TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/QUENCH-SOURCE-PROOF.txt Original file SHA-256: 83e14bc7bb6b17206c28ebcbe9b183436530dad43eacea260527c7dc7e838e8f Inclusive original lines: 1-118 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | L20261102E. The actual unpaired relative echo of the original Higgs family 000002 | 2 November2026. Same conditional circle source, no primitive world selection. 000003 | 000004 | 1. Why this is a separate reader 000005 | R080/D exponentiate a time-smeared quadratic force. Actual time-ordered 000006 | evolution of a changed h is different. Its unpaired phase cannot be replaced 000007 | by positivity of the paired graph norm. The old UV-INSERTION result already 000008 | proves implementability of the full coupled time and a logarithmically 000009 | divergent old-vacuum energy. We reuse those results; we now evaluate the 000010 | entire relative amplitude with its actually stated excitation normalization. 000011 | 000012 | Use the R079 positive occupation C+=expit(2ellh). Its positive-filled/hole 000013 | chart has excitation Hamiltonian K(h)=E-dGamma(h) per singular pair, 000014 | E=sqrt(p^2+s^2). K>=0, its ground energy is0, and its Gibbs covariance isC+. 000015 | It is NOT normal in the opposite negative-filled Fock chart without the 000016 | original charge/face transport. In the negative chart the corresponding 000017 | positive generator is dGamma(h)+E and all relative scalar signs reverse. 000018 | The paired graph contains both orderings; they are not silently exchanged. 000019 | 000020 | 2. Exact original finite pulse 000021 | The unchanged v252 quaternionic radial direction decomposes the actual 000022 | quark36 into three colors of six singular pairs, s the given Yu/Yd singular 000023 | values. Leptons and Majorana remain outside this reducing reader. 000024 | Let h0=p sigma_z+s sigma_x and he=p sigma_z+(1-epsilon)s sigma_x, 000025 | p=n+alpha, alpha1/4or3/4. This is a pulse in the ORIGINAL radial Higgs 000026 | direction, not a new interaction. Its durationT is a test parameter. 000027 | Put E0=sqrt(p^2+s^2), Ee=sqrt(p^2+(1-epsilon)^2s^2). 000028 | The actual interaction-picture one-particle unitary is 000029 | U_p=exp(iT h0)exp(-iT he). 000030 | Linear force centering gives the finite factor 000031 | 000032 | chi_center,p=exp[-i epsilon T tanh(ellE0)s^2/E0] 000033 | det(I-Cp+Cp U_p). (1) 000034 | 000035 | For the whole mode tower the second log coefficient has leading term 000036 | -i epsilon^2 T sigma^2 logN, sigma^2=3Tr(Yu*Yu+Yd*Yd). 000037 | More generally a smooth temporal profile has T replaced by||f||2^2. 000038 | This follows from the original covariance sigma^2 exp(2i|p|(t-s)): 000039 | int_0^infty R_f(tau)exp(iomega tau)dtau=i||f||2^2/omega+O(omega^-3). 000040 | The real second-order coefficient converges byD/R079. The scalar divergence 000041 | is not removed by linear centering. A fixed source-independent reference 000042 | denominator has no externalJ jet and cannot cancel it by definition. 000043 | 000044 | 3. Positive completion from the existing excitation rule 000045 | The actual relative group of the already specified ground-zero Hamiltonians 000046 | is exp(-iT K0)exp(iT Ke). Its exact scalar is fixed: 000047 | 000048 | chi_exc,p=exp[iT(Ee-E0)] det(I-Cp+Cp U_p). (2) 000049 | 000050 | No post hoc counterphase has been selected: (2) is the finite Fock trace 000051 | of these two GIVEN normalized Hamiltonians. The four-state Fock check 000052 | independently confirms both the scalar and its hole-chart sign. 000053 | 000054 | For the pure positive-filled reference define 000055 | d_p=(p^2+(1-epsilon)s^2)/(E0 Ee), delta_p=1-d_p. 000056 | The identity for the whole normalized amplitude is 000057 | 000058 | r_p=1+(delta_p/2)(exp(2iT Ee)-1), 000059 | 0<=delta_p<=epsilon^2 s^2/(2p^2). (3) 000060 | 000061 | The bound follows because t->(p,t)/sqrt(p^2+t^2) is Lipschitz with 000062 | constant1/|p|, and delta is half the squared difference of these unit 000063 | vectors. Hence |r_p-1|<=delta_p, UNIFORM for all realT. 000064 | The Gibbs block differs from the pure reference by trace norm at most 000065 | 4exp(-2ell|p|). Therefore 000066 | |chi_exc,p-1|<=epsilon^2s^2/(2p^2)+4exp(-2ell|p|). 000067 | The full product over ALL p, six pairs and three colors converges absolutely, 000068 | uniformly for epsilon in compact sets and allT. A finite zero factor is 000069 | allowed and gives zero amplitude. No finite nonzero amplitude is declared 000070 | to be unity, and no Gaussian replacement is made. 000071 | 000072 | 4. Full finite histories and actual unitaries 000073 | For every fixed finite list of the existing constant-H Hamiltonians, 000074 | their positive-energy projectors differ from the reference byO(1/|p|). 000075 | Corresponding unit ground vectors can be phased to distanceO(1/|p|). 000076 | Since each excitation group fixes its own ground vector, 000077 | ||(U_j-I)Omega_ref||=O(1/|p|) and 000078 | =O(1/p^2). 000079 | Expanding any finite ordered product, single terms have the latter bound; 000080 | terms with at least two factors have the sameO(1/p^2) by first/last 000081 | Cauchy-Schwarz and normbound2 for intervening factors. Thermal errors are 000082 | exponential. Thus all fixed finite history factors have an absolute 000083 | all-mode product, with the scalars from the SAME K(h) rule. 000084 | Tail estimates also give strong limits of unitary histories and adjoints 000085 | on the finite-excitation domain and hence everywhere. For a single 000086 | Hamiltonian, strong continuity follows by dominated tails and finite-mode 000087 | continuity; the Stone generator is the existing positive excitation one. 000088 | 000089 | The bound depends on the number of history pieces. It does not prove the 000090 | mesh->0 limit of arbitrary time-dependent Hamiltonian backgrounds. No 000091 | arbitrary time derivatives or energy moments are inferred: the old vacuum 000092 | may still lie outside Dom(K_e^(1/2)), exactly as the earlier UV proof says. 000093 | The smooth paired Dyson construction is separately completed in the other 000094 | proof and is not substituted for this unpaired amplitude. 000095 | 000096 | 5. Absolute geometry and reference stay attached 000097 | If the actual original generator includes a geometric vacuum scalar 000098 | Lambda(H), the relative echo must also retain its relative phase 000099 | exp[iT(Lambda(He)-Lambda(H0))] in this stated echo orientation. 000100 | The ground-zero EXCITATION rule does not derive or erase Lambda(H). 000101 | The entire original Zrel[0], reference denominator, eta and measure are 000102 | separate original data. REFERENCE.txt derives their anchored variation 000103 | from the actual full S, rather than assigning a freely chosen counterterm. 000104 | For an externalJ variation the reference is fixed inJ; for a genuine 000105 | background change its own bound field dependence must be differentiated. 000106 | 000107 | 6. Actual diagnostics 000108 | check_dyson_echo.py reads the original full96 constructor, verifies the 000109 | quark36 reduction and the exact color/singular coefficient. Both original 000110 | quarters, mode cutoffs up to2048 and an independent four-state Fock reader 000111 | are checked. All16 controls pass under -B -OO. Raw phase slope is 000112 | -1.1252590613773465 for the declared diagnostic pulse; the positive-filled 000113 | ground-zero echo atN2048 is approximately.5985873-.2841006i. These are 000114 | arithmetic probes, not predicted couplings, measured amplitudes or selected 000115 | physical scales. Equations(2)-(3) carry the all-mode theorem. 000116 | The existence of the UV Hamiltonfamily is reused. New here is its explicit 000117 | complete relative source evaluation and uniform whole-mode amplitude. 000118 | No fundamental TFPT, physical OS/3+1 or ledger completion is claimed.