TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/ZERO-FREE-JOINT.txt Original file SHA-256: 105feb74ea65381de2d66f6f45a3436afa00853072da50f7c4924877dfae3643 Inclusive original lines: 1-180 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 | INDEPENDENT REVIEW: NO FLAT FINITE TENSOR FACTOR IN THE WHOLE FREE SOURCE 000002 | 3 November 2026. joint_origin. Analytic proof. No execution or new model. 000003 | 000004 | VERDICT 000005 | 000006 | The proposed obstruction is correct. It excludes an exact flat rank-one 000007 | ground-state tensor factor of dimension d>2 and gap Delta>0 in the whole 000008 | unprojected positive heat source H=dGamma(E), E=|h|, even after an arbitrary 000009 | unitary field identification. It removes the linear-CAR restriction of the 000010 | earlier R105 application, for this stronger exact tensor-factor target. 000011 | 000012 | The target here is actual unitary equivalence of the oriented physical heat 000013 | operator Q(s)=exp(-sH), with its original input/output pairing and sewing. 000014 | It is not unitary equivalence of the doubled HS vector or its Liouvillian. 000015 | It is not merely the presence of a flat parent term in an interacting H. 000016 | 000017 | 1. EXACT HYPOTHESES 000018 | 000019 | Let E>=0 be selfadjoint on the original excitation one-particle space, 000020 | and assume Tr exp(-xE)0. Let 000021 | H=dGamma(E) on its full fermionic Fock space. This is exactly the given 000022 | Clifford source in SEED-FIELD.txt, not a newly chosen Hamiltonian. 000023 | 000024 | The hypothesis implies finite m=dim Ker E and a discrete positive spectrum 000025 | epsilon_j>0 with finite multiplicities, listed with those multiplicities. 000026 | There are only finitely many epsilon_j in every bounded interval. In 000027 | particular no infinite collection of zero or near-zero modes is hidden. 000028 | 000029 | Suppose for contradiction there is a unitary W onto C^d tensor K_R with 000030 | 000031 | W H W^* = Delta(I-P) tensor I + I tensor H_R, (1) 000032 | 000033 | where P is rank one, d>2, Delta>0, and H_R is selfadjoint. Equality means 000034 | the actual selfadjoint generator/closed form, or equivalently its whole 000035 | heat semigroup; it is not a formal equality of a selected matrix block. 000036 | 000037 | Since H>=0 and Delta(I-P) has a zero eigenvector, (1) forces H_R>=0. 000038 | This can be read by restriction to that reducing zero-factor subspace, 000039 | or by the joint spectral calculus; no subtraction of unbounded operators 000040 | on an unproved common domain is needed. 000041 | 000042 | 2. THE FULL FREE HEAT TRACE IS HOLOMORPHIC AND ZERO-FREE 000043 | 000044 | For Re z>0, occupation-number summation gives 000045 | 000046 | Z_free(z)=Tr exp(-zH) 000047 | =2^m product_j (1+exp(-z epsilon_j)). (2) 000048 | 000049 | The absolute occupation sum is Z_free(Re z)0 and are bounded by 000051 | 2^m exp(sum_j exp(-x epsilon_j)). Thus the full trace exists, and ordinary 000052 | absolute summation proves (2) also for complex z. 000053 | 000054 | On a compact subset of the right half-plane choose a>0 with Re z>=a. 000055 | The sum of |exp(-z epsilon_j)| is uniformly bounded by the convergent 000056 | sum exp(-a epsilon_j). Split off finitely many terms so the remaining 000057 | |exp(-z epsilon_j)|<=1/2. The analytic branch of log(1+w) on |w|<1 has 000058 | |log(1+w)|<=2|w| on this tail. Its logarithmic series therefore converges 000059 | normally, and its exponential is holomorphic and never zero. Every finite 000060 | prefix factor is also never zero, since |exp(-z epsilon_j)|<1. 000061 | 000062 | Consequently 000063 | 000064 | Z_free(z) is holomorphic and nonzero for Re z>0. (3) 000065 | 000066 | The finite zero modes contribute the nonzero constant 2^m. This is a proof 000067 | of nonvanishing of the INFINITE product; nonvanishing of each finite factor 000068 | alone would not have been sufficient. 000069 | 000070 | 3. THE REST TRACE CANNOT SUPPLY A COMPENSATING POLE 000071 | 000072 | At every real x>0 the supposed tensor decomposition gives 000073 | 000074 | Z_free(x)=[1+(d-1)exp(-x Delta)] Tr exp(-xH_R). 000075 | 000076 | Hence exp(-xH_R) is trace class for every x>0. Equivalently its trace-class 000077 | property follows by compression to the finite factor's zero eigenvector. 000078 | This conclusion does not assume a rest model or fit a rest density. 000079 | 000080 | For Re z>0 the rest trace 000081 | 000082 | Z_R(z)=Tr exp(-zH_R) 000083 | 000084 | is holomorphic. One may use its absolutely normally convergent eigenvalue 000085 | sum: on Re z>=a its terms are dominated by Tr exp(-aH_R); derivatives are 000086 | controlled on smaller compact sets by Tr exp(-(a/2)H_R). There are no poles 000087 | anywhere in this half-plane. The zero operator rest Hilbert space is excluded 000088 | by the nonzero original Fock space/unitary W. 000089 | 000090 | Trace invariance and tensor spectral calculus therefore give 000091 | 000092 | Z_free(z)=[1+(d-1)exp(-z Delta)] Z_R(z) (4) 000093 | 000094 | throughout Re z>0, either directly by absolute summation or by the identity 000095 | theorem from the real positive axis. No complex physical temperature is 000096 | postulated; complex z is a mathematical probe of the same real generator. 000097 | 000098 | At 000099 | 000100 | z_*=[log(d-1)+i pi]/Delta, 000101 | 000102 | one has Re z_*>0 and 1+(d-1)exp(-Delta z_*)=0. Since Z_R(z_*) is finite, 000103 | (4) gives Z_free(z_*)=0, contradicting (3). This proves the claim. 000104 | 000105 | For original DET16, d=65536 and Delta=2, so the same explicit point is 000106 | z_*=[log(65535)+i pi]/2. The phase in the cat projector does not change its 000107 | rank-one/flat-excited spectrum and has no effect on this argument. 000108 | 000109 | 4. A SINGLE EXACT POSITIVE STEP ALREADY SUFFICES 000110 | 000111 | Suppose at the actual fixed a>0 one has the exact operator equality 000112 | 000113 | W exp(-aH) W^*=exp(-a Delta(I-P)) tensor R. (5) 000114 | 000115 | The original heat is positive and injective. Its ground norm is one. 000116 | Compression to the rank-one ground factor implies R is positive, 000117 | injective and bounded with norm one. Define its actual selfadjoint spectral 000118 | logarithm H_R=-(1/a)log R>=0. No bounded inverse of R is required. 000119 | The unique selfadjoint logarithm of (5) gives (1). Positive fractional 000120 | powers then give the same full semigroup, with the SAME W. 000121 | 000122 | If the single step is normalized, its exact faithful density factorization 000123 | likewise yields the generator up to a real scalar by its selfadjoint log. 000124 | That scalar can be absorbed into H_R, or retained as a global offset. 000125 | The real heat traces of the rest at all positive times follow from the 000126 | already given free source, not from assuming arbitrary powers of a generic 000127 | trace-class operator are trace class. No new beta family is fitted. 000128 | 000129 | 5. SCOPE AND FIRST PLACES THE ARGUMENT WOULD NOT APPLY 000130 | 000131 | An energy offset multiplies Z by exp(-cz); positive time rescaling replaces 000132 | epsilon_j by a positive multiple. Neither produces a right-half-plane zero 000133 | or cancels the contradiction. The scalar factors allowed by one autonomous 000134 | heat semigroup are not arbitrary independently reset source normalizations. 000135 | 000136 | d>2 and Delta>0 are essential. At d=2 the flat factor is an ordinary one-mode 000137 | fermion and its zeros lie on Re z=0. At Delta=0 there is no positive gap. 000138 | For a more general two-level factor the same argument applies whenever the 000139 | excited multiplicity is greater than its ground multiplicity; this is not 000140 | needed for the original rank-one DET16 application. 000141 | 000142 | The result concerns the FULL unprojected trace. An actual Gauss, parity, 000143 | charge or other sector restriction replaces that trace by a projected trace 000144 | and need not retain the product (2). We do not assert that the actual 000145 | projected original source is zero-free. Keeping all components of a source 000146 | is a hypothesis here, not a license to remove its physical projection. 000147 | 000148 | Likewise integrating over fluctuating Phi, taking an infinite-volume state 000149 | without a trace-class heat, adding interactions, or tracing out a coupled 000150 | rest does not preserve this proof automatically. A flat parent term plus 000151 | noncommuting additional couplings does not imply the tensor factor (1). 000152 | This excludes exact equality to the literal tensor-factor source; it does 000153 | not exclude such a term, a compression, an effective approximation or a 000154 | different interacting source. Relative partition-function quotients alone 000155 | are not substituted for the actual heat trace in (2). 000156 | 000157 | An arbitrary unitary on HS that mixes the two face/unvectorization slots 000158 | does not automatically intertwine physical Q(s). The actual oriented field 000159 | legs and sewing must give the operator equivalence (1) or (5). This review 000160 | makes no identification of H with L_H-R_H or with a modular generator. 000161 | 000162 | EVIDENCE AND NOVELTY SCOPE 000163 | 000164 | Original source H and Tr exp(-x|h|)