TFPT scientific source excerpt Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/PAPER-COLLAR-REFERENCE.txt Original file SHA-256: de7f343439701bc7b86c46802387f0a69f835fceb93f6aa79c6a1375cf2508bc 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 | PAPER SOURCE RECONSTRUCTION: COLLAR, REFERENCE AND LOCAL CAR IMAGE 000002 | ================================================================= 000003 | 000004 | 25 October 2026. Read only audit of the current archived TFPT 4.5 sources. 000005 | Verdict: EXACT CATEGORICAL CORRECTION, ONE CONCRETE EMBEDDING PREMISE OPEN. 000006 | No Atlas, ledger or paper source was changed. 000007 | 000008 | 1. Result 000009 | --------- 000010 | 000011 | The previous requirement for one global unitary W identifying the complete 000012 | CAR source with a separately fixed P1 presentation is stronger than the 000013 | written operational seed definition. 000014 | 000015 | An object of Seedop is specified directly by 000016 | 000017 | S=(A_loc,tau_t,Theta,omega,[u_Sigma],D_coll) 000018 | 000019 | and by the four object conditions in the source. Unitary intertwiners occur 000020 | only as morphisms between complete objects. Therefore a global unitary to a 000021 | pre-existing P1 presentation is required to prove equivalence of two complete 000022 | seed presentations. It is not an object-membership condition. 000023 | 000024 | The source also explicitly defines the physical admissible local net by 000025 | restriction to an invariant admissible Hilbert subspace. This supports a 000026 | local boundary-subsystem route for the even CAR source. The relevant map is a 000027 | state, Euclidean-time and reflection covariant local *-monomorphism, followed 000028 | by the induced isometry of OS/GNS spaces. It need only be unitary onto the 000029 | closed cyclic image generated by the CAR observables. 000030 | 000031 | The archived paper does not construct the missing monomorphism from the 000032 | actual quotient CAR fold into its P1 admissible local net. Nor does it prove 000033 | that the resulting cyclic image carries the required elliptic collar and 000034 | finite relative action. This is the first remaining additional premise. 000035 | 000036 | 2. Exact source statements 000037 | -------------------------- 000038 | 000039 | Operational seed: 000040 | 000041 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000042 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex 000043 | lines 103--143 000044 | 000045 | Lines 105--119 define Seedop objects and then say separately that morphisms 000046 | are unitary intertwiners. Lines 122--143 define the forward functor 000047 | 000048 | C_seed : Seedop -> SBord^adm. 000049 | 000050 | The theorem starts from a complete seed. It is not an inverse reconstruction 000051 | of a seed from a boundary algebra or from a selected field. 000052 | 000053 | The continuation at lines 146--248 essentializes bordisms, defines the defect 000054 | filtration and pulls it back by 000055 | 000056 | D_seed(S)=D(C_seed(S)). 000057 | 000058 | The corollary at lines 227--243 applies only after a defect-minimal nontrivial 000059 | Seedop object has been supplied. It does not prove that C_seed is faithful, 000060 | that every minimal boundary datum has one seed preimage, or that the actual 000061 | CAR fold lies in that preimage. Physical operationality remains an axiom at 000062 | lines 245--248. 000063 | 000064 | Admissible boundary category: 000065 | 000066 | same file, lines 250--306. 000067 | 000068 | Its objects have an elliptic one-sided Dirac collar, reflection positivity, 000069 | finite relative action and nontrivial seam. Its morphisms are again unitary 000070 | intertwiners. Hence the category distinguishes direct admissibility from 000071 | unitary equivalence. 000072 | 000073 | Physical admissible net: 000074 | 000075 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000076 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex 000077 | lines 302--342. 000078 | 000079 | The source says that P_adm A P_adm is only shorthand for matrix elements. 000080 | The actual local net is 000081 | 000082 | A_adm(O)={ A|H_adm : A in A_M(O), 000083 | A H_adm subset H_adm, 000084 | A* H_adm subset H_adm }. 000085 | 000086 | Thus the paper itself uses a reducing local subsystem. It does not require a 000087 | global identification of every operator on the ambient Hilbert space. 000088 | 000089 | One-sided collar: 000090 | 000091 | same file, lines 42--80. 000092 | 000093 | The primitive one-sided datum contains D_+ and B_Sigma in product form 000094 | 000095 | D_+=gamma_n(partial_n+B_Sigma). 000096 | 000097 | The doubled reference/relative split is downstream of this one-sided datum. 000098 | The operational seed definition therefore requires a collar and finite 000099 | relative action, but not a separately displayed D_ref intertwiner as an 000100 | independent seed entry. 000101 | 000102 | Relative reflection positivity: 000103 | 000104 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000105 | 002-section-closure-theorems-from-the-admissibility-complex.tex 000106 | lines 256--316. 000107 | 000108 | Here 000109 | 000110 | Z_rel=Z_adm/Z_ref[0] 000111 | 000112 | and the reference sector is a fixed positive normalizing denominator. On a 000113 | Theta-compatible admissible sector, reflection positivity is inherited from 000114 | the admissible numerator. This does not establish finite spectral relative 000115 | action, but it shows that field-level OS positivity does not require global 000116 | operator equality with an independently presented D_ref. 000117 | 000118 | OS/CAR reconstruction: 000119 | 000120 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000121 | 008-section-relative-reflection-positivity-with-fermions.tex 000122 | lines 215--254. 000123 | 000124 | This theorem reconstructs the Hilbert space, positive-energy semigroup and 000125 | graded local field net from a full admissible Schwinger family satisfying RP, 000126 | temperedness, covariance, graded symmetry and clustering on P_adm. It remains 000127 | conditional on those hypotheses and does not identify the supplied CAR fold 000128 | with P_adm by itself. 000129 | 000130 | 3. Boundary-subsystem proposition 000131 | --------------------------------- 000132 | 000133 | Let A_CAR^+(O) be the bounded even positive-time local CAR algebra of the 000134 | actual fold. Let A_P^+(O) be the positive-time algebra of one complete 000135 | operational seed S_P. Assume a compatible family of injective unital 000136 | *-homomorphisms 000137 | 000138 | iota_O : A_CAR^+(O) -> A_P^+(O) (1) 000139 | 000140 | with the following properties: 000141 | 000142 | iota respects isotony and locality; 000143 | omega_P(iota(A))=omega_CAR(A); 000144 | iota(tau_CAR,t(A))=tau_P,t(iota(A)); 000145 | iota(Theta_CAR(A))=Theta_P(iota(A)); 000146 | iota carries the given seam action/class to the restricted P1 seam. 000147 | 000148 | Write q_CAR and q_P for the OS sesquilinear forms on positive-time histories. 000149 | Define on the OS quotient of the CAR histories 000150 | 000151 | V_iota [F]_CAR := [iota(F)]_P. (2) 000152 | 000153 | State and reflection covariance give 000154 | 000155 | q_P(iota(F),iota(G))=q_CAR(F,G). (3) 000156 | 000157 | Consequently every CAR null vector maps to a P1 null vector, (2) is well 000158 | defined and extends to an isometry 000159 | 000160 | V_iota : H_CAR -> K_CAR, 000161 | K_CAR=closure{iota(F) Omega_P}. (4) 000162 | 000163 | Time covariance gives 000164 | 000165 | V_iota exp(-t H_CAR)=exp(-t H_P)V_iota (5) 000166 | 000167 | on the reconstructed cyclic subsystem. Reflection and the seam action obey 000168 | the analogous restricted intertwining relations. This proof uses the OS 000169 | forms themselves. It does not assume that an arbitrary unbounded field acts 000170 | by left multiplication on the OS quotient. 000171 | 000172 | If K_CAR is invariant under every local image operator and its adjoint, its 000173 | restricted net has exactly the form used in the paper's definition of 000174 | A_adm(O). Equivalently, the orthogonal projection onto K_CAR reduces the 000175 | represented image net. Equations (1)--(5) are sufficient for the local CAR/F8 000176 | boundary mechanism and its state, Euclidean time and reflection histories. 000177 | 000178 | They do not yet make the CAR tuple a complete Seedop object. For that stronger 000179 | claim one must additionally exhibit a collar realization on the same image 000180 | such that: 000181 | 000182 | * K_CAR and the relevant collar domain are reducing for D_coll,P; 000183 | * the restricted operator is still a first-order elliptic collar generator 000184 | with reflection-regular positive side, rather than merely a self-adjoint 000185 | compression; 000186 | * the declared reference, eta/domain data and spectral profile give finite 000187 | relative action on that restricted seed; 000188 | * the nontrivial seam class is the same class carried by (1). 000189 | 000190 | Under these extra hypotheses the restricted data can be checked directly as 000191 | a new Seedop object. This is a direct object proof. The written Seedop 000192 | category has only unitary morphisms, so the inclusion iota itself is not a 000193 | Seedop morphism unless it is onto a complete equivalent object. 000194 | 000195 | 4. Application to the supplied fold 000196 | ------------------------------------ 000197 | 000198 | The first leg is already exact. For 000199 | 000200 | (F f)_j(x)=f((x+2 pi j)/8)/sqrt(8)