TFPT scientific source excerpt Original repository path: experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/OS-SOURCE-REPAIR.txt Original file SHA-256: 50bc771398e8a363a617c677f503f556c9f2adccf6e40cfcb460ca4eb9f55ba6 Inclusive original lines: 1-200 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 | OS SOURCE REPAIR: FULL MOMENTS, GRADED WORDS, AND THE PRIMITIVE SEAM BOUNDARY 000002 | ================================================================================ 000003 | 000004 | Scope and verdict 000005 | ----------------- 000006 | 000007 | This note repairs one precise mathematical defect in the archived TFPT 000008 | Osterwalder--Schrader argument. It does not construct a new physical source. 000009 | 000010 | PROVED here: 000011 | 000012 | 1. The connected or truncated distributions 000013 | 000014 | kappa_n = delta^n log Z |_{0} 000015 | 000016 | cannot themselves be used as the full Schwinger moment functional in the OS 000017 | Gram form. A two-variable Gaussian already gives a 2 by 2 counterexample. 000018 | 000019 | 2. The correct input is the full normalized moment functional 000020 | 000021 | S_n = Z[0]^{-1} delta^n Z |_{0}, 000022 | 000023 | equivalently the graded moment--cumulant expansion of exp(log Z). 000024 | 000025 | 3. A unital, reflection-intertwining boundary word map which preserves the 000026 | complete original Z-functional on all reflected word products induces an OS 000027 | isometry. This is a sufficient repair theorem. It does not require odd CAR 000028 | fields to belong to the physical observable algebra. 000029 | 000030 | OPEN after the repair: 000031 | 000032 | The cited TFPT source does not construct Z_adm from D_+, B_Sigma, the Calderon 000033 | projector, or the relative spectral/Chern--Simons expressions. It also does not 000034 | provide the state-preserving boundary word map required below. Thus this note 000035 | repairs the logical form of the OS step but does not close the primitive source 000036 | gap. 000037 | 000038 | 000039 | 1. The defect in the cited source 000040 | --------------------------------- 000041 | 000042 | The relevant original passages are: 000043 | 000044 | * documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000045 | 005-subsection-admissible-schwinger-functions-minkowski-loca.tex:144-161 000046 | defines W_rel = log Z_rel and explicitly calls 000047 | 000048 | S_n^T = delta^n W_rel |_{0} 000049 | 000050 | the truncated admissible Schwinger distributions. 000051 | 000052 | * The same file, lines 163-184, then applies the OS axioms directly to 000053 | {S_n^T} and says that the reconstructed theory has these truncated 000054 | distributions as its Schwinger functions. 000055 | 000056 | * documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000057 | 008-section-relative-reflection-positivity-with-fermions.tex:181-196 and 000058 | 217-254 repeats the same typing: it proves distributional properties of the 000059 | truncated family and supplies that family directly to the stated OS/CAR 000060 | reconstruction theorem. 000061 | 000062 | This is not the standard OS input. Reflection positivity is a positivity 000063 | condition on the full moment functional, or equivalently on the normalized 000064 | generating functional Z/Z[0]. It is nonlinear under passage from moments to 000065 | connected cumulants. Positivity of a reflected full Gram matrix is therefore 000066 | not positivity of a matrix formed by replacing its entries with cumulants. 000067 | 000068 | 000069 | 2. Small exact counterexample 000070 | ----------------------------- 000071 | 000072 | Let X be a centered real Gaussian variable smeared at positive Euclidean time, 000073 | and write Y = Theta X for its reflected copy. Choose the covariance 000074 | 000075 | E[X^2] = E[Y^2] = a > 0, 000076 | E[YX] = c > 0, 000077 | 000078 | with 0 < c <= a, so that the elementary covariance matrix 000079 | 000080 | [ a c ] 000081 | [ c a ] 000082 | 000083 | is positive semidefinite. Such a centered Gaussian field is reflection 000084 | positive on the one-dimensional positive-time test space. Explicitly, put 000085 | rho=c/a and expand a polynomial F in Hermite polynomials for the normalized 000086 | variable X/sqrt(a): F=sum_n f_n H_n. The bivariate Gaussian identity gives 000087 | 000088 | E[(Theta F)F] = sum_n n! rho^n |f_n|^2 >= 0. 000089 | 000090 | Thus positivity holds on the whole polynomial algebra, not merely at degree 000091 | one. 000092 | 000093 | Use the positive-time polynomial basis 000094 | 000095 | F_0 = 1, 000096 | F_1 = X^2. 000097 | 000098 | The genuine OS Gram matrix is 000099 | 000100 | G_ij = E[(Theta F_i) F_j]. 000101 | 000102 | Wick's rule gives 000103 | 000104 | E[X^2] = a, 000105 | E[Y^2 X^2] = E[Y^2]E[X^2] + 2E[YX]^2 000106 | = a^2 + 2c^2. 000107 | 000108 | Hence 000109 | 000110 | G_full = [ 1 a ], 000111 | [ a a^2 + 2c^2 ] 000112 | 000113 | and 000114 | 000115 | det G_full = 2c^2 > 0. 000116 | 000117 | Thus the full moment functional is positive on span{1,X^2}. 000118 | 000119 | Now make exactly the erroneous replacement of a full four-point entry by its 000120 | connected four-point cumulant. For a centered Gaussian, 000121 | 000122 | kappa_4(Y,Y,X,X) = 0, 000123 | 000124 | while the two-point cumulant is kappa_2(X,X)=a. The resulting cumulant 000125 | substitution is 000126 | 000127 | G_trunc = [ 1 a ], 000128 | [ a 0 ], 000129 | 000130 | with 000131 | 000132 | det G_trunc = -a^2 < 0. 000133 | 000134 | Therefore an RP Gaussian theory becomes nonpositive if truncated functions are 000135 | inserted as though they were full OS moments. Conversely, separate properties 000136 | of the cumulants cannot establish positivity of the full reflected word 000137 | functional without the moment reconstruction. 000138 | 000139 | This example is only a test of the theorem's mathematical typing. It introduces 000140 | no TFPT field, state, or physical source. 000141 | 000142 | 000143 | 3. Correct bosonic and fermionic reconstruction 000144 | ------------------------------------------------ 000145 | 000146 | Let the mixed source be mathcal J=(J,eta,bar eta), with J even and eta,bar eta 000147 | Grassmann odd. Assume Z[0] is nonzero and define 000148 | 000149 | W[mathcal J] = log Z[mathcal J] - log Z[0]. 000150 | 000151 | Connected distributions are derivatives of W. Full distributions are 000152 | 000153 | S_n(z_1,...,z_n) 000154 | = Z[0]^{-1} 000155 | delta_{z_1} ... delta_{z_n} Z[mathcal J] |_{mathcal J=0}. 000156 | 000157 | Equivalently, 000158 | 000159 | S_n(z_1,...,z_n) 000160 | = sum_{pi in Pi_n} epsilon(pi;z_1,...,z_n) 000161 | product_{B in pi} kappa_{|B|}(z_B). 000162 | 000163 | Here Pi_n is the set of set partitions. Inside each block and among blocks one 000164 | uses one declared canonical order. epsilon(pi;z_1,...,z_n) is the Koszul sign 000165 | of the permutation which moves the odd source derivatives into that block 000166 | order; even source derivatives contribute no sign. This is precisely the 000167 | coefficient identity Z/Z[0] = exp(W). 000168 | 000169 | For a purely fermionic Gaussian source, W has degree two. Thus connected 000170 | cumulants above degree two vanish, but the full even moments do not: 000171 | 000172 | S_{2m}(f_1,...,f_{2m}) 000173 | = Pf( K(f_i,f_j) )_{1 <= i,j <= 2m}. 000174 | 000175 | This Pfaffian is the Grassmann-signed sum over pairings. It is the fermionic 000176 | analogue of the a^2+2c^2 term in the preceding bosonic example. Reconstructing 000177 | with exp W therefore retains the complete Wick family which the direct use of 000178 | {S_n^T} discards. 000179 | 000180 | 000181 | 4. Minimal repaired OS theorem 000182 | ------------------------------ 000183 | 000184 | Let P_+ be the unital graded polynomial word algebra generated by admissible 000185 | Euclidean field insertions with support at positive time. Let Theta be the 000186 | antilinear graded reflection, including the declared order reversal and CAR 000187 | sign convention. Let omega_Z be the normalized full moment functional obtained 000188 | from Z/Z[0]. 000189 | 000190 | Assume: 000191 | 000192 | (R1) omega_Z is defined on every word (Theta F)G with F,G in P_+; 000193 | 000194 | (R2) full reflection positivity holds, 000195 | 000196 | omega_Z((Theta F)F) >= 0 for every F in P_+; 000197 | 000198 | (R3) positive Euclidean translations act on P_+, preserve the OS null space, 000199 | and the induced operators satisfy the symmetric contraction bounds and 000200 | strong continuity required for a self-adjoint positive generator;