TFPT scientific source excerpt Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/ORIGINAL-CAR-WARD-IDENTIFICATION.txt Original file SHA-256: c4ac90ce9f44039ebcf16c289e1b8d2ef661fafa266a8d5e42678b4ed60f8ed0 Inclusive original lines: 9-100 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. 000009 | Use the actual negative-energy-filled quarter representation, with CAR modes 000010 | a_r, r in Z, and occupation r<=-1. The orientation of the original QWZ top 000011 | edge is transported by r=-j, so r+1/4=1/4-j. This retains the physical 000012 | one-particle generator and the physical charge; it is not a replacement of 000013 | the state by an independently selected bosonic vacuum. 000014 | 000015 | For a residue k=0,...,7 and integer n>=1 put 000016 | 000017 | h_(k,-n) = sum_(r congruent k mod8) :a*_(r+8n) a_r:. 000018 | 000019 | The actual source stress mode, in folded normalization, is 000020 | 000021 | K = Lcal_-8/8 000022 | = sum_r c(r) :a*_(r+8) a_r:, 000023 | c(r)=(r+alpha+4)/8. 000024 | 000025 | On any fixed finite-energy vector these raising sums have only finitely many 000026 | nonzero terms. The CAR bilinear commutator therefore gives, without a 000027 | cutoff boundary term, 000028 | 000029 | [K,h_(k,-n)] 000030 | = sum_(r congruent k mod8) [c(r+8n)-c(r)] 000031 | :a*_(r+8n+8) a_r: 000032 | = n h_(k,-n-1). (1) 000033 | 000034 | There is no central term in (1), since both indices are raising indices. 000035 | On the actual sea, only r=k-8 crosses from an occupied to an empty level 000036 | in the first stress step. Hence 000037 | 000038 | K Omega = sum_k delta_k h_(k,-1) Omega, 000039 | delta_k=c(k-8)=(k+alpha)/8-1/2. (2) 000040 | 000041 | Equations (1)--(2) derive the shifted oscillator recursion directly from the 000042 | original CAR operators. They do not infer it from a matching dimension or 000043 | from a free choice of a Cartan vector. With y_n=sum_k delta_k h_(k,-n), 000044 | 000045 | K^s Omega = s! [z^s] exp(sum_(n>=1) y_n z^n/n) Omega. (3) 000046 | 000047 | This follows by induction using [K,y_n]=n y_(n+1). Every oscillator partition 000048 | at each level is retained. In particular, the pure mode-minus-one term of 000049 | K^8 Omega has coefficient one and is y_1^8 Omega. 000050 | 000051 | 2. The fixed original octic and its normalization 000052 | ------------------------------------------------ 000053 | 000054 | Retain exactly A=U_+^T O^T M from the existing original dictionary. Its 000055 | rows are Hermitian orthonormal and bilinearly isotropic. Put b=A delta and 000056 | 000057 | I8(x)=sum_a x_a^8 000058 | +14 sum_(a = 8! conjugate(F8hat(b)). (4) 000067 | 000068 | For alpha=1/4 the original, unfitted matrices give 000069 | 000070 | I8(b)=5416993/2^32 - i 584073/2^27, 000071 | 000072 | and therefore 000073 | 000074 | 000075 | =sqrt(35)(16250979+56071008 i)/128 != 0. (5) 000076 | 000077 | Both vectors in (5) have zero charge and original excitation energy64. 000078 | The factor relating (4) and (5) is exactly 8^8. Equation (5) is a response 000079 | component, not a statement that the stress history prepares a pure F8 state. 000080 | 000081 | 3. The microscopic sector used by this proof 000082 | -------------------------------------------- 000083 | 000084 | The existing QWZ theorem uses epsilon_j=1/4-j. Any excitation of energy at 000085 | most64 can involve only -63<=j<=64. Its coarse particle-plus-hole count is 000086 | at most256, since the smallest excitation energy is1/4. The existing window 000087 | 000088 | M_N=min(floor(N/8),floor(sqrt(N))) 000089 | 000090 | contains all these modes once N>=4096. On that entire energy sector the 000091 | projection Gamma(Pi_M) in the comparison contraction V_N is the identity. 000092 | The remaining map is an actual isometry into the original filled QWZ Fock 000093 | space and intertwines its finite particle/hole CAR words and adjoints. 000094 | 000095 | Each current or stress mode entering (1)--(5), acting on the relevant 000096 | finite-energy vectors, reduces to such a finite CAR word sum. The original 000097 | mode-convergence theorem therefore transports the energy64 Ward matrix 000098 | element. This claim does not depend on an unbounded point-field limit. 000099 | It also does not assert that a finite microscopic stress unitary has already 000100 | been shown to converge to every exponentiated continuum geometry history.