TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/ACTION-TYPE.txt Original file SHA-256: bea1f96bfb16206f6fb177be2fe489d09072d7825e86d3c4302d9eaadf4f8d41 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 | TYPE OF THE ORIGINAL WHOLE ACTION AND GENERATING FUNCTIONAL 000002 | 2 November2026. relational_logic. Current basis plan83, 000003 | Atlas4092044aa1e726b094e5c095. Canonical repository unchanged. 000004 | 000005 | RESULT 000006 | 000007 | The Original does NOT define S_adm as an already evaluated -log Z. 000008 | S_adm is explicitly the input action under the functional integral. The 000009 | primitive S_prim is an already REFERENCE-SUBTRACTED SPECTRAL action seed. 000010 | W_rel=log Z_rel and the Legendre effective action Gamma_ren are subsequent, 000011 | different objects. The stratified master functional M is different again: 000012 | it contains -log Z_rel plus additional displayed determinant/eta/transfer 000013 | and vacuum terms. No reviewed definition identifies M with S_adm or Gamma_ren. 000014 | 000015 | The word 'normalized' in the original Z_rel definition means REFERENCE 000016 | normalization, NOT necessarily Z_rel[0]=1. The actual normalized whole state 000017 | is generated by Z_rel[J]/Z_rel[0]. All n>=1 derivatives of log Z_rel already 000018 | have the correct connected normalization. Therefore the full original P1 000019 | state must not be replaced by a conditional CAR covariance and then be 000020 | accused of forgetting background weights that the full state already contains. 000021 | 000022 | A concrete constructive bridge is the actual full weight quotient on the 000023 | same original field basis: 000024 | Z_adm/Z_ref = omega_ref(w_adm/w_ref), 000025 | with all source legs and phases retained. When its original reference and 000026 | field map are given, the relative scalar is fixed, not a new arbitrary 000027 | choice. One may NOT reflexively write the exponent as 'S_prim minus S_ref', 000028 | because S_prim has already subtracted its own spectral reference. 000029 | The first unsupported identification is a specific substitution between 000030 | these levels, not a general need to invent a second measure or state. 000031 | 000032 | 1. EXACT ORIGINAL DEFINITIONS 000033 | 000034 | 1A. Primitive relative spectral action, BEFORE later field completion 000035 | 000036 | Source45 Boundary001:302–318 defines 000037 | S_prim[chi_seed] 000038 | =Tr f(D_rel/chi_seed)-Tr f(D_ref/chi_seed)+i pi Delta_eta/2. 000039 | It calls this the primitive Euclidean action / relative spectral seed. 000040 | The next sentence expressly puts gauge fixing, ghosts and carrier-compatible 000041 | bosonic fluctuations LATER, after carrier and admissibility closure. 000042 | The definition is not '-log of an evaluated path integral'. 000043 | Its displayed variable is chi_seed, with D_rel/D_ref, domain and spectral 000044 | profile in the fixed source data; not an explicit full independent field 000045 | integration tuple. 000046 | 000047 | The active SM010:270–280 likewise writes the relative spectral action 000048 | S=Tr f((D_rel+A_Sigma)/chi)-Tr f(D_ref/chi)+i pi Delta_eta/2, 000049 | and says that matter/gauge and gravitational data occur in the same trace. 000050 | The printed symbol S is still an action. The text's subsequent claim of a 000051 | spectral-action MEASURE does not turn the trace into log of that measure's 000052 | already integrated generating functional. 000053 | 000054 | 1B. S_adm is bare relative to the stated quantum integral 000055 | 000056 | Source45 QFT007:5–29 defines 000057 | exp W_k[J] 000058 | :=int_(H_adm) Dvarphi 000059 | exp[-S_adm[varphi]-/2+], 000060 | Gamma_k[phi]=sup_J{-W_k[J]}-/2, 000061 | Gamma_ren=lim_(k->0)Gamma_k. 000062 | This settles the direction. S_adm supplies the integrand; W and Gamma are 000063 | outputs. More decisively QFT007:119–142 states 000064 | Gamma_k[phi]=S_adm[phi]+hbar/2 STr log(S_adm''[phi]+R_k) 000065 | +sum_(l>=2)hbar^l Gamma_l,k^1PI[phi]. 000066 | The propagator and vertices are derivatives of this same input S_adm. 000067 | It would be a different operation to put Gamma_ren or -log Z back under 000068 | that same integral without specifying which fields had already been removed. 000069 | 000070 | S_adm can of course already be an effective action for some previously 000071 | integrated sector. 'Bare' here precisely means input relative to the fields 000072 | still integrated in THIS displayed W_k. It is not a claim that no earlier 000073 | coarse graining ever occurred. The field list and integration order decide 000074 | whether another determinant would be a new contribution or double counting. 000075 | 000076 | 1C. Relative generating functional and actual unit normalization 000077 | 000078 | QFT002:256–270 and QFT008:143–170 define exactly 000079 | Z_rel[J,eta,bar_eta]=Z_adm[J,eta,bar_eta]/Z_ref[0,0,0]. 000080 | The denominator is fixed and source independent. It is not a second 000081 | fluctuating measure mixed into the admissible numerator. 000082 | No equality Z_ref[0]=Z_adm[0] is stated there. Thus generally 000083 | Z_rel[0]=Z_adm[0]/Z_ref[0], 000084 | which need not equal1. The word 'normalized' denotes its declared relative 000085 | normalization. The corresponding normalized whole source is 000086 | q[J,eta,bar_eta]=Z_rel[J,eta,bar_eta]/Z_rel[0,0,0] 000087 | =Z_adm[J,eta,bar_eta]/Z_adm[0,0,0]. 000088 | It must not be silently identified with the unnormalized relative scalar. 000089 | 000090 | QFT005:144–162 sets W_rel=log Z_rel and defines truncated Schwinger 000091 | functions as all n>=1 derivatives with respect to the WHOLE source packet 000092 | mathcalJ=(J,eta,bar_eta). 000093 | The source-independent W_rel[0] cancels from these derivatives. Full 000094 | normalized moments are obtained from exp(W_rel[J]-W_rel[0]) or from all 000095 | connected distributions by their standard partition expansion. OS/CAR uses 000096 | this whole graded family under its original regularity, covariance, RP and 000097 | clustering hypotheses. It does not require another independently chosen 000098 | state once these complete same-algebra data are supplied. 000099 | 000100 | 2. THE ACTUAL FIELD LIST MUST BE READ AT ITS OWN LEVEL 000101 | 000102 | The reviewed exact definitions keep the full W_k field tuple aggregated as 000103 | varphi, with bosonic J and fermionic eta/bar_eta. The same source explicitly 000104 | includes geometric Hodge observables, gauge fixing/BRST and the matter 000105 | fields in its reconstruction, but does not expand varphi into a displayed 000106 | independent integration-coordinate list at QFT007:5–29. 000107 | 000108 | The primitive spectral expression contains D_rel,A_Sigma,chi and gravitational 000109 | data in the trace. It does not state that every one of these is an independent 000110 | variable of Dvarphi, nor that the finite96 labels are96 independent physical 000111 | complex Berezin variables. The actual chiral/selfdual pairing and constraints 000112 | already audited must remain. 000113 | 000114 | The explicit (U,psi,Phi) integral with Wilson gauge action, GW/overlap Dirac, 000115 | Higgs and seam/top terms occurs in tfpt_research_contracts017:153–185 as the 000116 | DEMAND of TFPT4D.LATTICE.ACTION.01. That displayed demanded construction is 000117 | not an already established identity for the earlier abstract Z_adm/Z_ref. 000118 | It must not be inserted as its missing field tuple by typography alone. 000119 | Conversely this does not erase the original abstract admissible source or 000120 | prove absence of other later constructions. It fixes the scope of these 000121 | specific definitions. 000122 | 000123 | The operational Seed Boundary002:103–123 supplies the full omega on its 000124 | quasilocal algebra as declared data. Using that source does not require a 000125 | new external field-coordinate representation. If an additional path-integral 000126 | realization is invoked, however, its coordinate fields and reference must 000127 | represent that SAME omega, not an arbitrary conditional source with a similar 000128 | covariance or action name. 000129 | 000130 | 3. THE TWO DIFFERENT 'ZETA/Z' OBJECTS 000131 | 000132 | The relative-object table Boundary003:1–25 explicitly separates 000133 | (i) mathcalZ_rel(s), whose residue / finite part define E_D and Gamma_D^(4), 000134 | (ii) Z_rel[J,eta,bar_eta], the reference-normalized generating functional. 000135 | Gamma_grav=-6chi_geo² Res_(s=1)mathcalZ_rel+FP_(s=0)mathcalZ_rel is a 000136 | spectral-residue functional. The symbol Gamma does not by itself make it 000137 | an already computed quantum1PI action. FP0 is not the derivative at0 that 000138 | would define a zeta logarithmic determinant. The earlier R044/R045 source 000139 | argument audit already resolved this distinction; it is reused here. 000140 | 000141 | Similarly v259 actually uses the heat test profile f(u)=exp(-u), not the 000142 | fermionic log-partition profile log(1+exp(-u)). Calling both 'KMS weight' 000143 | does not identify their traces. The actual field/operator arguments and 000144 | scale are retained. No automatic fermion-determinant cancellation follows. 000145 | 000146 | 4. WHAT THE PRIMITIVE COMPLETION MASTER DOES AND DOES NOT IDENTIFY 000147 | 000148 | Primitive-completion027:229–381 defines xi=(alpha,chi_geo,delta_ph,rho_vac) 000149 | and 000150 | M(xi)=-log Z_rel^Padm[xi] 000151 | +Re log det_zeta(D_xi²+mu²)+eta_Sigma,Padm(D_xi)/2 000152 | -log det(1-U6,xi)+Gamma_vac(rho_xi). 000153 | The lexicographic barrier restricts this to the stated minimal admissible 000154 | stratum; stationarity then selects the branch parameters. 000155 | 000156 | This M ALREADY contains the generating functional. It is not the same typed 000157 | input as S_prim or S_adm. It is a displayed variational assembly from that 000158 | source and further sector functionals. These exact passages contain no 000159 | identity M=S_adm, M=-log Z_total or M=Gamma_ren. In particular it would be 000160 | invalid to integrate exp(-M) once more over the very same fields merely 000161 | because it is called a master functional. 000162 | 000163 | Its factorization proof invokes commuting projectors and block diagonality. 000164 | Determinants/eta have the stated operator decomposition rules. A full 000165 | partition-function factorization additionally uses the actual product or 000166 | conditional integration architecture; shared-field integration retains its 000167 | mixed correlations as already proved in R026/R054 and the full source 000168 | blocking theorem. This does not invalidate a correctly specified stratified 000169 | variational functional. It prevents replacing the actual full source by an 000170 | unproved independent-sector product. 000171 | 000172 | Does its extra determinant DOUBLE COUNT the original fermion integral? 000173 | That cannot be decided from names. A genuine test must identify the SAME 000174 | kinetic operator, field legs, integration stage, regulator and reference in 000175 | both terms. If Z_rel has already integrated the same fermion determinant, 000176 | adding it a second time as a supposed missing quantum correction would 000177 | change the source. If it belongs to another constrained spectral-response 000178 | block or an explicitly prescribed normalization, that inference does not 000179 | follow. The exact reviewed definition alone does not state the equivalence 000180 | needed to label it a double count. No new determinant is added here. 000181 | 000182 | 5. CONSTRUCTIVE SAME-BASIS BRIDGE, WITH THE REFERENCE COUNTED ONCE 000183 | 000184 | At an actual finite original field regulator let the SAME field basis be nu, 000185 | with original full weights w_A(phi),w_R(phi), all domains, phases, Berezin 000186 | orders and source insertions retained. Write 000187 | Z_A=nu(w_A), Z_R=nu(w_R), 000188 | omega_R(X)=nu(w_R X)/Z_R. 000189 | Then, where the actual ratio and integrals are defined, 000190 | Z_A[J]/Z_R=omega_R((w_A/w_R) exp(J.fields)). (1) 000191 | For signed/complex weights this is the corresponding ordered algebraic 000192 | identity, not an assertion of positive probability. Fermion-source terms 000193 | must preserve their original Grassmann order; zeros require the actual 000194 | unnormalized polynomial rather than division on singular fibers. 000195 | 000196 | Equation(1) reconstructs both the relative zero-source scalar and every 000197 | mixed derivative from the supplied reference functional and actual relative 000198 | weight. No independent extra global measure is needed for this SAME source. 000199 | Equivalently one can use omega_A and the reciprocal ratio. This is the 000200 | proper constructive bridge for the already structured operational source.