TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/CONSTRUCTIVE-CRITERION.txt Original file SHA-256: 60bd3ed474d25b66e94096f621209b3afb86296446ba17bc748947f8b0198f84 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 | H: CONSTRUCTIVE CRITERION AT THE WHOLE ORIGINAL SOURCE CALLER 000002 | 2 November2026. Independent relational_logic review. Canonical unchanged. 000003 | Entry plan87, Atlas e99ef15ca0af3a2090099835. 000004 | 000005 | RESULT 000006 | 000007 | The existing structured P1 source and the WHOLE compatible weighted finite 000008 | sources need no second chosen measure. Exact consistency of those sources 000009 | already suffices to define their unique source functional on the original 000010 | local word algebra. With the original compatible graded RP kernels it also 000011 | constructs their common OS Hilbert space. On actually positive reduced 000012 | configuration measures, standard Borel projective extension gives the 000013 | corresponding measure. None of these statements requires the blocked 000014 | effective weight to retain a bare finite spectral ansatz. 000015 | 000016 | A further condition that can be removed is global reference dominance. 000017 | Uniform integrability of the FINITE DENSITIES against an original reference 000018 | is required if one claims that the limit has one integrable RN density 000019 | against that reference. It is not required for existence of the compatible 000020 | state/projective measure itself. This is different from uniform 000021 | integrability of a FIXED LOCAL WORD, which follows already from exact 000022 | marginality and its one finite-level integrability. 000023 | 000024 | The latter local conclusion and the full weighted marginal equation were 000025 | already proved in R080 and work/source_continuation/SOURCE.txt. They are 000026 | reused here. The constructive addition is the state/OS formulation for the 000027 | actual complex/fermionic caller and the separation of source existence, 000028 | bare-ansatz closure, and a globally dominated density representation. 000029 | 000030 | 1. THE ORIGINAL OBJECT MUST REMAIN THE FULL FUNCTIONAL 000031 | 000032 | Use the actual original finite field/source algebras A_n, their supplied 000033 | inclusion/blocking maps i_nm, all graded orders, the same reflected positive 000034 | time supports and original time maps. Let 000035 | 000036 | Z_n[F] = the complete original unnormalized relative source with insertion F, 000037 | z_n=Z_n[1], omega_n(F)=Z_n[F]/z_n. (1) 000038 | 000039 | The original fixed reference, eta, Pfaffian/Berezin orientation, geometric 000040 | weight, exterior fields and source legs belong to Z_n. Equation(1) does 000041 | not replace the original weight by exp(-S_prim) or square a Pfaffian. 000042 | It assumes the actual nonzero source normalization/branch supplied by the 000043 | caller; it does not declare z_n=1. 000044 | 000045 | For a genuine positive state on the original observable * algebra, the 000046 | necessary and sufficient consistency equation is 000047 | 000048 | omega_m(i_nm F)=omega_n(F), all original F in A_n, m>=n. (2) 000049 | 000050 | Given(2), define omega(F)=omega_n(F) on the algebraic union. This is well 000051 | defined. Any finite linear combination lies in one A_m, so positivity and 000052 | normalization follow there from the original state. This proves uniqueness 000053 | on that union. If the supplied observable algebras are C* algebras with 000054 | their original isometric inclusions, the norm-one positive functional 000055 | extends uniquely to their norm completion. No new CAR model or probability 000056 | law was selected. If P1 already supplies the full omega, its restrictions 000057 | satisfy(2) automatically. The finite weighted integral's identification 000058 | with those restrictions is then a representation identity, not another 000059 | state-existence axiom. 000060 | 000061 | The operational Seed ORIGINAL explicitly includes 000062 | (A_loc,tau_t,Theta,omega,[u_Sigma],D_coll) 000063 | and requires omega RP and clustering; see Boundary002:103–123. The task 000064 | must not replace that input by c3 alone or demand that its state be chosen 000065 | again after constructing its local restrictions. 000066 | 000067 | The actual later v1013 route does not require its finite free-boundary 000068 | Gibbs states to obey exact finite projectivity. Its original equilibrium 000069 | theorem extends those states to the common quasilocal algebra, obtains a 000070 | weak-* convergent subnet, and proves ground/KMS properties using the same 000071 | norm-convergent thermodynamic dynamics. The resulting SINGLE limit state 000072 | has exactly compatible local restrictions. Thus exact(2) is necessary 000073 | and sufficient to preserve a given entire finite marginal family, but is 000074 | not necessary for existence of a thermodynamic state obtained this way. 000075 | Demanding free-boundary Gibbs projectivity would add a false requirement 000076 | to this actual successful source route. Identification with the complete 000077 | original S/Z words uses their actual common caller, not a replacement 000078 | Hamiltonian chosen in this review. 000079 | 000080 | 2. THE GRADED OS VERSION DOES NOT REQUIRE A POSITIVE GHOST MEASURE 000081 | 000082 | For the actual Euclidean source with fermionic signs and complex eta, keep 000083 | the original graded RP convention. Let K_n(F,G) denote its actual reflected 000084 | positive-time word kernel. Require its given positivity and 000085 | 000086 | K_m(i_nm F,i_nm G)=K_n(F,G), (3) 000087 | 000088 | with i_nm preserving the original reflection, grading and positive-time 000089 | supports. Then [F]_n -> [i_nm F]_m is well defined and isometric: equality 000090 | of squared norms carries null vectors to null vectors. Its Hilbert direct 000091 | limit, or equivalently completion of the union modulo the common null 000092 | space, is the common OS Hilbert space. Any already supplied time/operation 000093 | maps intertwining these inclusions act by their original common rule; 000094 | their further continuity/domain properties are the ORIGINAL ones. 000095 | 000096 | Thus(3) is exactly necessary and sufficient for this isometric OS gluing. 000097 | For the WHOLE source require(2) for all its allowed words as well, rather 000098 | than trying to infer every unreflected correlator from one RP norm. 000099 | These statements retain all source coefficients and do not interpret a 000100 | Grassmann or complex eta weight as a positive classical probability. 000101 | They do not infer clustering from arbitrary RP finite kernels. When the 000102 | family is identified with the declared P1 source, its already given 000103 | clustering is retained by that identity, not added as a new selection law. 000104 | 000105 | 3. POSITIVE REDUCED CONFIGURATION VERSION: EXACT NECESSARY/SUFFICIENT TEST 000106 | 000107 | Only where the original reduction actually gives positive measures, write 000108 | mu_n=w_n nu_n/Z_n, 0L)dmu_m 000132 | =int |F|1_(|F|>L)dmu_n ->0, (5) 000133 | 000134 | uniformly over m>=n. Thus those local moments and tails are already closed. 000135 | No new generic moment/UI obligation is imposed for the SAME words. New 000136 | rescaled UV fields, a stronger smooth-support assertion, or infinitely 000137 | supported noncylinder insertions are different objects. This review does 000138 | not quietly substitute them for the original local-source task. 000139 | 000140 | 4. WHY A GLOBAL LIMIT DENSITY IS A STRONGER OPTIONAL REPRESENTATION 000141 | 000142 | Suppose the actually positive original reference family nu_n is also 000143 | consistent, and let r_n=dmu_n/dnu_n be the actual finite normalized density. 000144 | On its projective reference space, with F_n the original n-level sigma 000145 | algebra, (4) is exactly 000146 | 000147 | E_nu[r_m|F_n]=r_n, int r_n dnu=1. (6) 000148 | 000149 | This is a positive martingale. The target projective mu already exists by 000150 | the previous section. If one additionally demands mu=r_infinity nu on 000151 | F_infinity=sigma(union F_n), then precisely 000152 | 000153 | mu<r in L1 with mass1, so integration on every 000158 | cylinder identifies mu=r nu, and uniqueness extends equality to the 000159 | generated sigma algebra. Without UI the projective mu remains valid; it 000160 | need not be dominated by nu. Taking the pointwise martingale limit as the 000161 | WHOLE source could discard a singular component. That discard would be 000162 | the incorrect step, not evidence that no source exists. 000163 | 000164 | Primary confirmation: Eisenberg and Gan Shixin, Positive Martingales and 000165 | Their Induced Measures, Proc.AMS89(1983)680–684, General Theorem,p680, 000166 | equates UI/L1 convergence with extension absolutely continuous to the 000167 | reference; p681 separately notes existence of the standard projective 000168 | extension by Kolmogorov. Author-uploaded full text checked: 000169 | https://www.researchgate.net/publication/265318505_Positive_Martingales_and_Their_Induced_Measures 000170 | DOI10.1090/S0002-9939-1983-0718996-7. 000171 | For projective gluing itself, Kozen, Kolmogorov Extension, Martingale 000172 | Convergence, and Compositionality of Processes, LICS2016, section3, 000173 | Theorem1 and section3.2, gives the extension on the stated Radon spaces: 000174 | https://www.cs.cornell.edu/~kozen/Papers/KolmogorovDoob.pdf . 000175 | 000176 | 5. REPAIR THE OVERSTRONG BARE-ANSATZ REQUIREMENT AT THE ACTUAL CALLER 000177 | 000178 | Metrology005:137–178 prints 000179 | dmu_n proportional exp[-Tr f_star(D_Vn^2/Lambda^2)] dnu_Vn 000180 | and asserts projectivity using compatible blocked D_Vn and f_star. 000181 | The exact meaning required of 'compatible action' is(4), not merely 000182 | operator restriction. This clarification does not ask for another measure. 000183 | 000184 | The already present R080 full fiber kernel is the constructive route: 000185 | integrate the original eliminated fields with ALL their exterior source 000186 | legs, reference factors, eta, Jacobian and graded order. Its tower rule 000187 | composes consecutive integrations. In the positive subcase this defines 000188 | w_eff,n=E_nu_m[w_m|pi_mn], with its actual scalar. (8) 000189 | In the complex/Berezin case use its ordered unnormalized fiber integral. 000190 | 000191 | Equation(8) automatically retains induced higher vertices, memory and 000192 | conditional cumulants. It need not be expressible as the SAME small 000193 | spectral formula on the retained D_Vn. Demanding that additional ansatz 000194 | closure is stronger than constructing the full original source. If the 000195 | Original defines its blocked action by this actual exact fiber call, 000196 | (2)/(4) follows by the tower rule and closes the local-source continuation. 000197 | If it claims equality also to the compact displayed spectral ansatz, 000198 | that extra equality is a representation/readout theorem to check on that 000199 | same caller. Its possible failure would not reopen the given P1 state. 000200 |