Applies to the documented derived discrete stratum and c_u/c_d; it does not determine U_f* or absolute amplitudes.
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Result
The active TFPT text states that Q geometry, det R=8, and rigidity of c_u/c_d=55/117 are established on the derived stratum; the continuous wall point U_f* is needed only for absolute amplitude normalization. The ratio is not an open monodromy/amplitude calculation.
Required assumptions
P1/P2 remain declared starting axioms
This program checks which additional source, state, event, and observable structures follow from accepted TFPT principles; it does not revoke or replace P1/P2.
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.
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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.
Reuse and navigation correction for the existing conditional bundle theorem. No new global classification, physical source selection or ledger promotion.
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Result
The September22 lift and its later recovery supply all natural W_m=LambdaEven(E) tensor directsum(det E)^m, sum m=-2, for supplied unitary rank5 E. No global square-root bundle is needed. Supplied A_E induces the explicit connection A_Wm. The global mathematical lift and extension construction is therefore already done. Actual physical A_E, coupling Phi and the common original source intertwiner do not follow. The full current finding is also finding:recovered-natural-joint-lift-and-given-connection-already-exist.
Required assumptions
Existing natural joint lift with given split connection
Assume a GIVEN unitary E=E3+E2 and GIVEN block U3xU2 connection A_E. Use only the natural iota_m with sum m=-2 completely classified on22September and their derived connection A_Wm=rho_LambdaEven(A_E)+M_m tr A_E. This does not select A_E from P1;an independent nonabelian family connection is not silently assumed.
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.
Counterexample to the generic mathematical perturbation theorem within its stated boundary-condition type. The intended special TFPT perturbation is neither excluded nor constructed; its principal symbol, domain and actual anchor require verification. No ledger or paper promotion.
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Result
A=-i d/dx on the interval with periodic H1 domain is a self-adjoint elliptic modified APS realization: tangential operator0 and periodic trace diagonal a kernel Lagrangian. B=-A is symmetric first order with bounded coefficients but has exact A relative bound1; |n|<=a|n|+b forces a>=1. A+B=0 on Dom(A) is not self-adjoint. This disproves the original generic assertion. The valid replacement requires symmetry on Dom(A) and actual relative bound a<1. A sufficient reflection condition is a specified domain-preserving bundle lift with identical principal symbols, making D_rel bounded order0; reflection oddness alone does not lower order.
Required assumptions
Literal boundary-minimality and analytic-perturbation statements
Audit objects are the original boundary-source definitions and proofs: C_seed maps an operational seed forward to a spectral boundary tuple; defect order is lexicographic on N4; the self-adjointness statement asserts relative bound0 for every symmetric bounded-coefficient differential perturbation of order at most1. These source assertions are audited rather than assumed true. No special symbol cancellation, domain invariance or absolute scale is silently added.
Precise proof/definition audit of the cited arguments. No independence theorem, two complete P1 models or unconditional rescaling of undefined finite-block/action classes. No new physical model; historical problem status is preserved.
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Result
From d_seed=d o C_seed one obtains minimization only over im(C_seed). The original proof supplies neither attainment of the global minimum in that image nor classification of the equal-minimum fibre; well-foundedness alone gives no object uniqueness or unit spectral flow. Under explicit scale-covariant reference, action and cutoff conditions, B_lambda with lambda D and lambda B retain defects but have different spectra for a compact self-adjoint realization. This rescaling theorem remains conditional. An honest OS inverse needs complete supported Schwinger data with OS II assumptions, or a full Euclidean net/action, regular RP state and time-zero condition; arbitrary positive-time insertions do not automatically act on the OS quotient.
Required assumptions
Literal boundary-minimality and analytic-perturbation statements
Audit objects are the original boundary-source definitions and proofs: C_seed maps an operational seed forward to a spectral boundary tuple; defect order is lexicographic on N4; the self-adjointness statement asserts relative bound0 for every symmetric bounded-coefficient differential perturbation of order at most1. These source assertions are audited rather than assumed true. No special symbol cancellation, domain invariance or absolute scale is silently added.
Original P1, boundary, seed and local restriction definitions
Reviewed originals: P1 declares oriented seam, reflection-positive kernel, unit winding and c3; stronger T_boundary includes A_+ as data; operational seed takes A_loc, time, reflection, state, seam class and collar operator as inputs. A_adm(O) is A/A*-invariant restriction of supplied local insertions. Compare without P2/E8/target spinor assumptions and without presuming an original gauge/microscopic localization intertwiner.
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.
Original definition/source comparison, not logical independence or two complete alternative P1 models. Neither universal nonderivability nor incompatibility with every P1 seed proved. A specified full seed already includes its net as input. No new construction or numerical suite and no physical premise removed.
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Result
Literal scalar P1 names seam, reflection positivity, unit winding and c3; stronger boundary data supply A_+, and the operational seed explicitly takes A_loc as input. Admissible local net restricts supplied local insertions. These definitions do not identify primitive physical space and local algebras with the existing Gx=1 gauge net. Overall verdict: UNDECIDABLE DUE TO SPECIFIED DEFINITION GAP. With old local Z fixed, existing gauge isometry is not support-preserving because FZ=-ZF; the neutral pair has restricted compatibility by telescoped Gauss laws. Global Phi retains its old locality failure and cannot be composed with E without a domain/net identification.
Required assumptions
Original P1, boundary, seed and local restriction definitions
Reviewed originals: P1 declares oriented seam, reflection-positive kernel, unit winding and c3; stronger T_boundary includes A_+ as data; operational seed takes A_loc, time, reflection, state, seam class and collar operator as inputs. A_adm(O) is A/A*-invariant restriction of supplied local insertions. Compare without P2/E8/target spinor assumptions and without presuming an original gauge/microscopic localization intertwiner.
Conditional local parity-Gauss completion with common original time
ADDED comparison contract, NOT P1-derived: old four-state register with total-even matter; redundant Z2 link qubits and local Gx=(-1)^nx tau_(x-1)^x tau_x^x=1; gauge-invariant local observable net. Original QWZ hopping coefficients, common microscopic Hamiltonian and energy reference; no new electric energy. Sector-specific f(Z) subtractions must map to f(W) and can be nonlocal. No identification of Gauss space with original Padm.
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.
Originalarticle§1/§4 assumesB,mu_B1,c8,index4A=B^Z4 anddefinesomega_beta=Phi_beta^B|A. KMSextension existence is therebygiven; groupaverageE/invariantstateuniqueness applyonexistingB. Eprojectsneutralgrades,doesnotgeneratechargedlocalfields. Originalconclusion explicitlyretainsmicroscopicnonbilinear/lineidentificationgap. RotationalKMSclassification usesL0,notuncheckedH_source=8(L0+delta.p).
Required assumptions
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.
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.
Finite actual full-state/time control and analytic SEPARATE chosen-chiral comparison. No unproved transfer of Weyl sharp kernel to lattice parity continuum; actual filter not proved normal or spinorial asN->infinity. JOINT Bloch occupations not automatically separate chiralD8Cartans,no universal spinor-invisibility claim. Fixeds regularizes given chiral pair but removes neitherendpoint. NoP1 preparation selector,128/OPE/FE-GEN/ALG-EXH. Actual full-stripfilter usesH_N,thegivenchiralcomparisonK=H_cover/8; fixedpositive timeconversion preservescontinuity/compactness anddoesnotdivideprimitive spectralflow.
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Result
Actual e^(-sH_N)P_E Omega_N is pureSlater: Zp=B A^-1,Zp_s=e^(-s eps+)Zp e^(-s eps-),C_s=V_s(I+Zp_s*Zp_s)^-1V_s*. All finiteCAR words follow; exact normalized time kernel det(I+Zp_(s+t/2)*Zp_(s+t/2))/det(I+Zp_s*Zp_s). Nx32Ny8/bothquarters/all512modes checked independently by direct Slater/QR;8JOINT residues retained,both edge responses nonzero. SEPARATE given chiral Weyl comparison atd=pi has exact F_s(t)=[(1+r e^-t)(1-r)/((1-r e^-t)(1+r))]^4,r=e^-2s. s->0 gives norm1at0,kernel0fort>0; onlythis comparison limit explicitly violatesD.
Required assumptions
Actual neutral pair heat filter and separate chiral Weyl comparison
Diagnostic nonunitary attempt v_Ns=e^(-sH_N)P_E Omega_N with actual neutral eight-preimage parity pair and original full QWZ time/sea,s>0. No physical selection of this filter preparation claimed. Retain full Slater covariance,norm and time responses. Separately given chiral Weyl comparison usesK=H_cover/8,phi=f(8theta),Heisenberg oscillators; its sharp kernel is NOT identified with bare lattice parity limit without a separate proof. Joint Bloch-residue charges are not automatically separate chiral D8 Cartans.
Original neutral eight-preimage parity pair
Actual v1033 menu F_A=X² tensor(-1)^N_A and products F_B F_A*, plus the ALREADY GIVEN one-copy eightfold dictionary eta_j(u)=psi((u+2pi j)/8)/sqrt8. E is union of all8 preimages E_I of a base interval. Register factors cancel; no added Gauss rule or copy. Fold, free chiral source and P1/core-circle interface are not newly derived premises. Finite filled Fourier band is separately declared regulator comparison with a second Fermi edge.
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.
Retained localZ,full joint words and actual mass1width8Bloch matrix. Register-neutrality alone identifies no spinor; totalgrade and halfCartans distinct. New Gauss rule with globalWilsonZ changes premise and remains underived. AllN bound analytic,not extrapolated from finite energy/cusp controls; Ramond zero-mode mixture not replaced by Wick.
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Result
Remote locality to original evenZ gives exact||(Z-i^r)v_N||²=sum_q|i^(r+q)-i^r|²||v_qN||²,allnonzero-grade weights>=2; arbitrary register-changing vacuum components vanish in norm. Actual Bloch ground functione=-Tr|h|/2 has only cusp-|p| at0 and e'' of finite total variationV. Fourier bound|e_m|<=V/(2pi m²) and exact sampling give allN/all4flux |N(E0_alpha-E0_beta)/(2pi)|<=V/6. With ONE initial vacuum reference H_full>=-V/6,so off-grade components also vanish in full fixed-time kernel. No independently chosen flux energies.
Required assumptions
Retained local register with one common original vacuum reference
Original v1033 Z4 register Z remains even local and remote from the proposed localized bosonic field. Initial sea has definite quarter r1or3. Full joint words retain norm of remote commutator[Z,T_N]Omega_N,required to vanish. Register Hamiltonian stays block diagonal in Z with ONE common initial-vacuum subtraction,no independently chosen sector energies. Keep actual mass1width8 Bloch matrix with both edges and all massive modes. No new Gauss support rule.
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.
Constructed neutral pair and finite field action, NOT a new single spinor sector or analytic OPE-monodromy theorem. Eight-preimage support remains multilocal on source. No FE-GEN/ALG-EXH,128,P1-source/time/measure derivation. The reviewedRound5 pair misidentification retracted inRound6 after explicit counterexample; no uncritical review adoption.
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Result
T_E=product(F_B F_A*)=I4 tensor(-1)^N_E is an actual original register-grade0 operation. Periodic sigma(theta+2pi/8)=sigma(theta) acts on all8complex/16real folded fields as(-1)^1_I and retains r,h. SINGLE cover-half-flux argument C(-h)^8!=(-C(h))^8 does not apply to this NEUTRAL PAIR. Full original QWZ Nx32Ny8 mass1 control retains all256occupied modes perquarter; both edges changed, old seam phase retained. Finite-band comparison factorizes exactlyD_l(I)^8; odd-l half-circle blocks vanish, artificial second Fermi edge explicitly retained.
Required assumptions
Original neutral eight-preimage parity pair
Actual v1033 menu F_A=X² tensor(-1)^N_A and products F_B F_A*, plus the ALREADY GIVEN one-copy eightfold dictionary eta_j(u)=psi((u+2pi j)/8)/sqrt8. E is union of all8 preimages E_I of a base interval. Register factors cancel; no added Gauss rule or copy. Fold, free chiral source and P1/core-circle interface are not newly derived premises. Finite filled Fourier band is separately declared regulator comparison with a second Fermi edge.
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.