Immutable historical mathematical record · mathematical SHA-256 fccfde1380ab36d6710e8492fd84f0c606e9c0412c5b0425c138652436ba5968 · exact projection SHA-256 3b962afd9342b505766c4e0cffa66edf0ef33f7d645d33091a5bacbb76c6f8a0.

Historical mathematical dependency atlas

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

Graph source: frontiers/downstream_gate_20260925/GRAPH.json · SHA-256 8822e9618678321a342d69cd0b8ae6552de1b5d578c331de5072b2892ee9dd09.

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

• eng.d0-packet-allowlist-fix · engineering · AUTHOR_SIDE_CANDIDATE
• eng.hard-gate · integrity_control · ENGINEERING_CONTROL
• eng.main-pr87-crosswalk · engineering · HOLD
• hist.ALLCELL-FDZ-Q4 · predicate · OPEN_HISTORICAL
• hist.CH-LIFT · predicate · OPEN_ACTIVE
• hist.CL_ANTHROPIC_BUNDLE_2026-09-17_v5.zip · carrier · BLOCKED_ABSENT
• hist.ENV-RESCOV · predicate · OPEN_HISTORICAL
• hist.LOGQ-TAIL · predicate · OPEN_HISTORICAL
• hist.OBL-H5-JETMOD · predicate · OPEN_HISTORICAL
• hist.Piece-2-annulus · predicate · OPEN_ACTIVE
• hist.SYM-Fw-jet · predicate · OPEN_HISTORICAL
• hist.allcell_fdz_enclosures.json · carrier · BLOCKED_ABSENT
• hist.lemma_closed · register · FALSE
• hist.rnu_env.py · carrier · BLOCKED_ABSENT
• math.d1-component.congruence-erratum · reading_rule_component · PROVED_REVIEWED
• math.d1-component.marked-cylinder-cap · reading_rule_component · PROVED_REVIEWED
• math.d1-component.reconciliation-record · reading_rule_component · PROVED_REVIEWED
• math.d1-component.section9-replacement-v1_1 · reading_rule_component · PROVED_REVIEWED
• math.d5-component.annulus-proof · reading_rule_component · PROVED_REVIEWED
• math.d5-component.annulus-review · reading_rule_component · PROVED_REVIEWED
• math.d5-component.collar-proof · reading_rule_component · PROVED_REVIEWED
• math.d5-component.collar-review · reading_rule_component · PROVED_REVIEWED
• math.d5-component.i5-review-claude · reading_rule_component · PROVED_REVIEWED
• math.d5-component.i5-review-xai · reading_rule_component · PROVED_REVIEWED
• math.d5-component.intermediate-window-proof · reading_rule_component · PROVED_REVIEWED
• math.d5-component.offpin-second-moment-review · reading_rule_component · PROVED_REVIEWED
• math.d5-component.punctured-pin-continuum-check · reading_rule_component · PROVED_REVIEWED
• math.d5-component.punctured-pin-continuum-review · reading_rule_component · PROVED_REVIEWED
• math.d5-component.punctured-pin-proof · reading_rule_component · PROVED_REVIEWED
• math.d5-component.punctured-pin-review-111 · reading_rule_component · PROVED_REVIEWED
• math.d5-component.remote-window-proof · reading_rule_component · PROVED_REVIEWED
• math.d5-pin-neighborhood-first-moment · candidate · PROVED_REVIEWED
• math.lifetime-remainder · candidate · AUTHOR_SIDE_CANDIDATE
• math.p15-full-price · candidate · AUTHOR_SIDE_CANDIDATE
• math.p15-price-boundary · candidate · REFUTED
• math.rn-count-interface · candidate · AUTHOR_SIDE_CANDIDATE
• math.rn-fixed-annulus-window · candidate · AUTHOR_SIDE_CANDIDATE
• math.rn-fixed-remote-window · candidate · AUTHOR_SIDE_CANDIDATE
• math.rn-mesoscopic-reduction · candidate · AUTHOR_SIDE_REDUCTION
• math.rn-region.fixed-annulus-window · region · COVERED_BY_CANDIDATE
• math.rn-region.fixed-remote · region · COVERED_BY_CANDIDATE
• math.rn-region.intermediate-r-to-rho · region · PROVED_REVIEWED
• math.rn-region.mesoscopic-scaled-annulus · region · PROVED_REVIEWED
• math.rn-region.pin-collision · region · PROVED_REVIEWED
• math.rn-region.witness-collision · region · OPEN_ACTIVE
• math.rn-selector-region-crosswalk · inventory · ENGINEERING_CONTROL
• math.side24-coefficient · candidate · AUTHOR_SIDE_CANDIDATE
• math.uniform-matrix-cap-lifetime · candidate · PROVED_REVIEWED
• regional.fixed-annulus.high-jet-route · regional_supersession · SUPERSEDED_NONBLOCKING

Selected recorded mathematical sources

33 selected graph, node source/review and equation source files are retained in this mathematical-material-only projection.

• Math- · coefficients/side24_v1/PROOF.md
• Math- · frontiers/downstream_gate_20260925/GRAPH.json
• Math- · frontiers/downstream_gate_20260925/SELECTOR_REGION.json
• Math- · frontiers/downstream_gate_20260925/patches/d0_packet_allowlist.patch
• Math- · frontiers/full_price_20260924/PROOF.md
• Math- · frontiers/intermediate_window_20260928/PROOF.md
• Math- · frontiers/remote_window_20260924/PROOF.md
• Math- · frontiers/rn_annulus_bridge_20260925/PROOF.md
• Math- · frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md
• Math- · frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md
• Math- · frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md
• Math- · frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md
• Math- · imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md
• Math- · imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md
• Math- · imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md
• Math- · reviews/d1_chain_reconciliation_20260928/LEDGER.json
• Math- · reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md
• Math- · reviews/d1_section9_borel_repair_20260925/REPAIR.md
• Math- · reviews/d5_collar_count_20260928/REVIEW.md
• Math- · reviews/d5_i5_planar_f_20260928/REVIEW.md
• Math- · reviews/d5_intermediate_window_claude_20260928/REVIEW.md
• Math- · reviews/d5_local_collar_20260928/COLLAR_PROOF.md
• Math- · reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md
• Math- · reviews/d5_offpin_second_moment_20260928/REVIEW.md
• Math- · reviews/d5_punctured_pin_continuum_claude_20260929/REVIEW.md
• Math- · reviews/d5_punctured_pin_continuum_claude_20260929/pp_continuum_check.py
• Math- · reviews/d5_punctured_pin_nonauthor_20260928/REVIEW.md
• Math- · reviews/d5_reconciliation_20260929/RECONCILIATION.md
• Math- · reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md
• Math- · reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md
• main · experiments/universality/fold_structural_universality/PROOF.md
• main · experiments/universality/iid_short_bar_process/PROOF.md
• main · experiments/universality/periodized_gaussian_h0/PROOF.md

Recorded equation excerpts

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

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

    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)

    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)

    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)

    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)

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

    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)

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

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

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

    (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)

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

    [  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 ].

    |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)

    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)

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

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

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

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

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

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

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

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

    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].

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

    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.

    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)

    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)

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

    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].

    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)

    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)

    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)

    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)

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

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

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

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

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

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

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

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

    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)

    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)

    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)

    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)

    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).

    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)

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

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

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

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

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

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

    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,

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

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

    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)

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

    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)

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

    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)

    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)

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

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

    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)

    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)

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

    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)

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

    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)

    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.

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

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

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

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

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

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

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

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

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

    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)

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

    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.

    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)

    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)

    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)

    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)

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

    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)

    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)

    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)

    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)

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

    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)

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

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

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

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

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

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

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

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

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

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

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

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

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