a79e18ba6dcf7134666d63aef0c76d419800",
      "text": "    dmu_tail,v(t)/dmu_tail(t)\n       =p_(U_0 F_B)(v_0-U_0 t)/p_(U_0 F)(v_0).            (12)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:431-433",
      "line_end": 433,
      "line_start": 431,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "ef65a8308a353576465717aff0862b6900b37b3ee58026a9ce1e66133d436a01",
      "text": "    E_Qr[V_r(1-sigma_r)] -> 0,\n    E_Qr[V_r sigma_r] -> z_0>0,\n    Q_r^W(sigma_r=1) -> 1.                               (13)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:454-456",
      "line_end": 456,
      "line_start": 454,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "c0a87030a9e7b4bc80da9cf2801d7700ded8bc3f985e2953bba96ac2a4c94316",
      "text": "    integral_R p_O(b,0,b-ell,0)\n              E_Q[|det H_p det H_q|] db <=C,\n    0<=nu_bar,far(ell)<=nu_cand,far(ell)<=C.                (14)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:467-467",
      "line_end": 467,
      "line_start": 467,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "a29dff758858e46e112c673c561e2887ecbad2079a0fdf23a2148f839dd21e7c",
      "text": "    B_r=12 pi_r(v_r) E_Qr[V_r sigma_r].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:471-471",
      "line_end": 471,
      "line_start": 471,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "88dd5e32855a227b940628cc0ed2cf9455205860e252da558f650c91ef628256",
      "text": "    r^(d-1) dr, r^3 db dk, 12r^(-(d+3)) pi_r, r^2 E[V_r sigma_r],\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:476-478",
      "line_end": 478,
      "line_start": 476,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "ed3f3ff0b3dc45e78c20db0010d2c61cec437f76734157673ad8b4902a86e79d",
      "text": "    ell^(1/3) nu_bar,near(ell)\n       =integral 1{k>=ell/r_0^3}\n          B_((ell/k)^(1/3))(b,k,u)/(3k^(2/3)) db dk d sigma(u). (15)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:486-487",
      "line_end": 487,
      "line_start": 486,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "bd2fe453ef7701e0653f2335ec3c3390db07668cdf0e8011a498ea30e56f8c1d",
      "text": "    c_(d,L)=4 integral_(R,(0,infinity),S^(d-1))\n                    k^(-2/3) pi_0(v_0) z_0 db dk d sigma. (16)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:503-503",
      "line_end": 503,
      "line_start": 503,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "fce0d154a47b5e1364a503aca3b6bf1dfb659311840b672248b2b459839ac63e",
      "text": "    Cov(L_a F,L_b F)=sum_n q_n P_a(omega n) conjugate(P_b(omega n)),\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:509-512",
      "line_end": 512,
      "line_start": 509,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "1740857cde29938b127c1cdd36dfabb1796a52055cb348d8990a24cf432c4a49",
      "text": "    Cov(G_i,G_j)=omega^2 sum q_n n_i n_j,\n    Cov(H_ij,H_kl)=omega^4 sum q_n n_i n_j n_k n_l,\n    Cov(F_uuu,G_j)=-omega^4 sum q_n (n.u)^3 n_j,\n    Var(F_uuu)=omega^6 sum q_n (n.u)^6.                    (17)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:519-520",
      "line_end": 520,
      "line_start": 519,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "2de1ce4c5c15cd9f9731f3e6853365803cb0db13b262380f8867100b4bddce13",
      "text": "    tau_u^2=Var(F_uuu|G=0),\n    D_u=E[(det A_u)^2 1(A_u<0)|V_u=0].                    (18)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:527-529",
      "line_end": 529,
      "line_start": 527,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "2a84b671437cc6b2230b7a608445af2a67c44a8d805cc59ce9afd45ff5cf1ac4",
      "text": "    c_(d,L)=Gamma(7/6)/(24^(1/3) sqrt(pi))\n       *integral_(S^(d-1)) p_G(0) p_Vu(0) tau_u^(4/3)\n                                             D_u d sigma(u). (19)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:544-548",
      "line_end": 548,
      "line_start": 544,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "7d609b2b67b34313ec8744b24b5b1af38cc58bfa51ee205533d88bb3fc07f9be",
      "text": "    m_2=1-[sum_(j!=0) L^2 j^2 exp(-L^2 j^2/2)]/S_1 <1,\n    m_4=3+[sum_(j!=0) L^2 j^2(L^2 j^2-6)\n                                      exp(-L^2 j^2/2)]/S_1 >3,\n    m_6=15-[sum_(j!=0) L^2 j^2((L^2 j^2)^2-15L^2 j^2+45)\n                                      exp(-L^2 j^2/2)]/S_1 <15.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:554-554",
      "line_end": 554,
      "line_start": 554,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "a303a79ed173acfa28126b9503d010550b79fb31afc31e5509d88e024a531ab8",
      "text": "    0<tau_e1^2=m_6-m_4^2/m_2<6.                         (20)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:559-559",
      "line_end": 559,
      "line_start": 559,
      "path": "experiments/universality/periodized_gaussian_h0/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "e76cdd8a8602c069365c9003f02809fcd3562d7797ad773102600682324c3160",
      "text": "    Var(A|V=0)=m_4-m_2^4/m_4>8/3.                       (21)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:25-26",
      "line_end": 26,
      "line_start": 25,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "436b1fd68a6a3a180cc0aadcd4b687d89254198754eb8fd0b2ff3b82d504aaf9",
      "text": "    K_L(z)=sum_(j in Z^d) exp(-|z+Lj|^2/2)\n             /sum_(j in Z^d) exp(-|Lj|^2/2).                 (1)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:38-38",
      "line_end": 38,
      "line_start": 38,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "1afcf50fafc10dadb0f4cca39871dcb0269915946e8b498c2185540abb48b671",
      "text": "    b=F(M), ell=F(M)-F(S)>0.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:43-43",
      "line_end": 43,
      "line_start": 43,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "4ef759527c6edfd5d4dde4eb11475787957b980069f70a6dbd19f45b86050916",
      "text": "    E N_t ~ (3/2) L^d c_(d,L) t^(2/3), c_(d,L)>0.          (2)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:57-57",
      "line_end": 57,
      "line_start": 57,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "8f577e4faf2c79b89282fc5675c94f8dc4c7ada5c3a783c26a2f827a1e74a78d",
      "text": "    n t_n^(2/3) -> lambda, 0<lambda<infinity.               (3)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:68-68",
      "line_end": 68,
      "line_start": 68,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "afdd57122f0ff652775c1e9877218a21eed6bf0b68c2dade2f339073ece81515",
      "text": "    E[N_(Tt) 1{N_(Tt)>=2}] = o(t^(2/3)) for every fixed T. (4)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:94-95",
      "line_end": 95,
      "line_start": 94,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "f5c2a22d481a494471f09e5a9e4763eaf70f460506f8afba5441491a18c92087",
      "text": "    sigma(p_i(f),p_j(f),f)\n       1{0<f(p_i(f))-f(p_j(f))<=t}.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:106-106",
      "line_end": 106,
      "line_start": 106,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "0782374c0fbd0e7379f4d15db75b8db77eb8fc8f4f04aabbfd6ec019a4d03c44",
      "text": "    r=|h(M,S)|>0, u=h(M,S)/r in S^(d-1), k=ell/r^3>0.       (5)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:111-112",
      "line_end": 112,
      "line_start": 111,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "f9f1cc250bf9834607b604d47ec62ff3a064de002247df854fa0a70b92873208",
      "text": "    M=x-r u/2, S=x+r u/2,\n    F(M)=b, F(S)=b-k r^3, grad F(M)=grad F(S)=0.           (6)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:128-128",
      "line_end": 128,
      "line_start": 128,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "82419c609ef7ba74e08fd4246f2d0d6ea867c3aaba5941d7c32534fe2c70dd4c",
      "text": "    Xi_t=sum_(finite bars) delta_(ell/t, b, k, M, u)       (7)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:144-146",
      "line_end": 146,
      "line_start": 144,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "6f13b0b6306eee6597e75bd89d684993b3105d9a0ae58cc25a957f7a2627cbc7",
      "text": "    v_r=(b-k r^3/2,-k r^2,0,12k,0,...,0),\n    pi_r(v_r)=the U_r density at this target,\n    Q_r=the conditional full-field law given U_r=v_r.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:150-153",
      "line_end": 153,
      "line_start": 150,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "ee73ff5f60a3bc181e61785d2d1ab581564410ca218aeac08b047129e4a1a691",
      "text": "    W_r=|det H_M det H_S|\n          1{H_M negative definite, index(H_S)=d-1},\n    V_r=W_r/r^2,\n    z_0(b,k,u)=36 k^2 E_Q0[(det A_0)^2 1{A_0<0}].          (8)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:166-168",
      "line_end": 168,
      "line_start": 166,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "f6ed60d024394067c0ebceaab21051427b73acb6fb7badb168168277fb2443e6",
      "text": "    F_r -> F_0 globally in C4,\n    V_r -> V_0=36k^2(det A_0)^2 1{A_0<0},\n    E_Qr[V_r sigma_r] -> z_0(b,k,u),                      (9)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:172-174",
      "line_end": 174,
      "line_start": 172,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "de6135f23e81623c186052ad681de1d5645412de3670fc3ed04240e5ddfa9f37",
      "text": "    12 pi_r(v_r) E_Qr V_r <= H(b,k),\n    H(b,k)=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)],\n    integral_Rx(0,infinity) k^(-2/3) H(b,k) db dk<infinity. (10)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:186-192",
      "line_end": 192,
      "line_start": 186,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "b43248d8c1f698087e0bb7363d3153b6b92e7a562e2c196d9d81c40772cc3a36",
      "text": "    t^(-2/3) E[Xi_t h]_near\n      =integral_(a,b,k,x,u) a^(-1/3)\n         1{k>=ta/r_0^3}\n         [12 pi_r(v_r)/(3 k^(2/3))]\n         E_Qr[V_r sigma_r]\n         h(a,b,k,x-r u/2,u) da db dk dx d sigma(u),\n    r=(ta/k)^(1/3), 0<a<=T.                              (11)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:207-207",
      "line_end": 207,
      "line_start": 207,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "29ca27e1642c4f27f4c053fa5b09bc5e394f38660edabdd9243ea29fd535b03c",
      "text": "    ||h||infinity a^(-1/3) H(b,k)/(3k^(2/3)), 0<a<=T,      (12)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:212-214",
      "line_end": 214,
      "line_start": 212,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "b9ee77f9cc34cd664788acc2050482d6004efbc0b0f36a73f6e446756f381fa0",
      "text": "    t^(-2/3) E[Xi_t h] -> integral_E h d eta,              (13)\n    d eta=4 a^(-1/3) k^(-2/3) pi_0(v_0) z_0(b,k,u)\n                      da db dk dx d sigma(u).             (14)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:225-228",
      "line_end": 228,
      "line_start": 225,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "158f2157e298c5c4b0465d24402eb4e836eb1af7e5ccf59534149a6e2955200e",
      "text": "    beta(b,u)=4 integral_0^infinity\n                    k^(-2/3) pi_0(v_0) z_0(b,k,u) dk,\n    d eta_(a,b,x,u)=a^(-1/3) beta(b,u)\n                              da db dx d sigma(u).        (15)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:232-233",
      "line_end": 233,
      "line_start": 232,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "037979038392f3217fcf70fadc2912c07e3c9253b96be49bb22545039a2b9484",
      "text": "    eta{0<a<=T}=(3/2)L^d c_(d,L) T^(2/3),\n    eta_(a)(da)=L^d c_(d,L) a^(-1/3) da.                  (16)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:244-246",
      "line_end": 246,
      "line_start": 244,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "f651ff603c1ba9807e0c8effbe4330200d4081c9d4a056fbacfc76add523523b",
      "text": "    c_(d,L)=Gamma(7/6)/(24^(1/3) sqrt(pi))\n       * integral_S p_G(0) p_Vu(0) tau_u^(4/3)\n          E[(det A_u)^2 1{A_u<0}|V_u=0] d sigma(u),       (17)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:287-287",
      "line_end": 287,
      "line_start": 287,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "03bcffe206f5ed6eb8e64503e16b5d9ce4f6524166448f10938f624af3a74a90",
      "text": "    N_(Tt)(F_(r_t))=1 eventually as t->0.                 (18)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:294-297",
      "line_end": 297,
      "line_start": 294,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "40c5ce8a2503092ad6c7b4a20cb8cef622a23a49cc6aafd753744076fd39f464",
      "text": "    V_(r_t) sigma_(r_t)\n          1{N_(Tt)(F_(r_t))>=2} ->0 almost surely,\n    E_Q_(r_t)[V_(r_t) sigma_(r_t)\n          1{N_(Tt)>=2}] ->0.                             (19)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:309-309",
      "line_end": 309,
      "line_start": 309,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "40d17a687f23b47632ae490215205ad2f455ebebe43c1745f4a9f87595ca9fdc",
      "text": "    E[V_r sigma_r 1{N_(Tt)>=2}].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:317-317",
      "line_end": 317,
      "line_start": 317,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "050d7c2326fda9a7de3ef3791d2bb0b9c756325db4ec7c4b3cef893654d9e15a",
      "text": "    t^(-2/3) E[N_(Tt) 1{N_(Tt)>=2}]_near ->0.             (20)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:328-328",
      "line_end": 328,
      "line_start": 328,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "97acce61009d8caea83b50be72b6f4272ca05709705dccfa829d49e9abe3751f",
      "text": "    P(N_(Tt)=1) ~ (3/2)L^d c_(d,L) T^(2/3) t^(2/3).      (21)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:342-342",
      "line_end": 342,
      "line_start": 342,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "6f1e03b917761a9c99101eb97596ce73beccc35d6b02dccdf9de47020b9e0115",
      "text": "    S_n=sum_(j=1,...,n) Xi_(t_n)^(j).                     (22)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:347-347",
      "line_end": 347,
      "line_start": 347,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "24438bef38bb35aad342322501876efcbb9709fbbabb617ed84d838efb01b1d9",
      "text": "    q_t=E[1-exp(-Xi_t f)], m_t=E[Xi_t h].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:353-354",
      "line_end": 354,
      "line_start": 353,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "383aaac10127bc90524fc9b8aa2031646e680aa28c4b8d4c16f1adaaf3fea3cd",
      "text": "    0<=sum_i(1-exp(-f_i))-[1-exp(-sum_i f_i)]\n       <=N_(Tt) 1{N_(Tt)>=2}.                            (23)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:358-358",
      "line_end": 358,
      "line_start": 358,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "242d50f2e4cd4a53d9587706d3b9550467038d3be053717d579257b103b5a953",
      "text": "    q_t=t^(2/3) [integral_E (1-exp(-f)) d eta+o(1)].       (24)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:362-363",
      "line_end": 363,
      "line_start": 362,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "09a0d9b9983a3a581391689768e36207a1f738adb897bb9526243831189a1970",
      "text": "    E exp(-S_n f)=(1-q_(t_n))^n\n      ->exp[-lambda integral_E(1-exp(-f))d eta].          (25)\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:374-374",
      "line_end": 374,
      "line_start": 374,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "ff6198cdfeaecf6769e7e3ca76c004eb0dbd689b93b71341edc8e1d8a5c2e58e",
      "text": "    n P(N_(Tt_n)>=2)=o(n t_n^(2/3))->0.\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:402-402",
      "line_end": 402,
      "line_start": 402,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "15324b381015545d709aefa9374885a5a514b4130a8cee5c6813004c5fcfb894",
      "text": "    lambda L^d c_(d,L) a^(-1/3) da,\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:421-422",
      "line_end": 422,
      "line_start": 421,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "63d6ac33ca86a537ca6bd90a7d8fbee88d896c0038fd0ec86ebd1597b52c6cc0",
      "text": "    B_i(z)=product_(j!=i) theta_(p_j)(z)^2,\n    theta_p(z)=sum_l[1-cos((2pi/L)(z_l-p_l))].\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:503-504",
      "line_end": 504,
      "line_start": 503,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "9ee39cfc0180059d5e19299925788ebed1378d1a19a736b7ed73b300a6167a52",
      "text": "    E Z_t=t^(2/3) eta,\n    E[N_t 1{N_t>=2}]=m t^(2/3).\n"
    },
    {
      "commit": "f3b27d3cae5c56c65c47187244073dd280b8a2cd",
      "id": "d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:509-510",
      "line_end": 510,
      "line_start": 509,
      "path": "experiments/universality/iid_short_bar_process/PROOF.md",
      "repository": "d6g8k5htny-coder/main",
      "sha256": "8a11ff7d4b33a36ad8a9b4d043922528958e7d78f0980b2db431e3983a9643c3",
      "text": "    exp{-(lambda m/2)\n          [1-(integral exp(-f) dP)^2]}.\n"
    }
  ],
  "graph": {
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        "disposition": "research-integrity control; scientific effect NONE",
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          "relation": "regionally_supersedes_historical_high_jet_route",
          "required": false,
          "to": "regional.fixed-annulus.high-jet-route"
        },
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        },
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          "relation": "requires_evidence",
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          "to": "math.d5-component.collar-proof"
        },
        {
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          "relation": "requires_evidence",
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        },
        {
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          "relation": "requires_evidence",
          "required": true,
          "to": "math.d5-component.annulus-proof"
        },
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        {
          "from": "math.rn-region.witness-collision",
          "relation": "requires_evidence",
          "required": true,
          "to": "math.d5-component.offpin-second-moment-review"
        }
      ],
      "nodes": {
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          "notes": "Portable fix verified locally; Cursor cannot push to main (403); awaiting agent/owner with write",
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          "review_source": "reviews/d1_chain_reconciliation_20260928/LEDGER.json",
          "scope": "replaces parent section 9 (Borel elder mark in weighted pair Kac-Rice), C1-C6",
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        },
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          ],
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        },
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          "kind": "reading_rule_component",
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        },
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          "kind": "reading_rule_component",
          "layer": "D5",
          "source": "reviews/d5_i5_planar_f_20260928/REVIEW.md"
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          "review_basis": [
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          "kind": "reading_rule_component",
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          "source": "reviews/d5_punctured_pin_continuum_claude_20260929/REVIEW.md"
        },
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          "fingerprint": "f972e50bc07674ebce9d971d1a1bd4dba2f8036dde37ecb57d8a44a35650f76a",
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          "review_basis": [
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            },
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              "review": "d5_punctured_pin_continuum_claude_20260929",
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          "kind": "reading_rule_component",
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          "kind": "reading_rule_component",
          "layer": "D5",
          "review_basis": [
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              "review": "STATUS D4 (main#76, external)",
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          "source": "frontiers/remote_window_20260924/PROOF.md"
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          "classification": "PROVED_REVIEWED",
          "controlling": false,
          "kind": "candidate",
          "layer": "D5",
          "scope": "D5-a..D5-f of RECONCILIATION.md section 2; existential constants; fixed T; compact marks with k >= k_- > 0; d = 2",
          "source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md"
        },
        "math.lifetime-remainder": {
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          "kind": "candidate",
          "layer": "D2",
          "notes": "Imports named #63 interfaces (D1-A-E); not parent collapse; does not consume Theorem A (1.1)",
          "review_disposition": "ACCEPT",
          "review_issue": 67,
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          "review_source": "https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841782206",
          "scope": "Theorem R at its stated existential O(1)-remainder scope; no numerical constants or radii",
          "source": "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md"
        },
        "math.p15-full-price": {
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          "kind": "candidate",
          "layer": "D6",
          "notes": "Full probability cube under demand>=2; review open. Price-boundary refutation is a scope boundary, not a required premise.",
          "review_issue": 74,
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        },
        "math.p15-price-boundary": {
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          "fingerprint": "498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b",
          "kind": "candidate",
          "layer": "D6",
          "notes": "Unrestricted same-palette transformed-price extension is false",
          "source": "frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md"
        },
        "math.rn-count-interface": {
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          "kind": "candidate",
          "layer": "D4",
          "notes": "Probability-to-count interface; not an expected-count rate",
          "review_issue": 67,
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        },
        "math.rn-fixed-annulus-window": {
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          "fingerprint": "081abc13c5e66342c13df2af2bd6b114f320486f",
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          "notes": "Source-bound xAI/Grok technical ACCEPT at exact scoped theorem only. No global D5/RN/JETMOD, k->0, d>=3, pin-neighborhood, intermediate-scale, collision, or elder-selection closure.",
          "review_disposition": "ACCEPT",
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          "review_source": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md",
          "scientific_status_unchanged": true,
          "scope": "d=2; fixed L; fixed scaled annulus 1<A0<=|y|<=B0<infinity; compact positive gaps k_->0; all frames; between-pin height window",
          "source": "frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md"
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        "math.rn-fixed-remote-window": {
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          "kind": "candidate",
          "layer": "D4",
          "notes": "Cubic mean on fixed spatial exclusion and between-pin height window",
          "review_issue": 76,
          "source": "frontiers/remote_window_20260924/PROOF.md"
        },
        "math.rn-mesoscopic-reduction": {
          "classification": "AUTHOR_SIDE_REDUCTION",
          "controlling": false,
          "fingerprint": "Math-PR7-draft",
          "kind": "candidate",
          "layer": "D5",
          "notes": "Legacy author-side PR7 route. Regionally superseded/nonblocking for the reviewed fixed-annulus between-pin height-window subregion; still unresolved for scopes outside that theorem.",
          "source": "external:Math-PR7"
        },
        "math.rn-region.fixed-annulus-window": {
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          "controlling": false,
          "coverage_source": "math.rn-fixed-annulus-window",
          "fingerprint": "fixed-d2-annulus-between-pin-window-reviewed",
          "kind": "region",
          "layer": "D5",
          "notes": "Covered by math.rn-fixed-annulus-window only at its exact declared scope.",
          "review_disposition": "ACCEPT_AT_SCOPED_SOURCE"
        },
        "math.rn-region.fixed-remote": {
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