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  <screens|<\hidden>
    <cqft-title-slide|<math|\<Phi\><rsup|4><rsub|3>> on the torus
    I:<next-line>the FBSDE in force coordinates and renormalisation|Joint
    work with Sarah-Jean Meyer, with the use of AI|SISSA lectures,
    Lecture<nbsp>6 \V notes: Lecture<nbsp>6, ŸŸ10\U12>
  </hidden>|<\hidden>
    <mytitle|Where we are>

    \;

    <with|font-series|bold|Lecture<nbsp>4.> Interacting field at resolution
    <math|t> = solution of the FBSDE

    <\equation*>
      \<Phi\><rsub|t>=<big|int><rsub|0><rsup|t><wide|Q|\<dot\>><rsub|s>*F<rsub|s><space|0.17em>\<mathd\>s+<big|int><rsub|0><rsup|t><wide|Q|\<dot\>><rsub|s><rsup|1/2>\<mathd\>W<rsub|s>,<space|2em>F<rsub|t>=\<bbb-E\><rsub|t>*<around*|[|-D*V<rsub|T><around|(|\<Phi\><rsub|T>|)>|]>.
    </equation*>

    In the <strong|forecast coordinate> <math|\<psi\><rsub|t>=\<bbb-E\><rsub|t>*\<Phi\><rsub|T>>
    the forward equation has no drift and the trees are integrated exactly;
    the backward half is the <strong|force equation>.

    <with|font-series|bold|Lectures<nbsp>6\U7.> Construct
    <math|\<Phi\><rsup|4><rsub|3>> on <math|\<bbb-T\><rsup|3>> with this
    FBSDE:

    <\enumerate>
      <item><strong|deterministic renormalisation>: an explicit approximate
      force <math|<wide|\<cal-F\>|\<bar\>><rsub|q>> and its defect (today);

      <item><strong|a priori estimates> for the exact FBSDE in the
      coordinates of <math|<wide|\<cal-F\>|\<bar\>><rsub|q>>, uniform in the
      lattice (Lecture<nbsp>7);

      <item><strong|removing the cutoffs> and identifying the limit
      (Lecture<nbsp>7).
    </enumerate>

    <with|par-mode|right|<small|<with|color|dark grey|Notes:
    Lectures<nbsp>6\U7 and the Supplement; Bailleul\UChevyrev\UG. (in
    prep.)>>>
  </hidden>|<\hidden>
    <mytitle|What is proved>

    \;

    For every coupling <math|\<lambda\>\<gtr\>0>, every reference mass
    <math|q<rsub|0>\<gtr\>0> and every finite part <math|r> of the
    counterterm, on a fixed torus, with no smallness, and an explicit
    counterterm with arbitrary finite part:

    <\itemize>
      <item>the whole lattice sequence converges to a measure <math|\<nu\>>,
      an explicit density over an explicit SDE;

      <item>the forecasts converge to the solution of an FBSDE from
      <math|Y<rsub|q<rsub|0>>=0>, driven by the limit of the lattice
      innovations, and <math|\<psi\><rsub|q>=m<rsub|q><around|(|\<Phi\><rsub|q>|)>>;

      <item>uniqueness in law for its scalar form, a variational formula with
      no infinite constant, quartic tails, integration by parts.
    </itemize>

    <strong|Not proved:> the vector FBSDE alone, strong solutions, OS axioms
    in full, infinite volume.

    <strong|Guiding principle.> Do not estimate <math|\<Phi\>> and <math|F>.
    Estimate the error of an explicit approximate forecast.

    The results of Lectures<nbsp>6\U7 are new: joint work with S.-J. Meyer +
    AI.
  </hidden>|<\hidden>
    <vspace*|2.5cm>

    \;

    <cqft-part-title|The problem and the exact FBSDE>

    <vspace|0.3cm>

    <with|color|dark grey|Lecture<nbsp>6>

    \;
  </hidden>|<\hidden>
    <mytitle|Lattice and the mass-regulator profile>

    \;

    <math|\<Lambda\><rsub|N>=<around|(|N<rsup|-1>*\<bbb-Z\>/\<bbb-Z\>|)><rsup|3>>,
    <math|<big|int>f=N<rsup|-3>*<big|sum><rsub|x>f<around|(|x|)>>,
    <math|\<Delta\>=\<Delta\><rsub|N>> nearest neighbour. Resolution
    parameter <math|q=q<rsub|0>+t<rsup|2>\<in\><around|[|q<rsub|0>,\<infty\>|)>>:

    <\equation*>
      K<rsub|q>=q-\<Delta\>,<space|2em>C<rsub|q>=K<rsub|q><rsup|-1>,<space|2em>C=C<rsub|q<rsub|0>>=<around|(|q<rsub|0>-\<Delta\>|)><rsup|-1>.
    </equation*>

    In Lecture<nbsp>3 notation (<math|T=\<infty\>>, and
    <math|<wide|Q|\<dot\>>=\<partial\><rsub|q>*Q<rsub|q>>):

    <\equation*>
      Q<rsub|q>=C-C<rsub|q>,<space|1em><wide|Q|\<dot\>>=C<rsub|q><rsup|2>,<space|1em><wide|Q|\<dot\>><rsup|1/2>=C<rsub|q>,<space|1em>S<rsub|q>=C<rsub|q>,<space|1em>\<cal-R\><rsub|q>=q-q<rsub|0>.
    </equation*>

    <\myframed>
      <strong|Free resolvents (uniform in <math|N>)>

      <math|C<rsub|q>\<geqslant\>0> as a kernel,
      <math|C<rsub|q>*<with|math-font|Bbb|1>=q<rsup|-1>*<with|math-font|Bbb|1>>,
      <math|<around|\<\|\|\>|C<rsub|q>|\<\|\|\>><rsub|p\<to\>p>=q<rsup|-1>>,

      <\equation*>
        a<rsub|2><around|(|q|)>=<around|(|C<rsub|q><rsup|2>|)><around|(|x,x|)>\<lesssim\>q<rsup|-1/2>\<wedge\>N<rsup|3>*q<rsup|-2>,<space|2em><math-up|Tr>C\<asymp\>N.
      </equation*>
    </myframed>

    The <strong|bubble> <math|a<rsub|2><around|(|q|)>> is the
    coincident-point variance of the shell at resolution <math|q>.
  </hidden>|<\hidden>
    <mytitle|The cutoff measure>

    <center|<math|U<around|(|\<varphi\>|)>=<frac|\<lambda\>|12><big|int>\<varphi\><rsup|4>>,
    <math|V<rsub|N>=U+<frac|\<mu\><rsub|N>|2><big|int>\<varphi\><rsup|2>>,
    <math|\<nu\><rsub|N><around|(|<space|0.17em>\<mathd\>\<varphi\>|)>=\<cal-Z\><rsub|N><rsup|-1>\<mathe\><rsup|-V<rsub|N><around|(|\<varphi\>|)>>\<mu\><rsup|C><around|(|<space|0.17em>\<mathd\>\<varphi\>|)>>.>

    So <math|D<rsup|2>U<around|(|y|)>=\<lambda\> y<rsup|2>>: the coupling is
    the curvature, the same <math|\<lambda\>> as in Lecture<nbsp>4.\ 

    The Wick ordering is now part of <math|\<mu\><rsub|N>>.

    <strong|Reference mass> <math|q<rsub|0>\<gtr\>0> (constants
    <math|\<in\>ker \<Delta\>> on the torus; <math|r> may be negative). Only
    <math|q<rsub|0>+\<mu\><rsub|N>> matters: fix <math|q<rsub|0>>, vary
    <math|r>.

    <\myframed>
      <strong|Main result, informal.> Let

      <\equation*>
        \<mu\><rsub|N>=r-\<lambda\><math-up|Tr>
        C+<frac|2|3>\<lambda\><rsup|2><big|int> C<rsup|3>.
      </equation*>

      Then (a) <math|\<nu\><rsub|N>\<to\>\<nu\>> in <math|H<rsup|-s>>,
      <math|s\<gtr\><frac|1|2>>, along the whole sequence; (b) the forecasts
      converge in law to a solution of a limiting FBSDE; (c) <math|\<nu\>>
      has a variational formula, an integration by parts formula and quartic
      tails.
    </myframed>

    These come from bounds on the error <math|Z> of an approximate forecast,
    uniform in <math|N>: <strong|UV stability>.
  </hidden>|<\hidden>
    <mytitle|The Polchinski process as a filter>

    By Lecture<nbsp>4, in the profile <math|S<rsub|q>=C<rsub|q>>,
    <math|\<cal-R\><rsub|q>=q-q<rsub|0>>; at fixed <math|N> everything is a
    theorem. The Polchinski process solves

    <\equation*>
      \<mathd\>\<Phi\><rsub|q>=C<rsub|q><rsup|2>F<rsub|q><space|0.17em>\<mathd\>q+C<rsub|q><space|0.17em>\<mathd\>W<rsub|q>,<space|1em>F<rsub|q>=\<bbb-E\><rsub|q><around|[|-D*V<rsub|N><around|(|\<Phi\><rsub|\<infty\>>|)>|]>,<space|1em>\<Phi\><rsub|q<rsub|0>>=0,
    </equation*>

    and <math|\<Phi\><rsub|q>\<to\>\<Phi\><rsub|\<infty\>>\<sim\>\<nu\><rsub|N>>.

    <\itemize>
      <item>a noisy observation of its endpoint,
      <math|\<Phi\><rsub|q>=<around|(|q-q<rsub|0>|)>C<rsub|q>\<Phi\><rsub|\<infty\>>+C<rsub|q>B<rsub|q-q<rsub|0>>>,
      with <math|B> independent of <math|\<Phi\><rsub|\<infty\>>>;

      <item>forecast

      <\equation*>
        \<psi\><rsub|q>=\<bbb-E\><rsub|q>\<Phi\><rsub|\<infty\>>=\<Phi\><rsub|q>+C<rsub|q>F<rsub|q>=m<rsub|q><around|(|\<Phi\><rsub|q>|)>,
      </equation*>

      covariance <math|<math-up|Cov><rsub|q>\<gtr\>0>;

      <item><math|W> is the innovation: <math|\<mathd\>\<psi\><rsub|q>=<math-up|Cov><rsub|q><space|0.17em>\<mathd\>W<rsub|q>>;

      <item>noise: <math|\<omega\><rsub|q>=K<rsub|q>\<Phi\><rsub|q>-<around|(|q-q<rsub|0>|)>\<psi\><rsub|q>=\<bbb-E\><rsub|q>B<rsub|q-q<rsub|0>>>,
      sub-Gaussian whatever the interaction (L5).
    </itemize>
  </hidden>|<\hidden>
    <mytitle|One site>

    \;

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D3125F31A98BBC79C3A4169D517CC7A4967A34BF605299F8EFF22B2635ADE6574CAAB2979E38A975DDF5C6494D5AC39E39A932389E39A9DA367CE2A4B6E103F3CE49552CF099932A37EF8993CA40ADCB2F9CD4307C28DC38A95DE1D3674E6AF4917CE7A4BA2EBD7352832F11774E6A572EF13327358A6F7AE7A496EE3CDA3B27D5EFFBC4492D63F153AF9C547FFE3B27D5EB969F38A9512AFF8E49D58272C7A4F6E170D61B26B52AD3E28E496562DDC2AA5E31A9AA2F7EC6A46A03FC09935A161FF486496DAEF19F30A9A21E3F51524B7C9392AA03F49E21A929BD0D495596CB1324B5ABAAF61992AA332D9E21A94A667B86A4BAF67AA2A4D6B468A8174A2AD3EE5D7EA5A4162D7C774A6AE80E7FBD53526B5938D42B265545284F98D4B8609A774C6A8D6F6352158B7DC2A43A2EFE89922A14C01325956784B87CA7A42EE3E28992EA8AE70993AA40F533263539F7F486496D6EEC3C61521D997DE7A4F27847975F38A9A53B3FF5CE4955BEC53327B5F4B739A9AE689F38A93D2D7EEACE49F5F3179F39A9A5BBF8C2496DFEA9EE9C5455663F73527546CD1327D591F04F9CD4EC68D23B27B52C1CEA15935ADFA4A40E872A3F61529D5C7BC3A48ABBF38C4955F1C53326B51FF22B2635C97679C2A1862F103B2495E8D3F0F8F50313E6F1C71BD47407A87EB6FEC5AD30216871856B8BC4DD5216BEB9C49A04BBC12F1F11378924AC3E7EF9F2BB9788FF7E1020B092E2C27A19F32EB9C9818BFFECB397EF7D843F352C3A54CC473F7FF9F007F1311F1FFDE2E53BBFFDEEE3A35FBF30403C9891C3DCFEC016DFF98DFFA2F3BC45EEB30CFFC5C7FE0B1254B8415F5DFE30392C29165FB1EA7DFDC1E7F60B7CC468F916E72F7EBAAEC4DA073EF3BAF767F60B3A86851937E73DFE713D548B8CF9B5E3A13E5D3727A04D07987DCDA77DF716EF3EED3FD92F328BD488E4387BEA57EF75E17F78EFE6EF3DD5873F48FEBDFED55A58427C66B86DFD3EFBEFFF62751D35EDFE94CDBFC2F1176B047C8387F99D5F3344D8291A3C1F5840F31C67F901C3CF1A87D57FB00FF4E84F4DDF7985EC0F14ED177FFDD1CB8F5E7E82D1CEA4FCC73FE0773F7F304F7E325EC130633097CCFE9538E825276EF68934CC8C0AA3A1D44737C318138FBBDFE322D3865EBEC96C3AFEF2F193F5088F1FDEA7A3EDE5D85F1927993B56759B8D1C2AFD78772CFAF6F6FF670D9A1AD5A5957D4A37A71DADCB03A68C9AC7D53C0D0371D1E3BCB58EC7B5D717655E6966E57EF49EFDFBEFF86F68D860F16300611FC3A6698D7754D72F3E59D79AC3F876C75FFC7CFD052B82F74BBDFB8B8FFD1718614CA3E480D875D01B37FFFBEF9E43C194EEEB30DE345724EEE5F2D2DC6332AA1A6C6DD675F8BFCA178E0272B5D26D14B8EC320A0ED94529FFF14B8C0212D5FE2C83209D83E0776BA6468B8C59B8E0CF39087EBB744724907DF0C01AFDE27FBCF7493F779D38F0345DD5E87FE25BFFDEFFA20EEE797224FB2FFED9FF2231779ED1DA7951C8856962C5F037EB174B21C388B6E372FEF4987DF705BFF4087C269E3FBE02F17C1B81C51CCDEB0874D965041EB2AF3C022D9DF39D21988E61F27FD79B3345241A15F84F0F41C210E037C242A48D786B7D0E6F69788BDE90C347A7F69B8EEF3F9F024D7FDEB9F30BFF05C37B248C1F0BF2BB63EEDD99F0EEDCF9D99A22CC5199E59C6DEF4E9177E7E7BB73E71CF0BF7B04AB80C32FC2FA91611F4CCE6A878B9E63EA679F3D3EFC41FDE0C71FFDCD7FFD8FBFFDCD7FFECD277FF34F3FFDE00FBFF9E4AF79DDEFFF1673E7472F1FFE657CFCF2730CD35FCA76F6318C7FBFC0D56074889B476E1557963286C3245EFF3CEDE8ED4F4EE1AF5E5E7E81FB24DE6799CFBE64F86CC1F420E99DF1C965A16F127C3D4EB4EF614AFD719F588960046BC3B0AFA229DBC4E294CBFD66709FE3AD19DEE3300A977D344237F8130662E604B111EC97B8DA52A67CEC3D5E327337499F28C7F36F92F79E3F97E66DC852FE365F20DE5FA058513BEB39FCF937C1BB8FAFFD38A623B3FCE35B7CFCF4F4F8D50AD9E1D71FCFBF49DE7B8162672F9000C6BCA46FF1F9F3FDF92BD90164B5E463A66D92779F9F1B9F6CC3ED98FE6DBE40D95F00CEF60BA7E23020CDE1406F922207FA7B10EA90AE58CE66A7E4689683A10C99E575A883537236B3E33C08E33C9A6D92B319A12E831C94A3D926399B65E68DC13B6A47B34DB2359BB06699AA7CB43A0467233BA83AD3D858AD36C9D68CBC064B963C9A9D92ADD9B4C2F5F3C14EC1D988C07596649E1AE8109C8D9ACE00CCE7736D92B359B75DE17CF6EB29381B19656B328560B5DA245BB369277DC6F3E937C9D96C662BBEABE747DA245B33CB4B8B3D9ECD4EC9D68C3BD6F040CE9B6E92A359E18E71ACDC6338D4DE29399B1123D20DB2B99A6D92B3198B4707A98647B34D723663C8320FE6C3AF669B646BC654F241F7FB68764ACE664CB1E67135ABD1F1EFAD0937731237E78E46A7646B66DB60E4791CCD4EC9D98C04FF56981970A8DA537236238134459A68ABD926399B75A637145AA0ABD926399B31676B18477135DB245B33710BC2D9B59BE46CC6BC99609507ABD926D99A31812486ED0B6C92AD1911D099E95947B3537234ABC4E144182AC7D4DB2547B3FB1ACBC4C7C84D19C66C1EF3F54FC4137FEF166726D8B359A9FE1707E2DE8D0EBE1B997C37D6F787F7E26DEF46E83E79EF525F3D2CFA65A365DBA2BBD9AAD54E5E6ED31624238E5FA46F1A9F0434B39D3C1516E31093C76CF7F10DD7609E70C09D89C4D8EBFB8B70BB7A0416A70BEBC768A7C0D3DFC10421AC7CB903F183EF7FFCEB9FFEDD1F7EFCF1EF3FF9C5079F7DF29B3F7C0E57E0F1A3B7BA65D29360E6FAB55F4EF197E8986184E76088F3FEED744CF937EF18716B0AF7B52F3DB3C9BF44D788566D3593A97C53C3ED4BF64DFAB7EF9B4ABC3217D15BDF9CF237FB8675095BDF542300C1321F449B7CCB5E9DED26AD57D2037D95A3F75EBE6E28D4DCDE914ACAA2E6560231B2BBC38327C2A8249EF54C9658687226830B72CC6DDFE1E289DB194A8927C959AA82CBB98029D333B278B32DF91496AE13862B584036749981BC296719AAA575987C1A6EC91243DB14FF9172A68C65CF5FC572EFB7AD44A9592E3D13974A2CA92DF98CB1E83A3C4D2CADFBE6CE2454E5F22491F9292E8DC9F42BD9A6A4F538B5E944BBA213ADF292B35A56C9362CDC3EC40312DF476502A8D56C516E85D6BE635F9CC19399081747CDDAA3440FEB6850CA79CE899519B00870F4385D0ED7AC6555259431745022E5DCA4F7F2F41E8A18DA944F3B9BC16B2466B78C891C0C1E5DBD1682F935B3B9BC16E51C71D3768A204431E37349A52261FAC922994972AC03F1FABD84D19B5CCEEF199D574300A62ECF9C2D1E83F250A17AD6D9C394F30CB7EA253F3D0FDBE2CFCC6E2B415BF9C6CC8D96B84079CFD9C8142CB19949E92A99CC31562E8BB142FE9925526046B1F862A822872854CB4AA178D4A80AA16A79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    One site, <math|\<nu\>\<propto\>\<mathe\><rsup|-\<lambda\>*\<varphi\><rsup|4>/12>\<mu\><rsup|1/q<rsub|0>>>,
    <math|\<lambda\>=6>, <math|q<rsub|0>=1>. Left: one path of the
    observation <math|\<Phi\><rsub|q>> and of the forecast
    <math|\<psi\><rsub|q>>, both converging to the hidden
    <math|\<Phi\><rsub|\<infty\>>>. Right, over 200 paths: the forecast error
    <math|\<sim\>q<rsup|-1/2>>, the force term
    <math|\<psi\><rsub|q>-\<Phi\><rsub|q>=C<rsub|q>F<rsub|q>\<sim\>q<rsup|-1>>,
    and the error <math|Z<rsub|q>> of the approximate forecast from
    <math|<wide|\<cal-F\>|\<bar\>>=<frac|\<lambda\>|3> y<rsup|3>>, below the
    bound <math|2C<rsub|\<ast\>><around|(|\<lambda\>/3|)><rsup|1/2>q<rsup|-3/2>>
    of Theorem<nbsp>10.12 (see "Integrating the cube" below).
  </hidden>|<\hidden>
    <mytitle|The exact force in the forecast coordinate>

    Since <math|\<Phi\><rsub|q>\<mapsto\>\<psi\><rsub|q>> is invertible,
    define <math|\<cal-F\><rsub|q>> by

    <\equation*>
      \<Phi\><rsub|q>=\<psi\><rsub|q>+C<rsub|q>\<cal-F\><rsub|q><around|(|\<psi\><rsub|q>|)>
    </equation*>

    and <math|H<rsup|<math-up|ex>><rsub|q><around|(|y|)>=<around|(|K<rsub|q>+D\<cal-F\><rsub|q><around|(|y|)>|)><rsup|-1>>.
    It is the <math|\<cal-F\>> of Lecture<nbsp>4
    (<math|x=\<psi\>+S\<cal-F\><around|(|\<psi\>|)>>), and
    <math|F<rsub|q>=-\<cal-F\><rsub|q><around|(|\<psi\><rsub|q>|)>>.

    <\myframed>
      <strong|The force equation with a mass regulator>

      <\equation*>
        \<partial\><rsub|q>\<cal-F\><rsub|q>+<frac|1|2><math-up|Ctr><rsub|<around|(|H<rsup|<math-up|ex>><rsub|q>|)><rsup|2>>D<rsup|2>\<cal-F\><rsub|q>=0,<space|1em><math-up|Cov><rsub|q>=H<rsup|<math-up|ex>><rsub|q><around|(|\<psi\><rsub|q>|)>.
      </equation*>
    </myframed>

    \;

    Itô on <math|\<Phi\>=\<psi\>+C\<cal-F\><around|(|\<psi\>|)>>. Its
    integrated form is Wetterich's equation, which we do not need. An
    identity about the unknown <math|\<nu\><rsub|N>>, not a construction.\ 

    <with|font-series|bold|Strategy:> an approximate force
    <math|<wide|\<cal-F\>|\<bar\>>> gives approximate forecast coordinates;
    the error should be driven by the failure of
    <math|<wide|\<cal-F\>|\<bar\>>> to solve this equation.
  </hidden>|<\hidden>
    <mytitle|Approximate forecast and error>

    \;

    <math|<wide|\<cal-F\>|\<bar\>><rsub|q>> odd,
    <math|D<wide|\<cal-F\>|\<bar\>><rsub|q>> symmetric,
    <math|K<rsub|q>+D<wide|\<cal-F\>|\<bar\>><rsub|q>\<geqslant\><wide|c|\<bar\>><rsub|q>I>,
    <math|<wide|c|\<bar\>><rsub|q>\<gtr\>0>. Recall
    <math|\<psi\><rsub|q>=\<bbb-E\><rsub|q><around*|[|\<Phi\><rsub|\<infty\>>|]>>,
    <math|<math-up|Cov><rsub|q>=Var<rsub|q><around*|(|\<Phi\><rsub|\<infty\>>|)>>,

    <\equation*>
      H<rsub|q>=<around|(|K<rsub|q>+D<wide|\<cal-F\>|\<bar\>><rsub|q>|)><rsup|-1>,<space|1em>
      \<cal-D\><rsub|q>=\<partial\><rsub|q><wide|\<cal-F\>|\<bar\>><rsub|q>+<frac|1|2><math-up|Ctr><rsub|H<rsub|q><rsup|2>>D<rsup|2><wide|\<cal-F\>|\<bar\>><rsub|q>,<space|1em>
      b<rsub|q>=-H<rsub|q>\<cal-D\><rsub|q>.
    </equation*>

    <\equation*>
      Y<rsub|q>+C<rsub|q><wide|\<cal-F\>|\<bar\>><rsub|q><around|(|Y<rsub|q>|)>=\<Phi\><rsub|q>,<space|1em>
      Z<rsub|q>=\<psi\><rsub|q>-Y<rsub|q>,<space|1em>\<Xi\><rsub|q>=<math-up|Cov><rsub|q>-H<rsub|q><around|(|Y<rsub|q>|)>.
    </equation*>

    \;

    <\myframed>
      <strong|Exact, at fixed> <math|N>

      <\equation*>
        <space|0.17em>\<mathd\> Y<rsub|q>=<around|[|b<rsub|q><around|(|Y<rsub|q>|)>+H<rsub|q><around|(|Y<rsub|q>|)>Z<rsub|q>|]><space|0.17em>\<mathd\>
        q+H<rsub|q><around|(|Y<rsub|q>|)><space|0.17em>\<mathd\>
        W<rsub|q>,<space|2em>Y<rsub|q<rsub|0>>=0
      </equation*>

      <\equation*>
        <space|0.17em>\<mathd\> Z<rsub|q>=-<around|[|b<rsub|q><around|(|Y<rsub|q>|)>+H<rsub|q><around|(|Y<rsub|q>|)>Z<rsub|q>|]><space|0.17em>\<mathd\>
        q+\<Xi\><rsub|q><space|0.17em>\<mathd\> W<rsub|q>,<space|1em>
        Z<rsub|\<infty\>>=\<Phi\><rsub|\<infty\>>-Y<rsub|\<infty\>>.
      </equation*>
    </myframed>

    \;

    Proof: Itô on the coordinate equation, with the Polchinski process and
    <math|\<mathd\>C<rsub|q>=-C<rsub|q><rsup|2>\<mathd\>q>: the martingale
    part gives <math|H>, the drift gives <math|H*Z+b>.
  </hidden>|<\hidden>
    <mytitle|Reading the system>

    \;

    <\itemize>
      <item><math|Y>: forward and explicit once <math|Z> is known;
      coefficients <math|b,H> built from <math|<wide|\<cal-F\>|\<bar\>>>.

      <item><math|Z>: backward. The terminal condition
      <math|q*Z<rsub|q>\<to\>0> is <strong|proved> at fixed <math|N>
      (Lecture<nbsp>7), because <math|<wide|\<cal-F\>|\<bar\>><rsub|q>\<to\>>
      bare force.

      <item>The only source of <math|Z> is the drift <math|b>, the force
      defect, transported by <math|exp <big|int>H>.
    </itemize>

    \;

    <\myframed>
      <strong|UV stability>

      The explicit force <math|<wide|\<cal-F\>|\<bar\>>> computes the
      forecast up to an error <math|Z> which is small uniformly in <math|N>.
    </myframed>

    \;

    Needed: (i) a defect integrable against the backward propagator
    (<strong|renormalisation>); (ii) global coercivity of
    <math|<wide|\<cal-F\>|\<bar\>>> (<strong|large fields>); (iii) a priori
    estimates (Lecture<nbsp>7).
  </hidden>|<\hidden>
    <mytitle|The structure of the construction>

    \;

    \;

    <with|par-mode|center|<\script-output|tikz|default>
      \\documentclass[tikz]{standalone}

      \\usepackage[T1]{fontenc}

      \\usepackage[sc]{mathpazo}

      \\usepackage{amsmath,amssymb,mathtools,bm}

      \\usepackage[mathscr]{eucal}

      \\usetikzlibrary{decorations.pathreplacing,arrows.meta,positioning}

      \\usepackage[dvipsnames]{xcolor}

      \\definecolor{navy}{HTML}{1B2A41}

      \\definecolor{accent}{HTML}{B5402E}

      \\definecolor{SNavy}{HTML}{1B2A49}

      \\definecolor{STeal}{HTML}{1F7A72}

      \\definecolor{SGold}{HTML}{B8862B}

      \\begin{document}

      \\begin{tikzpicture}[scale=0.82,transform shape,

      box/.style={draw=navy,rounded corners=2pt,align=center,

      inner sep=4pt,font={},text width=4.0cm},

      arr/.style={-{Stealth[length=2.2mm]},thick,draw=navy!80}]

      \\node[box] (ren) at (0,0) {Renormalisation:\\\\ seed $+$ 2
      corrections,\\\\ mass path};

      \\node[box] (clip) at (5.4,0) {Clipping: global\\\\ coercivity,
      defect\\\\ $b=b^{\\mathrm b}+b^{\\mathrm l}$};

      \\node[box] (fbsde) at (10.8,0) {Exact FBSDE for\\\\ $(Y,Z)$ at fixed
      $N$};

      \\node[box] (fwd) at (2.2,-2.4) {Forward coercivity:\\\\ large fields
      $J$ by\\\\ noise tails $+E$ (L7)};

      \\node[box] (bwd) at (8.6,-2.4) {Backward\\\\ representation:\\\\
      energy $E$ by $b^{\\mathrm b}$,\\\\ boundary,
      $\\frac{C\\lambda^2}{q_\\ast}J$ (L7)};

      \\node[box] (bd) at (0,-4.8) {Boundary $qZ_q\\to0$\\\\ at fixed $N$
      (L7)};

      \\node[box] (uv) at (5.4,-4.8) {UV stability:\\\\
      $\\sup_N\\mathbb{E}\\\| Z_q\\\| ^2\\le Cq^{-5/2}$};

      \\node[box] (lim) at (10.8,-4.8) {Compactness, limit\\\\ FBSDE, DS
      (L7)};

      \\draw[arr] (ren)--(clip); \\draw[arr] (clip)--(fbsde);

      \\draw[arr] (fbsde)--(bwd); \\draw[arr] (clip)--(fwd);

      \\draw[arr] ([yshift=0.15cm]fwd.east) -- ([yshift=0.15cm]bwd.west);

      \\draw[arr] ([yshift=-0.15cm]bwd.west) -- ([yshift=-0.15cm]fwd.east);

      \\draw[arr] (bd)--(uv); \\draw[arr] (fwd)--(uv); \\draw[arr]
      (bwd)--(uv); \\draw[arr] (uv)--(lim);

      \\end{tikzpicture}

      \\end{document}
    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    <small|<with|color|dark grey|<math|J>: large-field functional, <math|E>:
    energy (both Lecture<nbsp>7); <math|b<rsup|<math-up|b>>,b<rsup|<math-up|l>>>:
    bounded- and large-field parts of the drift.>>
  </hidden>|<\shown>
    <mytitle|Why not Polchinski with a frozen cube?>

    \;

    <strong|Programme:> keep <math|-<frac|\<lambda\>|3>\<varphi\><rsup|3>>
    unclipped, clip the higher orders (result: <math|f>), pay in the source
    of <math|R=F-f>. Test it on one site: <math|K<rsub|q>=q>, <math|S=1/q>.

    <\equation*>
      F<rsub|q><around|(|x|)>=-<frac|\<lambda\>|3>\<bbb-E\><rsub|x>\<Phi\><rsub|\<infty\>><rsup|3>=q<around|(|m<rsub|q><around|(|x|)>-x|)>,<space|1em><around|\||m<rsub|q><around|(|x|)>|\|>\<leqslant\>2<around|(|\<lambda\>/3|)><rsup|-1/4>+2<around|(|3q<around|\||x|\|>/\<lambda\>|)><rsup|1/3>.
    </equation*>

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      \\end{scope}

      \\begin{scope}[xshift=6.6cm,x=0.72cm,y=0.42cm]

      \\draw[black!45] (0,0)--(6,0) (0,0)--(0,8);

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      \\node[anchor=south east,black!60,font=\\scriptsize] at (6,3.2292)
      {bound, Theorem 10.12};

      \\node[anchor=north east,font=\\scriptsize] at (6,1.5568) {$\\lvert
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      \\end{scope}

      \\end{tikzpicture}

      \\end{document}
    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    So <math|3R/<around|(|\<lambda\> x<rsup|3>|)>\<to\>1>: the remainder
    <strong|cancels the cube> (the dip: <math|R> changes sign, Wick term
    below, cube above). Field equation: <math|q*x=<around|(|q-q<rsub|0>|)>m<rsub|q>+\<omega\><rsub|q>>,
    only linear confinement.
  </shown>|<\hidden>
    <mytitle|The price of clipping>

    \;

    <math|f<rsup|c>=f<rsub|1>+\<cal-C\>*f<rsub|2>>, with
    <math|f<rsub|1>=-<frac|\<lambda\>|3><around|(|x<rsup|3>+3S*x|)>>,
    <math|f<rsub|2>> the first tree, <math|\<kappa\><rsub|q>> a smooth clip
    at <math|<around|\||x|\|>\<sim\>r<rsub|q>>,
    <math|\<cal-C\>*f<rsub|2>=\<kappa\><rsub|q>'\<cdot\>
    f<rsub|2>\<circ\>\<kappa\><rsub|q>>. For
    <math|<around|\||x|\|>\<geqslant\>2r<rsub|q>>, the Burgers defect is
    exactly

    <\equation*>
      \<cal-S\><rsub|q>=S<rsup|2>f<rsub|1>f<rsub|1>'\<approx\><frac|\<lambda\><rsup|2>|3>S<rsup|2>x<rsup|5>,
    </equation*>

    of order <math|<around|(|q/\<lambda\>|)><rsup|1/2>> at the edge of the
    clip: not integrable in <math|q>.

    <\equation*>
      <space|0.17em>\<mathd\> R<rsub|q>=-<around|[|\<cal-S\><rsub|q>+S<rsup|2><around|(|f<rsup|c>|)>'R<rsub|q>|]><space|0.17em>\<mathd\>
      q+\<Xi\><rsup|R><rsub|q><space|0.17em>\<mathd\>
      W<rsub|q>,<space|1em><text|bracket>\<approx\><frac|\<lambda\><rsup|2>|3>S<rsup|2>x<rsup|5>-\<lambda\>
      S<rsup|2>x<rsup|2>R.
    </equation*>

    The source and the dissipation cancel for the true
    <math|R\<sim\><frac|\<lambda\>|3> x<rsup|3>>: separate estimates give
    nothing.

    Tails independent of <math|R>: on one site from the posterior moment, but
    not uniform on the lattice (negative bare mass).

    More orders do not help: the tree series has radius
    <math|<frac|2|3>r<rsub|q>>.
  </hidden>|<\hidden>
    <mytitle|Integrating the cube>

    \;

    Solve the transport part of the flow exactly:

    <\equation*>
      \<partial\><rsub|q>f+S<rsup|2>f*f<rprime|'>=0,<space|1em>
      f<rsub|\<infty\>>=-<frac|\<lambda\>|3>
      x<rsup|3><space|1em>\<Rightarrow\><space|1em>
      f<rsub|q><around|(|x|)>=-<frac|\<lambda\>|3> y<rsup|3>,<space|1em>
      q*y+<frac|\<lambda\>|3> y<rsup|3>=q*x.
    </equation*>

    <\equation*>
      sup<rsub|x><around|\||F<rsub|q><around|(|x|)>-f<rsub|q><around|(|x|)>|\|>\<leqslant\>2C<rsub|\<ast\>><around|(|<frac|\<lambda\>|3q>|)><rsup|1/2>,<space|1em>
      C<rsub|\<ast\>>\<less\><frac|1|2>.
    </equation*>

    <\itemize>
      <item><math|y<rsub|q><around|(|\<Phi\><rsub|q>|)>=Y<rsub|q>>:
      <strong|this is the forecast coordinate>.
      <math|F<rsub|q>-f<rsub|q>=K<rsub|q>Z<rsub|q>>, and <math|Z> is the
      posterior mean minus the mode.

      <item>The defect is <math|q*b<rsub|q><around|(|y|)>>: bounded for all
      fields. The trees are in <math|x\<mapsto\> y>; the cube does not flow
      in <math|y>, so only loops are clipped.

      <item>The cube is coercive in <math|y>: <math|q*y+<frac|\<lambda\>|3>
      y<rsup|3>=q*\<Phi\><rsub|q>>.
    </itemize>

    \;
  </hidden>|<\hidden>
    <vspace*|2.5cm>

    \;

    <cqft-part-title|Renormalisation: why two of them>

    <vspace|0.3cm>

    <with|color|dark grey|Lecture<nbsp>6>

    \;
  </hidden>|<\hidden>
    <mytitle|Approximating the exact force>

    \;

    We seek an explicit force with a small <strong|defect>, not the full
    solution of the nonlinear flow.

    <\equation*>
      \<partial\><rsub|q>\<cal-F\><rsub|q>=-<frac|1|2><math-up|Ctr><rsub|<around|(|H<rsup|<math-up|ex>><rsub|q>|)><rsup|2>>D<rsup|2>\<cal-F\><rsub|q>,<space|2em>H<rsup|<math-up|ex>><rsub|q>=<around|(|K<rsub|q>+D\<cal-F\><rsub|q>|)><rsup|-1>.
    </equation*>

    At infinite resolution: <math|\<cal-F\><rsub|\<infty\>>=U'+\<mu\><rsub|N>y>,
    with <math|U'<around|(|y|)>=<frac|\<lambda\>|3>y<rsup|3>>.

    <\enumerate>
      <item>Insert the bare cube into the response and the vertex: a
      <strong|one-loop seed>.

      <item>Feed each approximation back into the equation: <strong|two
      correction steps>.

      <item>Estimate the remaining defect in the backward equation for
      <math|Z>.
    </enumerate>

    \;

    <\myframed>
      First separate the linear flow: it produces the divergent mass
      counterterm and cannot be treated as a small nonlinear error.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Why the linear term must be separated>

    Insert <math|U'> into the loop, with

    <\equation*>
      G<rsub|q><around|(|y|)>=<around|(|K<rsub|q>+\<lambda\>y<rsup|2>|)><rsup|-1>:
    </equation*>

    <\equation*>
      -<frac|1|2><math-up|Ctr><rsub|G<rsub|q><rsup|2>>D<rsup|2>U'<around|(|y|)><around|(|x|)>=-\<lambda\>y<around|(|x|)>G<rsub|q><around|(|y|)><rsup|2><around|(|x,x|)>.
    </equation*>

    Its linear Taylor coefficient is

    <\equation*>
      -\<lambda\>a<rsub|2><around|(|q|)>y<around|(|x|)>,<space|2em>a<rsub|2><around|(|q|)>=C<rsub|q><rsup|2><around|(|x,x|)>.
    </equation*>

    In three dimensions <math|a<rsub|2><around|(|q|)>> has continuum size
    <math|q<rsup|-1/2>>: its integral to infinity diverges. At fixed lattice
    cutoff it is finite, but grows with <math|N>.

    <\myframed>
      This is the <strong|tadpole>. Further insertions also produce the
      logarithmic <strong|sunset>. Keep the linear flow in a running mass;
      iterate only the nonlinear remainder.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|The projections <math|\<Pi\>> and <math|\<Pi\><rprime|'>>>

    For a smooth odd force <math|f>, and a translation-invariant operator
    <math|L=<around|\<langle\>|L|\<rangle\>>I+L<rsup|\<circ\>>> (constant
    mode plus centred part):

    <\equation*>
      <around|(|\<Pi\>f|)><around|(|y|)>=f<around|(|y|)>-Df<around|(|0|)>y,<space|2em><around|(|\<Pi\><rprime|'>f|)><around|(|y|)>=f<around|(|y|)>-<around|\<langle\>|Df<around|(|0|)>|\<rangle\>>y.
    </equation*>

    <\itemize>
      <item><math|\<Pi\>f> starts at <strong|cubic order>;
      <math|\<Pi\><rprime|'>f=\<Pi\>f+<around|(|Df<around|(|0|)>|)><rsup|\<circ\>>y>
      removes only the <strong|scalar> part of the linear jet.

      <item>The divergences (tadpole, sunset) are <strong|constant modes>.
      The centred part of the linear flow is integrable at infinity.
    </itemize>

    <\equation*>
      \<cal-F\><rsub|q>=U'+M<rsub|q>y+\<ell\><rsub|q>,<space|2em>\<Pi\>\<ell\><rsub|q>=\<ell\><rsub|q>,<space|1em>M<rsub|q>=\<mu\><rsub|q>I+\<Theta\><rsub|q>.
    </equation*>

    <\myframed>
      <math|\<ell\>> and the centred kernel <math|\<Theta\>>: integrate from
      <math|\<infty\>>, by the same finite iteration. Scalar mass
      <math|\<mu\>>: solve exactly, fixed at an auxiliary large scale
      <math|q<rsub|\<ast\>>> (with <math|g<rsub|q<rsub|\<ast\>>>> small).
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Scales and power counting>

    \;

    <\equation*>
      g<rsub|q>=\<lambda\>*q<rsup|-1/2><around|(|<text|dimensionless
      coupling>|)>,
    </equation*>

    <\equation*>
      r<rsub|q>=<around|(|q/\<lambda\>|)><rsup|1/2><around|(|\<lambda\>*r<rsub|q><rsup|2>=q:<text|nonlinearity>\<approx\><text|regulator>|)>.
    </equation*>

    Free field at resolution <math|q>: free field <math|X<rsub|q>>,
    <math|<math-up|Var>X<rsub|q><around|(|x|)>\<lesssim\>q<rsup|1/2>>,
    typical size <math|q<rsup|1/4>=g<rsub|q><rsup|1/2>*r<rsub|q>\<ll\>r<rsub|q>>.

    Rooted norm of a vertex <math|T<around|(|x<rsub|1>,\<ldots\>,x<rsub|k>|)>>
    and field ball:

    <\equation*>
      s<around|(|T|)>=max<rsub|i> sup<rsub|x<rsub|i>>
      <big|int><around|\||T|\|>*<big|prod><rsub|j\<ne\>i>\<mathd\>x<rsub|j>,<space|2em><math-ss|S><rsub|k><around|(|f|)>=sup<rsub|\<cal-B\><rsub|q>>
      s*<around|(|D<rsup|k-1>*f|)>,<space|2em>\<cal-B\><rsub|q>=<around|<left|{|2>|<around|\<\|\|\>|y|\<\|\|\>><rsub|\<infty\>>\<less\>2*r<rsub|q>|<right|}|2>>.
    </equation*>

    Natural size of the <math|k>-leg vertex of the <strong|loop corrections>:

    <\equation*>
      <math-ss|S><rsub|k>\<lesssim\>g<rsub|q>*<space|0.17em>q*<space|0.17em>r<rsub|q><rsup|2-k>*<space|2em><around|(|k=4:\<lambda\>*g<rsub|q>|)>.
    </equation*>
  </hidden>|<\hidden>
    <mytitle|The loop estimate>

    \;

    \;

    Root one leg of a closed loop and follow the loop: two of its propagators
    are estimated in <math|L<rsup|2>> (this gives the bubble
    <math|a<rsub|2><around|(|q|)>>), all the others in operator norm (each
    <math|q<rsup|-1>>).

    \;

    <\myframed>
      <strong|Loop estimate>

      A closed loop with <math|p> vertices:

      <\equation*>
        s<around|(|<text|loop>|)>\<leqslant\>C*a<rsub|2><around|(|q|)>*<space|0.17em>q<rsup|-<around|(|p-1|)>>*<big|prod><rsub|i>s<around|(|T<rsub|i>|)>
      </equation*>

      (two propagators in <math|L<rsup|2>>, the others in operator norm).
      After the <math|q> integration a one-loop term is of size
      <math|g<rsub|q>*q*r<rsub|q><rsup|2-k>>; every further loop gains a
      factor <math|g<rsub|q>>.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Computing the one-loop seed>

    Project the bare loop before integrating:

    <\equation*>
      \<Pi\><around*|[|-\<lambda\>y<around|(|x|)>G<rsub|q><around|(|y|)><rsup|2><around|(|x,x|)>|]>=-\<lambda\>y<around|(|x|)><around|[|G<rsub|q><around|(|y|)><rsup|2><around|(|x,x|)>-a<rsub|2><around|(|q|)>|]>.
    </equation*>

    Use <math|\<partial\><rsub|q>G<rsub|q>=-G<rsub|q><rsup|2>> and
    <math|\<partial\><rsub|q>C<rsub|q>=-C<rsub|q><rsup|2>>. With zero
    boundary value at infinity:

    <\myframed>
      <\equation*>
        \<ell\><rsup|<around|[|1|]>><rsub|q><around|(|y|)><around|(|x|)>=\<lambda\>y<around|(|x|)><around|[|G<rsub|q><around|(|y|)><around|(|x,x|)>-C<rsub|q><around|(|x,x|)>|]>.
      </equation*>
    </myframed>

    The leading term is the bubble:

    <\equation*>
      -\<lambda\><rsup|2>y<around|(|x|)><big|int>C<rsub|q><around|(|x,z|)><rsup|2>y<around|(|z|)><rsup|2>\<mathd\>z.
    </equation*>

    Four-leg size: <math|<math-ss|S><rsub|4><around|(|\<ell\><rsup|<around|[|1|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1/2>=\<lambda\>g<rsub|q>>.
    This is the seed <math|\<ell\><rsup|<around|[|1|]>>>: the superscript
    counts loops.
  </hidden>|<\hidden>
    <mytitle|The seed in graphs>

    \;

    \;

    <with|par-mode|center|<\script-output|tikz|default>
      \\documentclass[tikz]{standalone}

      \\usepackage[T1]{fontenc}

      \\usepackage[sc]{mathpazo}

      \\usepackage{amsmath,amssymb}

      \\usetikzlibrary{calc,decorations.markings}

      \\tikzset{

      \ blob/.style={circle,draw,thick,fill=#1,minimum size=7mm,inner
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      \ blob/.default={black!12},

      \ vtx/.style={circle,fill=black,inner sep=1.5pt},

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      \ full/.style={double,double distance=1.3pt,thick},

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      west)--(path picture bounding box.south east) (path picture bounding
      box.north east)--(path picture bounding box.south west);}},

      }

      % external legs: \\legs{node}{angles}

      \\newcommand\\legs[2]{\\foreach \\a in {#2}{\\draw[thick]
      (#1.\\a)--++(\\a:2.6mm);}}

      \\newcommand\\rootleg[2]{\\draw[line width=1.4pt,-latex]
      (#1.#2)--++(#2:3.6mm);}

      \\begin{document}

      \\begin{tikzpicture}[font=\\small,baseline]

      \\def\\ring#1#2{% centre x, j

      \ \ \\begin{scope}[xshift=#1 cm]

      \ \ \\draw[prop] (0,0) circle (.48);

      \ \ \\foreach \\k in {1,...,#2}{\\pgfmathsetmacro\\t{90+360*(\\k-1)/#2}

      \ \ \ \ \\node[vtx] at (\\t:.48) {};

      \ \ \ \ \\draw[thick] (\\t:.48)--++(\\t+22:.42); \\draw[thick]
      (\\t:.48)--++(\\t-22:.42);}

      \ \ \\end{scope}}

      \\node at (-.75,0) {$\\ell^{[1]}_q(y)\\ =\\ D\\,\\big[$};

      \\node at (.35,0) {$-\\tfrac14$}; \\ring{1.6}{2}

      \\node at (3.05,0) {$+\\tfrac16$}; \\ring{4.3}{3}

      \\node at (5.75,0) {$-\\tfrac18$}; \\ring{7.0}{4}

      \\node at (8.8,0) {$+\\ \\cdots\\ \\big]$};

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      \\end{document}

      \;
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    \;

    The seed as one-loop graphs: lines <math|C<rsub|q>>, a vertex
    <math|\<lambda\> y<rsup|2>> with its two external fields at each corner.
    The <math|j>-th ring has <math|2j> legs and coefficient
    <math|<around|(|-1|)><rsup|j+1>/<around|(|2j|)>>; <math|D> marks one leg,
    giving the force. The series sums to a logarithm, with radius of
    convergence <math|r<rsub|q>>. The quartic ring is the bubble.
  </hidden>|<\hidden>
    <mytitle|Splitting the flow and deriving the iteration>

    For <math|\<cal-F\>=U'+M*y+\<ell\>>, the loop map

    <\equation*>
      \<cal-T\><rsub|q><around|[|M,\<ell\>|]>=-<frac|1|2><math-up|Ctr><rsub|R<rsup|2>>D<rsup|2><around|(|U'+\<ell\>|)>,<space|1em>R=<around|(|K<rsub|q>+M<rsub|q>+D<around|(|U'+\<ell\>|)>|)><rsup|-1>.
    </equation*>

    Three equations of different nature:

    <\equation*>
      \<partial\><rsub|q>\<ell\>=\<Pi\>\<cal-T\><around|[|M,\<ell\>|]>,<space|1em>\<partial\><rsub|q>\<Theta\>=<around|(|D\<cal-T\><around|[|M,\<ell\>|]><around|(|0|)>|)><rsup|\<circ\>>,<space|1em>\<partial\><rsub|q>\<mu\>=<around|\<langle\>|D\<cal-T\><around|[|M,\<ell\>|]><around|(|0|)>|\<rangle\>>.
    </equation*>

    <strong|Fix a trial scalar path <math|\<mu\>>> (determined at the end as
    a fixed point, "Closing the mass path"). Level masses
    <math|M<rsup|<around|[|j|]>>=\<mu\>I+\<Theta\><rsup|<around|[|j|]>>>,
    sources <math|\<frak-S\><rsup|<around|[|j|]>>=D\<cal-T\><around|[|M<rsup|<around|[|j|]>>,\<ell\><rsup|<around|[|j|]>>|]><around|(|0|)>>,
    zero-field response
    <math|R<rsup|<around|[|j|]>><rsub|0,q>=<around|(|K<rsub|q>+M<rsup|<around|[|j|]>><rsub|q>|)><rsup|-1>>:

    <\myframed>
      <\equation*>
        \<ell\><rsup|<around|[|j+1|]>><rsub|q>=-<big|int><rsub|q><rsup|\<infty\>>\<Pi\>\<cal-T\><rsub|s><around|[|M<rsup|<around|[|j|]>>,\<ell\><rsup|<around|[|j|]>>|]>\<mathd\>s,<space|1em>\<Theta\><rsup|<around|[|j+1|]>><rsub|q>=-<big|int><rsub|q><rsup|\<infty\>><around|(|\<frak-S\><rsup|<around|[|j|]>><rsub|s>|)><rsup|\<circ\>>\<mathd\>s,<space|1em>j=1,2.
      </equation*>
    </myframed>

    With <math|\<ell\><rsup|<around|[|0|]>>=0>,
    <math|M<rsup|<around|[|0|]>>=0>, the case <math|j=0> is the seed step
    without mass, <math|\<partial\><rsub|q>\<ell\><rsup|<around|[|1|]>>=\<Pi\>\<cal-T\><around|[|0,0|]>>,
    and <math|\<Theta\><rsup|<around|[|1|]>>=0> (the tadpole is local).
    Output: <math|<around|(|\<ell\><rsup|<around|[|3|]>>,\<Theta\><rsup|<around|[|3|]>>|)><around|[|\<mu\>|]>>.
  </hidden>|<\hidden>
    <mytitle|Centre before taking absolute values>

    A centred kernel <math|\<Theta\><around|(|z|)>=\<Theta\><around|(|0,z|)>>
    is measured by its first moment <math|v<rsub|1><around|(|\<Theta\>|)>=<big|int>d<around|(|0,z|)><around|\||\<Theta\><around|(|z|)>|\|>\<mathd\>z>,
    <math|d> the torus distance. For an even kernel,
    <math|\<Theta\><with|math-font|Bbb|1>=0> means that the diagonal cancels
    the off-diagonal mass. With <math|<around|(|\<sigma\><rsub|z>f|)><around|(|x|)>=f<around|(|x+z|)>>,
    <math|\<delta\><rsub|z><rsup|2>=\<sigma\><rsub|z>+\<sigma\><rsub|-z>-2I>,

    <\equation*>
      C<rsub|q>\<Theta\>=<frac|1|2><big|int>\<Theta\><around|(|z|)>\<delta\><rsub|z><rsup|2>C<rsub|q>\<mathd\>z.
    </equation*>

    <strong|The absolute row norm discards the cancellation.> For a lattice
    step <math|h> in direction <math|e>, take

    <\equation*>
      \<Theta\><rsub|h>=<frac|a|2h><around|(|\<sigma\><rsub|he>+\<sigma\><rsub|-he>-2I|)>,<space|2em>v<rsub|1><around|(|\<Theta\><rsub|h>|)>=a,<space|1em><around|\<\|\|\>|\<Theta\><rsub|h>|\<\|\|\>><rsub|<math-up|row>>=<frac|2a|h>.
    </equation*>

    The row norm diverges as <math|h\<to\>0>; the first moment stays bounded.

    <\myframed>
      Estimate the signed product <math|C<rsub|q>\<Theta\>> first. The
      quantity needed by the loop is the kernel after resolvent smoothing,
      not the absolute mass of <math|\<Theta\>> itself.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Smoothing turns cancellation into a moment bound>

    The resolvent smooths on length scale <math|q<rsup|-1/2>>:

    <\equation*>
      <around|\<\|\|\>|\<delta\><rsub|z><rsup|2>C<rsub|q>|\<\|\|\>><rsub|<math-up|row>>\<leqslant\>C*q<rsup|-1/2>d<around|(|0,z|)>.
    </equation*>

    Scale check: integrate the heat-kernel translation bound
    <math|C*d<around|(|0,z|)>t<rsup|-1/2>> against
    <math|\<mathe\><rsup|-qt>\<mathd\>t>.

    \;

    <\myframed>
      <\equation*>
        <around|\<\|\|\>|C<rsub|q>\<Theta\>|\<\|\|\>><rsub|<math-up|row>>\<leqslant\>C*q<rsup|-1/2><big|int>d<around|(|0,z|)><around|\||\<Theta\><around|(|z|)>|\|>\<mathd\>z=C*q<rsup|-1/2>v<rsub|1><around|(|\<Theta\>|)>.
      </equation*>
    </myframed>

    \;

    The iteration (next slides) supplies exactly the compensating moment:

    <\equation*>
      v<rsub|1><around|(|\<Theta\><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1/2><space|1em>\<Longrightarrow\><space|1em>
      q<rsup|-1/2>v<rsub|1><around|(|\<Theta\><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1>=g<rsub|q><rsup|2>.
    </equation*>

    <small|A bound by <math|d<around|(|0,z|)><rsup|2>> loses a logarithm; one
    distance power, with <math|q<rsup|-1/2>>, suffices (Lemma<nbsp>19.3 of
    the Supplement).>
  </hidden>|<\hidden>
    <mytitle|The small factor in a resolvent comparison>

    The field-dependent response also receives this cancellation. With
    <math|V=\<lambda\>y<rsup|2>> and <math|G=<around|(|K<rsub|q>+V|)><rsup|-1>>,

    <\equation*>
      G\<Theta\>=<around|(|I-GV|)>C<rsub|q>\<Theta\>,<space|2em><around|\<\|\|\>|I-GV|\<\|\|\>><rsub|\<infty\>\<to\>\<infty\>>\<leqslant\>2.
    </equation*>

    The bounded Neumann series for the full <math|R> then gives

    <\equation*>
      <around|\<\|\|\>|R\<delta\>M|\<\|\|\>><rsub|\<infty\>\<to\>\<infty\>>\<leqslant\>C\<rho\><rsub|q>,<space|1em>\<rho\><rsub|q>=<frac|<around|\||\<delta\>\<mu\><rsub|q>|\|>|q>+q<rsup|-1/2>v<rsub|1><around|(|\<delta\>\<Theta\><rsub|q>|)>.
    </equation*>

    Holding the nonlinear correction fixed,
    <math|\<delta\>R=-R\<delta\>M<wide|R|^>> (<math|<wide|R|^>> the response
    at mass <math|M-\<delta\>M>) has row norm
    <math|\<lesssim\>\<rho\><rsub|q>/q> and row <math|L<rsup|2>> norm
    <math|sup<rsub|x><around|\<\|\|\>|\<delta\>R<around|(|x,\<cdot\>|)>|\<\|\|\>><rsub|L<rsup|2>>>
    <math|\<lesssim\>\<rho\><rsub|q><sqrt|a<rsub|2><around|(|q|)>>>.

    <\myframed>
      A marked propagator gains the relative factor <math|\<rho\><rsub|q>>:
      for <math|\<Theta\><rsup|<around|[|2|]>>> in the second correction,
      <math|\<rho\><rsub|q>\<lesssim\>g<rsub|q><rsup|2>>; for two scalar
      paths, see "Mass differences" below.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Centring the source removes local terms>

    Separate the source at level <math|j> into tadpole and correction vertex:

    <\equation*>
      \<frak-S\><rsup|<around|[|j|]>><rsub|q>=-\<lambda\><math-up|Tr><around|(|<around|(|R<rsup|<around|[|j|]>><rsub|0,q>|)><rsup|2>|)>I+E<rsub|q>,<space|1em>E<rsub|q>=-<frac|1|2><math-up|Ctr><rsub|<around|(|R<rsup|<around|[|j|]>><rsub|0,q>|)><rsup|2>>T<rsup|<around|[|j|]>><rsub|q>,<space|1em>T<rsup|<around|[|j|]>><rsub|q>=D<rsup|3>\<ell\><rsup|<around|[|j|]>><rsub|q><around|(|0|)>.
    </equation*>

    Write <math|E<rsub|q><with|math-font|Bbb|1>=e<rsub|q><with|math-font|Bbb|1>>.
    The constant mode contains the whole local contribution:

    <\equation*>
      <around|\<langle\>|\<frak-S\><rsup|<around|[|j|]>><rsub|q>|\<rangle\>>=-\<lambda\><math-up|Tr><around|(|<around|(|R<rsup|<around|[|j|]>><rsub|0,q>|)><rsup|2>|)>+e<rsub|q>,<space|2em><around|(|\<frak-S\><rsup|<around|[|j|]>><rsub|q>|)><rsup|\<circ\>>=E<rsub|q>-e<rsub|q>I.
    </equation*>

    The tadpole cancels; so does any term <math|c<rsub|q>I>. Off-diagonal
    entries stay unchanged. Since <math|d<around|(|0,0|)>=0>,

    <\equation*>
      v<rsub|1><around|(|<around|(|\<frak-S\><rsup|<around|[|j|]>><rsub|q>|)><rsup|\<circ\>>|)>=v<rsub|1><around|(|E<rsub|q>|)>.
    </equation*>

    <\myframed>
      Centring removes the local obstruction. The remaining spatial estimate
      must come from the first moment of the nonlinear vertex.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|The iteration produces the first-moment bound>

    <\with|font-size|0.9>
      Four-leg vertex <math|T<rsup|<around|[|j|]>><rsub|q>> of
      <math|\<ell\><rsup|<around|[|j|]>>>, proved below level by level
      (<math|\<Theta\><rsup|<around|[|2|]>>> needs only the bubble
      <math|T<rsup|<around|[|1|]>>>, <math|\<Theta\><rsup|<around|[|3|]>>>
      only <math|T<rsup|<around|[|2|]>>>, at
      <math|M<rsup|<around|[|1|]>>=\<mu\>I>):

      <\equation*>
        s<around|(|T<rsup|<around|[|j|]>><rsub|q>|)>+<sqrt|q>v<rsub|1><around|(|T<rsup|<around|[|j|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1/2>,<space|1em><text|hence><space|1em>v<rsub|1><around|(|T<rsup|<around|[|j|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1>.
      </equation*>

      Keep the distance between the uncontracted legs of
      <math|T<rsup|<around|[|j|]>><rsub|q>>; since
      <math|sup<rsub|u,v><around|\||<around|(|R<rsup|<around|[|j|]>><rsub|0,q>|)><rsup|2><around|(|u,v|)>|\|>\<lesssim\>q<rsup|-1/2>>,

      <\equation*>
        v<rsub|1><around|(|<around|(|\<frak-S\><rsup|<around|[|j|]>><rsub|q>|)><rsup|\<circ\>>|)>=v<rsub|1><around|(|E<rsub|q>|)>\<lesssim\>q<rsup|-1/2>v<rsub|1><around|(|T<rsup|<around|[|j|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-3/2>.
      </equation*>

      Integrate with zero boundary value at infinity:

      <\equation*>
        v<rsub|1><around|(|\<Theta\><rsup|<around|[|j+1|]>><rsub|q>|)>\<leqslant\><big|int><rsub|q><rsup|\<infty\>>v<rsub|1><around|(|<around|(|\<frak-S\><rsup|<around|[|j|]>><rsub|s>|)><rsup|\<circ\>>|)>\<mathd\>s\<leqslant\>C\<lambda\><rsup|2>q<rsup|-1/2>.
      </equation*>

      <\myframed>
        Uniformly in <math|N>: <math|\<eta\><rsub|q><around|(|\<Theta\><rsup|<around|[|2|]>>|)>\<lesssim\>g<rsub|q><rsup|2>>,
        and one more <math|g> for the difference,
        <math|\<eta\><rsub|q><around|(|\<Theta\><rsup|<around|[|3|]>>-\<Theta\><rsup|<around|[|2|]>>|)>\<lesssim\>g<rsub|q><rsup|3>>,
        with <math|\<eta\><rsub|q><around|(|\<Theta\>|)>=q<rsup|-1/2>v<rsub|1><around|(|\<Theta\>|)>>.
      </myframed>
    </with>
  </hidden>|<\hidden>
    <mytitle|First correction: insert the seed>

    Set <math|R<rsup|<around|[|j|]>>=<around|(|K+M<rsup|<around|[|j|]>>+DU'+D\<ell\><rsup|<around|[|j|]>>|)><rsup|-1>>
    and <math|\<delta\><rsup|<around|[|2|]>>=\<ell\><rsup|<around|[|2|]>>-\<ell\><rsup|<around|[|1|]>>>.
    Subtract the bare loop from the loop containing the seed:

    <\equation*>
      \<delta\><rsup|<around|[|2|]>><rsub|q>=<frac|1|2><big|int><rsub|q><rsup|\<infty\>>\<Pi\><around*|[|<math-up|Ctr><rsub|<around|(|R<rsup|<around|[|1|]>>|)><rsup|2>-G<rsup|2>>D<rsup|2>U'+<math-up|Ctr><rsub|<around|(|R<rsup|<around|[|1|]>>|)><rsup|2>>D<rsup|2>\<ell\><rsup|<around|[|1|]>>|]><rsub|s>\<mathd\>s.
    </equation*>

    <strong|Two changes:> the propagators and the vertex.

    <\equation*>
      R<rsup|<around|[|1|]>>-G=-R<rsup|<around|[|1|]>><around|(|\<mu\>+D\<ell\><rsup|<around|[|1|]>>|)>G,<space|2em>M<rsup|<around|[|1|]>>=\<mu\>I.
    </equation*>

    The insertion <math|<around|(|\<mu\>+D\<ell\><rsup|<around|[|1|]>>|)>/q>
    is <math|O<around|(|g<rsub|q>|)>>. One extra loop gives

    <\equation*>
      <math-ss|S><rsub|4><around|(|\<delta\><rsup|<around|[|2|]>><rsub|q>|)>\<lesssim\><big|int><rsub|q><rsup|\<infty\>>g<rsub|s>s<rsup|1/2>r<rsub|s><rsup|-4>\<mathd\>s=\<lambda\><rsup|3>q<rsup|-1>=\<lambda\>g<rsub|q><rsup|2>.
    </equation*>

    All factors inside the integral are evaluated at scale <math|s>.
  </hidden>|<\hidden>
    <mytitle|Second correction: insert the first correction>

    Put <math|\<delta\><rsup|<around|[|3|]>>=\<ell\><rsup|<around|[|3|]>>-\<ell\><rsup|<around|[|2|]>>>.
    Subtract the first two Picard equations:

    <\equation*>
      \<delta\><rsup|<around|[|3|]>><rsub|q>=<frac|1|2><big|int><rsub|q><rsup|\<infty\>>\<Pi\><around*|[|<math-up|Ctr><rsub|<around|(|R<rsup|<around|[|2|]>>|)><rsup|2>-<around|(|R<rsup|<around|[|1|]>>|)><rsup|2>>D<rsup|2><around|(|U'+\<ell\><rsup|<around|[|1|]>>|)>+<math-up|Ctr><rsub|<around|(|R<rsup|<around|[|2|]>>|)><rsup|2>>D<rsup|2>\<delta\><rsup|<around|[|2|]>>|]><rsub|s>\<mathd\>s.
    </equation*>

    The marked insertion is now the first correction itself, nonlinear and
    centred:

    <\equation*>
      R<rsup|<around|[|2|]>>-R<rsup|<around|[|1|]>>=-R<rsup|<around|[|2|]>><around|(|\<Theta\><rsup|<around|[|2|]>>+D\<delta\><rsup|<around|[|2|]>>|)>R<rsup|<around|[|1|]>>,
    </equation*>

    where <math|\<Theta\><rsup|<around|[|2|]>>> costs
    <math|\<eta\><rsub|q><around|(|\<Theta\><rsup|<around|[|2|]>>|)>\<lesssim\>g<rsub|q><rsup|2>>
    (above).

    <\equation*>
      <math-ss|S><rsub|4><around|(|\<delta\><rsup|<around|[|3|]>><rsub|q>|)>\<lesssim\><big|int><rsub|q><rsup|\<infty\>>g<rsub|s><rsup|2>s<rsup|1/2>r<rsub|s><rsup|-4>\<mathd\>s=<frac|2|3>\<lambda\><rsup|4>q<rsup|-3/2>.
    </equation*>

    <\myframed>
      Four-leg sizes: seed <math|\<lambda\>g<rsub|q>>, first correction
      <math|\<lambda\>g<rsub|q><rsup|2>>, second correction
      <math|\<lambda\>g<rsub|q><rsup|3>>.
    </myframed>

    <small|For squared propagators use <math|A<rsup|2>-B<rsup|2>=A<around|(|A-B|)>+<around|(|A-B|)>B>;
    no commutativity is assumed.>
  </hidden>|<\hidden>
    <mytitle|Following a spatial distance through a loop>

    At zero field, set <math|W<rsub|q><around|(|T|)>=s<around|(|T|)>+<sqrt|q>v<rsub|1><around|(|T|)>>.
    Mark two external legs and follow their distance around the loop.

    <\itemize>
      <item>A displacement inside a vertex is controlled by that vertex's
      first moment.

      <item>A displacement along a propagator is controlled by its weighted
      <math|L<rsup|1>> or <math|L<rsup|2>> bound. The triangle inequality
      distributes the external distance over these pieces.
    </itemize>

    For <math|p> vertices and <math|p> propagators:

    <\equation*>
      W<rsub|q><around|(|<text|loop>|)>\<lesssim\>q<rsup|-1/2>q<rsup|-<around|(|p-1|)>><big|prod><rsub|i=1><rsup|p>W<rsub|q><around|(|T<rsub|i>|)>.
    </equation*>

    <\myframed>
      The weight <math|<sqrt|q>> makes a distance of order
      <math|q<rsup|-1/2>> dimensionless: the weighted loop has the same scale
      powers as the unweighted loop.
    </myframed>

    <small|Keep within-vertex moments: smoothing the propagators cannot
    supply spatial localisation for an arbitrary nonlocal vertex.>
  </hidden>|<\hidden>
    <mytitle|Integrating preserves the moment weight>

    A loop difference with relative size <math|\<varepsilon\><rsub|s>> has a
    marked vertex or propagator. With <math|\<delta\>\<cal-T\><rsub|s>=\<cal-T\><rsub|s><around|[|M,\<ell\>|]>-\<cal-T\><rsub|s><around|[|M,<wide|\<ell\>|^>|]>>,
    the weighted comparison gives

    <\equation*>
      W<rsub|s><around|(|\<mathD\><rsup|k-1>\<delta\>\<cal-T\><rsub|s><around|(|0|)>|)>\<lesssim\>\<varepsilon\><rsub|s>s<rsup|1/2>r<rsub|s><rsup|-k>.
    </equation*>

    For <math|s\<geqslant\>q>, <math|<sqrt|q>\<leqslant\><sqrt|s>>:

    <\equation*>
      W<rsub|q><around*|(|<big|int><rsub|q><rsup|\<infty\>>A<rsub|s>\<mathd\>s|)>\<leqslant\><big|int><rsub|q><rsup|\<infty\>><around|(|s<around|(|A<rsub|s>|)>+<sqrt|q>v<rsub|1><around|(|A<rsub|s>|)>|)>\<mathd\>s\<leqslant\><big|int><rsub|q><rsup|\<infty\>>W<rsub|s><around|(|A<rsub|s>|)>\<mathd\>s.
    </equation*>

    <\myframed>
      First correction: <math|\<varepsilon\><rsub|s>\<lesssim\>g<rsub|s>>.
      Second correction: <math|\<varepsilon\><rsub|s>\<lesssim\>g<rsub|s><rsup|2>>.
    </myframed>

    <math|\<Pi\>> subtracts only a linear force, so it leaves all
    <math|k\<geqslant\>4> vertices unchanged. The field supremum also
    propagates, since <math|\<cal-B\><rsub|q>\<subseteq\>\<cal-B\><rsub|s>>.
  </hidden>|<\hidden>
    <mytitle|The four-leg moment after two corrections>

    Write <math|T<rsup|<around|[|j|]>><rsub|q>=\<mathD\><rsup|3>\<ell\><rsup|<around|[|j|]>><rsub|q><around|(|0|)>>.
    The seed is the bubble:

    <\equation*>
      W<rsub|q><around|(|T<rsup|<around|[|1|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2><big|int><around|(|1+<sqrt|q>d<around|(|0,z|)>|)>C<rsub|q><around|(|z|)><rsup|2>\<mathd\>z\<lesssim\>\<lambda\><rsup|2>q<rsup|-1/2>.
    </equation*>

    For the increments, integrate the weighted loop bounds:

    <\equation*>
      W<rsub|q><around|(|T<rsup|<around|[|2|]>><rsub|q>-T<rsup|<around|[|1|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|3><big|int><rsub|q><rsup|\<infty\>>s<rsup|-2>\<mathd\>s=\<lambda\><rsup|3>q<rsup|-1>,
    </equation*>

    <\equation*>
      W<rsub|q><around|(|T<rsup|<around|[|3|]>><rsub|q>-T<rsup|<around|[|2|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|4><big|int><rsub|q><rsup|\<infty\>>s<rsup|-5/2>\<mathd\>s=<frac|2|3>\<lambda\><rsup|4>q<rsup|-3/2>.
    </equation*>

    <\myframed>
      For <math|j=1,2,3>, <math|s<around|(|T<rsup|<around|[|j|]>><rsub|q>|)>+<sqrt|q>v<rsub|1><around|(|T<rsup|<around|[|j|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1/2><around|(|1+g<rsub|q>+g<rsub|q><rsup|2>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1/2>>.
    </myframed>

    <small|The <math|10\<to\>8\<to\>6> leg budget supplies the higher
    vertices needed by both comparisons. Thus
    <math|v<rsub|1><around|(|T<rsup|<around|[|j|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1>>.>
  </hidden>|<\hidden>
    <mytitle|The mass>

    \;

    \;

    \;

    Impose the <strong|scalar> part of the linear flow for the final
    approximation <math|M=M<rsup|<around|[|3|]>>=\<mu\>I+\<Theta\><rsup|<around|[|3|]>>>,
    with <math|R<rsup|<around|[|3|]>><rsub|0,q>=<around|(|K<rsub|q>+M<rsub|q>|)><rsup|-1>>:

    <\equation*>
      \<partial\><rsub|q>\<mu\><rsub|q>=\<beta\><rsub|q>=<around|\<langle\>|\<frak-S\><rsup|<around|[|3|]>><rsub|q>|\<rangle\>>,<space|1em>\<frak-S\><rsup|<around|[|3|]>><rsub|q>=-\<lambda\><math-up|Tr><around|(|<around|(|R<rsup|<around|[|3|]>><rsub|0,q>|)><rsup|2>|)>I-<frac|1|2><math-up|Ctr><rsub|<around|(|R<rsup|<around|[|3|]>><rsub|0,q>|)><rsup|2>>D<rsup|3>\<ell\><rsup|<around|[|3|]>><rsub|q><around|(|0|)>,
    </equation*>

    with <math|\<mu\><rsub|q<rsub|\<ast\>>>=b<rsub|\<ast\>>>,
    <math|<around|\||b<rsub|\<ast\>>|\|>\<leqslant\> B<rsub|0>\<lambda\>
    q<rsub|\<ast\>><rsup|1/2>>, at an <strong|auxiliary scale>
    <math|q<rsub|\<ast\>>>. The centred part of <math|\<frak-S\><rsup|<around|[|3|]>>>
    is not imposed. Bare mass <math|\<mu\><rsub|N>=<math-up|lim><rsub|q\<to\>\<infty\>>\<mu\><rsub|q>>:
    an output of the flow. <math|\<beta\>> depends on the whole path
    <math|\<mu\>> through <math|\<ell\><rsup|<around|[|3|]>><around|[|\<mu\>|]>>:
    a fixed point ("Closing the mass path").
  </hidden>|<\hidden>
    <mytitle|The defect after the second correction>

    The bounded-field approximation:

    <\equation*>
      \<cal-F\><rsup|o><rsub|q>=U'+M<rsub|q>y+\<ell\><rsup|<around|[|3|]>><rsub|q>,<space|2em>\<partial\><rsub|q>\<mu\>=<around|\<langle\>|D\<cal-T\><rsub|q><around|[|M,\<ell\><rsup|<around|[|3|]>>|]><around|(|0|)>|\<rangle\>>.
    </equation*>

    The approximations use the preceding level,
    <math|\<partial\><rsub|q>\<ell\><rsup|<around|[|3|]>>=\<Pi\>\<cal-T\><around|[|M<rsup|<around|[|2|]>>,\<ell\><rsup|<around|[|2|]>>|]>>,
    <math|\<partial\><rsub|q>\<Theta\><rsup|<around|[|3|]>>=<around|(|\<frak-S\><rsup|<around|[|2|]>>|)><rsup|\<circ\>>>.
    Hence

    <\equation*>
      \<cal-D\><rsup|o><rsub|q>=\<Pi\><rprime|'><around*|(|\<cal-T\><rsub|q><around|[|M<rsup|<around|[|2|]>>,\<ell\><rsup|<around|[|2|]>>|]>-\<cal-T\><rsub|q><around|[|M<rsup|<around|[|3|]>>,\<ell\><rsup|<around|[|3|]>>|]>|)>:
    </equation*>

    nonlinear part and a <strong|centred linear part>
    <math|\<cal-E\><rsup|c><rsub|q>y>; the scalar part vanishes by the mass
    equation.

    <\myframed>
      One loop with <math|\<delta\><rsup|<around|[|3|]>>> and
      <math|\<Theta\><rsup|<around|[|3|]>>-\<Theta\><rsup|<around|[|2|]>>>
      inserted, <strong|without a scale integration>:

      <\equation*>
        <math-ss|S><rsub|4><around|(|\<cal-D\><rsup|o><rsub|q>|)>\<lesssim\>\<lambda\><rsup|5>q<rsup|-3>,<space|2em><around|\<\|\|\>|\<cal-E\><rsup|c><rsub|q>|\<\|\|\>><rsub|<math-up|row>>\<lesssim\>g<rsub|q><rsup|4>.
      </equation*>
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Why stop after two corrections?>

    The constant and scalar linear jets of the defect vanish, so on the field
    ball

    <\equation*>
      <around|\||b<rsub|s>|\|>\<lesssim\>s<rsup|-1><around|(|<math-ss|S><rsub|4><around|(|\<cal-D\><rsub|s>|)>r<rsub|s><rsup|3>+<around|\<\|\|\>|\<cal-E\><rsup|c><rsub|s>|\<\|\|\>><rsub|<math-up|row>>r<rsub|s>|)>,<space|2em>Z<rsub|q>\<approx\><big|int><rsub|q><rsup|\<infty\>><frac|s|q>b<rsub|s>\<mathd\>s.
    </equation*>

    Both terms have the same size; <math|s/q> is the backward transport. The
    cubic source:

    <center|<data-table|<tformat|<table|<row|<cell|corrections>|<cell|<math|<math-ss|S><rsub|4><around|(|\<cal-D\><rsub|s>|)>>>|<cell|backward
    source integral>>|<row|<cell|0: seed>|<cell|<math|\<lambda\><rsup|3>s<rsup|-2>>>|<cell|<math|\<lambda\><rsup|3/2>q<rsup|-1><big|int><rsub|q><rsup|\<infty\>>s<rsup|-1/2>\<mathd\>s=\<infty\>>>>|<row|<cell|1>|<cell|<math|\<lambda\><rsup|4>s<rsup|-5/2>>>|<cell|<math|\<lambda\><rsup|5/2>q<rsup|-1><big|int><rsub|q><rsup|\<infty\>>s<rsup|-1>\<mathd\>s=\<infty\>>>>|<row|<cell|2>|<cell|<math|\<lambda\><rsup|5>s<rsup|-3>>>|<cell|<math|\<lambda\><rsup|7/2>q<rsup|-1><big|int><rsub|q><rsup|\<infty\>>s<rsup|-3/2>\<mathd\>s=2\<lambda\><rsup|7/2>q<rsup|-3/2>>>>>>>>

    <\myframed>
      One correction is logarithmically borderline. <strong|Two corrections
      beyond the seed suffice> for the bounded-field source estimate.
    </myframed>

    <small|Each step costs two field derivatives: keep the iteration finite.
    Full estimates: Lecture<nbsp>7.>
  </hidden>|<\hidden>
    <mytitle|The corrections in graphs>

    <with|par-mode|center|<\script-output|tikz|default>
      \\documentclass[tikz]{standalone}

      \\usepackage[T1]{fontenc}

      \\usepackage[sc]{mathpazo}

      \\usepackage{amsmath,amssymb}

      \\usetikzlibrary{calc}

      \\tikzset{

      line/.style={thick},

      full/.style={double,double distance=1.3pt,thick},

      vtx/.style={circle,fill=black,inner sep=1.4pt},

      blob/.style={circle,draw,thick,fill=black!15,minimum size=4.2mm,inner
      sep=0pt},

      root/.style={line width=1.1pt,-latex},

      cross/.style={circle,draw,thick,fill=white,inner sep=0pt,minimum
      size=2.4mm,

      path picture={\\draw[thick] (path picture bounding box.north
      west)--(path picture bounding box.south east)

      (path picture bounding box.north east)--(path picture bounding
      box.south west);}},

      lab/.style={font=\\footnotesize,align=center},

      }

      % \\base{x}{y}: outer loop of bare lines G_q at (x,y), cross on top,
      root leg below

      \\newcommand\\loopc[2]{%

      \\draw[line] (#1,#2) circle (.45);

      \\node[cross] at (#1,#2+.45) {};}

      \\newcommand\\base[2]{\\loopc{#1}{#2}\\draw[root]
      (#1,#2-.45)--++(0,-.38);}

      % \\lineins{x}{y}{depth}: a correction inserted on the right line

      \\newcommand\\lineins[3]{%

      \\node[vtx] at (#1+.45,#2) {};

      \\draw[line] (#1+.67,#2) circle (.22);

      \\ifnum#3<gtr>1 \\node[vtx] at (#1+.89,#2) {}; \\draw[line] (#1+.99,#2)
      circle (.1);\\fi

      \\node[vtx] at (#1,#2-.45) {};}

      % \\vtxins{x}{y}{depth}: a correction as the vertex

      \\newcommand\\vtxins[3]{%

      \\draw[line] (#1,#2-.67) circle (.22);

      \\draw[root] (#1,#2-.89)--++(0,-.3);

      \\ifnum#3<gtr>1 \\node[vtx] at (#1+.22,#2-.67) {}; \\draw[line]
      (#1+.32,#2-.67) circle (.1);\\fi}

      \\begin{document}

      \\begin{tikzpicture}[font=\\small]

      % ---------------- (a) one step

      \\node[anchor=west] at (-1.2,1.3) {\\textbf{(a)} one step of the
      iteration};

      \\node at (.35,0) {$\\partial_q\\ell^{[j+1]}_q\\;=\\;-\\tfrac12\\,\\Pi$};

      \\draw[full] (2.35,.08) circle (.5);

      \\node[cross] at (2.35,.58) {};

      \\node[blob] (b) at (2.35,-.42) {};

      \\draw[root] (b.south)--++(0,-.35);

      \\begin{scope}[xshift=3.7cm,font=\\footnotesize]

      \\node[blob,minimum size=3mm] at (0,.55) {}; \\node[anchor=west] at
      (.25,.55) {vertex $D^2(U'+\\ell^{[j]})$};

      \\draw[full] (-.18,.05)--(.18,.05); \\node[anchor=west] at (.25,.05)
      {line $R=(K_q+M^{[j]}_q+D(U'+\\ell^{[j]}))^{-1}$};

      \\node[cross] at (0,-.45) {}; \\node[anchor=west] at (.25,-.45)
      {$\\partial_qK_q=1$, the derivative of the scale};

      \\draw[root] (-.18,-.95)--(.18,-.95); \\node[anchor=west] at (.25,-.95)
      {root: the output slot of the force};

      \\end{scope}

      \\node[lab,anchor=west] at (3.45,-1.45) {$\\ell^{[j]}$, $j\\geq 1$,
      enters twice: in the lines and in the vertex};

      % ---------------- (b) expansion in bare lines

      \\begin{scope}[yshift=-3.4cm]

      \\node[anchor=west] at (-1.2,1.25) {\\textbf{(b)} expanded in bare
      lines $G_q=(K_q+DU')^{-1}$ \\ (single lines; external field legs not
      drawn)};

      % seed

      \\base{0}{0}\\node[vtx] at (0,-.45) {};

      \\node[lab] at (0,-1.6) {seed $\\delta^{[1]}=\\ell^{[1]}$\\\\one loop,
      $\\lambda g_q$};

      % delta 1

      \\node at (1.15,0) {$\\longrightarrow$};

      \\base{2.15}{0}\\lineins{2.15}{0}{1}

      \\node at (3.35,0) {$+$};

      \\loopc{4.2}{0}\\vtxins{4.2}{0}{1}

      \\node[lab] at (3.2,-1.6) {$\\delta^{[2]}$: $\\ell^{[1]}$
      inserted\\\\two loops, $\\lambda g_q^2$};

      % delta 2

      \\node at (5.25,0) {$\\longrightarrow$};

      \\base{6.25}{0}\\lineins{6.25}{0}{2}

      \\node at (7.6,0) {$+$};

      \\loopc{8.45}{0}\\vtxins{8.45}{0}{2}

      \\node[lab] at (7.4,-1.6) {$\\delta^{[3]}$: $\\delta^{[2]}$
      inserted\\\\three loops, $\\lambda g_q^3$};

      % stop

      \\node at (9.75,0) {$\\Longrightarrow$};

      \\base{10.95}{0}\\node[vtx] at (10.95,-.45) {};

      \\node[vtx] at (11.4,0) {}; \\node[blob,fill=black!30,minimum
      size=5.2mm] at (11.66,0) {};

      \\node[font=\\scriptsize] at (11.66,0) {$\\delta^{\\![3]}$};

      \\node[lab] at (11.15,-1.6) {defect $\\mathcal D^{\\rm o}_q$, no
      $q$-integral\\\\$\\mathsf S_4(\\mathcal D^{\\rm
      o}_q)\\lesssim\\lambda^5q^{-3}$};

      \\end{scope}

      \\end{tikzpicture}

      \\end{document}
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    <small|<with|color|dark grey|Each step inserts the previous correction
    into the loop: one more loop, one more <math|g<rsub|q>>. Two steps give a
    defect of order <math|g<rsub|q><rsup|3>>.>>
  </hidden>|<\hidden>
    <mytitle|Closing the mass path>

    \;

    Given a trial <strong|scalar> path <math|\<mu\>>, compute the two steps
    of the iteration, corrections and centred kernels, then

    <\equation*>
      \<beta\><rsub|q><around|[|\<mu\>|]>=<around|\<langle\>|D\<cal-T\><rsub|q><around|[|M<rsup|<around|[|3|]>><around|[|\<mu\>|]>,\<ell\><rsup|<around|[|3|]>><around|[|\<mu\>|]>|]><around|(|0|)>|\<rangle\>>,<space|2em>\<cal-A\><around|[|\<mu\>|]><rsub|q>=b<rsub|\<ast\>>+<big|int><rsub|q<rsub|\<ast\>>><rsup|q>\<beta\><rsub|s><around|[|\<mu\>|]>\<mathd\>s.
    </equation*>

    \;

    \;

    <\myframed>
      The mass is the <strong|fixed point>
      <math|\<mu\>=\<cal-A\><around|[|\<mu\>|]>>: a scalar function of
      <math|q>, not an operator-valued path. It is the only fixed point we
      solve; the centred kernels are explicit outputs of two Picard steps.
    </myframed>

    \;

    \;

    This is an equation for the whole path: the corrections and centred
    kernels at <math|q> depend on <math|\<mu\>> at later scales.
  </hidden>|<\hidden>
    <mytitle|What the fixed point gives>

    \;

    For <math|q<rsub|\<ast\>>> large, uniformly in <math|N>:

    <\itemize>
      <item>a unique scalar path in the ball
      <math|<around|\<\|\|\>|\<mu\>|\<\|\|\>><rsub|\<ast\>>\<leqslant\>R>
      (norm on the next slide), so <math|<around|\||\<mu\><rsub|q>|\|>\<lesssim\>\<lambda\>
      q<rsup|1/2>>;

      <item>centred kernels: <math|v<rsub|1><around|(|\<Theta\><rsup|<around|[|j|]>><rsub|q>|)>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1/2>>;

      <item>increments: <math|\<cal-N\><rsub|k,q><around|(|\<ell\><rsup|<around|[|j|]>>-\<ell\><rsup|<around|[|j-1|]>>|)>\<lesssim\>g<rsub|q><rsup|j>q*r<rsub|q><rsup|2-k>>,
      <math|j=1,2,3>, <math|4\<leqslant\>k\<leqslant\>12-2j>, where

      <\equation*>
        \<cal-N\><rsub|k,q><around|(|f|)>=<math-ss|S><rsub|k><around|(|f|)>+<sqrt|q>v<rsub|1><around|(|D<rsup|k-1>f<around|(|0|)>|)>;
      </equation*>

      <item>defect: <math|<math-ss|S><rsub|4><around|(|\<cal-D\><rsup|o><rsub|q>|)>\<lesssim\>\<lambda\><rsup|5>q<rsup|-3>>,
      <math|<around|\<\|\|\>|\<cal-E\><rsup|c><rsub|q>|\<\|\|\>><rsub|<math-up|row>>\<lesssim\>g<rsub|q><rsup|4>>.
    </itemize>

    <\myframed>
      <strong|Explicit counterterm.> <math|\<mu\><rsub|N>=r-\<lambda\><math-up|Tr>
      C+<frac|2|3>\<lambda\><rsup|2><big|int> C<rsup|3>> is reached by a
      unique boundary value <math|b<rsub|\<ast\>,N>>, and
      <math|b<rsub|\<ast\>,N>> converges as <math|N\<to\>\<infty\>>. The
      measure does not depend on <math|q<rsub|\<ast\>>>.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Two distinct iterations>

    <strong|Inner recursion:> for each trial scalar path <math|\<mu\>>, start
    from the seed and compute exactly two steps.

    <\equation*>
      \<mu\>\<longmapsto\><around|(|\<ell\><rsup|<around|[|2|]>>,\<Theta\><rsup|<around|[|2|]>>|)><around|[|\<mu\>|]>\<longmapsto\><around|(|\<ell\><rsup|<around|[|3|]>>,\<Theta\><rsup|<around|[|3|]>>|)><around|[|\<mu\>|]>\<longmapsto\>\<beta\><around|[|\<mu\>|]>\<longmapsto\>\<cal-A\><around|[|\<mu\>|]>.
    </equation*>

    \;

    <strong|Outer contraction:> start at <math|\<mu\><rsup|<around|(|0|)>><rsub|q>=b<rsub|\<ast\>>>;
    iterate <math|\<mu\><rsup|<around|(|n+1|)>>=\<cal-A\><around|[|\<mu\><rsup|<around|(|n|)>>|]>>,
    in the norm <math|<around|\<\|\|\>|\<mu\>|\<\|\|\>><rsub|\<ast\>>=sup<rsub|q\<geqslant\>q<rsub|\<ast\>>><around|\||\<mu\><rsub|q>|\|>/<around|(|\<lambda\><sqrt|q>|)>>:

    <\equation*>
      <around|\<\|\|\>|\<cal-A\><around|[|\<mu\>|]>-\<cal-A\><around|[|<wide|\<mu\>|^>|]>|\<\|\|\>><rsub|\<ast\>>\<leqslant\>C*g<rsub|q<rsub|\<ast\>>><around|\<\|\|\>|\<mu\>-<wide|\<mu\>|^>|\<\|\|\>><rsub|\<ast\>>.
    </equation*>

    \;

    <\myframed>
      Every outer step recomputes the same two-level force. Its depth stays
      <math|2>, so no additional field derivatives are consumed. At the fixed
      point the scalar linear flow is exact; the defect, nonlinear and
      centred linear, remains.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Mass differences gain a small insertion>

    Compare two scalar paths with the same <math|b<rsub|\<ast\>>>,
    <math|d=<around|\<\|\|\>|\<mu\>-<wide|\<mu\>|^>|\<\|\|\>><rsub|\<ast\>>>;
    <math|\<delta\>> denotes output differences,
    <math|\<delta\>M<rsup|<around|[|j|]>>=\<delta\>\<mu\>I+\<delta\>\<Theta\><rsup|<around|[|j|]>>>.

    <\equation*>
      R<rsup|<around|[|j|]>>-<wide|R|^><rsup|<around|[|j|]>>=-R<rsup|<around|[|j|]>><around|(|\<delta\>M<rsup|<around|[|j|]>>+D\<delta\>\<ell\><rsup|<around|[|j|]>>|)><wide|R|^><rsup|<around|[|j|]>>.
    </equation*>

    The relative scalar and centred insertions obey

    <\equation*>
      <frac|<around|\||\<delta\>\<mu\><rsub|q>|\|>|q>\<leqslant\>g<rsub|q>d,<space|2em>\<eta\><rsub|q><around|(|\<delta\>\<Theta\><rsup|<around|[|j|]>>|)>\<lesssim\>g<rsub|q><rsup|3>d,
    </equation*>

    the latter by integrating the differences of the centred sources from
    infinity. Loop comparisons through the two levels give
    <math|\<cal-N\><rsub|k,q><around|(|\<delta\>\<ell\><rsup|<around|[|j|]>>|)>\<lesssim\>g<rsub|q><rsup|2>q*r<rsub|q><rsup|2-k>d>,
    <math|j=2,3>, <math|4\<leqslant\>k\<leqslant\>12-2j> (the seed does not
    depend on <math|\<mu\>>).

    <\myframed>
      The resulting source differences are

      <\equation*>
        <around|\||\<delta\>\<beta\><rsub|q>|\|>\<lesssim\>\<lambda\><rsup|2>q<rsup|-1>d,<space|2em>v<rsub|1><around|(|\<delta\><around|(|\<frak-S\><rsup|<around|[|j|]>><rsub|q>|)><rsup|\<circ\>>|)>\<lesssim\>\<lambda\><rsup|3>q<rsup|-2>d.
      </equation*>
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Integration leaves the factor <math|g<rsub|q<rsub|\<ast\>>>>>

    \;

    <strong|Scalar update:> the common boundary value <math|b<rsub|\<ast\>>>
    cancels. With <math|x=q/q<rsub|\<ast\>>\<geqslant\>1>,

    <\equation*>
      <around|\||\<delta\>\<cal-A\><around|[|\<mu\>|]><rsub|q>|\|>\<leqslant\>C\<lambda\><rsup|2>d<big|int><rsub|q<rsub|\<ast\>>><rsup|q><frac|\<mathd\>s|s>=C\<lambda\><rsup|2>log
      x*d,
    </equation*>

    <\equation*>
      <frac|<around|\||\<delta\>\<cal-A\><around|[|\<mu\>|]><rsub|q>|\|>|\<lambda\><sqrt|q>>\<leqslant\>C*g<rsub|q<rsub|\<ast\>>><frac|log
      x|<sqrt|x>>d\<leqslant\><frac|2C|\<mathe\>>g<rsub|q<rsub|\<ast\>>>d.
    </equation*>

    The centred kernels are not part of the unknown: their variation is
    already contained in <math|\<delta\>\<beta\>>.

    \;

    <\myframed>
      Contraction constant <math|C*g<rsub|q<rsub|\<ast\>>>>. Fix an invariant
      ball, then choose <math|q<rsub|\<ast\>>> so that
      <math|C*g<rsub|q<rsub|\<ast\>>>\<less\>1/2>, uniformly in <math|N>.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Why the derivative budget stays fixed>

    A loop needs <math|k+2> legs to control a <math|k>-leg vertex. Every
    evaluation of <math|\<cal-A\>> uses the same budget:

    <\equation*>
      \<ell\><rsup|<around|[|1|]>><text|:
      >10\<longrightarrow\>\<ell\><rsup|<around|[|2|]>><text|:
      >8\<longrightarrow\>\<ell\><rsup|<around|[|3|]>><text|:
      >6\<longrightarrow\><text|defect: >4.
    </equation*>

    <small|A <math|k>-leg force vertex has <math|k-1> field derivatives. The
    mass source uses only the four-leg vertex of
    <math|\<ell\><rsup|<around|[|3|]>>>, the centred sources those of
    <math|\<ell\><rsup|<around|[|1|]>>>, <math|\<ell\><rsup|<around|[|2|]>>>.>

    <\itemize>
      <item>A path difference inserts <math|\<delta\>M<rsup|<around|[|j|]>>>
      in a resolvent. It is field-independent:
      <math|D<rsub|y>\<delta\>M<rsup|<around|[|j|]>>=0>. The comparison uses
      the same field-derivative ranges as the construction.

      <item><math|\<cal-A\>> returns a <strong|scalar path>. At the next
      outer step, restart from the original seed and perform the same two
      steps. We do not use the previous <math|\<ell\><rsup|<around|[|3|]>>>
      as a new seed.
    </itemize>

    <\myframed>
      The outer iteration contracts in one fixed space of scalar paths. The
      finite Lipschitz estimates transfer convergence to the corrections;
      their derivative budget does not shrink with the outer index.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Two mass renormalisations>

    \;

    The two divergent self-energies of Appendix<nbsp>A.2 of Lecture<nbsp>5:

    <\itemize>
      <item><strong|Tadpole>: <math|-\<lambda\>*<big|int><rsub|q<rsub|\<ast\>>><rsup|\<infty\>><math-up|Tr>R<rsub|0,q><rsup|2><space|0.17em>\<mathd\>q\<approx\>-\<lambda\><math-up|Tr>C\<sim\>-\<lambda\>N>.
      Wick ordering: <math|<frac|\<lambda\>|12>*\<varphi\><rsup|4>-<frac|1|2>*\<lambda\>*c*\<varphi\><rsup|2>=<frac|\<lambda\>|12>*H<rsub|4>*<around|(|\<varphi\>;c|)>+>const.

      <item><strong|Sunset>: contracting the bubble in
      <math|D<rsup|3>\<ell\><rsup|<around|[|3|]>><around|(|0|)>> with
      <math|R<rsub|0><rsup|2>> and integrating gives the two-loop
      self-energy,

      <\equation*>
        <frac|2|3>*\<lambda\><rsup|2>*<big|int>C<rsup|3>\<approx\><frac|\<lambda\><rsup|2>|24*\<pi\><rsup|2>>*log
        N<space|1em><around|(|=2*\<gamma\><rsub|T>*<text| of Lecture 5>|)>.
      </equation*>
    </itemize>

    Power counting: nothing else. In the flow: the integrals from
    <math|q=\<infty\>> of all vertices with <math|k\<geqslant\>4> legs
    converge; only the constant mode of the linear jet needs a boundary
    condition at a finite scale.

    <math|\<mu\><rsub|q>> at fixed <math|q> has a limit as
    <math|N\<to\>\<infty\>>; <math|\<mu\><rsub|N>> does not. <strong|Never
    interchange the two limits.>
  </hidden>|<\hidden>
    <mytitle|The two mass renormalisations in graphs>

    \;

    <with|par-mode|center|<\script-output|tikz|default>
      \\documentclass[tikz]{standalone}

      \\usepackage[T1]{fontenc}

      \\usepackage[sc]{mathpazo}

      \\usepackage{amsmath,amssymb}

      \\usetikzlibrary{calc,decorations.markings}

      \\tikzset{

      blob/.style={circle,draw,thick,fill=#1,minimum size=7mm,inner sep=0pt},

      blob/.default={black!12},

      vtx/.style={circle,fill=black,inner sep=1.5pt},

      prop/.style={thick},

      full/.style={double,double distance=1.3pt,thick},

      ins/.style={circle,fill=black,inner sep=1.5pt},

      cross/.style={circle,draw,thick,fill=white,inner sep=0pt,minimum
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      west)--(path picture bounding box.south east) (path picture bounding
      box.north east)--(path picture bounding box.south west);}},

      }

      % external legs: \\legs{node}{angles}

      \\newcommand\\legs[2]{\\foreach \\a in {#2}{\\draw[thick]
      (#1.\\a)--++(\\a:2.6mm);}}

      \\newcommand\\rootleg[2]{\\draw[line width=1.4pt,-latex]
      (#1.#2)--++(#2:3.6mm);}

      \\begin{document}

      \\begin{tikzpicture}[font=\\small,baseline]

      \\node at (-.1,0) {$\\partial_q \\mu_q\\ =$};

      \\coordinate (a) at (1.75,-.1);

      \\node[vtx] (v) at (a) {};

      \\draw[thick] (v)--++(200:.5); \\draw[thick] (v)--++(340:.5);

      \\draw[prop] (v) .. controls +(40:1.1) and +(140:1.1) ..
      node[cross,pos=.5]{} (v);

      \\node at (1.75,-.75) {\\scriptsize tadpole, $D^4U=2\\lambda$};

      \\node at (.85,0) {$-\\tfrac12$};

      \\node at (2.75,0) {$-\\ \\tfrac12$};

      \\node[blob,minimum size=9.5mm] (h) at (3.85,-.2) {\\scriptsize
      $D^3\\ell^{[3]}$}; \\legs{h}{210,330}

      \\draw[prop] (h.60) .. controls +(60:1.1) and +(120:1.1) ..
      node[cross,pos=.5]{} (h.120);

      % contains

      \\node at (5.0,0) {$\\ni$};

      \\node[vtx] (a) at (5.75,-.15) {}; \\node[vtx] (b) at (7.15,-.15) {};

      \\draw[prop] (a) to[bend left=40] (b); \\draw[prop] (a) to[bend
      right=40] (b);

      \\draw[thick] (a)--++(180:.4); \\draw[thick] (b)--++(0:.4);

      \\draw[prop] (a) .. controls +(75:1.3) and +(105:1.3) ..
      node[cross,pos=.5]{} (b);

      \\node at (6.45,-.95) {\\scriptsize sunset};

      % integrated

      \\begin{scope}[yshift=-2.2cm]

      \\node at (-.1,0) {$\\mu_N\\ \\approx$};

      \\node at (.95,0) {$-$};

      \\node[vtx] (v) at (1.65,-.1) {};

      \\draw[thick] (v)--++(200:.5); \\draw[thick] (v)--++(340:.5);

      \\draw[prop] (v) .. controls +(40:1.1) and +(140:1.1) .. (v);

      \\node[font=\\scriptsize] at (1.65,-.7)
      {$\\lambda\\,\\mathrm{Tr}\\,C\\sim\\lambda N$};

      \\node at (2.75,0) {$+$};

      \\node[vtx] (a) at (3.6,0) {}; \\node[vtx] (b) at (5.0,0) {};

      \\draw[prop] (a) to[bend left=45] (b); \\draw[prop] (a)--(b);
      \\draw[prop] (a) to[bend right=45] (b);

      \\draw[thick] (a)--++(180:.4); \\draw[thick] (b)--++(0:.4);

      \\node[font=\\scriptsize] at (4.3,-.7) {$\\tfrac23\\lambda^2\\!\\int
      C^3\\sim\\lambda^2\\log N$};

      \\node[font=\\scriptsize,align=left,right] at (5.9,0) {the $q$-integral
      removes\\\\ the cross: $\\int\\! C_q^2\\,dq$ closes a line};

      \\end{scope}

      \\end{tikzpicture}

      \\end{document}
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    <small|<with|color|dark grey|The cross is
    <math|\<partial\><rsub|q>K<rsub|q>=1>, as before; with the two lines
    through it, it gives <math|<around|(|R<rsup|<around|[|3|]>><rsub|0,q>|)><rsup|2>>.
    The bubble in <math|D<rsup|3>\<ell\><rsup|<around|[|3|]>><around|(|0|)>>
    produces the sunset.>>
  </hidden>|<\hidden>
    <vspace*|2.5cm>

    \;

    <cqft-part-title|The flow at large fields>

    <vspace|0.3cm>

    <with|color|dark grey|Lecture<nbsp>6>

    \;
  </hidden>|<\hidden>
    <mytitle|The large-field problem>

    \;

    All bounds hold on <math|\<cal-B\><rsub|q>> only. The free field exceeds
    <math|r<rsub|q>> with small, nonzero probability, and nothing a priori
    controls the interacting forecast. We need
    <math|<wide|\<cal-F\>|\<bar\>><rsub|q>> defined and uniformly coercive
    for all fields.

    <\myframed>
      <strong|Scalar model>

      <\equation*>
        \<ell\><rsup|<around|[|1|]>><rsub|q><around|(|y|)>=\<lambda\>
        y<around|[|<around|(|q+\<lambda\> y<rsup|2>|)><rsup|-1>-q<rsup|-1>|]>
      </equation*>

      Its power series has radius exactly <math|r<rsub|q>> (singularities at
      <math|\<pm\>\<mathi\> r<rsub|q>>). Half of the truncations destroy
      monotonicity. The exact function is harmless,
      <math|<around|\||\<ell\><rsup|<around|[|1|]>><rsub|q><rprime|'>|\|>\<leqslant\><frac|9|8>\<lambda\>/q>.
    </myframed>

    \;

    Perturbative control for <math|<around|\||y|\|>\<lesssim\> r<rsub|q>>;
    outside, a non-perturbative replacement that keeps the structure the
    estimates use: coercivity, carried by the bare cube.

    One site: <math|q*y+<frac|\<lambda\>|3> y<rsup|3>=q*\<Phi\><rsub|q>>,
    noise coefficient <math|H<rsub|q><around|(|y|)>=<around|(|q+\<lambda\>
    y<rsup|2>|)><rsup|-1>\<leqslant\> q<rsup|-1>\<wedge\><around|(|\<lambda\>
    y<rsup|2>|)><rsup|-1>>. Large fields are more elliptic and less noisy.
  </hidden>|<\hidden>
    <mytitle|Clipping>

    \;

    <math|\<tau\>> smooth odd, <math|\<tau\><around|(|u|)>=u> on
    <math|<around|[|-1,1|]>>, constant outside <math|<around|[|-2,2|]>>,
    <math|0\<leqslant\>\<tau\>'\<leqslant\>1>;

    <\equation*>
      \<kappa\><rsub|q><around|(|y|)>=r<rsub|q>\<tau\><around|(|y/r<rsub|q>|)>,<space|1em><wide|\<ell\>|~><rsub|q>=D\<kappa\><rsub|q>\<cdot\>\<ell\><rsup|<around|[|3|]>><rsub|q>\<circ\>\<kappa\><rsub|q>,<space|1em><wide|\<cal-F\>|\<bar\>><rsub|q>=U'+M<rsub|q>y+<wide|\<ell\>|~><rsub|q>.
    </equation*>

    The cubic and linear parts are not clipped, and <math|D\<kappa\><rsub|q>>
    keeps the force a gradient. Globally:
    <math|s<around|(|D<rsup|j-1><wide|\<ell\>|~><rsub|q>|)>\<leqslant\>
    C*g<rsub|q>q*r<rsub|q><rsup|2-j>> (<math|j=2,3,4>),
    <math|s<around|(|D\<partial\><rsub|q><wide|\<ell\>|~><rsub|q>|)>\<leqslant\>
    C*g<rsub|q>>.

    <\myframed>
      <strong|Global coercivity>

      For all <math|y>: <math|c<around|(|K<rsub|q>+\<lambda\>
      y<rsup|2>|)>\<leqslant\> H<rsub|q><around|(|y|)><rsup|-1>\<leqslant\>
      C<around|(|K<rsub|q>+\<lambda\> y<rsup|2>|)>>,

      <\equation*>
        <around|\<\|\|\>|H<rsub|q><around|(|y|)>|\<\|\|\>><rsub|p\<to\>
        p>\<leqslant\><frac|1+C<around|(|g<rsub|q>+g<rsub|q><rsup|2>|)>|q>,<space|1em><math-up|diag>H<rsub|q><around|(|y|)><rsup|2>\<leqslant\>
        C*a<rsub|2><around|(|q|)>.
      </equation*>
    </myframed>

    Neumann series for <math|H=<around|(|I+G<around|(|M+D<wide|\<ell\>|~>|)>|)><rsup|-1>G>,
    <math|G=<around|(|K<rsub|q>+\<lambda\> y<rsup|2>|)><rsup|-1>\<leqslant\>
    C<rsub|q>>.
  </hidden>|<\hidden>
    <mytitle|The complete defect>

    <\with|font-size|0.9>
      \;

      <math|b=b<rsup|<math-up|b>>+b<rsup|<math-up|l>>>,
      <math|b<rsup|<math-up|b>>=-H*<space|0.17em>D*\<kappa\>*<space|0.17em>\<cal-D\><rsup|<math-up|o>>*<around|(|\<kappa\>*y|)>>,
      <math|t<rsub|q><around|(|y|)>=y*<with|math-font|Bbb|1><rsub|<around|\||y|\|>\<gtr\>r<rsub|q>>>.

      <\proposition*>
        <dueto|uniformly in <math|y>, <math|N>, <math|p>>

        <\equation*>
          <around|\<\|\|\>|b<rsup|<math-up|b>><rsub|q>|\<\|\|\>><rsub|p>\<leqslant\>C*\<lambda\><rsup|7/2>*q<rsup|-5/2>,<space|2em><around|\<\|\|\>|b<rsup|<math-up|l>><rsub|q>|\<\|\|\>><rsub|p>\<leqslant\>C*\<lambda\>*q<rsup|-3/2><around*|(|<around|\<\|\|\>|t<rsub|q>|\<\|\|\>><rsub|p>+r<rsub|q><rsup|-1><around|\<\|\|\>|y<rsup|2>*<with|math-font|Bbb|1><rsub|<around|\||y|\|>\<gtr\>r<rsub|q>>|\<\|\|\>><rsub|p>|)>,
        </equation*>

        <\equation*>
          <around|\<\|\|\>|b<rsup|<math-up|l>><rsub|q>|\<\|\|\>><rsub|2><rsup|2>\<leqslant\>C*\<lambda\><rsup|4>*q<rsup|-5><around|\<\|\|\>|t<rsub|q>|\<\|\|\>><rsub|6><rsup|6>.
        </equation*>
      </proposition*>

      Sources of <math|b<rsup|<math-up|l>>> (four groups): the mass term
      <math|M<rprime|'>*y-J*M<rprime|'>*\<kappa\>*y> (<math|J=D*\<kappa\>>,
      <math|M<rprime|'>=\<partial\><rsub|q>*M>; linear part unclipped, flow
      computed at <math|\<kappa\>*y>); the motion of the clip; cubic vertex
      at <math|y> vs correction at <math|\<kappa\>*y>; chain-rule terms with
      <math|\<kappa\><rprime|''>,\<kappa\><rprime|'''>>. All vanish if
      <math|<around|\<\|\|\>|y|\<\|\|\>><rsub|\<infty\>>\<leqslant\>r<rsub|q>>.

      <\myframed>
        <strong|The point>

        <math|b<rsup|<math-up|l>>> is as large as the whole drift, but
        <strong|supported on large fields>. It is controlled by how often the
        actual dynamics visits them: <math|<big|int>q<rsup|-3/2>*\<lambda\><around|\<\|\|\>|t<rsub|q><around|(|Y<rsub|q>|)>|\<\|\|\>><rsub|6><rsup|6><space|0.17em>\<mathd\>q>
        (Lecture<nbsp>7).
      </myframed>
    </with>
  </hidden>|<\hidden>
    <mytitle|The coordinate equation>

    \;

    With <math|U<rprime|'>=<frac|\<lambda\>|3>*y<rsup|3>>, <strong|up to a
    small Lipschitz perturbation, the forecast solves the cubic elliptic
    equation driven by the field at resolution <math|q>>:

    <\equation*>
      K<rsub|q>*\<Phi\><rsub|q>=K<rsub|q>*<around*|(|Y<rsub|q>+\<varrho\><rsub|q><around|(|Y<rsub|q>|)>|)>+<frac|\<lambda\>|3>*Y<rsub|q><rsup|3>,
    </equation*>

    <\equation*>
      \<varrho\><rsub|q><around|(|y|)>=C<rsub|q>*<around*|(|M<rsub|q>*y+<wide|\<ell\>|~><rsub|q><around|(|y|)>|)>,<space|1em><around|\<\|\|\>|D*\<varrho\><rsub|q>|\<\|\|\>><rsub|p\<to\>p>\<leqslant\>C*<around|(|g<rsub|q>+g<rsub|q><rsup|2>|)>.
    </equation*>
  </hidden>|<\hidden>
    <mytitle|The force below the auxiliary scale>

    \;

    The expansion needs <math|g<rsub|q>> small, so
    <math|<wide|\<cal-F\>|\<bar\>><rsub|q>> is built for <math|q\<geqslant\>
    q<rsub|\<ast\>>>. Below <math|q<rsub|\<ast\>>> nothing needs
    constructing: freeze it,

    <\equation*>
      <wide|\<cal-F\>|\<bar\>><rsub|q><around|(|y|)>=<wide|\<cal-F\>|\<bar\>><rsub|q<rsub|\<ast\>>><around|(|y|)>+<around|(|q<rsub|\<ast\>>-q|)>y.
    </equation*>

    Then <math|K<rsub|q>+D<wide|\<cal-F\>|\<bar\>><rsub|q>=K<rsub|q<rsub|\<ast\>>>+D<wide|\<cal-F\>|\<bar\>><rsub|q<rsub|\<ast\>>>>:
    coercivity is automatic, <math|H<rsub|q>=H<rsub|q<rsub|\<ast\>>>>, and
    <math|y<rsub|q><around|(|x|)>=y<rsub|q<rsub|\<ast\>>><around|(|C<rsub|q<rsub|\<ast\>>>K<rsub|q>x|)>>.

    The defect is explicit: <math|\<cal-D\><rsub|s><around|(|y|)>=-y+<frac|1|2><math-up|Ctr><rsub|H<rsub|q<rsub|\<ast\>>><rsup|2>>D<rsup|2><wide|\<cal-F\>|\<bar\>><rsub|q<rsub|\<ast\>>><around|(|y|)>>,
    with <math|<around|\<\|\|\>|\<cal-D\><rsub|s><around|(|y|)>|\<\|\|\>><rsub|p>\<leqslant\>
    C<around|\<\|\|\>|y|\<\|\|\>><rsub|p>>.

    \;

    So the system holds on <math|<around|[|q<rsub|0>,\<infty\>|)>> with
    <math|Y<rsub|q<rsub|0>>=0>: the initial value is deterministic, with no
    large mass. <math|q<rsub|\<ast\>>> stays only in the proofs, where the
    expansion and the closure of the loop need <math|g<rsub|q>> small.
  </hidden>|<\hidden>
    <mytitle|Summary of today>

    \;

    \;

    \;

    At every lattice:

    <\itemize>
      <item><math|\<mu\><rsub|N>>, <math|M<rsub|q>> and a globally coercive
      <math|<wide|\<cal-F\>|\<bar\>><rsub|q>>, all deterministic;

      <item>the exact FBSDE for <math|<around|(|Y,Z|)>>;

      <item>a source whose bounded-field part is integrable (two steps), and
      whose large-field part is controlled by sixth powers of <math|Y> on
      large fields.
    </itemize>

    <with|font-series|bold|Next:> estimates on <math|<around|(|Y,Z|)>>
    uniform in <math|N>.
  </hidden>>
</body>

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