TFPT scientific source excerpt Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/MATRIX-KERNEL-PROOF.txt Original file SHA-256: ff2db6b22553834ea847c7986c19000f2d19b2e43aa6ff00c592db97abd7e988 Inclusive original lines: 6-136 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. 000006 | An actual finite selfadjoint original Hamiltonian and its actual boson 000007 | coordinate resolution determine a complete matrix-valued path integration. 000008 | No additional physical boson measure, interaction, state or time is chosen. 000009 | The construction retains the entire inner fiber, absolute scalar, original 000010 | boundary state or physical selector, reference, and every source insertion. 000011 | It applies to the existing finite Hamiltonian branch and passes bounded 000012 | word limits wherever its already proved thermodynamic construction does. 000013 | It does not identify that branch with primitive spectral S_adm, derive an 000014 | unknown primitive H_B, or prove a positive scalar boson path law. 000015 | 000016 | The practical reduction is that a proposed original S_adm strip kernel can 000017 | be compared infinitesimally, with every matrix coefficient retained. An 000018 | evaluated same-source strip identity then implies all time-ordered source 000019 | identities. It must not be replaced by naming a kernel or action. 000020 | 000021 | 1. Actual coordinates, entire fibers, exact discrete integration 000022 | -------------------------------------------------------------- 000023 | Let H=H* act on the ACTUAL finite regulated source Hilbert space Hcal. 000024 | Let {E_q:q in Q} be a finite orthogonal resolution of identity belonging 000025 | to the actual bosonic coordinate chart. If no such physical chart is 000026 | identified, these formulas are an operator representation, not an 000027 | identification with geometric/Higgs coordinates. No new chart is selected 000028 | by the theorem. Put H_q=Ran E_q and H_yx=E_y H E_x:H_x->H_y. 000029 | Fibers can have different dimensions. All matter, multiplicity, rest and 000030 | constraint legs not resolved by E_q remain in H_q. 000031 | 000032 | Define the concrete whole strip kernel 000033 | K_t(y,x)=E_y exp(-tH) E_x:H_x->H_y. (1) 000034 | For any positive partition t=sum_j delta_j, 000035 | K_t(q_M,q_0)=sum_(q_1,...,q_(M-1)) 000036 | K_deltaM(q_M,q_(M-1)) ... K_delta1(q_1,q_0). (2) 000037 | Proof: insert the original identity sum_q E_q=I between the original 000038 | semigroup factors. No marginal normalization, scalar logarithm or 000039 | positivity inference is used. Equation (2) is exact at each partition. 000040 | 000041 | For actual bounded source insertions A_j, use their full rectangular 000042 | blocks A_j(y,x)=E_y A_j E_x. Insert these blocks in (2) at the original 000043 | times. Summing the intermediate coordinates is exactly the original 000044 | operator product. In particular insertions need not be diagonal or 000045 | individually commute with the coordinate projections. The theorem does 000046 | not add charged observables to the physical source algebra. 000047 | 000048 | For the actual unnormalized pure boundary vector Omega, put 000049 | Omega_q=E_q Omega. The whole open source is the sum of 000050 | 000051 | over ALL endpoint and intermediate coordinates. Endpoints are not made 000052 | periodic. A supplied mixed boundary functional is contracted in its 000053 | original matrix form instead. A coordinate Fock vacuum is not substituted 000054 | for the physical OS boundary state. 000055 | 000056 | For the ACTUAL thermal selector P on the original common space, 000057 | Tr(P exp(-beta H)) 000058 | =sum_(q_0,...,q_M) Tr_(H_q0)[ 000059 | P_(q0,qM) K_deltaM(q_M,q_(M-1)) ... K_delta1(q_1,q_0)], (3) 000060 | where P_(x,y)=E_x P E_y:H_y->H_x. Thus the product inside the trace maps 000061 | H_q0 to itself even when the fiber dimensions differ. P is kept once at 000062 | the boundary; no extra P is inserted between strips or charged letters. 000063 | For P=I the boundary block is delta_(q0,qM) I_q0, yielding closed loops. 000064 | For general P it need not be a closed coordinate loop. 000065 | 000066 | Equation (3) is an algebraic identity without [P,H]=0. Its interpretation 000067 | as the previously constructed physical thermal Gibbs source uses the 000068 | ACTUAL reducing physical P, [P,H]=0, nonzero physical space, and the 000069 | actual admissible insertions from PHYSICAL-THERMAL-SOURCE.txt. The 000070 | coordinate E_q themselves need not be separately physical observables: 000071 | they can resolve the original ambient operator before the one final P. 000072 | If working directly on physical space, use its actual coordinate map; 000073 | do not compress an ambient resolution and assume it remains a PVM. 000074 | 000075 | Use (3) separately for actual numerator and actual reference callers, 000076 | when both have this representation. Preserve their original quotient. 000077 | The theorem does not postulate a new reference Hamiltonian or set Z_ref=1. 000078 | Identity shifts in H remain in every strip and hence in the absolute Z. 000079 | 000080 | 2. Explicit continuous-time matrix integration without a Markov premise 000081 | --------------------------------------------------------------------- 000082 | For every ordered pair y!=x with H_yx!=0 choose a COMPUTATIONAL positive 000083 | rate a_yx>0. The deterministic default a_yx=||H_yx|| uses only H. 000084 | Set lambda_x=sum_(y!=x) a_yx and a_yx=0 for a zero block. Let X_t be 000085 | the finite-state continuous-time Markov chain with these jump rates. 000086 | The process is merely a reference representation for the source. 000087 | 000088 | Start U_0=I_x on H_x. While X_t=z, evolve the rectangular carried map by 000089 | dU_t/dt=-(H_zz-lambda_z I_z) U_t. (4) 000090 | At a jump z->y multiply it on the LEFT by 000091 | J_yz=-H_yz/a_yz:H_z->H_y. (5) 000092 | After any path from x to y, U_t:H_x->H_y, so no equal-fiber assumption 000093 | is needed. Then 000094 | E_x[1_(X_t=y) U_t]=K_t(y,x). (6) 000095 | 000096 | Proof A, forward equation. Let F_yx(t) denote the expectation on the 000097 | left. A no-jump increment at y contributes 000098 | -(H_yy-lambda_y I)F_yx-lambda_y F_yx=-H_yy F_yx. 000099 | A jump z->y contributes a_yz J_yz F_zx=-H_yz F_zx. 000100 | Thus dot F_yx=-sum_z H_yz F_zx, with F_yx(0)=delta_yx I_x. 000101 | Finite-dimensional ODE uniqueness gives F=exp(-tH), proving (6). 000102 | 000103 | Proof B, path density. A path x_0=x,...,x_n=y with jump times 000104 | 0