Exact counterexample only to naive operator/projector averaging on the specified actual flat Craig background. Does not exclude whole-source sums or derived blocking or prove original TFPT/GW equality.29 checks of the addendum pass,no Lean validation or ledger promotion.
Read the exact result, assumptions and evidence
Result
Actual Craig Eq16 on the periodic flat one-site4torus with temporal SU3 center z0^k=diag(omega^k,omega^k,omega^k,1,1) embedded in SU5 gives triality-one D0=0,D1=3I/2+i sqrt3 gamma4/2,D2=3I/2-i sqrt3 gamma4/2. All plaquettes areI,HW²=I and all three GW/projector identities hold. Mean D=I has GW residual gamma5;mean hatPminus=I/2 has projector defect-I/4. Projection must transport whole sources with their own operators,field spaces and measures. Operator averaging is not valid source averaging.
Required assumptions
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.
Compatible GW operator and Berezin domain
Mathematical scope: finite gamma5 Hermitian GW operator with coefficient1 and actual compatible ordered fermion spaces. Inverse formulas apply only on a common invertible domain; fulfillment by the original TFPT source is not assumed.
Actual center rule on the named existing CAR realization and explicit conditional transfer application;no primitive carrier/Padm/field identity established. Remaining selectors,dynamic bosons,reference and R068 whole diagonally neutral Slater source retain their types. No extra96 CAR duplication,new source or physical promotion.
Read the exact result, assumptions and evidence
Result
On the existing original Splus16 CAR representation the Z3 center selector has rank21856 in the65536D exterior space,degree-one rank4 and degree-two rank42. No one-particle projection can generate it by exterior lifting: degree one forces p0,whose whole rank is16 and degree-two rank6. All84 original v252 matrix directions commute with color center. On any genuinely compatible conditional quadratic CAR transfer the whole center kernel is exactly Kc(barv,u)=1/3 sum exp(barv z^k R_b u). Thermal terms have boundary conditions psi(beta)=-z^-k psi(0) and explicit Green inverses. Selector coefficients are boundary operations,not thereby derived local bulk vertices.
Required assumptions
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.
Exact counterexample to the specified unblocked ordinary product identification at the stated momentum. No exclusion of derived nonunitary blocking, other regulators, continuum equivalence or TFPT; no already produced TFPT lattice kernel assumed.
Read the exact result, assumptions and evidence
Result
At the same free p=(pi/2,0,0,0),a1 the ordinary product gives K=i gamma1 pi/2, actual Craig D=1+i gamma1 and Cayley-required D(1-D/2)^-1=2i gamma1. The fixed Cayley identity fails. Nor can a two-sided unitary chiral map identify the distinct singular values pi/2 and sqrt2. A_-^dagger A_-=I2/2 yields Berezin Jacobian magnitude2 per two-dimensional Weyl fiber with unchanged anti-leg. This example and an independently formed product action do not compute the whole original TFPT measure identity.
Required assumptions
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.
Compatible GW operator and Berezin domain
Mathematical scope: finite gamma5 Hermitian GW operator with coefficient1 and actual compatible ordered fermion spaces. Inverse formulas apply only on a common invertible domain; fulfillment by the original TFPT source is not assumed.
Whole original finite matrix content on the already conditional ordinary4D product consumer. No derivation of an original fermion term or actual interacting S_adm; no arbitrary gauge connection check or physical/ledger promotion.
Read the exact result, assumptions and evidence
Result
On the existing conditional ordinary Spin4 product branch, the whole symbolic original v252 Yukawa/Majorana family gives D=-i gamma.nabla tensor1+gamma5 tensorPhi and M_e=E_e^T(C4 tensor KF)^dagger D E_e. Both total Gamma fibers have rank192. The symmetric derivative coefficient becomes skew after integration by parts; the whole mass block is skew with squared norm2Tr96Phi^2. The isolated symmetric KO6 pairing therefore does not kill the spacetime pairing.26 exact symbolic checks pass. The added fermionic product prescription remains unbound to the whole original S_adm Hessian.
Required assumptions
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.
Formulas and precisely unresolved application only. Gaussian lemma requires actual compatible Grassmann spaces; original higher terms/Pfaffians are not replaced by an added full96 determinant. Numerical curvature is a finite diagnostic without claiming Craig threshold coverage or an all-small-field proof. No CHIRAL4D/TOE promotion.
Read the exact result, assumptions and evidence
Result
Full finite ordered Gaussian Berezin elimination gives Kp=KT+barB alpha B, Da=alpha-alpha B Kp^-1 barB alpha and necessarily detKp/detalpha plus the original source contact term. The concrete Craig equation is Da^-1=alpha^-1+B KT^-1 barB=DC[U]^-1 on the inverse domain, with zero modes handled by source minors. After a genuine line isomorphism dtheta=jC-jT must hold with matching holonomy and original component factors. The reviewed original sources contain no evaluated identity of these concrete4D operators/lines over all integrated backgrounds. P_adm is not automatically the Craig plaquette indicator. Existing S/Z/OS/Gamma and whole old Clifford source maps are retained.
Required assumptions
Compatible GW operator and Berezin domain
Mathematical scope: finite gamma5 Hermitian GW operator with coefficient1 and actual compatible ordered fermion spaces. Inverse formulas apply only on a common invertible domain; fulfillment by the original TFPT source is not 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.
Exact exclusion only of the simple globally invertible Cayley chart. No universal overlap/TFPT no-go and no mandatory lattice/continuum route.
Read the exact result, assumptions and evidence
Result
For a free periodic even4D torus Craig Eq16 gives D=2 and A=0 at p=(pi,0,0,0), with exactly trivial plaquettes. The Cayley field redefinition is not invertible on this permitted whole space. Finite GW operators with balanced gamma5 trace and nonzero index also have2 modes. Other sectorwise maps, genuine blocking, other fixed spin boundary conditions, and TFPT itself are not excluded.
Required assumptions
Compatible GW operator and Berezin domain
Mathematical scope: finite gamma5 Hermitian GW operator with coefficient1 and actual compatible ordered fermion spaces. Inverse formulas apply only on a common invertible domain; fulfillment by the original TFPT source is not assumed.
Compatible finite field spaces or common invertible domain only; no global field map, locality/RP derivation, or physical measure identity.16 symbolic controls and direct Craig operator checks executed; no Lean validation.
Read the exact result, assumptions and evidence
Result
For gamma5 Hermitian GW D, A=1-D/2 intertwines modified and ordinary chirality. On an invertible domain Kreq=D A^-1 is chirally odd and anti-Hermitian and Kreq A=D. Source legs, Yukawa insertions and the actual Majorana pairing must all be transported. The restricted chiral Berezin Jacobian cannot be discarded: an exact algebraic example yields different background factors4/5 and1/2. This does not identify original TFPT K with Kreq.
Required assumptions
Compatible GW operator and Berezin domain
Mathematical scope: finite gamma5 Hermitian GW operator with coefficient1 and actual compatible ordered fermion spaces. Inverse formulas apply only on a common invertible domain; fulfillment by the original TFPT source is not 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.
The reviewed decision record says which approach can be reused, which failed in its scope and what a new attempt would have to change.
What is resolved, and what remains
Open task
Compare original4D fermion Hessian and measure line with Craig
Derive from the same original S_adm pairing a valid local covariant reduction to DC[U], preserving physical projection, dynamic boson weights, zero modes, reference and original line factor, or exclude the specified map by a source word.
Next investigation: Retain R112 classical action/boundary/bosonic variation and new whole center transfer. Determine actual original field images of Pprim/PTheta/Pgap and joint dynamic boson/reference call. Compare the whole source kernel including boundary selectors with Craig;do not use a one-particle projection for whole Psing or average Craig operators/projectors. Distinguish linear bulk quantum-source identity from projected boundary operations;do not force a Gaussian whole source without proof. Original bare Hessian and physical source identity remain uncomputed.
Original historical attempts linked to this question
These are original research records. Their reported verdicts are distinct from the reviewed results above.
craig original source bridge 20261103
Does Craig overlap measure represent the original TFPT full graded source, and what exact operator and measure maps are required?
Recorded verdict: PARTIAL_WITH_EXACT_LOCAL_TRANSPORT_AND_GLOBAL_CHART_OBSTRUCTION
Historical inventory; not independently reviewed by this export.