Current research sources

Pinned source observation: . Compare this timestamp with the present time; observation is not scientific revalidation.

main cefdb56e87a1787be2874ec260a70a730bbbddd7 · Math- 6e502c1f8a5632eb90670fd9fc0d52e421c015a2

Current status source · Current research guide · Current proof index

Current mathematical atlas · Current static graph and complete source inventory · Exact current source observation payload. Scientific effect: NONE.

Historical mathematical material (original pins and observation).

Current static mathematical dependency atlas

49 graph nodes · 55 edges · 42 required edges · 13 context edges. These are current graph inventory counts.

Graph source: frontiers/downstream_gate_20260925/GRAPH.json · SHA-256 8822e9618678321a342d69cd0b8ae6552de1b5d578c331de5072b2892ee9dd09. The upstream graph may be unchanged while its repository advances.

Every current node has static source/evidence reading and required/context neighbor navigation at the fresh observation pins. Graph-to-artifact binding is unresolved; artifact roles and custody are unknown. No classification is promoted here.

Complete current public source inventory

3373 indexed files at the pinned repository commits. Legacy, quarantine, recovery, scratch and operational paths are excluded from this discovery population.

Recorded equation excerpts

108 exact source excerpts are retained as a separate collection. No equation-to-graph dependency mapping is inferred.

Recorded equation excerpt 1

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:36-36.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 36–36.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 14bdb5565ba3798cf9971db88f8a6a05181c04b4554cec282908b431a866cdca.

    ell=F(M)-F(S)>0.

Recorded equation excerpt 2

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:64-65.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 64–65.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 96336ce51ad939e663aa6ae70020560393cfa3ac5fb8d1e56c4dd77c2b7c0bfb.

    nu_bar(ell) ~ C ell^(-1/3),                 ell down to 0,
    E N_bar((0,t])/V_X ~ (3/2) C t^(2/3),       t down to 0.        (1)

Recorded equation excerpt 3

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:78-81.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 78–81.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 96c36b3913251532b200ef4ccc306cc104de89d468fe1e427e0c96d3aa5255cb.

    O_(p,q)=(F(p),grad F(p),F(q),grad F(q)),
    Delta_(p,q)=|det Hess F(p) det Hess F(q)|,
    tau_(p,q)=1{Hess F(p)<0} 1{Hess F(q) has index d-1},
    W_(p,q)=Delta_(p,q) tau_(p,q).                           (2)

Recorded equation excerpt 4

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:91-94.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 91–94.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 1ddcdb23e55dde4e216c194505c6a9aca2cbadbaff8d16cbe92d98184f4dfae2.

    E sum_(p!=q,grad F(p)=grad F(q)=0) w(p,q,F)
      =integral_(p!=q) p_(grad F(p),grad F(q))(0)
                           E[Delta_(p,q) w | both gradients=0]
                           dvol(p) dvol(q).                (3)

Recorded equation excerpt 5

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:100-102.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 100–102.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: a3b3a5e2e73ef5fa32f83f7a80b08376dc005329afef759285299757d20a0341.

    K_sel(p,q,b,s)=p_O(b,0,s,0) E_Q[W_(p,q) sigma(p,q,F)],
    nu_bar(ell)=V_X^(-1) integral_(p!=q) integral_R
                       K_sel(p,q,b,b-ell) db dvol(p) dvol(q).   (4)

Recorded equation excerpt 6

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:123-123.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 123–123.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: fc59fa26a7f0bf4ea4aa72e4418117f1ca012141ff1ed89d6269512a09c8029a.

    min_(closure B_i) f > lambda for each ball in the chain.

Recorded equation excerpt 7

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:129-130.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 129–130.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 440e97281fec90eb96ee8361db9905a4904f0cfdac66edb836da5c5f553b1858.

    D_f(p)=sup{lambda in Q: lambda<f(p),
         exists z in D with f(z)>f(p) connected to p in {f>lambda}}, (5)

Recorded equation excerpt 8

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:140-140.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 140–140.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 37d81641db6b3ed8c9a4f25da286a82067f679cdd68cb194f08ba01fd32e70c2.

    D_f(p)=f(q)<f(p);

Recorded equation excerpt 9

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:175-175.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 175–175.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: d863ddd34922bd121b351a5c379d22393bdd8ee24010ed07c47f3ec09a0b578e.

    M=exp_x(-r u/2), S=exp_x(r u/2).

Recorded equation excerpt 10

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:181-181.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 181–181.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: f8ea374207247bc48d971eb33da0fd7f8948610f22b8664f74cd1c4236a0bddd.

    (h_M,h_S,g_uM,g_uS,(g_iM,g_iS)_(i=1,...,d-1)).

Recorded equation excerpt 11

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:185-189.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 185–189.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 89fad618b01d7180fb22e8dca36254517e9967ec6acd9362532446c154b301ab.

    (h_M+h_S)/2,
    (h_S-h_M)/r,
    (g_uS-g_uM)/r,
    (6/r^2)[g_uM+g_uS-2(h_S-h_M)/r],
    ((g_iM+g_iS)/2,(g_iS-g_iM)/r)_(i=1,...,d-1).           (6)

Recorded equation excerpt 12

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:193-193.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 193–193.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: b32e048523f61fda15fae865445dd011bf1c74a38327fbf2adbfaa7201adbc36.

    v_r=(b-k r^3/2,-k r^2,0,12k,(0,0)_(i=1,...,d-1)).     (7)

Recorded equation excerpt 13

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:197-200.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 197–200.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 5d23671b1f78d638199a93c186d27676d7a6b3fd9f03f489c250ad4b420e9427.

    [  1/2       1/2         0        0    ]
    [ -1/r       1/r         0        0    ]
    [   0         0        -1/r      1/r   ]
    [ 12/r^3   -12/r^3      6/r^2    6/r^2 ].

Recorded equation excerpt 14

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:206-207.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 206–207.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 7b23021ae3f9bb3449d29a14f64ab2a445da7d89168d5a2477a6c5916b6c83b5.

    |det(O -> U_r)|=12 r^(-(d+3)),
    p_O(b,0,b-k r^3,0)=12 r^(-(d+3)) pi_r(v_r).            (8)

Recorded equation excerpt 15

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:215-216.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 215–216.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 3da423efc94d33fa5078b2ccc398fe516250e3adc04c5c11ff0f273dd20bf655.

    U_0=(F,F_u,F_uu,F_uuu,(F_vi,F_uvi)_(i=1,...,d-1)),
    v_0=(b,0,0,12k,(0,0)_(i=1,...,d-1)).                 (9)

Recorded equation excerpt 16

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:236-237.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 236–237.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 8c51a12327a8a47a5873e97b2de337a93662b05fb2af39e053a63d1be4b086b7.

    H_i=[ r alpha_i   r beta_i^T ]
        [ r beta_i        A_i   ].                       (10)

Recorded equation excerpt 17

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:244-245.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 244–245.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: d0469d9143feea2b2b78c80795e4a0aace6f310cc208070ea969fc722e8a86ec.

    alpha_M -> -6k, alpha_S -> 6k,
    beta_M,beta_S bounded, A_M,A_S -> A_0.

Recorded equation excerpt 18

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:253-253.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 253–253.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 827660c5815e583fbc2b5d4d5425157d62121603b0a60e596212af3db0b8721a.

    det H_i/r=alpha_i det A_i-r beta_i^T adj(A_i) beta_i.   (11)

Recorded equation excerpt 19

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:263-263.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 263–263.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: ef4ac4cf405f71283c311f4b8563317ee79a3a9f6ce3d7100e7cf6890909ee8b.

    V_r:=W_r/r^2 -> V_0:=36k^2 (det A_0)^2 1{A_0<0}.     (12)

Recorded equation excerpt 20

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:356-356.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 356–356.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 49cdbb048798c3dc08325c3a1fa8df4ac2c54e9871e21ef5a0a4268e55dfec18.

    Psi(x,h)=(exp_x(-h/2),exp_x(h/2))

Recorded equation excerpt 21

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:364-364.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 364–364.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 746831f8045cc028d30f79d6aa6c31a27ca15b3e590a4320abf7eaf75b08e0d2.

    (a,b) -> (a-b/2,a+b/2),

Recorded equation excerpt 22

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:371-372.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 371–372.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: b7abcfa13c8cdb05e42b9ec202d14fd735f73d52c32bdce0e25ef384b1e8525c.

    dvol(M) dvol(S)
       =J(x,r u) dvol(x) r^(d-1) dr d sigma_x(u).          (13)

Recorded equation excerpt 23

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:383-383.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 383–383.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 99284f9becb38af43180a679059d0f2f21d48e30c6249bafe8143f40269c58e9.

    B_r(b,k,x,u)=12 pi_r(v_r) E_Qr[V_r sigma].

Recorded equation excerpt 24

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:392-395.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 392–395.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: fc3dd6ec0e426953fbda8ed278650d8654dd6fa7062de740107508ad003c62ec.

    spatial polar volume         r^(d-1) dr,
    height change s=b-k r^3       r^3 db dk,
    original observation density 12 r^(-(d+3)) pi_r(v_r),
    selected determinant weight  r^2 E_Qr[V_r sigma].

Recorded equation excerpt 25

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:400-400.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 400–400.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 24d2b5ba4fedf67cb92e99654219eed050a864d2ebcc256eef9e39cf2c453b80.

    (d-1)+3-(d+3)+2=1.

Recorded equation excerpt 26

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:409-414.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 409–414.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: e149cda777474fe0cb62496945987abd266f46c64dfc5bf12d5a1f2fe811a139.

    ell^(1/3) nu_near(ell)
      =(1/(3 V_X)) integral_(SX,R,(0,infinity))
          1{k>=ell/r_0^3} k^(-2/3)
          J(x,(ell/k)^(1/3)u)
          B_((ell/k)^(1/3))(b,k,x,u)
          dvol(x) d sigma_x(u) db dk.

Recorded equation excerpt 27

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:425-427.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 425–427.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 3f3da91457480b8153b70dfdb13abd226e18e1616e1426aeecf9a59ea7e0bca3.

    C=(4/V_X) integral_(SX,R,(0,infinity))
                    k^(-2/3) pi_0(v_0) z_0(b,k,x,u)
                    dvol(x) d sigma_x(u) db dk.             (14)

Recorded equation excerpt 28

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:456-457.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 456–457.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 313067933b5b820f803d7f87d2749dc78297a9ffac11619582e20584b9485680.

    odd:  G=grad F (dimension d), T=F_uuu (one scalar);
    even: F, V=Hess F(u,.) (dimension d), A=Hess F|_(u-perp). (15)

Recorded equation excerpt 29

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:469-470.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 469–470.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: af8f2dd62e4a2f73d27cd6eb7bf8bef2dbd8f0a07a2548cc671d9c3f5df60cf9.

    tau_(x,u)^2=Var(T|G=0)>0,
    D_(x,u)=E[(det A)^2 1{A<0} | V=0].                   (16)

Recorded equation excerpt 30

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:475-476.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 475–476.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: aff0fd1f16c08fb3f5750537f25e458a34859effdc777a9ca9163dadf88301ce.

    pi_0(v_0)=p_(F,V)(b,0) p_G(0) phi_tau(12k),
    z_0=36k^2 E[(det A)^2 1{A<0} | F=b,V=0].

Recorded equation excerpt 31

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:481-483.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 481–483.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 422065f9723f0a971af704214075f65641ae1e8dd3d3af036fe466bf503e2cf7.

    integral_R p_(F,V)(b,0)
        E[(det A)^2 1{A<0} | F=b,V=0] db
      =p_V(0) D_(x,u).                                    (17)

Recorded equation excerpt 32

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:490-492.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 490–492.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 9de9189be21cd9ab881fbdc050238eb47cb35d752b5afa30312794c5064457ed.

    144 integral_0^infinity k^(4/3) phi_tau(12k) dk
      =12^(-1/3) integral_0^infinity t^(4/3) phi_tau(t) dt
      =Gamma(7/6) tau^(4/3)/(24^(1/3) sqrt(pi)).            (18)

Recorded equation excerpt 33

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:498-501.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 498–501.

Recorded source file identity: Git blob 483d779cfdfdcea0bd8d0ec47c9fdaeff485fd4b · 43927 bytes.

Exact excerpt SHA-256: 378b606e2f08aeb1f7967f71685b1f06977ccc1a39ea1aa5c50a5465913ce337.

    C=Gamma(7/6)/(24^(1/3) sqrt(pi) V_X)
       *integral_SX p_G(0) p_V(0) tau_(x,u)^(4/3)
                                             D_(x,u)
                    dvol(x) d sigma_x(u).                  (19)

Recorded equation excerpt 34

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:545-547.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 545–547.

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Exact excerpt SHA-256: 7705c6501a5fca0a754db98232b6a96b0b20533854778e7d65966a1437422e1d.

    xi_r=m_r+P_r Z,
    m_r=L_r^T (L_r L_r^T)^(-1) v_r,
    P_r=I-L_r^T (L_r L_r^T)^(-1) L_r,                     (20)

Recorded equation excerpt 35

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:564-565.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 564–565.

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Exact excerpt SHA-256: b3a65afb9a2cd7397f152707ac217074b42427c3091f1d64a01d72e39df1d302.

    |det H_i|/r<=C(1+||F||C3)^d,
    W_r/r^2<=C(1+||F||C3)^(2d).                          (21)

Recorded equation excerpt 36

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:574-575.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 574–575.

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Exact excerpt SHA-256: 4a6e1dbb96211588c9716b88cc1b08038d87f8e3685c11c3b288f7ad505f7da2.

    12 pi_r(v_r) E_Qr[V_r]
      <=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)].                (22)

Recorded equation excerpt 37

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:614-615.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 614–615.

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Exact excerpt SHA-256: 1b03e4499184f59bbc69f91197987730f5ea153c4ff8c4d5657b3fbdbe8eb839.

    integral_R p_O(b,0,b-ell,0) E_Q[W] db <=C,
                                          0<ell<=1.       (23)

Recorded equation excerpt 38

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:663-663.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 663–663.

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Exact excerpt SHA-256: af6f4ebbb02b3daf07519e7b1371ec64465b3dfa3d1a8428348d57a52f57af3a.

    R=S/sqrt(E S^2), F=R H, E S^2=alpha/(alpha+2).

Recorded equation excerpt 39

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:672-674.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 672–674.

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Exact excerpt SHA-256: e0044907b9bd2ac914e708f3adffec553d39cbdd13f7d102b7a25e73e762ee4a.

    E N_F((0,t])
      >=E N_H([a,b]) P(R<=t/b)
      =c t^alpha, c>0.                                   (24)

Recorded equation excerpt 40

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:701-702.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 701–702.

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Exact excerpt SHA-256: 862d38aab3f507807ba9e43637ccac4d789390cf9e6df783c614fc2d65dbc674.

    s=gamma+m-a+w,
    r^s B_r dr db dk d(other parameters).

Recorded equation excerpt 41

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:706-706.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 706–706.

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Exact excerpt SHA-256: 4a10e3337fe06ce0b1f5172df4df52622c710398910965b831dd893ad1c5178c.

    e=(s+1)/m-1=(gamma+1-a+w)/m,

Recorded equation excerpt 42

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/fold_structural_universality/PROOF.md:710-710.

Source: d6g8k5htny-coder/main · experiments/universality/fold_structural_universality/PROOF.md · lines 710–710.

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Exact excerpt SHA-256: 47e43a9edc4e1d6a0f94e18bc28496d5f1ffd87cc36abaebdaaa6364bd08a603.

    (1/m) integral k^(-(s+1)/m) B_0 db dk d(other parameters). (25)

Recorded equation excerpt 43

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:27-28.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 27–28.

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Exact excerpt SHA-256: 1eb8a9e1501ed77070323e44383a290f528cf626e7dc1cd43655ab9796d1d60b.

    K_24(z)=sum_(j in Z^d) exp(-|z+24j|^2/2)
             /sum_(j in Z^d) exp(-|24j|^2/2), d=2 or 3.     (1)

Recorded equation excerpt 44

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:35-36.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 35–36.

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Exact excerpt SHA-256: 9ffa9cde7d2a3308707c92a7aa11d07af0541b74cd6f8c33f99a7dd9299729bf.

    K_L(z)=sum_(j in Z^d) exp(-|z+Lj|^2/2)/S_(d,L),
    S_(d,L)=sum_(j in Z^d) exp(-|Lj|^2/2).                 (2)

Recorded equation excerpt 45

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:46-47.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 46–47.

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Exact excerpt SHA-256: 36940c390000f5ac920f53d120753265ec47a416e615f813dc45de4c86380200.

    q_n=exp(-omega^2 |n|^2/2)/Z_(d,L),
    Z_(d,L)=sum_(j in Z^d) exp(-omega^2 |j|^2/2), n in Z^d. (3)

Recorded equation excerpt 46

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:60-61.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 60–61.

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Exact excerpt SHA-256: 951483ec419a87fc721af18b7431ab4faac11f0fe4e2e521c4097ed8479ede15.

    nu_bar(ell) ~ c_(d,L) ell^(-1/3),
    E N_bar((0,t])/L^d ~ (3/2)c_(d,L) t^(2/3),             (4)

Recorded equation excerpt 47

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:91-93.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 91–93.

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Exact excerpt SHA-256: d1871f81cb314a86ab3c7297dcddcc1db9a01912b2690036225354430c02a8fb.

    a_n=(1/(L^d S_(d,L))) integral_Rd
                   exp(-|z|^2/2) exp(-i omega n.z) dz
        =(2pi)^(d/2)/(L^d S_(d,L)) exp(-omega^2 |n|^2/2).

Recorded equation excerpt 48

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:107-108.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 107–108.

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Exact excerpt SHA-256: ebac1b40bf522e88710ea716fe8cd3231e497af78fb72603cb3146738e987230.

    F(z)=sqrt(q_0) C_0+sum_pairs sqrt(2q_n)
                   [C_n cos(omega n.z)+S_n sin(omega n.z)]. (5)

Recorded equation excerpt 49

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:112-112.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 112–112.

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Exact excerpt SHA-256: 2128f471adb90388553ff476bee5507eb19fd4332a1826d57b841deac942b395.

    A_q:=sqrt(q_0)+sum_pairs sqrt(2q_n)(1+omega|n|)^q<infinity.

Recorded equation excerpt 50

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:122-122.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 122–122.

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Exact excerpt SHA-256: 343b3689959a209dda7d184ac6ff631cf1182d94dbb16eecc9cd0aead5394896.

    E ||F||Cq^p<infinity for each integer q>=0 and finite p>0. (6)

Recorded equation excerpt 51

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:135-135.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 135–135.

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Exact excerpt SHA-256: 125b39093b6c904e585d238648c4dc41df75ce7af5cd029e7aa33f6c0c33c8c2.

    theta_a(z)=sum_(j=1,...,d) [1-cos(omega(z_j-a_j))].

Recorded equation excerpt 52

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:150-151.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 150–151.

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Exact excerpt SHA-256: 772504a6b6a50b571b49230ed4c9d2e09abb4b43646d2821af9897f66d57421e.

    h_p,0=B_p,
    h_p,j=B_p sin(omega(z_j-p_j))/omega, j=1,...,d.

Recorded equation excerpt 53

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:164-164.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 164–164.

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Exact excerpt SHA-256: 49ab4f82844d210220ce5ce6019a318c4407b7b9674170eb46bc45866f9e3a11.

    B_p=theta_x^2 theta_q^2, B_q=theta_x^2 theta_p^2.

Recorded equation excerpt 54

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:206-206.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 206–206.

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Exact excerpt SHA-256: 0685f78c72e580e97b81b992c03d6e47a1ed91c75269bd170d7ccc6e8b2533e8.

    F_r(z)=F(z)+c_r(z) Sigma_r^(-1)(v_r-U_r F).             (7)

Recorded equation excerpt 55

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:219-220.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 219–220.

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Exact excerpt SHA-256: f32f811b7132d01c2313bfed81a937c23d5b4c891d96dc4dbe23053bf619aef4.

    U_r F -> U_0 F in every finite Lp and almost surely,
    sup_(x,R,0<r<=1) |U_r F|<=C ||F||C3,

Recorded equation excerpt 56

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:232-233.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 232–233.

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Exact excerpt SHA-256: 6897123b1cdea3ed6e4f26c4326ff2a990dc68dc7f2662a79f451dd51c79b79b.

    F_0=F+c_0 Sigma_0^-1(v_0-U_0 F),
    v_0=(b,0,0,12k,0,...,0).                               (8)

Recorded equation excerpt 57

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:247-247.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 247–247.

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Exact excerpt SHA-256: 608635032c2841e28adef5806ca46f599f8e25423b037d42bea7bf1c5dd2fb31.

    ||F_r||Cq<=||F||Cq+C||F||C3+C(|b|+k).                 (9)

Recorded equation excerpt 58

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:263-265.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 263–265.

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Exact excerpt SHA-256: 414777ebfb710f368e5877f9d25ac90849e6baa384defbe28ad92a3d673da6f2.

    12 pi_r(v_r) E_Qr[W_r/r^2]
      <=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)],
    integral_Rx(0,infinity) k^(-2/3) RHS db dk<infinity.    (10)

Recorded equation excerpt 59

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:295-295.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 295–295.

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Exact excerpt SHA-256: b037af6593c60d088625358817b407a831488f572649a5ca4fd7cc172b0f6e71.

    a(p,eta)=-B(p)^(-1)[grad T(p)+C(p)eta].

Recorded equation excerpt 60

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:317-319.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 317–319.

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Exact excerpt SHA-256: 1f278150e56ea0423a83ce3ec806f656e21a1be9acc29d7e6f2511ad34e6d89a.

    E sum_(p!=q,grad F(p)=grad F(q)=0) w(p,q,F)
      =integral_(p!=q) p_(grad F(p),grad F(q))(0)
          E[|det H_p det H_q| w | both gradients=0] dp dq. (11)

Recorded equation excerpt 61

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:368-369.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 368–369.

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Exact excerpt SHA-256: 98605b759509fe4fc52d23097a10a79e18ba6dcf7134666d63aef0c76d419800.

    dmu_tail,v(t)/dmu_tail(t)
       =p_(U_0 F_B)(v_0-U_0 t)/p_(U_0 F)(v_0).            (12)

Recorded equation excerpt 62

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:431-433.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 431–433.

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Exact excerpt SHA-256: ef65a8308a353576465717aff0862b6900b37b3ee58026a9ce1e66133d436a01.

    E_Qr[V_r(1-sigma_r)] -> 0,
    E_Qr[V_r sigma_r] -> z_0>0,
    Q_r^W(sigma_r=1) -> 1.                               (13)

Recorded equation excerpt 63

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:454-456.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 454–456.

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Exact excerpt SHA-256: c0a87030a9e7b4bc80da9cf2801d7700ded8bc3f985e2953bba96ac2a4c94316.

    integral_R p_O(b,0,b-ell,0)
              E_Q[|det H_p det H_q|] db <=C,
    0<=nu_bar,far(ell)<=nu_cand,far(ell)<=C.                (14)

Recorded equation excerpt 64

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:467-467.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 467–467.

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Exact excerpt SHA-256: a29dff758858e46e112c673c561e2887ecbad2079a0fdf23a2148f839dd21e7c.

    B_r=12 pi_r(v_r) E_Qr[V_r sigma_r].

Recorded equation excerpt 65

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:471-471.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 471–471.

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Exact excerpt SHA-256: 88dd5e32855a227b940628cc0ed2cf9455205860e252da558f650c91ef628256.

    r^(d-1) dr, r^3 db dk, 12r^(-(d+3)) pi_r, r^2 E[V_r sigma_r],

Recorded equation excerpt 66

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:476-478.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 476–478.

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Exact excerpt SHA-256: ed3f3ff0b3dc45e78c20db0010d2c61cec437f76734157673ad8b4902a86e79d.

    ell^(1/3) nu_bar,near(ell)
       =integral 1{k>=ell/r_0^3}
          B_((ell/k)^(1/3))(b,k,u)/(3k^(2/3)) db dk d sigma(u). (15)

Recorded equation excerpt 67

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:486-487.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 486–487.

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Exact excerpt SHA-256: bd2fe453ef7701e0653f2335ec3c3390db07668cdf0e8011a498ea30e56f8c1d.

    c_(d,L)=4 integral_(R,(0,infinity),S^(d-1))
                    k^(-2/3) pi_0(v_0) z_0 db dk d sigma. (16)

Recorded equation excerpt 68

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:503-503.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 503–503.

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Exact excerpt SHA-256: fce0d154a47b5e1364a503aca3b6bf1dfb659311840b672248b2b459839ac63e.

    Cov(L_a F,L_b F)=sum_n q_n P_a(omega n) conjugate(P_b(omega n)),

Recorded equation excerpt 69

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:509-512.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 509–512.

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Exact excerpt SHA-256: 1740857cde29938b127c1cdd36dfabb1796a52055cb348d8990a24cf432c4a49.

    Cov(G_i,G_j)=omega^2 sum q_n n_i n_j,
    Cov(H_ij,H_kl)=omega^4 sum q_n n_i n_j n_k n_l,
    Cov(F_uuu,G_j)=-omega^4 sum q_n (n.u)^3 n_j,
    Var(F_uuu)=omega^6 sum q_n (n.u)^6.                    (17)

Recorded equation excerpt 70

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:519-520.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 519–520.

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Exact excerpt SHA-256: 2de1ce4c5c15cd9f9731f3e6853365803cb0db13b262380f8867100b4bddce13.

    tau_u^2=Var(F_uuu|G=0),
    D_u=E[(det A_u)^2 1(A_u<0)|V_u=0].                    (18)

Recorded equation excerpt 71

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:527-529.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 527–529.

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Exact excerpt SHA-256: 2a84b671437cc6b2230b7a608445af2a67c44a8d805cc59ce9afd45ff5cf1ac4.

    c_(d,L)=Gamma(7/6)/(24^(1/3) sqrt(pi))
       *integral_(S^(d-1)) p_G(0) p_Vu(0) tau_u^(4/3)
                                             D_u d sigma(u). (19)

Recorded equation excerpt 72

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:544-548.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 544–548.

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Exact excerpt SHA-256: 7d609b2b67b34313ec8744b24b5b1af38cc58bfa51ee205533d88bb3fc07f9be.

    m_2=1-[sum_(j!=0) L^2 j^2 exp(-L^2 j^2/2)]/S_1 <1,
    m_4=3+[sum_(j!=0) L^2 j^2(L^2 j^2-6)
                                      exp(-L^2 j^2/2)]/S_1 >3,
    m_6=15-[sum_(j!=0) L^2 j^2((L^2 j^2)^2-15L^2 j^2+45)
                                      exp(-L^2 j^2/2)]/S_1 <15.

Recorded equation excerpt 73

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:554-554.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 554–554.

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Exact excerpt SHA-256: a303a79ed173acfa28126b9503d010550b79fb31afc31e5509d88e024a531ab8.

    0<tau_e1^2=m_6-m_4^2/m_2<6.                         (20)

Recorded equation excerpt 74

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/periodized_gaussian_h0/PROOF.md:559-559.

Source: d6g8k5htny-coder/main · experiments/universality/periodized_gaussian_h0/PROOF.md · lines 559–559.

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Exact excerpt SHA-256: e76cdd8a8602c069365c9003f02809fcd3562d7797ad773102600682324c3160.

    Var(A|V=0)=m_4-m_2^4/m_4>8/3.                       (21)

Recorded equation excerpt 75

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:25-26.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 25–26.

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Exact excerpt SHA-256: 436b1fd68a6a3a180cc0aadcd4b687d89254198754eb8fd0b2ff3b82d504aaf9.

    K_L(z)=sum_(j in Z^d) exp(-|z+Lj|^2/2)
             /sum_(j in Z^d) exp(-|Lj|^2/2).                 (1)

Recorded equation excerpt 76

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:38-38.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 38–38.

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Exact excerpt SHA-256: 1afcf50fafc10dadb0f4cca39871dcb0269915946e8b498c2185540abb48b671.

    b=F(M), ell=F(M)-F(S)>0.

Recorded equation excerpt 77

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:43-43.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 43–43.

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Exact excerpt SHA-256: 4ef759527c6edfd5d4dde4eb11475787957b980069f70a6dbd19f45b86050916.

    E N_t ~ (3/2) L^d c_(d,L) t^(2/3), c_(d,L)>0.          (2)

Recorded equation excerpt 78

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:57-57.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 57–57.

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Exact excerpt SHA-256: 8f577e4faf2c79b89282fc5675c94f8dc4c7ada5c3a783c26a2f827a1e74a78d.

    n t_n^(2/3) -> lambda, 0<lambda<infinity.               (3)

Recorded equation excerpt 79

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:68-68.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 68–68.

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Exact excerpt SHA-256: afdd57122f0ff652775c1e9877218a21eed6bf0b68c2dade2f339073ece81515.

    E[N_(Tt) 1{N_(Tt)>=2}] = o(t^(2/3)) for every fixed T. (4)

Recorded equation excerpt 80

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:94-95.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 94–95.

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Exact excerpt SHA-256: f5c2a22d481a494471f09e5a9e4763eaf70f460506f8afba5441491a18c92087.

    sigma(p_i(f),p_j(f),f)
       1{0<f(p_i(f))-f(p_j(f))<=t}.

Recorded equation excerpt 81

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:106-106.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 106–106.

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Exact excerpt SHA-256: 0782374c0fbd0e7379f4d15db75b8db77eb8fc8f4f04aabbfd6ec019a4d03c44.

    r=|h(M,S)|>0, u=h(M,S)/r in S^(d-1), k=ell/r^3>0.       (5)

Recorded equation excerpt 82

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:111-112.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 111–112.

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Exact excerpt SHA-256: f9f1cc250bf9834607b604d47ec62ff3a064de002247df854fa0a70b92873208.

    M=x-r u/2, S=x+r u/2,
    F(M)=b, F(S)=b-k r^3, grad F(M)=grad F(S)=0.           (6)

Recorded equation excerpt 83

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:128-128.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 128–128.

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Exact excerpt SHA-256: 82419c609ef7ba74e08fd4246f2d0d6ea867c3aaba5941d7c32534fe2c70dd4c.

    Xi_t=sum_(finite bars) delta_(ell/t, b, k, M, u)       (7)

Recorded equation excerpt 84

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:144-146.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 144–146.

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Exact excerpt SHA-256: 6f13b0b6306eee6597e75bd89d684993b3105d9a0ae58cc25a957f7a2627cbc7.

    v_r=(b-k r^3/2,-k r^2,0,12k,0,...,0),
    pi_r(v_r)=the U_r density at this target,
    Q_r=the conditional full-field law given U_r=v_r.

Recorded equation excerpt 85

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:150-153.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 150–153.

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Exact excerpt SHA-256: ee73ff5f60a3bc181e61785d2d1ab581564410ca218aeac08b047129e4a1a691.

    W_r=|det H_M det H_S|
          1{H_M negative definite, index(H_S)=d-1},
    V_r=W_r/r^2,
    z_0(b,k,u)=36 k^2 E_Q0[(det A_0)^2 1{A_0<0}].          (8)

Recorded equation excerpt 86

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:166-168.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 166–168.

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Exact excerpt SHA-256: f6ed60d024394067c0ebceaab21051427b73acb6fb7badb168168277fb2443e6.

    F_r -> F_0 globally in C4,
    V_r -> V_0=36k^2(det A_0)^2 1{A_0<0},
    E_Qr[V_r sigma_r] -> z_0(b,k,u),                      (9)

Recorded equation excerpt 87

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:172-174.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 172–174.

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Exact excerpt SHA-256: de6135f23e81623c186052ad681de1d5645412de3670fc3ed04240e5ddfa9f37.

    12 pi_r(v_r) E_Qr V_r <= H(b,k),
    H(b,k)=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)],
    integral_Rx(0,infinity) k^(-2/3) H(b,k) db dk<infinity. (10)

Recorded equation excerpt 88

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:186-192.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 186–192.

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Exact excerpt SHA-256: b43248d8c1f698087e0bb7363d3153b6b92e7a562e2c196d9d81c40772cc3a36.

    t^(-2/3) E[Xi_t h]_near
      =integral_(a,b,k,x,u) a^(-1/3)
         1{k>=ta/r_0^3}
         [12 pi_r(v_r)/(3 k^(2/3))]
         E_Qr[V_r sigma_r]
         h(a,b,k,x-r u/2,u) da db dk dx d sigma(u),
    r=(ta/k)^(1/3), 0<a<=T.                              (11)

Recorded equation excerpt 89

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:207-207.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 207–207.

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Exact excerpt SHA-256: 29ca27e1642c4f27f4c053fa5b09bc5e394f38660edabdd9243ea29fd535b03c.

    ||h||infinity a^(-1/3) H(b,k)/(3k^(2/3)), 0<a<=T,      (12)

Recorded equation excerpt 90

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:212-214.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 212–214.

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Exact excerpt SHA-256: b9ee77f9cc34cd664788acc2050482d6004efbc0b0f36a73f6e446756f381fa0.

    t^(-2/3) E[Xi_t h] -> integral_E h d eta,              (13)
    d eta=4 a^(-1/3) k^(-2/3) pi_0(v_0) z_0(b,k,u)
                      da db dk dx d sigma(u).             (14)

Recorded equation excerpt 91

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:225-228.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 225–228.

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Exact excerpt SHA-256: 158f2157e298c5c4b0465d24402eb4e836eb1af7e5ccf59534149a6e2955200e.

    beta(b,u)=4 integral_0^infinity
                    k^(-2/3) pi_0(v_0) z_0(b,k,u) dk,
    d eta_(a,b,x,u)=a^(-1/3) beta(b,u)
                              da db dx d sigma(u).        (15)

Recorded equation excerpt 92

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:232-233.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 232–233.

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Exact excerpt SHA-256: 037979038392f3217fcf70fadc2912c07e3c9253b96be49bb22545039a2b9484.

    eta{0<a<=T}=(3/2)L^d c_(d,L) T^(2/3),
    eta_(a)(da)=L^d c_(d,L) a^(-1/3) da.                  (16)

Recorded equation excerpt 93

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:244-246.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 244–246.

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Exact excerpt SHA-256: f651ff603c1ba9807e0c8effbe4330200d4081c9d4a056fbacfc76add523523b.

    c_(d,L)=Gamma(7/6)/(24^(1/3) sqrt(pi))
       * integral_S p_G(0) p_Vu(0) tau_u^(4/3)
          E[(det A_u)^2 1{A_u<0}|V_u=0] d sigma(u),       (17)

Recorded equation excerpt 94

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:287-287.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 287–287.

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Exact excerpt SHA-256: 03bcffe206f5ed6eb8e64503e16b5d9ce4f6524166448f10938f624af3a74a90.

    N_(Tt)(F_(r_t))=1 eventually as t->0.                 (18)

Recorded equation excerpt 95

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:294-297.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 294–297.

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Exact excerpt SHA-256: 40c5ce8a2503092ad6c7b4a20cb8cef622a23a49cc6aafd753744076fd39f464.

    V_(r_t) sigma_(r_t)
          1{N_(Tt)(F_(r_t))>=2} ->0 almost surely,
    E_Q_(r_t)[V_(r_t) sigma_(r_t)
          1{N_(Tt)>=2}] ->0.                             (19)

Recorded equation excerpt 96

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:309-309.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 309–309.

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Exact excerpt SHA-256: 40d17a687f23b47632ae490215205ad2f455ebebe43c1745f4a9f87595ca9fdc.

    E[V_r sigma_r 1{N_(Tt)>=2}].

Recorded equation excerpt 97

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:317-317.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 317–317.

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Exact excerpt SHA-256: 050d7c2326fda9a7de3ef3791d2bb0b9c756325db4ec7c4b3cef893654d9e15a.

    t^(-2/3) E[N_(Tt) 1{N_(Tt)>=2}]_near ->0.             (20)

Recorded equation excerpt 98

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:328-328.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 328–328.

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Exact excerpt SHA-256: 97acce61009d8caea83b50be72b6f4272ca05709705dccfa829d49e9abe3751f.

    P(N_(Tt)=1) ~ (3/2)L^d c_(d,L) T^(2/3) t^(2/3).      (21)

Recorded equation excerpt 99

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:342-342.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 342–342.

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Exact excerpt SHA-256: 6f1e03b917761a9c99101eb97596ce73beccc35d6b02dccdf9de47020b9e0115.

    S_n=sum_(j=1,...,n) Xi_(t_n)^(j).                     (22)

Recorded equation excerpt 100

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:347-347.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 347–347.

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Exact excerpt SHA-256: 24438bef38bb35aad342322501876efcbb9709fbbabb617ed84d838efb01b1d9.

    q_t=E[1-exp(-Xi_t f)], m_t=E[Xi_t h].

Recorded equation excerpt 101

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:353-354.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 353–354.

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Exact excerpt SHA-256: 383aaac10127bc90524fc9b8aa2031646e680aa28c4b8d4c16f1adaaf3fea3cd.

    0<=sum_i(1-exp(-f_i))-[1-exp(-sum_i f_i)]
       <=N_(Tt) 1{N_(Tt)>=2}.                            (23)

Recorded equation excerpt 102

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:358-358.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 358–358.

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Exact excerpt SHA-256: 242d50f2e4cd4a53d9587706d3b9550467038d3be053717d579257b103b5a953.

    q_t=t^(2/3) [integral_E (1-exp(-f)) d eta+o(1)].       (24)

Recorded equation excerpt 103

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:362-363.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 362–363.

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Exact excerpt SHA-256: 09a0d9b9983a3a581391689768e36207a1f738adb897bb9526243831189a1970.

    E exp(-S_n f)=(1-q_(t_n))^n
      ->exp[-lambda integral_E(1-exp(-f))d eta].          (25)

Recorded equation excerpt 104

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:374-374.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 374–374.

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Exact excerpt SHA-256: ff6198cdfeaecf6769e7e3ca76c004eb0dbd689b93b71341edc8e1d8a5c2e58e.

    n P(N_(Tt_n)>=2)=o(n t_n^(2/3))->0.

Recorded equation excerpt 105

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:402-402.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 402–402.

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Exact excerpt SHA-256: 15324b381015545d709aefa9374885a5a514b4130a8cee5c6813004c5fcfb894.

    lambda L^d c_(d,L) a^(-1/3) da,

Recorded equation excerpt 106

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:421-422.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 421–422.

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Exact excerpt SHA-256: 63d6ac33ca86a537ca6bd90a7d8fbee88d896c0038fd0ec86ebd1597b52c6cc0.

    B_i(z)=product_(j!=i) theta_(p_j)(z)^2,
    theta_p(z)=sum_l[1-cos((2pi/L)(z_l-p_l))].

Recorded equation excerpt 107

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:503-504.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 503–504.

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Exact excerpt SHA-256: 9ee39cfc0180059d5e19299925788ebed1378d1a19a736b7ed73b300a6167a52.

    E Z_t=t^(2/3) eta,
    E[N_t 1{N_t>=2}]=m t^(2/3).

Recorded equation excerpt 108

Original excerpt identity: d6g8k5htny-coder/main:experiments/universality/iid_short_bar_process/PROOF.md:509-510.

Source: d6g8k5htny-coder/main · experiments/universality/iid_short_bar_process/PROOF.md · lines 509–510.

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Exact excerpt SHA-256: 8a11ff7d4b33a36ad8a9b4d043922528958e7d78f0980b2db431e3983a9643c3.

    exp{-(lambda m/2)
          [1-(integral exp(-f) dP)^2]}.