TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/REGIONAL-SOURCE-PARTNER-PROOF.txt Original file SHA-256: 08ce8ea751041c7541d12c4af105f056d87eae6dab013ef42f87394bb1ab3cf9 Inclusive original lines: 1-184 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 | UR.UNIFIED.REGIONAL-SOURCE-PARTNER.01 000002 | 31 October2026. R061. Actual support-resolved partner of the given source. 000003 | 000004 | 1. Original source and decisive question 000005 | Keep the GIVEN free elliptic quarter CAR source, not a chosen new state: 000006 | H=L2(S1,dtheta/(2pi)), e_r=exp(ir theta), alpha=1/4 or3/4, 000007 | beta=4pi, C e_r=c_r e_r, c_r=(1+exp(beta(r+alpha)))^-1, 000008 | X=sqrt(C(1-C)), x_r=(2cosh(beta(r+alpha)/2))^-1. 000009 | In the already established transported positive two-face chart, 000010 | P=[[1-C,X],[X,C]] on H_L directsum H_R is a projection. 000011 | The original Green half-period and graded HS realization supply this P and 000012 | ALL its free CAR words already. R060 supplies the right interval C_I and 000013 | its local modular generator. Neither result is rederived or counted new. 000014 | The new test asks whether the canonical partner of a local right source can 000015 | be identified with ONLY a proper local piece of the original other face. 000016 | Success requires the actual support map and a charged two-insertion word, 000017 | followed by the actual positive partner construction when that shortcut fails. 000018 | This is the existing free annulus, not the full primitive P1/E8 measure. 000019 | 000020 | 2. Entire proper paired-interval class 000021 | For nonempty proper intervals I on R and J on L, let iota_I/iota_J extend 000022 | by zero, P_I=iota_I iota_I*, Q_I=1-P_I, and similarly for J. Both intervals 000023 | have nonempty open complements. Put S=iota_J directsum iota_I and 000024 | R_JI=S* P S. This is the ACTUAL physical two-face regional covariance. 000025 | The existing general projection defect identity gives exactly 000026 | R_JI-R_JI^2=S* P(1-SS*)P S >=0. (1) 000027 | On a right-only vector (0,f), f in L2(I), its quadratic form is 000028 | ||Q_J X iota_I f||^2+||Q_I C iota_I f||^2. (2) 000029 | The FIRST term is strictly positive for EVERY nonzero f. Indeed x_r>0 and 000030 | x_r decays exponentially as exp(-2pi|r|). For any L2 input, Xf extends 000031 | holomorphically to |Im theta|<2pi, hence is real analytic on the whole circle. 000032 | If Q_J Xf=0, its boundary values vanish on the open complement of J. 000033 | Analytic uniqueness implies Xf=0 everywhere; injectivity of X gives f=0. 000034 | The proof applies equally to proper finite unions with an open complement. 000035 | Thus the full regional R_JI is NEVER a projection. This is not inferred 000036 | from a finite Fourier lattice or from truncating a compactly supported field. 000037 | No positive uniform lower gap in (2) is claimed. 000038 | 000039 | The whole global P remains a pure quasifree covariance; its regional 000040 | restriction is mixed as a state on the corresponding regional CAR algebra. 000041 | A one-particle unitary cannot turn R_JI into a pure canonical double, since 000042 | unitary conjugation preserves the projection identity. More generally a 000043 | state-preserving algebra isomorphism preserves purity. Hence the original 000044 | proper left interval ALONE cannot be identified as the entire canonical 000045 | purification partner of the right interval with this state/dictionary. 000046 | No failure of geometric OS reflection, RP, local CAR, or real-time causality 000047 | follows. Pure global states routinely have mixed local restrictions. 000048 | This does not assert that every nonprojector Gaussian restriction is quantum 000049 | entangled with a particular omitted factor in a chosen tensor representation. 000050 | 000051 | 3. Positive partner INSIDE the original whole source 000052 | Partition the SAME physical one-particle source as U directsum E, where 000053 | U=right L2(I), E=ALL left H directsum right Q_I H. 000054 | In this order write P=[[C_I,B*],[B,D]], with 000055 | C_I=iota_I* C iota_I, 000056 | Bf=(X iota_I f, Q_I C iota_I f). 000057 | Projection gives B*B=C_I(1-C_I)=A^2, A=sqrt(C_I(1-C_I)). 000058 | Both endpoint kernels of C_I vanish, as already proved in R060. Thus 000059 | A is injective with dense range. Polar decomposition defines the UNIQUE 000060 | bounded isometry T:U->E satisfying 000061 | B=T A. (3) 000062 | The expression B A^-1 is used only on ran(A), then extended by this bounded 000063 | isometry. No bounded inverse, spectral gap, finite diagonalization, or new 000064 | physical Hilbert space is assumed. T is fixed by the original P and I. 000065 | The remaining projection identity DB=B(1-C_I), and dense ran(A), imply 000066 | DT=T(1-C_I). 000067 | Consequently U directsum T U is REDUCING for the actual P, with pullback 000068 | [[C_I,A],[A,1-C_I]]. (4) 000069 | This is the canonical pure local CAR double as an actual subsource of the 000070 | original whole source. Its mixed/charged words follow from the already 000071 | established quasifree state and the CAR isometry. No new physical state is 000072 | selected, no postselection is performed, and no extra particle is added. 000073 | The range of T cannot be contained in just any proper left J with zero 000074 | right-complement component: otherwise range(B)=range(T A) would too, 000075 | contradicting (2). This is a statement about the WHOLE partner map, not a 000076 | claim that every individual T-vector has a specified minimum outside norm. 000077 | An abstract HS purification was already known; (2)-(4) specify where its 000078 | canonical one-particle partner sits in this ACTUAL spatial environment. 000079 | 000080 | 4. Bounded constructive evaluation, rather than unbounded inversion 000081 | For epsilon>0 use the original source operators to define 000082 | T_epsilon=B(A^2+epsilon)^-1/2. 000083 | These are contractions, and T_epsilon* T_epsilon=A^2/(A^2+epsilon). 000084 | They converge strongly to T because ker(A)=0. On u=A g an explicit bound is 000085 | ||(T-T_epsilon)u||<=sqrt(epsilon)||g||. (5) 000086 | Indeed for a>=0, a(1-a/sqrt(a^2+epsilon))<=sqrt(epsilon). 000087 | Dense ran(A) and contractivity extend strong convergence to all U. 000088 | There is no claimed uniform operator-norm convergence when the spectrum 000089 | approaches0. A fixed bounded CAR word converges in norm under these input 000090 | maps by ||a(f)||=||f|| and the telescoping product bound. On normalized 000091 | inputs in ran(A), errors sum the corresponding bounds(5). Thus original 000092 | full-word queries have a controlled deterministic evaluation route with 000093 | the original covariance, not a search over arbitrary mirror embeddings. 000094 | At finite epsilon the input map is not an exact CAR isometry; it is an 000095 | approximation to the field arguments, not a new exact representation. 000096 | 000097 | 5. Actual charged words with the entire sea retained 000098 | Take I=J=(-pi/2,pi/2), u=sqrt(2)1_I and h=sqrt(2)1_(Icomplement). 000099 | These are normalized orthogonal L2 CAR test vectors; CAR fields are bounded 000100 | for L2 inputs. They are NOT claimed to be smooth current test profiles. 000101 | Their Fourier coefficients are u0=h0=1/sqrt(2), 000102 | u_r=sqrt(2)sin(rpi/2)/(pi r) for r!=0, h_r=-u_r there. 000103 | Set c=, x=, a=sqrt(c(1-c)). The actual charged two-insertion 000104 | word has =x, while the canonical pure one-probe partner 000105 | has cross a. These are individual odd charged insertions with a neutral 000106 | total word; no nonzero isolated odd expectation is asserted. 000107 | The actual physical two-probe covariance in order(R,L) is 000108 | M=[[c,x],[x,1-c]], det(M)=c(1-c)-x^2>0. (6) 000109 | Its complete four-field word is =det(M). The canonical paired 000110 | source has =0. Both states are free; this difference is not 000111 | an interacting Wick defect or a substitute for R054. 000112 | 000113 | The entire occupied sea contribution to c is exactly1/4. Only c_r-1_(r<0) 000114 | is numerically summed, exponentially convergent. X has no sea singularity. 000115 | For |r|>R its norm remainder is bounded by 000116 | [exp(-beta(R+1+alpha)/2)+exp(-beta(R+1-alpha)/2)] 000117 | /(1-exp(-beta/2)); (7) 000118 | the occupation-correction bound uses beta instead of beta/2 throughout. 000119 | The ACTUAL outside witnesses are exactly 000120 | yC==c_0-c, yX==x_0-x. 000121 | Equation(2) for the FULL interval has lower bound |yC|^2+|yX|^2. 000122 | For alpha1/4 this is about0.06211773; for alpha3/4 about0.05964212. 000123 | The finite-probe determinant in(6), about0.18711052 and0.18123952, also 000124 | contains omitted modes INSIDE I. It must not all be called outside leakage. 000125 | All numbers are dimensionless source diagnostics, not physical probabilities 000126 | of signalling, rates or predictions of spacetime dimension. 000127 | 000128 | 6. Explicit partner of this ONE original bounded probe 000129 | There is also an exactly executable specialization of (3) to U0=span(u), 000130 | whose environment is ALL left H plus right u-perp (including other modes 000131 | inside I). In the original face order it is the normalized vector 000132 | t=((X u)/a, (C u-c u)/a). 000133 | It is orthogonal to right u, has norm1, and the actual projection satisfies 000134 | P(right u)=c(right u)+a t, 000135 | P t=a(right u)+(1-c)t. (8) 000136 | Its covariance is therefore exactly[[c,a],[a,1-c]]. The source itself fixes 000137 | this partner and all its words. This probe specialization is not asserted 000138 | to equal T_I u for the WHOLE interval subsystem; the subsystem cuts differ. 000139 | Its left norm weight is /a^2; the other weight is1 minus that. 000140 | The outside-pair weight has the genuine lower bound (|yC|^2+|yX|^2)/a^2. 000141 | Independent4D Fock matrices verify charged words and four-field orderings 000142 | for both actual physical probes and the derived paired subsource. 000143 | Per-Fourier-mode identities verify(8) algebraically for ALL modes. The 000144 | normalization uses entire sea moments and is not a finite-cut projection. 000145 | 000146 | 7. Original quarter and reflection typing retained 000147 | The actual geometric quarter maps the vertical annulus cut to the horizontal 000148 | one, with s'=-s, U_Q C_+ U_Q*=1-C_- and U_Q X_+ U_Q*=X_-. 000149 | Thus diag(U_Q,U_Q)P_+diag(U_Q*,U_Q*) has blocks 000150 | [[C_-,X_-],[X_-,1-C_-]], i.e. the labeled complementary chart, not P_- 000151 | with unchanged labels. The support intervals are transported at the same 000152 | time. For any such ACTUAL source/support intertwiner V, polar uniqueness 000153 | gives T'=V_E T V_U*, in the transported chart. No extra face swap is inserted 000154 | as a new physical operation and no internal quarter time is inferred. 000155 | The OS field reflection includes its original charge/order/Clifford phases. 000156 | The covariance partner (3) is not a declaration that geometric Theta equals 000157 | the modular conjugation of a regional algebra. Full graded HS signs and 000158 | the positive product-Clifford source remain as already proved. 000159 | 000160 | 8. Scientific decision 000161 | There is a consequential connection to the already proved R054. Under its 000162 | EXPLICIT positive normalized nonconstant radial Higgs-mixture premise, the 000163 | actual integrated quark state has fourth-word defect24 Var_mu(m)>0 and is 000164 | not quasifree. By the existing full-projector CAR rigidity theorem, it cannot 000165 | be a marginal of any pure PROJECTOR-covariance CAR double with unchanged 000166 | linear field embedding: such a full state, and every linear CAR restriction, 000167 | would be quasifree. Therefore this free P-based linear partner construction 000168 | cannot replace that integrated interacting source using only its averaged C. 000169 | A general nonGaussian GNS/purification remains possible; the primitive free 000170 | source BEFORE Higgs integration is not excluded. This is a consequence of 000171 | R054 plus the prior rigidity theorem, not new independent physical evidence. 000172 | The original free source now gives a support-resolved canonical partner, 000173 | actual charged words, and a complete exclusion of its proper-other-face-only 000174 | shortcut. This advances the regional field/sewing comparison, not the 000175 | primitive P1 measure, full E8 field dictionary or physical4D reconstruction. 000176 | Keep full word/Q memory, R035/R037/R043/R054-R060, Z0/ref/eta/H0, clocks, 000177 | Yukawa/Flavor/Alpha. No new metric, coupling, current state or Gaussianization. 000178 | Primary comparison:Botero/Reznik,quant-ph/0404176,Eq2 and its finite-mode 000179 | Gaussian pairing theorem, https://arxiv.org/pdf/quant-ph/0404176 . 000180 | Its finite theorem is context, not an infinite-interval proof; Sections2-4 000181 | above supply that bounded-operator proof directly for the given source. 000182 | Verdict:EXACT_SCOPED_REGIONAL_PARTNER_IN_ORIGINAL_FREE_SOURCE; 000183 | PROPER_OTHER_FACE_ONLY_CANONICAL_PARTNER_EXCLUDED;PHYSICAL_ORIGIN_OPEN. 000184 | Experiment contract only,no paper/Lean/ledger or empirical promotion.