TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/LOCAL-SOURCE-RECORD-SPLIT-PROOF.txt Original file SHA-256: db077a0dee8f0f057c75891534afd6bbaf8439a080ac721c891f13f9436a95e8 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 | R065. ORIGINAL SOURCE RECORDS: CANONICAL LOCAL IMPLEMENTATION AND EXTERIOR TEST 000002 | 2026-11-01 | conditional mathematical bridge; physical F1 remains in progress 000003 | 000004 | PURPOSE 000005 | R064 computes original record action but does not identify physical events. 000006 | The next actual test compares the already available original current-module 000007 | map with genuine local operations in that SAME net. No new spatial graph, 000008 | oscillator, Yukawa, primitive state or fitted event distribution is introduced. 000009 | A conditional local implementation is obtained. Its state price is then 000010 | tested immediately; it is not substituted for a fundamental origin theorem. 000011 | 000012 | ORIGINAL DATA REUSED 000013 | UR.COMPILER.CURRENT_DESCENDANT.16, PROOF.txt, supplies an explicit isometry 000014 | E0: C10_D5 vector tensor Sym2(C4_A3) -> H_E8, at full conformal L0 weight2. 000015 | Let Hc=C100 denote its domain. No particular D5 vector is chosen. 000016 | The actual unrefined source has B_z=U_z/sqrt60, U_z=Sym2(r_z), 000017 | r_z=I-zz*/2 for the given sixty Gaussian E8 rays with norm-square4. 000018 | Set u_z=exp(-i pi/4)r_z in SU4. The existing compact SU4 current action 000019 | rho on the GIVEN E8_1 vacuum net satisfies 000020 | 000021 | rho(u_z) E0 = -i E0 (I10 tensor U_z). 000022 | J_0(X) E0 = E0 (I10 tensor dGamma2(X)). 000023 | L0 E0=2E0. 000024 | 000025 | The all-mode current, descendant, glue and net statements are conditional 000026 | original inputs; their presence is not raw P1 or physical-state selection. 000027 | R064 and the existing Gamma/Hodge current identity are reused. The neutral 000028 | A3 Bell ten is not the charged P2 particle ten. Both actual D5 companion and 000029 | all charged extension sectors remain in H_E8. 000030 | 000031 | 1. THE EXISTING NET ALREADY HAS THE REQUIRED SPLIT INCLUSIONS 000032 | In the given lattice E8_1 vacuum net, for every beta>0, 000033 | Tr exp(-beta L0)=Theta_E8(exp(-beta)) product_(n>=1)(1-exp(-beta n))^-8 000034 | is finite. The theta Gaussian lattice sum converges. The logarithm of the 000035 | oscillator product is bounded by 000036 | 8/(1-q) sum_(k>=1) q^k/k, q=exp(-beta)<1. 000037 | Thus the full infinite net, not a finite coordinate algebra, satisfies the 000038 | trace-class criterion. The standard split theorem applies to any strictly 000039 | nested proper intervals closure(I) subset J of this GIVEN circle net. 000040 | There is a nonzero collar. The zero-collar limit is not assumed split. 000041 | Vacuum Reeh-Schlieder gives cyclic/separating Omega for A(I),A(J). 000042 | The relative commutant A(I)' intersect A(J) contains a nonempty collar 000043 | interval algebra; Omega is cyclic there and separating as a vector for A(J). 000044 | Hence (M1=A(I) subset M2=A(J),Omega) is a standard split inclusion. 000045 | 000046 | Primary theorem: Doplicher/Longo, Standard and split inclusions of von 000047 | Neumann algebras, Invent.Math.75(1984)493-536, doi10.1007/BF01388641. 000048 | Precise primary-author exposition read: R.Longo, Lectures on Conformal Nets, 000049 | PartII, https://www.mat.uniroma2.it/longo/Lecture-Notes_files/LN-Part2.pdf, 000050 | sections7.1.2-7.1.6, Theorems7.1.6/7.1.13 and7.3.3. 000051 | These are known operator-algebra results; the TFPT application is below. 000052 | 000053 | 2. CANONICAL LOCALIZER: A FULL ALGEBRA MAP, NOT A COMPRESSION 000054 | For this standard inclusion let W:H->H tensor H be the canonical standard 000055 | implementation of the multiplication isomorphism. With A in M1,B in M2', 000056 | 000057 | W A B W* = A tensor B. 000058 | Psi(X)=W*(X tensor I)W. 000059 | F=Psi(B(H)), M1 subset F subset M2. 000060 | 000061 | Psi is a normal unital injective *-homomorphism, so it preserves all bounded 000062 | products, adjoints, Kraus relations and ordered words. It fixes M1 pointwise. 000063 | F is an INFINITE type-I intermediate factor in the existing type-III local 000064 | algebras. It is not a finite-dimensional replacement of the net or a 000065 | point-local algebra. No arbitrary finite matrix W is used in the checker. 000066 | 000067 | W is fixed by the inclusion and Omega's standard form, not by requiring 000068 | W Omega=Omega tensor Omega. Instead W eta=Omega tensor Omega for the 000069 | canonical product-state vector eta. Ordinarily eta is not Omega. 000070 | Psi is NOT a coordinate change preserving the original spatial assignment: 000071 | for B in M2' its image lies in M1' intersect M2, generally not at B's old 000072 | location. Localizing the entire global algebra inside J does not reconstruct 000073 | a physical spatial net from finite matrices. 000074 | 000075 | For the existing compact SU4 symmetry rho(g), rho(g)Omega=Omega and the 000076 | inclusion is invariant. Standard-form functoriality gives exactly 000077 | W rho(g)=(rho(g) tensor rho(g))W. 000078 | Consequently V(g)=Psi(rho(g)) is a genuine strongly continuous group 000079 | representation in M2, with 000080 | Ad V(g)|M1=Ad rho(g)|M1 and [V(g),M2']=0. 000081 | No projective phase is discarded. For an existing Möbius transformation d 000082 | the canonical construction is covariant: 000083 | V_(dI,dJ)(g)=U(d)V_(I,J)(g)U(d)*. 000084 | This transports the already given geometry and conformal time. It does not 000085 | identify the event parameter of the old .15 path, a physical TFPT clock, the 000086 | quarter thermal flow or a 3+1 spacetime with that conformal time. 000087 | 000088 | 3. ALL ORIGINAL RECORD WORDS HAVE AN EXACT LOCAL REALIZATION 000089 | Define a mathematical encoded source subspace, using the existing exterior 000090 | vacuum and retaining the WHOLE D5 companion: 000091 | E_L psi = W*(E0 psi tensor Omega). 000092 | It is an isometry. Define the bounded local branch operators 000093 | 000094 | Bhat_z = i V(u_z)/sqrt60 in A(J). 000095 | 000096 | They obey sum Bhat_z*Bhat_z=I on the FULL Hilbert space and 000097 | Bhat_z E_L=E_L (I10 tensor B_z). 000098 | By multiplication, this intertwines every finite ordered original record 000099 | word, including all native relative phases and record probabilities. 000100 | It also transports coherent word sums with their actual specified 000101 | coefficients; no repeated projection onto Bell10 is made. 000102 | The record carrier, readout basis and uniform weights are the original 000103 | specified instrument, not selected by the split theorem. 000104 | 000105 | For any given finite source corner observable X=E0 Xc E0*, the observable 000106 | Xhat=Psi(X) is bounded and localized in F subset A(J), and 000107 | E_L* Xhat E_L=Xc. 000108 | Thus R064's same preparation Q_IZ^+/3 and effect Q_XI^+, with any D5 density, 000109 | have the exact LOCAL encoded responses1/3 and1 under the two orders of 000110 | original events0 and13. No new detector matrices are fitted. 000111 | Infinite-dimensional split existence, affiliation and covariance follow 000112 | from the stated primary theorem and the proof; they are NOT a numerical 000113 | finite-matrix certification. 000114 | 000115 | Full E8 charges and center remain important: 000116 | (i V(u_z))^2 = -V(-iI4). 000117 | This is identity on the encoded A3 class2 descendant, but is not identity 000118 | on every E8 glue sector. The local construction does not replace this 000119 | charged action by a fictitious full-source reflection. Glue class2 is not 000120 | the P2 five-slot axiom. The selected/stopped Petersen reader and its frame 000121 | can be transported as their given bounded corner instruments, not silently 000122 | identified with a full unconditioned source or with the deployed cusp clock. 000123 | 000124 | 4. FIRST ORIGIN TEST: THE ORIGINAL EVENT IS NOT A LOCAL OPERATION ON ITS 000125 | UNCHANGED GLOBAL CODE AND EXTERIOR OBSERVABLES 000126 | The same original E0 source occupies a fixed energy grade. Therefore for a 000127 | smooth current smear h with Fourier mean h0, 000128 | E0* J_X(h) E0 = h0 (I10 tensor dGamma2(X)). 000129 | Indeed J_n changes L0 by -n, so every n!=0 has zero compression on grade2. 000130 | Take h supported outside proposed J, h0>0, and retain the actual original 000131 | state E0(sigma_D5 tensor rho10)E0*, rho10=Q_IZ^+/3 as in R064. 000132 | 000133 | Before an event, its exterior J_IZ(h) expectation is2h0. 000134 | After actual original global event13, r13=I-J4/2, it is0. 000135 | After ignoring the label and applying the original C60 it is(2/5)h0. 000136 | The matrix coefficients2,0,2/5 are checked directly and exactly below; 000137 | the current support and all-mode relation are analytic, not a Fourier 000138 | cutoff approximation. They use an actual original probe/state, not a toy 000139 | spatial model. 000140 | 000141 | Every nonselective localized normal CP operation with Kraus operators K_a 000142 | in A(J), sum K_a*K_a=I, fixes each bounded exterior B in A(J)': 000143 | sum K_a* B K_a = B. 000144 | Consequently the exterior expectation MUST remain2h0. The statement for 000145 | the current is obtained by differentiating the exterior current Weyl 000146 | expectation in these finite-energy original source states; no unjustified 000147 | unbounded Kraus rearrangement is required. 000148 | Thus neither literal global event13 nor the global unread C60 is that local 000149 | operation on the unchanged original code/exterior dictionary. 000150 | This excludes that exact identification, not the global clock or P1 theory. 000151 | 000152 | A local branch of constant effect pI on the ENTIRE physical Hilbert space 000153 | satisfies E_branch*(B)=pB and has unchanged conditional exterior values. 000154 | Merely a scalar COMPRESSED effect on the finite code is not enough for this 000155 | conclusion: off-code products can matter. State-dependent postselection 000156 | can steer exterior conditionals and is not excluded; its actual records, 000157 | probabilities, reference and joint source must then be retained. 000158 | 000159 | 5. THE CANONICAL ENCODING DOES NOT SOLVE STATE SELECTION 000160 | The positive construction in section3 passes all original finite record 000161 | tests after the stated encoding, but changes actual exterior data. 000162 | E_L rho E_L* has vacuum restriction on M2', independent of logical rho. 000163 | The original E0 source instead has exterior stress expectation2h0 for any 000164 | outside stress smear h, in vacuum-subtracted conformal conventions. 000165 | The encoded state has exterior stress0. Hence the encoding is not a 000166 | state-preserving presentation of that actual original source. 000167 | 000168 | This is not just the particular choice of an exterior vacuum. Every fixed 000169 | outer state in a product encoding gives the same exterior for all logical 000170 | inputs. Original Q_IZ^+ and Q_IZ^- source inputs are distinguished outside J 000171 | by J_IZ(h) expectations+2h0 and-2h0. A nonselective operation localized in J 000172 | cannot replace this WHOLE original source family by such a fixed-exterior 000173 | encoding while preserving those outside answers. 000174 | 000175 | Canonical eta and Omega are likewise different: eta factorizes M1 and M2' 000176 | with their vacuum marginals, whereas the original vacuum has nonzero 000177 | separated current two-point correlations. No physical reset, product 000178 | preparation or source measure is inferred from the standard-form theorem. 000179 | The actual vacuum restriction to F corresponds to the trace-class faithful 000180 | density Tr_2 W|Omega>