Only the specified M4 endpoint channel, positive integer powers, same system and homogeneous GKSL time. Intermediate covariance is not assumed. Positive GNS Hilbert transfer and classical B recovery semigroup remain valid; larger systems and time-dependent dynamics are not excluded. Contract remains PARTIAL and physical source derivation open; source review is not a fresh kernel/numerical run.
Read the exact result, assumptions and evidence
Result
For Phi(X)=7X/12+sum_{j=1}^5 Gamma_j X Gamma_j/12 on M4 with the specified Clifford matrices and 1+5+10 spectral blocks, no Phi^n for positive integer n is the exponential of a time-homogeneous GKSL generator on the same M4. The proof covers all logarithm branches using block traces and Clifford twirling; the forced canonical coefficient beta=n log(3/4)/16 is negative. This is a different channel from the shared S3 register hierarchy.
Required assumptions
Fixed marked M4 Clifford channel
Five specified Hermitian Gamma_j on M4 satisfy {Gamma_i,Gamma_j}=2delta_ij I. The endpoint is Phi=7id/12+sum_j Ad(Gamma_j)/12 with spectrum 1,1/3,2/3 on orthogonal blocks 1,5,10. Comparison class is any homogeneous GKSL generator on the same M4 with exp(L)=Phi^n for positive integer n.
The entire specified finite shared-event family on the same registers; time-homogeneous GKLS generator in dimensionless step units. Source proof reviewed, without independent external peer review or fresh proof execution. Single-register continuation, enlarged environments, time-dependent dynamics and conformal/modular source time remain separate; no proof of fundamental discrete time or physical q selection.
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Result
For the specified family Psi_N,q=sum_g w_g(q) Ad(u_g tensor N), 0<=q<=1/3, a consistent time-homogeneous GKLS continuation on the same registers exists iff 2/9<=q<=1/4. Necessity already follows on two registers using fixed sector projections, simple block spectra and conditional Choi positivity of the symmetric part of every possible generator; intermediate S3 covariance is not assumed. Explicit nonnegative rates construct the continuation for every N. q=0 is noninvertible; q=1/3 has a Choi-rank jump from five to six.
Required assumptions
Specified shared S3 register family and homogeneous GKLS time
The specified six native words u_g act in each register by the same S3 representation; shared event weights w(q)=(1/2+q/6,q/6,q/6,1/18-q/6,2/9-q/6,2/9-q/6), 0<=q<=1/3. Seek a consistent time-homogeneous GKLS semigroup on the same finite registers with unit time step.
Conditional on charged CAR/Wick identification and the same graph covariance. No P1/P2 selection of physical statistics, normal length, marking or events. Existing quarter holonomy is retained; Clifford quarter-turn algebra is not a holonomy derivation.
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Result
The second face of the supplied pure quasifree graph state has n=1−A=(1+exp(2ell B))⁻¹. Thus log((1−n)n⁻¹)=2ell B, without separately inserted occupations or a second time generator. This realizes the already known faithful KMS inverse through the full collar boundary relation.
Required assumptions
Complete data of both product-collar faces
A selfadjoint B occurs in an exact normal-independent product equation. The full orthogonal L2 Cauchy graph projection of both labeled faces, their actual fiber identification and independently known positive normal length ell are supplied. Selection of these data from P1/P2 is not presumed.
Charged quasifree state for the complete graph covariance
Additionally identify the actual field algebra with charged CAR(H direct_sum H) and the state with the gauge-invariant quasifree state whose occupation covariance is G_ell. This is an extra physical premise, not a consequence of the geometric projection calculation.
Exact spectral calculus for a supplied full product collar and known normal length. Without ell only ell B; no independent derivation of the original P1 kernel.
Read the exact result, assumptions and evidence
Result
The actual product equation gives T=exp(−ell B) and G=[[A,C],[C,1−A]], A=(1+exp(−2ell B))⁻¹. B=log(A/(1−A))/(2ell) reconstructs signs, magnitudes and operator frame. For unbounded T, T is the closure of C A⁻¹; the raw composition is defined only on D(T²). The executable reader needs no supplied B and rejects sign-only projections.
Required assumptions
Complete data of both product-collar faces
A selfadjoint B occurs in an exact normal-independent product equation. The full orthogonal L2 Cauchy graph projection of both labeled faces, their actual fiber identification and independently known positive normal length ell are supplied. Selection of these data from P1/P2 is not presumed.
Applies on a faithful finite KMS sector or the same gapped transfer sector; physical TFPT transfer remains unidentified.
Read the exact result, assumptions and evidence
Result
For known nonzero beta and faithful finite KMS covariance 0<C_beta<I, H=beta^-1 log((I-C_beta)C_beta^-1) is injectively determined. Alternatively, a full operator T=exp(-tau |H|), known tau, and [T,sgn H]=0 reconstruct the gapped H=sgn(H)(-log T)/tau. A population or eigenvalue list is insufficient; physical identification of TFPT transfer with this T must be shown separately.
Required assumptions
Known KMS scale or full transfer
Positive dynamics reconstruction assumes either known nonzero beta and faithful finite covariance 0<C_beta<I, or the same full transfer operator T=exp(-tau |H|), known tau, and commutation with the sign; a spectrum alone or physical identification of the TFPT channel is insufficient.
Exact finite-dimensional counterexample for zero-temperature sign projection; refutes neither modular uniqueness for a fixed standard pair nor a complete TFPT source derivation.
Read the exact result, assumptions and evidence
Result
For C=(I+sgn H)/2, H_a=diag(1,a,-1,-a), a>0, gives the same fixed projector C and rho_0 but different positive energies. Thus [rho,C]=0⇒[rho,H]=0 is false without an extra premise. The counterexample concerns only inference from state/sign covariance to complete dynamics.
Required assumptions
Only sign covariance is given
The conditional counterexample assumes C=(I+sgn H)/2 and [rho,C]=0, without knowing positive magnitude |H| or a complete transfer.
This inverse is analytically settled; do not repeat Hamiltonian search when complete G is supplied.
Consequence: For source selection supply the actual original graph kernel, fiber comparison and normal length; G generated from a guessed B is not a provenance derivation.
The original source identifies the same complete transfer operator with scale, sign, domain, and state.
Next investigation: Retain both finite endpoint traces, fiber comparison and normal length from the original source. Evaluate the full graph with the existing inverse reader. Do not identify the geometric deck lift with 2P_C−I. Establish CAR state identification and actual marking action from the same source; do not count a matrix generated from guessed B as a P1 derivation.
Builds on: Reuse the sign-projector counterexample; Reuse conditional full-transfer reconstruction; Reuse complete collar reconstruction
Resolved in scope
Reuse complete collar reconstruction
The actual full graph relation determines the same signed operator with correct domains.
Builds on: Reuse conditional full-transfer reconstruction
The sign projector loses |H|; unique dynamics returns only with faithful KMS covariance or the same complete transfer T. This ties the time question to the open joint selection of tau_t, state, and operator in the target problem.
Joint reconstruction replaces independent time/state fitting by selection of one full original graph kernel plus physical field identification. Marking needs its local algebra; inversion does not settle source selection.
Original historical attempts linked to this question
These are original research records. Their reported verdicts are distinct from the reviewed results above.
seam full collar reconstruction 20261015
Which original seam information determines one common field, state and time action without confusing geometric reflection with boundary polarization?
Recorded verdict: EXACT_CONDITIONAL_RECONSTRUCTION_WITH_SOURCE_CORRECTION
Historical inventory; not independently reviewed by this export.
Can any homogeneous Lindblad logarithm realize the supplied full marked source channel, and which continuous source-operator completions preserve the actual marked transfer words?
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.
UR.SOURCE.OPERATOR_TIME.01: Do the affine E8 current Ward/OPE data determine the RR/W Hamiltonian jets, and do the direct affine-grade-time or global linear RR-field identifications hold?
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.