TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/DET16-WHOLE-SOURCE-REDUCTION.txt Original file SHA-256: fc43fcfef453bb278b8f5c28123078331a36d1e4350672b45c47a5d99c992897 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 | R104. Exact original DET16 source and its whole-word reduction 000002 | 2 November2026. Conditional original Hamiltonian application. 000003 | 000004 | 1. SOURCE AND DECISION 000005 | 000006 | L20261102G/H already give the full original geometric package transport 000007 | and a complete physical thermal word source for their stated Hamiltonian 000008 | class. They are retained. No new universal CAR, Gibbs-reference, bosonic 000009 | measure or a->0 requirement is introduced. R103's source-type exclusions 000010 | and the existing E8 minimal-extension theorem remain unchanged. 000011 | 000012 | The actual v1013 table declares a cell with four physical edge fermions, 000013 | a separate mirror Fock factor, two outgoing links and a four-state clock. 000014 | Only its DET row acts on the mirror factor: 000015 | 000016 | H_m=2(I-P), P=|phi>=(|0>+exp(i vartheta)|F>)/sqrt(2), 000018 | n=16, d=2^16=65536, vartheta=the original v1002.PHI=pi/7. 000019 | 000020 | The other six table rows act on the rest. In particular the assembly 000021 | source explicitly couples the seam to the PHYSICAL edge Q density. 000022 | For this literal table, H_Lambda=H_R,Lambda+sum_x H_m,x. Its rest factor 000023 | has dimension16*4*4=256 per cell. This is an operator factorization of 000024 | the declared table, not a new physical choice or a test of all other 000025 | Hamiltonians. v1007 explicitly contains mirror hopping and falls outside 000026 | this factorization. v1034 likewise distinguishes source parent classes; 000027 | no statement here supersedes its locality or mirror-coupling boundaries. 000028 | 000029 | Original sources reviewed: 000030 | paper-source:4cc6102a923009e972617edc, source001:32-118; 000031 | documents/articles--2026-08-30--tfpt_thermodynamic_dynamics_en/sections/ 000032 | 001-section-the-concrete-uniformly-local-class.tex. 000033 | verification/v1002_spin10_mirror_projector.py:204-267,311-315; 000034 | experiments/tfpt-discovery/tfpt2p1d_master_assembly_probe.py:26-35,66-73. 000035 | The latter is its declared2+1D/DET4 provenance, not a full4D proof. 000036 | 000037 | 2. COMPLETE THERMAL MIRROR SOURCE, INCLUDING Z[0] 000038 | 000039 | For beta>=0 put a=exp(-2 beta), b=1-a. Since P^2=P, 000040 | 000041 | E_beta=exp(-beta H_m)=a I+b P, 000042 | Z_m=Tr E_beta=d a+b=1+(d-1)exp(-2 beta), 000043 | rho_m=E_beta/Z_m. (1) 000044 | 000045 | For EVERY mirror operator A, 000046 | 000047 | omega_beta(A)=[a Tr A+b]/Z_m. (2) 000048 | 000049 | This is the entire finite source state, not a covariance fit. The scalar 000050 | identity part of H_m is retained. Removing it from a commutator is valid; 000051 | removing it from this unnormalized source would change Z_m and energy. 000052 | The same original reference source, if present, contributes its actual 000053 | corresponding factor; no equality of reference factors is assumed. 000054 | 000055 | The native original CAR trace gives for all i,j 000056 | 000057 | =delta_ij/2, =0, 000058 | =1/4+b/(4Z_m), i!=j, 000059 | =b exp(i vartheta)/(2Z_m). (3) 000060 | 000061 | Here Omega_16=|0>0 000062 | the density-pair excess is positive: the source is not quasifree even 000063 | though its full EQUAL-TIME normal and anomalous covariance matrices equal 000064 | their infinite-temperature values. Its temporal two-point functions are 000065 | not claimed equal. A covariance-logit inverse cannot reconstruct this 000066 | interacting thermal Hamiltonian just from (3)'s covariance. 000067 | This is a concrete original application of the already known full-word 000068 | requirement, not a new global no-Gaussian statement for every TFPT source. 000069 | 000070 | At different fixed vartheta, words shorter than16 cannot connect |0> and 000071 | |F>, so they do not see the relative phase. The16-letter determinant word 000072 | does. A simultaneous original field/frame transport can change the phase 000073 | convention; this is not a new physical parameter prediction. 000074 | 000075 | 3. ALL GEOMETRICALLY ORDERED MIRROR WORDS 000076 | 000077 | Let A_0,...,A_(k-1) be actual mirror CAR letters in their written order, 000078 | and let nonnegative heat intervals t_0,...,t_k sum to beta. The complete 000079 | unnormalized Euclidean coefficient is 000080 | 000081 | W=Tr[E_t0 A_0 E_t1 A_1 ... A_(k-1) E_tk]. (4) 000082 | 000083 | Each E_t is exactly a_t I+b_t P. Expanding this identity is a proof for 000084 | every finite word, but its literal2^(k+1) expansion is unnecessary. 000085 | For each term with projector cuts use 000086 | 000087 | Tr(P B_1 P B_2 ... P B_r)=product_j . (5) 000088 | 000089 | The blocks B_j keep the actual order and original fermion signs. The 000090 | P-free term is exp(-2 beta) times the ordinary CAR Fock trace. 000091 | 000092 | 4. MATRIX-FREE CYCLIC RENEWAL, POLYNOMIAL IN WORD LENGTH 000093 | 000094 | By cyclicity combine the last and first heat intervals. Write the resulting 000095 | k cyclic intervals as delta_i, each immediately before A_i. For i. (6) 000100 | 000101 | For any nonempty set of selected P cuts, multiplying the consecutive 000102 | cyclic edge weights gives exactly that term of (4): every selected cut 000103 | contributes its b, every unselected interval its a, every block its 000104 | matrix element. No reordering of CAR letters is made. 000105 | 000106 | There is a unique smallest cut i in {0,...,k-1}. Fix it, put D_i=1, and 000107 | compute for j=i+1,...,k-1 000108 | 000109 | D_j=sum_(h=i)^(j-1) D_h e(h,j). 000110 | 000111 | Its complete closing contribution is sum_(h=i)^(k-1)D_h e(h,i+k). 000112 | Sum over i and add the P-free term. Every nonempty subset of cuts is 000113 | counted exactly once by its unique ordered path and smallest cut. 000114 | The empty word is Z_m itself. Divide by Z_m only for normalized answers. 000115 | 000116 | For a CAR-letter block, requires only applying that word to 000117 | the two bit states0 andF. For its full Fock trace, work right to left: 000118 | the first action on each used mode forces its initial occupation. Check 000119 | the resulting word on that initial bit choice. If it is zero or changes 000120 | the initial state, the trace is zero. Otherwise its sign is independent 000121 | of unused spectator bits: every acted mode flips an even number of times, 000122 | so the total Jordan-Wigner exponent of any spectator bit is even. 000123 | Thus Tr B=sign*2^(16-number_of_used_modes). 000124 | 000125 | There are O(k^2) possible edges, each calculated in O(k), and the dynamic 000126 | path sums cost O(k^3). Stored edges cost O(k^2). These are exact arithmetic 000127 | operation counts at the fixed original16-mode factor. Floating evaluation 000128 | retains ordinary numerical precision limits; no uniform relative-error 000129 | claim for arbitrarily tiny answers or arbitrary bit-complexity is made. 000130 | The evaluator uses no Hilbert-space matrix for this renewal calculation. 000131 | It does not claim efficient input access to arbitrary unspecified operators. 000132 | 000133 | 5. ATTACHMENT TO THE WHOLE ORIGINAL HAMILTONIAN SOURCE 000134 | 000135 | In the declared decoupled ambient table, for rest-only source legs, 000136 | 000137 | Z_Lambda^amb[J_R]=Z_R,Lambda[J_R] product_x Z_m. 000138 | 000139 | This is exact partial trace, at every volume; the interacting H_R is NOT 000140 | solved or discarded. Its normalized rest words factor, while its actual 000141 | unnormalized source retains all mirror free-energy factors. The free-box 000142 | KMS subnet therefore has these same finite mirror thermal factors together 000143 | with the rest subnet state. This is not a unique-phase theorem for H_R. 000144 | 000145 | The physical Gauss projection must not be replaced by this ambient product. 000146 | Suppose the ACTUAL supplied gauge action factors as 000147 | U(g)=U_R(g) tensor product_x U_m,x(g_x), and use its SAME original 000148 | P_G=|G|^(-|V_int|)sum_g U(g). For gauge-compatible rest source legs, 000149 | 000150 | Z_phys[J_R]=|G|^(-|V_int|) sum_g 000151 | Tr_R[U_R(g) exp(-beta H_R) original_ordered_source[J_R]] 000152 | * product_x z_m,x(g_x), 000153 | z_m(g)=a Tr U_m(g)+b. (7) 000154 | 000155 | Proof: substitute the actual group projector into the full trace, retain 000156 | its order, then take the mirror partial traces using (1). This performs 000157 | exact reduction of the65536-dimensional factor. It does NOT replace the 000158 | sum by independent projected Hilbert factors or solve the256-dimensional 000159 | rest lattice. Nonfactorizing gauge actions require their actual map. 000160 | 000161 | For the expressly given canonical second quantization U_m=Gamma(r), 000162 | with det r=1 and no extra implementer phase, both0 andF are invariant. 000163 | Exterior-algebra trace gives Tr Gamma(r)=det(I+r), hence 000164 | 000165 | z_m(g)=a det(I+r(g))+b. (8) 000166 | 000167 | Actual Spin10 central q=1,i,-1,-i acts as q^N on this native spinor Fock 000168 | factor. It fixes phi, and its four weights are 000169 | a(1+q)^16+b: a*65536+b, a*256+b, b, a*256+b. 000170 | The native code verifies them directly. This is not an identified v1013 000171 | gauge embedding or an equality to the P1 clock just because both have 000172 | order4. That actual mirror gauge representation was not printed in the 000173 | reviewed v1013 table. 000174 | 000175 | With mirror source legs, replace z_m by the entire alternating heat/word 000176 | trace (4), with its actual U_m insertion. Keep the original graded tensor 000177 | ordering and gauge carriers for mixed even observables. The untwined 000178 | renewal routine does not pretend to implement these unknown gauge maps. 000179 | 000180 | 6. GAP, THERMAL WEIGHT AND THE NEXT ORIGINAL SOURCE COMPARISON 000181 | 000182 | In this ambient factor the ground probability is p0=1/Z_m; it is not 000183 | determined by the energy gap alone. The65535 excited states retain their 000184 | entropy. Diagnostics beta=1,2pi,8pi give about0.000112737,0.813972,1. 000185 | These are not a selected physical temperature, KMS circumference or P1 000186 | clock calibration. No thermal no-mirror claim follows by deleting that 000187 | weight. At independent ambient cells p0^|Lambda| tends to0 for finite 000188 | beta while each local state still has its specified p0; global ground 000189 | overlap is not a criterion for local IR decoupling. Coupled original 000190 | mirror models and their stability theorems keep their own hypotheses. 000191 | 000192 | This result evaluates an original complete factor and its non-Gaussian 000193 | word source. It makes the literal Hamiltonian process substantially more 000194 | tractable without choosing another model or scanning new constants. 000195 | It does not derive the frozen coefficients, the actual full master-field 000196 | dictionary, IR universality, a physical4D limit or the fundamental solution. 000197 | 000198 | The next decisive comparison uses the ACTUAL original field/gauge map: 000199 | does the common S_adm/Z_rel insertion produce this same rank-one vertex, 000200 | its coefficient, its16-field coherence and the resulting weights in (7)?