TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/FULL-SOURCE-REGULATOR-PROOF.txt
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000001 | ORIGINAL-FULL-SOURCE-REGULATOR-20261030
000002 | F1 continuation, 30 October 2026. Original source consumer, not a new model.
000003 | 
000004 | 1. Original question and the actual caller
000005 | 
000006 | R022/R023 preserve ordered words and the exact original L/J memory process.
000007 | Here follow the ALREADY ORIGINAL complete functional into the same process.
000008 | Do not replace it by a Gaussian, a new Hamiltonian or a free Yukawa family.
000009 | Existing flavor/alpha/clock results remain at their proved scopes.
000010 | 
000011 | Original v42 sections/035, lines3-67, defines
000012 |  Z_k[J]=exp W_k[J]=int_Hadm Dphi exp(-S_adm[phi]-V_k[phi]+J.phi),
000013 |  V_k=phi.R_k.phi/2,
000014 |  Gamma_k=Legendre(W_k)-V_k, and t=log k.
000015 | Lines115-172 explicitly retain higher S_adm derivatives and full1PI vertices.
000016 | The original relative source and master functional additionally retain their
000017 | reference, determinant and eta data. The full moment parent and the need to
000018 | keep Z[0] were ALREADY recovered; they are reused, not claimed as new results.
000019 | 
000020 | Types: P_adm is the original structural admissibility selection. A configuration
000021 | readout pi:phi->p, a positive-time algebra, and the state projection P in R022
000022 | are different objects. [R_k,P_adm]=0 implies no commutation with arbitrary pi/P.
000023 | The68 states of the L/J response are not68 field coordinates of S_adm.
000024 | No such dictionary is inserted. RG t=log k is not physical time.
000025 | 
000026 | 2. Exact whole-source flow, retaining the null-source component
000027 | 
000028 | First consider finitely regulated commuting bosonic fields. Let the original
000029 | measure rho(dphi)=exp(-S_adm[phi])Dphi, its domain and normalization be fixed.
000030 | Assume required integrals and differentiated moments exist; Z_k[0]!=0.
000031 | The same equations are formal coefficientwise when a convergent realization
000032 | is not supplied. An infinite-dimensional measure is not constructed here.
000033 | 
000034 | For constant-in-J symmetric R define D_R=(1/2)sum R_ij d_Ji d_Jj. Then
000035 |   partial_t Z_k = -D_(partial_t R_k) Z_k.
000036 | Proof: differentiating the ORIGINAL integrand inserts
000037 |  -phi.(partial_t R_k).phi/2. A source derivative inserts the same field.
000038 | This does not discard S_adm^(3), S_adm^(4), or any higher moment.
000039 | Constant-coefficient D_R commute, so formally
000040 |   Z_R=exp(-D_R)Z_0,  Z_0=exp(D_R)Z_R.
000041 | An analytic reading of the second formula is NOT justified by finite field
000042 | number alone. Prefer the exact integral identity below to a divergent Taylor
000043 | series. In the continuum convergence/domains and regulator removal must hold.
000044 | 
000045 | Write q_k[J]=Z_k[J]/Z_k[0] and ell_k=log Z_k[0] on a fixed nonzero branch.
000046 | The closed pair equations are
000047 |   partial_t q_k = -D_dotR q_k + a_k q_k,
000048 |   partial_t ell_k = -a_k,
000049 |   a_k=(D_dotR q_k)[0].
000050 | Indeed q_k[0]=1 stays fixed while the second equation retains the discarded
000051 | scalar. For complex/fermionic sources keep Z_k[0] itself and its actual phase;
000052 | a logarithm may have zeros or branch information, including eta contributions.
000053 | 
000054 | For positive finite rho, let mu_R=rho exp(-V_R)/Z_R[0]. Then exactly
000055 |   <O>_0 = <O exp(V_R)>_R / <exp(V_R)>_R,
000056 |   Z_0[0]=Z_R[0] <exp(V_R)>_R.
000057 | No moment-series convergence is needed if the stated integrals are finite.
000058 | For R>=0 and finite rho with integrable |O| the right side is finite: its
000059 | reweighted integral is the original one. This is source reweighting, not a
000060 | performed physical intervention or a proposed measurement probability.
000061 | 
000062 | At a given original configuration readout p=pi(phi), disintegration gives
000063 |   b_R(p)=E_R[exp(V_R)|p],
000064 |   <f(p)>_0=E_R[f(p)b_R(p)]/E_R[b_R(p)].
000065 | For a general O the numerator instead uses E_R[O exp(V_R)|p].
000066 | The scalar denominator is kept. Sequential conditional integration satisfies
000067 | the tower identity. Equivalently retain the entire unnormalized induced kernel
000068 | For an originally specified base factorization rho(dp,dq)=m_P(dp) r(p,dq),
000069 | including its actual Jacobian and normalization, this means
000070 |  K_R(p;J_q)=int_fiber(p) r(p,dq) exp(-V_R(p,q)+J_q.q),
000071 |  Z_R[J_p,J_q]=int m_P(dp) exp(J_p.p) K_R(p;J_q).
000072 | It contains the exterior source legs and every conditional higher cumulant.
000073 | Setting J_q=0 too early generally loses them. With signed/Berezin weights
000074 | use unnormalized ordered fiber integration, not probability conditioning.
000075 | 
000076 | This is the nonlinear source-level counterpart of the memory principle in
000077 | R022. All cross terms of R remain inside the fiber integral. We neither
000078 | identify configuration fibers with Hilbert-state subspaces nor assume the
000079 | original domain is a product without its actual Jacobian/Berezin ordering.
000080 | 
000081 | If only a correction using a proposed weight w(p) is allowed, it reproduces
000082 | ALL bounded retained observables exactly iff b_R(p) is proportional to w(p)
000083 | almost everywhere. This follows from equality of their normalized measures.
000084 | Factorization can make b_R constant even with a nonzero exterior regulator.
000085 | Coupling/exterior R alone does not prove nonclosure in every special source.
000086 | 
000087 | 3. What the full null-source factor contributes to the TFPT readouts
000088 | 
000089 | Set W_norm=log q_k, with the same Legendre convention as the original caller.
000090 | Then exactly Gamma_k=Gamma_norm,k-ell_k. Hence a geometry variation at fixed
000091 | field chart gives delta Gamma_k=delta Gamma_norm,k-delta ell_k. The identity
000092 | holds for the actual relative source too, including its reference variation.
000093 | The source contains determinants and eta separately; these are not replaced
000094 | by a single Gaussian vacuum loop.
000095 | 
000096 | At zero source the mean is phi_bar=delta_J W_k|0, which need NOT be zero.
000097 | The correct equation is Gamma_k[phi_bar]+V_k[phi_bar]=-ell_k. One may write
000098 | Gamma_k[0]=-ell_k only under the additional zero-mean hypothesis. No existing
000099 | Higgs or flavor input is changed to enforce it. For k=0, V_0=0, the value at
000100 | the actual mean field is -ell_0.
000101 | 
000102 | The normalized field functional alone loses an overall background-dependent