Standard coherent reconstruction, implemented and applied exactly here;1–5 modes. No P1/P2-selected field dictionary, geometric locality or original Φ-dependence. The minimum's term is an objective artifact. T00, reference eigenvalue, matrix logarithm and Grassmann logarithm remain distinct.
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Result
The full coherent CAR kernel is invertible and sews exactly. Its4^n duality values determine finite gluing; T00·exp(-S_nil) reconstructs all link vertices. Existing five-mode certificates give actions through degree10. The charged interacting control retains actual quarter time and absolute sea factors.
Required assumptions
Complete kernel in a fixed CAR dictionary
Given: full even T in a fixed labeled finite CAR Fock basis and explicit boundary/trace condition. Link logarithm needs T00≠0; finite selfadjoint matrix logarithm needs T>0. Physical identification, length and original Φ-dependence are not presumed derived from P1/P2.
Exact independently checked inverse-data control. Equivalent when all fields are conjugated together; locality of U is unproved. Not a physical alternative candidate or full P1/P2 countermodel.
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Result
For all alpha,beta, H and UHU* retain the full thermal partition, charge, quarter phase, sea energy and joint eta/sea/Ward identity. U fixes vacuum and one-particle sector and hence all vacuum two-field time responses. Yet the fixed neutral pair response at t=log2 is 1/2 versus13/50 after removing the same common sea factor.
Required assumptions
Fixed fields in the four-mode diagnostic
Given: the first two particle/hole pairs of one chiral CAR source, energies (alpha,1+alpha,1-alpha,2-alpha), the same Hurwitz sea energy and fixed field labels. U rotates only neutral pairs p0h0/p1h1 by 3/5,4/5. Spatial locality or P1/P2 selection of U is not assumed; this is not the internal P2 V16 carrier.
Only the general proof step is excluded. The a=1 control is not an admissible elliptic Dirac operator or a P1/P2 countermodel.
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Result
The inference in original source01:397-405 is invalid: P=aA has relative bound |a|. The independent quarter/reflection control A=sigma_x p_theta+sigma_y p_r, P=a I p_r retains the quarter domain; at a=1, A+P on H1 is not closed. Repair: an actual relative bound<1, bounded symmetric perturbation, or a separate theorem for the total elliptic operator and boundary domain.
Required assumptions
Audit of the general perturbation argument
Given: a selfadjoint first-order anchor with H1 graph norm and a symmetric first-order perturbation with bounded coefficients on the same domain. The quarter/reflection control uses the explicit compact torus operator in section11.2, not a claimed original P1/P2 source.
Source audit and reuse of the known geometric Green identity. No derivation of fermionic statistics or physical operator from P1/P2. the reviewed source checked against current Ludewig/Roos primary text1909.04212v4: algebraic twist also needs closed harmonic determinant factor; published2020 coherence proof correction is in v4. This is no new original source derivation or established error in direct TFPT collar proofs.
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Result
P1/P2 does not fully specify the primitive kernel's field arguments. The archived reconstruction starts with algebra, time flow, state and collar operator; CAR and anticommuting sources are later premises. For a supplied product Dirac operator, the classical Green boundary form follows from normal symbol and boundary metric. Identification with the quantum field amplitude does not follow.
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.
Conditional absolute extension of the supplied single chiral source. Regularization is input; neither subtraction nor bulk/eta cancellation of the central phase is proved from P1/P2. Angular determinant and thermal partition sum are different functionals. The finite excitation-product control already retains the full regularized sea term. The response is twist-conjugate spectral charge, not a new physical event law.
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Result
The declared Abel/zeta subtraction gives E_sea=-1/96. At beta=2ell=8log4, q_beta=1/65536 and Z_beta[0]=exp(-beta E_sea) times the complete particle/hole product. Its two-mode factor is2^(1/6)69649/65536. Adjoint field action stays i/-i; the absolute time-loop operator additionally carries exp(-i pi/48). Existing tail bounds apply at the same beta. Under the same Hurwitz prescription E_sea=eta(0)^2/8-1/24 and d_alpha E_sea=-eta(0)/2. The logarithmic alpha derivative of the unnormalized thermal partition sum agrees with the occupation response including the sea term, checked by an independent Fock trace and explicit derivative tail bound. The resulting actual reference amplitude2^(1/12) passes the existing CLI moment search with206 source moments; the same data with reference amplitude1 are infeasible. This is a numerical software control under the same supplied regularization.
Required assumptions
Given chiral source in both quarter domains
D_alpha=-i∂theta on f(theta+2pi)=exp(2pi i alpha)f(theta), filled negative spectrum and relative particle/hole excitation time; alpha=1/4,3/4. This compares two conjugate domains, not opposite charges in one fixed domain or all P1 sources.
Charged CAR algebra and original negative sea
Assume the actual field algebra is charged CAR(H) and its reference state is the pure quasifree sea with occupation 1−C. Relative charge and positive excitation time are normal ordered against this sea. This physical identification is an additional premise.
Explicit Abel subtraction of chiral sea
For the already given single chiral particle/hole source, use the exponential Abel sum of negative sea energies and subtract only its -1/epsilon² term. This is a declared absolute regularization; its selection and compensation of the central loop phase by an original eta/bulk contribution are not presumed consequences of P1/P2.
Given angular quarter operator and zeta cut
Given a single circle operator D_alpha=-i partial_theta with f(2pi)=exp(2pi i alpha)f(0), 0<alpha<1, complex-fermion zeta quantization and negative eigenvalue argument +pi. The quarter source has alpha=1/4; alpha_ref=3/4 is a diagnostic comparison, not archival D_ref. Its angular Green propagation is not positive occupation covariance or already physical time evolution.
Supplied zero-mode-free angular circle operator with fixed spectral cut. Not the full physical TFPT source operator, not thermal occupation, and no identification of the diagnostic reference branch with D_ref. Transport periodic operator and source frames together when varying alpha.
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Result
For supplied D_alpha, zeta quantization gives det D_alpha=1-exp(2pi i alpha), exactly1-i at alpha1/4 with eta(0)=1/2. The same operator gives G_-=i/(h-1), G_+=ih/(h-1), jumpi and retained holonomyh=i. The joint source derivative contains partial_alpha log det=pi cot(pi alpha)+i pi; the extra i pi is the eta contact term. The general determinant formula also retains zeta(K²;0).
Required assumptions
Given angular quarter operator and zeta cut
Given a single circle operator D_alpha=-i partial_theta with f(2pi)=exp(2pi i alpha)f(0), 0<alpha<1, complex-fermion zeta quantization and negative eigenvalue argument +pi. The quarter source has alpha=1/4; alpha_ref=3/4 is a diagnostic comparison, not archival D_ref. Its angular Green propagation is not positive occupation covariance or already physical time evolution.
Exact for supplied free K, measure and source coupling. The example integrates a hole mode, not a geometric collar interior. Coherent source contraction K^-1 is not normal-ordered occupation. No general no-go for a separate bosonic spectral action plus fermion determinant and no original P1/P2 quantization.
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Result
Direct Grassmann integration gives det K exp(bareta K^(-1) eta). Schur reduction jointly retains det K_ii and the inverse effective boundary/mode matrix. Two collars of the actual lowest quarter particle/hole pair under the supplied fermionic trace give det K=69649/65536 and occupations1/17,1/4097. Direct identification of the primitive spectral action with integrated complex fermions requires x f'(x)+1=0, hence a logarithmic profile; arbitrary f is insufficient.
Required assumptions
Explicit complex-fermion source quantization
Given the actual invertible free kinetic operator K, its reference, Grassmann source couplings and oriented Berezin measure with integral exp(-barpsi psi)=1. A mode partition is not a spatial boundary decomposition without geometric identification. Neither comparison part D_rel nor a general spectral profile is automatically identified with this K or -log det K; this quantization is not presumed derived from P1/P2.
Software correction with supplied lambda; no original absolute amplitude selection. The scaled control matrix is not a new physical quarter source. The added unit test concerns one fixed OS system; unnormalized unit norms may vary across changing geometries/sources. No independently proved pointwise scalar identity or general origin route is excluded. Distinguish source derivatives from geometric derivatives and reuse the existing coupled-reference result.
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Result
Search and both entry paths retain T Omega=lambda Omega with explicit positive lambda, matching bounds and cache identity. A rational scaled control kernel passes at lambda=1/16 and contradicts the same moments at lambda=1. Pure Fock completion accepts only vacuum_effect, lambda=1 and ceiling1; other classes remain inconclusive. omega(1)=1 normalizes only the conditional state and does not reconstruct Z[0]. In the actual original OS call, unital translation fixes [1] and its matrix element is time independent. This does not identify it with the nonstationary old cap vector as proposed by The reviewed. Retain the original reflected insertion caller omega_rel(Theta F tau_t G); a cap insertion must identify the actual old sea vector. The reference cancels from Zrel[J]/Zrel[0], not from geometric variation of logZrel[0].
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.
Complete construction within the already supplied free source. No P1/P2 selection or infinite particle-hole unitary in empty Fock space. Convergence is in a common HS/Choi space, not of raw filled Slater vectors in empty Fock space. Relative partition sums do not recover the divergent absolute sea scalar. A single chiral D_alpha branch; the entire doubled graded collar is not identified with the same time/loop generator.
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Result
In the supplied sea-relative chiral CAR model the same particle-hole map produces charge N_+-N_-, positive energy and W=exp(2pi iH)=i^(N_+-N_-). The original signed two-mode amplitude diag(1,1/4,64,16) becomes diag(1,1/4,1/64,1/256) through that map and the separate sea factor64. The infinite weighted l2 domain, geometric trace tails and explicit operator-norm/Choi-HS bounds control state and time beyond finite windows.
Required assumptions
Given chiral source in both quarter domains
D_alpha=-i∂theta on f(theta+2pi)=exp(2pi i alpha)f(theta), filled negative spectrum and relative particle/hole excitation time; alpha=1/4,3/4. This compares two conjugate domains, not opposite charges in one fixed domain or all P1 sources.
Charged CAR algebra and original negative sea
Assume the actual field algebra is charged CAR(H) and its reference state is the pure quasifree sea with occupation 1−C. Relative charge and positive excitation time are normal ordered against this sea. This physical identification is an additional premise.
Canonical fermionic quantization of supplied full boundary data. A geometric Cauchy projector is not automatically a physical CAR covariance; boundary pairing is also a premise. The absolute scalar, partition determinant and phase are not recovered from normalized G. Original action permits higher interaction vertices; general OS sewing from complete Schwinger functions remains separate.
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Result
If the full two-face graph G is the actual CAR covariance, positivity and CAR force its state ray without a separate Wick assumption. The specified oriented input/output pairing reconstructs Q proportional to the direct sum of Lambda^n R; multi-field amplitudes are minors and Q(R2R1)=Q(R2)Q(R1). Complex frame changes and noncommuting composition are checked. Vacuum and one-particle responses alone are insufficient.
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.
Actual CAR boundary covariance and oriented amplitude pairing
The full geometric two-face graph G is additionally identified as the actual CAR state covariance. The first labeled fiber is input/dual, the second output; the source realizes the specified fermionic boundary-state/operator pairing. These are physical identification premises, not consequences of matrix sizes or classical Cauchy projection. Normalized data determine only the operator ray and not the absolute partition factor.
Executable finite software and source controls; no physical source selection. Historical min/max results are calibration endpoints, not preferred solutions. The actual charged-pair SCS status is optimal_inaccurate; the exact matrix construction independently proves feasibility, not clean numerical optimality.
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Result
The default is feasibility without optimizing state or response. The explicitly duplicated lowest-mode calibration gives1/16; the actual lowest quarter-source particle/hole pair gives1/256. A global floor1/4 is an artificial restriction. Thermal GNS transfer with spectrum1,1,1/4,4 passes at bound4; bound1 is a class mismatch. An unsupported pure Fock completion remains inconclusive.
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.
Declared finite CAR/transfer class. Five-mode completion additionally assumes a minimal irreducible realization with inferred pure occupation state; no uniqueness inferred from truncated ranks. Floor and cluster gap 1/4 are search bounds, not quarter holonomy or physical scale. Finite clustering is not a local OS net. No original selection of marking, statistics or events. Historical extremal calibration; since2026-10-16 implementation defaults to feasibility with explicit transfer classes. The original boundary/carrier/S/Z/OS/Gamma chain exists. This optional finite search class is not a necessary global TFPT origin prerequisite; R101 extends the search without physical promotion of this historical construction.
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Result
The new moment search takes no supplied H, rho or target K. A one-mode result has an exact flat GNS certificate. Two additional minimal five-mode completions replay all 2245 moments through degree4. Checking the original source requirements adds finite clustering: a previous result with 27 stationary directions fails; two replacements each have exactly one stationary direction. Both yield finite unitary field time from the same T and first differ at degree5 by 1/4 versus 3/4. An independent offline verifier confirms full kernels and separation.
Required assumptions
Declared finite positive kernel search class
Given up to five CAR modes, an extra selfadjoint transfer operation, 0<=T<=I, T Omega=Omega and internal CAR quarter invariance. Five-mode completions additionally use a minimal irreducible Fock representation with pure occupations inferred from searched moments. Objectives, positive floor and cluster gap are declared search choices; neither these nor the representation are presumed consequences of P1/P2.