TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/QUENCH-LOGIC-REVIEW.txt Original file SHA-256: 2e9686bbc5895b41559859cec9ff23b376c78ea518a9e8150a063000c526a804 Inclusive original lines: 1-189 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: CANONICAL POSITIVE-EXCITATION QUENCH SOURCE 000002 | 2 November2026. relational_logic. Analytical review, no duplicate native run. 000003 | 000004 | VERDICT 000005 | 000006 | The proposed whole-mode quench closure is correct after the original face / 000007 | occupation / time orientation is stated explicitly. It supplies a concrete 000008 | absolute convergent product for the RELATIVE EXCITATION echo and finite 000009 | constant-background histories. The scalar is fixed by the already existing 000010 | positive excitation Hamiltonian with ground energy0. It is not an arbitrary 000011 | counterphase introduced to force convergence. 000012 | 000013 | The general existence and implementability of each full constant coupled 000014 | Hamiltonian were already proved in UV-INSERTION. Those results are reused. 000015 | The explicit source amplitude, its nonperturbative mode product, the exact 000016 | finite formula and uniform convergence are a constructive additional 000017 | application. They do not determine the full TFPT geometric vacuum scalar 000018 | Lambda(H), a boson distribution, or a continuous-time refinement theorem 000019 | for arbitrary smooth H(t). 000020 | 000021 | 1. REQUIRED SIGN / CHART CONVENTION 000022 | 000023 | For each actual radial original singular pair write 000024 | h0=p sigma_z+s sigma_x, 000025 | h1=p sigma_z+s1 sigma_x, s1=(1-epsilon)s, 000026 | E0=sqrt(p²+s²), E1=sqrt(p²+s1²). 000027 | Both original p=n+alpha quarter sets are retained. 000028 | 000029 | The current R079/D native covariance is C+=expit(+beta h0), beta=2ell. 000030 | It fills the POSITIVE h0 band at zero temperature. Use the corresponding 000031 | hole-chart positive excitation Hamiltonian 000032 | K_j=E_j-dGamma(h_j), 000033 | whose ground energy is exactly0. Then 000034 | exp(-iT K0)exp(+iT K1) 000035 | =exp[iT(E1-E0)] Gamma(exp(+iT h0)exp(-iT h1)). (1) 000036 | This is the proposed scalar and matrix ordering. 000037 | 000038 | By contrast the literal negative-sea chart in UV-INSERTION has 000039 | C-=expit(-beta h0), H_j=E_j+dGamma(h_j). 000040 | Its corresponding echo exp(+iT H0)exp(-iT H1) has scalar 000041 | exp[iT(E0-E1)]. 000042 | The two are related through the actual face/adjoint/charge chart, not by 000043 | calling C+ a normal state in the literal negative-sea Fock representation. 000044 | C+ tends P+, C- tends P-; their literal difference is not HS. The parent 000045 | has explicitly corrected the theorem to the hole chart. With that correction 000046 | there is no remaining sign conflict. 000047 | 000048 | 2. EXACT FINITE PAIR FORMULA 000049 | 000050 | Let n0,n1 be the unit Pauli directions of h0,h1 and put 000051 | d_p=n0 dot n1=(p²+s s1)/(E0 E1), delta_p=1-d_p. 000052 | The pure positively filled reference gives 000053 | <+0| exp(iT h0)exp(-iT h1) |+0> 000054 | =exp(iT E0)[cos(T E1)-i d_p sin(T E1)]. 000055 | Multiplying by the exact scalar from(1) gives 000056 | r_p=1+(delta_p/2)[exp(2iT E1)-1]. (2) 000057 | Therefore |r_p-1|<=delta_p. For fixed nonzero p, the direction angle is 000058 | arctan(s/p) in the appropriate half-plane. Its derivative in s has modulus 000059 | |p|/(p²+s²)<=1/|p|. Hence 000060 | delta_p<=epsilon²s²/(2p²). (3) 000061 | This bound is uniform for ALL real T, not only compact time intervals. 000062 | It also covers sign changes of s1; no adiabatic approximation is used. 000063 | 000064 | For the actual faithful thermal C+, let q_p=expit(-beta E0). 000065 | The four Fock probabilities are 000066 | P_empty=P_double=q_p(1-q_p), 000067 | P_positive=(1-q_p)², P_negative=q_p². 000068 | The trace-norm distance from the positive one-particle pure state is 000069 | 2[1-(1-q_p)²]<=4q_p<=4exp(-beta|p|). 000070 | Every echo is unitary. Thus its thermal factor F_p obeys 000071 | |F_p-1|<=epsilon²s²/(2p²)+4exp(-beta|p|). (4) 000072 | The scalar is included in F_p. This proves the proposed all-mode absolute 000073 | product, uniformly over epsilon in compact sets and in all real T. 000074 | 000075 | The exact thermal factor can be evaluated by the actual two-dimensional 000076 | Klich determinant 000077 | F_p=exp[iT(E1-E0)] det[I-C_p+C_p U_p], 000078 | U_p=exp(iT h0)exp(-iT h1). 000079 | This determinant includes the empty and doubly occupied sectors, not merely 000080 | a pure-vacuum overlap. Zero factors are allowed. The product exists in the 000081 | precise sense sum_p|F_p-1||=O(1/p²). 000129 | The second statement follows by decomposing Omega_ref into its ground-line 000130 | and orthogonal components for K_j; R_j kills the first and preserves its 000131 | orthogonal complement. In the expansion of a fixed finite history product, 000132 | each term with at least two R factors has first and last factors bounded 000133 | on Omega_ref byO(1/|p|), while intermediate factors have norm<=2. 000134 | Consequently its pure-reference expectation is1+O(1/p²), uniformly in all 000135 | history times. The thermal correction is exponentially small as in(4). 000136 | Thus every finite history has an absolute convergent mode product. 000137 | 000138 | The same estimate gives Cauchy vacuum vectors for cutoff history operators. 000139 | Their one-particle CAR automorphisms converge on fixed fields. On the dense 000140 | CAR-polynomial vacuum domain, covariance of the action then gives strong 000141 | convergence; adjoint histories obey the same estimates. The limits are 000142 | unitary, in agreement with the already established common-Fock Hamiltonians. 000143 | Multiplication yields the full finite-history word functional, not a new 000144 | independent Hilbert-space construction. 000145 | 000146 | 5. LIMITS THAT MUST REMAIN EXPLICIT 000147 | 000148 | (a) The bound for histories depends on their finite length. This proof is 000149 | not a uniform mesh-to-zero theorem for arbitrarily fine piecewise-constant 000150 | approximations to a general smooth H(t). Such a refinement could require 000151 | additional control of domains and geometric/phase connections. 000152 | 000153 | (b) The existence of all finite unitary histories does not imply all time 000154 | moments or arbitrary source derivatives. The old vacuum can remain outside 000155 | Dom(K_j^(1/2)), although its group action and echo are defined. For instance 000156 | the derivative at T=0 can interchange limits incorrectly: in(2), 000157 | delta_p E1 is generically of order epsilon²s²/(2|p|), whose sum diverges. 000158 | Uniform convergence of the source itself does not repair that derivative. 000159 | 000160 | [OMITTED: non-scientific conversation or execution metadata] 000161 | [OMITTED: non-scientific conversation or execution metadata] 000162 | [OMITTED: non-scientific conversation or execution metadata] 000163 | [OMITTED: non-scientific conversation or execution metadata] 000164 | [OMITTED: non-scientific conversation or execution metadata] 000165 | [OMITTED: non-scientific conversation or execution metadata] 000166 | 000167 | (d) The C+ hole-chart group and a full-graph simultaneous diag(U,U) 000168 | stationary cancellation are distinct constructions. The latter can be 000169 | Liouvillian rather than positive OS time. Their conclusions should not 000170 | be merged. 000171 | 000172 | 6. NOVELTY / REUSE JUDGMENT 000173 | 000174 | This is a genuine positive completion of the specified relative quench 000175 | amplitude, not merely a repetition that some implementer exists. It evaluates 000176 | the canonical source scalar, proves the full product nonperturbatively, 000177 | gives explicit finite factors and retains every original singular channel. 000178 | The underlying existence and phase-normalization RULE are old results and 000179 | must be cited as such. No new universal TFPT source axiom or physical OS 000180 | identification is introduced. 000181 | 000182 | [OMITTED: non-scientific conversation or execution metadata] 000183 | [OMITTED: non-scientific conversation or execution metadata] 000184 | [OMITTED: non-scientific conversation or execution metadata] 000185 | [OMITTED: non-scientific conversation or execution metadata] 000186 | [OMITTED: non-scientific conversation or execution metadata] 000187 | [OMITTED: non-scientific conversation or execution metadata] 000188 | [OMITTED: non-scientific conversation or execution metadata] 000189 | [OMITTED: non-scientific conversation or execution metadata]