Fixed phases and corresponding invariant starts; real time retains conjugation/time reversal. No assertion about all P1/P2 sources.
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Result
C and K may separately break; A=CK preserves H for all fixed link phases and T_p→T_−p. Invariant starts retain equal Euclidean opposite-charge returns, including quarter phases/flux.
Required assumptions
Full original charge-field source
Finite Round20 lattice/scalar space with the original bilinear E8 cocycle, complete integer charges, onsite T and balanced W, coercive quadratic Hdiag and supplied hopping. No P1/P2 selection or local chiral field identification is presumed.
Fixed link phases and A-invariant start
Arbitrary fixed unitary link phases in the same hopping, real original Hdiag/cocycle and initial density invariant under A=C times complex conjugation. Positive Euclidean T return responses are considered; no general real-time equality is assumed.
Fixed nonzero chiral quarter twist
The comparison target is the stipulated Kδ Fock space with original nonzero quarter twist, normalized ±Bp primary states, fixed charge marking and the same positive time. Equality of all these responses is required.
Only the fixed marked response identification at these starts and the same time.
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Result
The native ratio is one; the fixed quarter twist gives exp(−2tδ·Bp), which differs from one on at least one original basis direction.
Required assumptions
Full original charge-field source
Finite Round20 lattice/scalar space with the original bilinear E8 cocycle, complete integer charges, onsite T and balanced W, coercive quadratic Hdiag and supplied hopping. No P1/P2 selection or local chiral field identification is presumed.
Charge-conjugation-invariant start
The initial density is invariant under linear C:n→−n, including full and C0 Gibbs starts of unchanged H. Compare actual adjoint-paired, opposite marked T responses at the same time.
Fixed nonzero chiral quarter twist
The comparison target is the stipulated Kδ Fock space with original nonzero quarter twist, normalized ±Bp primary states, fixed charge marking and the same positive time. Equality of all these responses is required.
Exclusion only of the affine lift model at fixed K,Q and oscillator level; covariant tuple transport remains possible.
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Result
An isometric affine lattice lift Ap+r with A²=−I and unchanged oscillator level would require r=Aδ−δ and hence −2δ=(I+A)r in the E8 lattice. Adjacent coordinate differences of2δ are1/4; this fails exactly. Also Jq≠q.
Required assumptions
Stipulated E8 Fock space with marked clock tuple
Given the original ordered CAR/E8 cocycle, E8 Fock representation, Kδ(p,N)=|p|²/2+δ·p+N, original q and full uncompensated quarter transition. J is the actual Gaussian lattice operator, not the cyclotomic Galois operator. A covariant comparison transports δ,q,τ together; a fixed marked source holds them fixed.
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.
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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.
Exact finite-dimensional counterexample for zero-temperature sign projection; refutes neither modular uniqueness for a fixed standard pair nor a complete TFPT source derivation.
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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.
Only if the same marked family connection is irreducible and beta preserves it in parallel; without this added premise the 2+2 packet is not excluded. A noncommuting common source-derived connection remains open.
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Result
Under the explicitly additional premise that beta preserves the fixed marked irreducible SU(3) family connection, its commutant is scalar and beta must have the 3+1 form diag(b,b,b,-3b). Thus a beta packet with eigenvalue multiplicities 2+2 is excluded under that premise only; no original parallelity rule follows.
Required assumptions
Common parallelity to irreducible family connection
Only for the commutant conclusion, assume additionally that the marked SU(3) family connection is irreducible and beta preserves it in the same marked frame. This is not an accepted P1/P2 axiom and must be derived from the source.
Canonical source Gibbs/parity probe with retained Cartans and source winding
In existing conditional one-copy eightfold dictionary retain all8namedCartans,pinZ8,originalsea,H_source=8K,Q,globalPhi:D8+v→D8+c andactual SOURCE W1with[Q,W1]=W1. PrimitiveP1circle identification remains missing. CanonicalGibbs traces forEVERYbeta>0 are mathematicalfaithfulword/operator probes,not selectedphysicaltemperature. Retainbothoriginal limitingedges/parities; distinguishTOTAL-even andLEFT-even. No independent edgefluxregisters or microscopic pairedpumpfactorization asserted. ChangedphysicalCartan/netdictionary still needsderivation.
Given eightfold source fold with explicitly changed odd target action
Given existing one-copy CAR source,sea,fold,p in Z8,Heisenberg oscillators,Q=sum p,H=8K,K=N+p²/2+delta.p with delta_k=(k+alpha)/8-1/2,and actual SOURCE windingT1=Rp+e0,alpha1/4or3/4. Test explicit Phi=I on D8,rho=Pp+r on odd v with oscillator-P to odd spinor extension. Odd target Cartan action is explicitly changed by this comparison; inclusion in the original local net is separate. Neither folding,SOURCE/P1-core circle identification nor bosonic statistics is derived from P1.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Given holomorphic KMS inclusion with distinct local/global domains
Testing the cited abstract theorem assumes already existing holomorphic conformal c8netB,mu_B1,A=B^Z4,irreducible localindex4 andinternalgauge commutingwithrotations. omega_beta is DEFINED as Gibbs(B)restriction. Distinguish globalGNS KMSmodulargroup,L0rotations andlocalmodulargroups ofrestrictedfaithfulstates. These premises do not constructmicroscopicBfromP1; originalH_source=8(L0+delta.p) isnot silentlyidentifiedwithL0orphysicalbeta_angle2pi.
Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only unchanged full compact source/sea/time/observable comparison and full norm/time kernels. B is an operator property independent of v_N: disjoint-state limits do not automatically violate B; with fixedA,B,C they must failD. Divergent mean alone is not escape. No classification of all local scaling nets/P1 realizations or derivation of8copies. Originally derived changed net/Gauss/operator/time comparison outside scope; elementary spectral compactness,not a claimed new abstract theorem.
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Result
Arbitrary v_N=T_N Omega_N in fixed full strip obey tail_E<=[||v_N||²-<v_N,e^(-tH_N)v_N>]/[1-e^(-tE)]. Finite nonzero norm and limiting-kernel continuity at0 imply energy tightness. Entire finite spectral windows yield norm-subsequence compactness; fixed low-window operator matrix elements transfer all retained bounded-observable states into old global folium. Real original time also suffices: tail_E<=2[n_N-a^-1 int_0^a ReG_N(t)dt] forE>=2/a. No bounded mean energy,||T_N|| or word degree needed. Under fixed D8/even-CAR dictionary no additional s/c spinors; right edge only adds multiplicity.
Required assumptions
Full old observable map and continuous original time for nonunitary insertions
Fixed full quarter-QWZ carrier has compatible entire finite-dimensional low spectral windows. Every retained bounded A_N is uniformly bounded and converges on them to the original operator action,including odd intermediate states. This is an operator premise independent of prepared v_N=T_N Omega_N. Arbitrary T_N,including divergent renormalization,nonunitarity and growing word degree,are required to have finite nonzero norm and a limiting ACTUAL original positive-time or real unitary-time kernel continuous at0 with that norm. No bounded first energy moment,different charged Hamiltonian or new P1 realization assumed. Spinor consequence additionally retains the given unchanged D8/even-CAR dictionary.
Unchanged full quarter-QWZ excitation source
Current v1033 QWZ cylinder at mass1,width8,ONE complex CAR copy,alpha1/4 or3/4,full negative sea and H_N=N/(2pi)[dGamma(h_N)-Tr(P_Nh_N)]. Retain both edges,massive modes,charge sectors and physical energy/vacuum reference; compare actual low eigenmodes. No P1/P2 selection of this realization,changed stress or executed quotient.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only normalized smooth UNITARY vector-state sequence on fixed compact source circle. Nonunitary sums/field intertwiners,microscopically derived charged covariance cocycles and decompactified time outside scope. Local normality away from removed point is not finite global original energy. Individuallambda_i±1/2 distinct from totalQ=sumlambda_i; two-point exponent/energy alone no128/FE-GEN/ALG-EXH proof.
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Result
Original Weyl/CCR time formula andphi(theta)=f(8theta) give exactDeltaK=8sum_n>0 n²|f_n|²=(2/pi)int|f'|². Any pi->0 return squeezed intoeps costsDeltaK>=2pi/eps. Normalized W(phi)Omega states therefore have no bounded-energy separation limit under same global K reference; scalar normalization leaves Rayleigh quotient unchanged. Class lower bound,not profile scan. Known conditional K_lambda=K_0+lambda.p+lambda²/2+delta.lambda retained; its positive charged sectors are NOT excluded.
Required assumptions
Smooth unitary pair return at fixed compact source time
Given original chiral current/Weyl source, sea and K=Hcover/8 with delta_k=(k+alpha)/8-1/2. Periodic real phi(theta)=f(8theta), normalized UNITARY vector state W(phi)Omega; pi->0 return squeezed into base width eps on fixed source circle, same original energy reference. No assertion about general nonunitary sums, charged covariant representations, sector energy connections or decompactified time. Conditional Sugawara sector energy already fixed once lambda is given.
Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only smooth vector-phase class and unchanged mixed Ward action in stated complete Dirac contract. No all-spinor/P1 no-go,arbitrary global-loop-holonomy theorem or infinite determinant theorem. C/1-C are grading components on one face. Chiral source, changed current representation and other original operations outside scope.
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Result
Implementable commuting vector phases obey[J_i,vec(h),V_vec(f)]=0 because opposite grading cocycles cancel and noncentral term vanishes. At pi interval profile separate chiral components have opposite half endpoint residues; full vector sum0. Thus not a spinor pair withlambda_i=±1/2 under SAME8vector Cartan currents,including scalar renormalization and mixed-word limits. Finite exact control for any Hermitian C andunitaryg: chi_(1-C)(g)=detg conjchi_C(g),normal-order-corrected complete amplitude|chi_C|².
Required assumptions
Complete vector phases with unchanged Cartan Ward action
Only original complete Dirac/CAR Lagrangian source with normal-commuting implementable smooth profiles H_i h and fI, common analytic domains and regular Weyl representation. Target comparison retains same8 vector Cartan currents and mixed Ward words. Two grading components C,1-C on ONE face are neither strip edges nor collar faces. Finite complementary determinants are algebraic controls only; no unregulated infinite determinant or automatic QWZ/Dirac source identification.
Complete Dirac Cauchy source with Lagrangian normal action
Complete orthogonal Dirac Cauchy projection C on the entire boundary module, with actual normal unitary Clifford action N satisfying N C N*=I-C. Explicitly assume charged CAR quantization with pure quasifree covariance C. Bounded current generators X,Y commute with N and have Hilbert-Schmidt commutators with C. The existing compact product collar B=diag(A_alpha,-A_alpha) on both components and both faces is the concrete control; its smooth scalar boundary multipliers are implementable. Statistics and physical measure selection are not derived from P1; no automatic extension of the unrenormalized trace to higher boundary dimensions.
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Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Exclusion only for fixed charge/connection dictionary. No blanket SO16 or spinor-sector no-go. Eight independent copies, physically derived charge conjugation or changed field/domain connections separate. Source/P1-circle, base-deck/clock, chiral fermionic measure and local field products still require derivation; no new internal Z8^f/Arf branch. Neutral-pair refinement: eight-preimage parity pairs T_E with canceled register and unchangedh are NOT excluded by this single-sector comparison.
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Result
Original one-copy eightfold: C(-h)^8=-hI, while desired uniform spinor twist (-C(h))^8=hI. No complex charge-preserving intertwiner of given holonomies. Both-quarter h=±i countercontrol: real SO16 O=realify(diag((-1)^j))K DOES conjugate holonomies but reverses Jcomplex and original U1 charge. Uniform BASE Z2 spin lift is invisible to cover holonomy at even degree; global parity×deck relabel still creates no localized charged intertwiner.
Required assumptions
Fixed charge-preserving connection in original one-copy eightfold
Given one-copy fold has transition C(h), C(h)^8=hI. Available register half flux changes cover holonomy h to -h. Compare uniform spinor-OPE twist -C(h) in same complex U1 charge dictionary and connection, up to charge-preserving fiber conjugacy/periodic gauge transformation. Real SO16 charge conjugation at h=±i, changed connections/domains and eight independent copies lie outside exclusion class. Source circle/P1 core and cover/P1 deck are not identified.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.