TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C57-ORIGINAL-Z-AND-OS-UNIT-REVIEW.txt Original file SHA-256: a034806a62492d8b2283329ded13da4a3c8c4c97067c5a6031044daeb37e0e6e Inclusive original lines: 1-125 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 | C57: ORIGINAL RELATIVE GENERATOR, OS UNIT AND THE EXISTING CAP 000002 | 30 October 2026. Independent original-source and analytic review. 000003 | No original edits, Atlas mutation, new state, new model or checker run. 000004 | 000005 | Verdict 000006 | The original OS construction supplies a genuine two-boundary matrix-element 000007 | caller, but its unit class is translation invariant. Substituting F=G=1 000008 | therefore does NOT identify the original zero-source generator with the 000009 | already proved, generally varying old-sea return amplitude det_F C. That 000010 | amplitude remains valid. A source-specific boundary insertion or boundary 000011 | condition identification, absent from the cited OS call, would be needed 000012 | to insert its actual old sea vector. An equality of unnormalized scalars 000013 | at individual geometric parameters is not excluded by this unit test. 000014 | 000015 | 1. Original definitions, read directly 000016 | Q2 = documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000017 | 002-section-closure-theorems-from-the-admissibility-complex.tex 000018 | 256-264: Z_rel[J,eta,bareta]=Z_adm[J,eta,bareta]/Z_ref[0,0,0]. 000019 | 280-297: the relative reflected form is the admissible form divided by 000020 | this fixed positive denominator. This statement is at fixed source data. 000021 | SHA256 61ecee7727bdec47936bbffc028d3dbac5260a025b6dbcce30124a60d635fc9c 000022 | research-source:e825505d56a04218d72ae44e, registered lines256-310. 000023 | Q8 = same source directory, sections/ 000024 | 008-section-relative-reflection-positivity-with-fermions.tex 000025 | 143-151 repeats the quotient and explicitly calls its denominator SOURCE 000026 | independent. The displayed formula suppresses geometric parameters; 000027 | this suppression is no assertion that the denominator ignores them. 000028 | 217-231 states a conditional OS/CAR reconstruction theorem from the 000029 | admissible Schwinger family, its positivity and the other stated axioms. 000030 | 233-247 uses (F,G)_OS=_rel, the null quotient, completion and 000031 | positive Euclidean translation to construct T_t=exp(-t H_adm). 000032 | SHA256 898aa879cf6a3eb6464d3180274bc2fc028408aeec9676af70a862dad9c3f4dc 000033 | research-source:f62e926f379876bbdf8e994d, lines143-169; 000034 | paper-source:023f8b7e09f96c131ee29d4f, lines215-255. 000035 | 000036 | These actual sources were read through the Atlas. The earlier reflection 000037 | positivity hypotheses are retained; this note does not reprove them. 000038 | 000039 | 2. The first C57 identification, tested on the actual OS unit 000040 | At one fixed original system write omega_rel for the functional used in 000041 | its OS form, and N=omega_rel(1)>0. Translation on its observable algebra 000042 | is unital: tau_t(1)=1. Consequently 000043 | T_t[1]=[1], 000044 | ([1],T_t[1])_OS=N, 000045 | Omega=[1]/sqrt(N), H_adm Omega=0, 000046 | =1. 000047 | If omega_rel is the raw generating functional, N=Z_rel[0]. If the original 000048 | Schwinger functional is normalized separately, N=1. Either convention 000049 | gives a t-independent unit matrix element. The ratio alone must not be 000050 | silently conflated with a separately normalized state convention. 000051 | 000052 | This is a fixed-system time test. Varying the collar geometry ell, hence 000053 | the functional omega_ell and its unit norm N_ell, is a different operation. 000054 | N_ell can vary. The test does not prohibit a separately established 000055 | identity Z_rel[0;ell]=det_F C(ell); it shows that F=G=1 and OS translation 000056 | do not establish that identity or its parameter derivative. 000057 | 000058 | 3. What the existing cap actually computes, without rerunning it 000059 | Reuse FIXED-CAP-STATE.txt:14-120, 000060 | research-source:f6396a970e05d81113556da9, 000061 | SHA256 072ad02ddf3a1298079647e939f7d74c175421d3dd0c622ee097ce39e0c0e451. 000062 | For the given coupled source, its old sea Omega0 is a definite vector in 000063 | the same Fock representation as the coupled sea Omega_h. With coupled 000064 | excitation energy H_exc>=0 and H_exc Omega_h=0, the established identity is 000065 | A_cap(ell)==det_F C(ell). 000066 | For each existing odd singular pair C_p=a_p+(1-a_p)exp(-2ell E_p), 000067 | a_p=(1+|p|/E_p)/2, E_p=sqrt(p^2+m_p^2). 000068 | For ell>0 the exact distinguishing response is 000069 | A_cap'(ell)=-||H_exc^(1/2)exp(-ell H_exc/2)Omega0||^2. 000070 | It is strictly negative if an actual nonzero pair mass excites Omega0. 000071 | The derivative at ell=0 is not asserted: the old vector can fail the 000072 | coupled energy form domain there. No extra conjugate-pair count is added. 000073 | 000074 | Thus Omega0 is generically not the OS unit vacuum for a system whose 000075 | Hamiltonian is identified with this H_exc. At Phi=0 the two seas agree, 000076 | so the unit test alone cannot distinguish their common trivial amplitude. 000077 | C57's constant-pair derivative in u=m^2/k is a valid CAP derivative: 000078 | -partial_u log A_cap|_0 000079 | =sum_(actual p,pair j) k_j(1-exp(-2ell|p|))/(4p^2). 000080 | It has not thereby become a derivative of the original Z_rel or MASTER. 000081 | 000082 | 4. The positive original caller and the actual boundary-vector question 000083 | The original translation construction gives, for its actual positive-time 000084 | admissible insertion classes, 000085 | ([F],T_t[G])_OS=omega_rel(Theta F tau_t G). 000086 | After state normalization this expression is divided by N. This is the 000087 | existing caller for both boundary insertions, reflected products and time; 000088 | there is no need to invent another state for the full operational Seed. 000089 | The cap could enter this caller if an actual source map identifies 000090 | [F0]/||[F0]|| <-> Omega0, H_adm <-> H_exc, 000091 | with the same original reference, domain and boundary conditions. Then 000092 | its proper call is F=G=F0, not automatically the unit. More generally a 000093 | specified source boundary functional can supply that vector without a 000094 | single polynomial representative. Density of OS polynomial vectors alone 000095 | does not identify a particular F0, its locality or its admissibility. 000096 | 000097 | Neither Q2:256-264 nor Q8:143-151,217-255 specifies old Phi=0 APS sea 000098 | boundary data as such an insertion or equates Z_ref[0] with this vector. 000099 | The targeted search in these QFT/boundary sources found no such explicit 000100 | Omega0 cap call. This is the scope of the read, not a corpus-wide absence 000101 | claim. A quotient by a scalar reference partition function is not itself 000102 | that missing boundary-vector identification. 000103 | 000104 | 5. Source and parameter derivatives: executable original consequences 000105 | Keep the original fields/sources, bound reference and all original data. 000106 | For W_rel=log Z_adm[J;xi]-log Z_ref[0;xi], source independence implies 000107 | delta_J^n W_rel=delta_J^n log Z_adm, n>=1. 000108 | The actual normalized generating functional satisfies exactly 000109 | Zhat[J;xi]=Z_rel[J;xi]/Z_rel[0;xi] 000110 | =Z_adm[J;xi]/Z_adm[0;xi]. 000111 | This cancellation holds before a xi derivative of normalized correlators. 000112 | It does not remove the reference from the zero-source free energy: 000113 | partial_xi W_rel[0;xi] 000114 | =partial_xi log Z_adm[0;xi]-partial_xi log Z_ref[0;xi]. 000115 | With fixed integration measure/domain/regulator this is the corresponding 000116 | difference of negative expected action variations. If those data move, 000117 | their original measure, projector/domain, chi, reference and eta responses 000118 | must also be included. No field-independent addition is selected here. 000119 | 000120 | [OMITTED: non-scientific conversation or execution metadata] 000121 | [OMITTED: non-scientific conversation or execution metadata] 000122 | [OMITTED: non-scientific conversation or execution metadata] 000123 | [OMITTED: non-scientific conversation or execution metadata] 000124 | [OMITTED: non-scientific conversation or execution metadata] 000125 | [OMITTED: non-scientific conversation or execution metadata]