Source provenance and exact algebra under each stated normalization, carrier, primitive sector and defect response contract. No general nonderivability theorem, no retraction of the canonical circle defect line, marked D5 response or positive root uniqueness. No new physical selector or ledger promotion. Historical linear, exponential, quadratic and resummed openings are different calculation contracts; similar coefficients do not prove identical physical actions.
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Result
Older originals give explicit factor provenance: V1.06 uses g=n/(8pi²), n=1 and g=8c3². ImprovedCubic of18September2025 gives linear backreaction1-2alpha; v2.7 derives exp(-alpha J) under additive defect coupling with J=d SF=2. In the later carrier gamma=tr_EY²=3/9+2/4=5/6 and B=rankE3/rankE2=3/2, so B gamma=5/4; family trace tr(S+)Y²=10/3 gives b1=41/10 for three families and one Higgs. The rank-one determinant yields (1-q)^(-5/4); its response q+(5/4)q²+(45/32)q³+... embeds the earlier two-defect coefficient in a fixed series. These factors must not be dismissed collectively as unexplained fits. Full identification remains narrower: v2.7 declares the CFE potential coefficients as supplied scheme inputs; v4.2 supplies P_loc'=alpha³-2c3³alpha² as a condition and defines Gamma by integrating the desired response. The master proof identifies its U(1) block with the already-defined EM action; it does not evaluate an original coupled Dirac determinant to obtain that F there.
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.
Model results remain conditional
Calculations in a fixed E8 net, quotient, family packet, or readout apply only under their stated inputs and do not select those inputs.
Source-reviewed status/type correction, not a new general nonderivability theorem or gate. P1/P2 remain accepted starting principles; being axioms is not an objection. The line is derived on the old circle, not an arbitrary vector to select again. v382 replay26October:6/6; literalTrue status checks do not prove open functional origin. Ledger Numerical and full physical status remain unchanged.
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Result
Coefficient identities and the unique alpha root are retained. The old circle canonically supplies Pi and det_F(1-qPi)=1-q; for trace-class C-P the same relative factor survives exactly in every faithful KMS polarization class on that same circle. Actual full positive projection covariance fixes the whole CAR state without an extra Gaussian premise. The finite Quillen chain is beyond its older open summary: v985 resolves the finite mu4 channel swap; v995 closes finite KMS rigidity and state extension; v1014 checks the finite determinant-line restriction isomorphism and records reference A0. Do not reopen those steps. What remains is the original local EM source/operator map and relative real renormalized determinant norm producing the seam factor with response exponent5/4, coupling q(alpha), local variation terms and normalization in the same channel. Abstract continuum/Bismut-Freed identification and matching to the physical Thomson coupling do not follow from finite calculations. Chiral phase cancellation does not refute real EM susceptibility. EM.WARD.01 and ALPHA.QUILLEN.EXACT.01 retain their existing physical residual.
Required assumptions
Specified EM function with positive seam response
F(x)=x³−2c³x²+(4/5)c⁶M log(p+q(1−q)^(−5/4)), c=1/(8π), p=1/(6π), q=48c⁴exp(−2x), M>0; original budget M=41. The function and inputs are premises here.
Model results remain conditional
Calculations in a fixed E8 net, quotient, family packet, or readout apply only under their stated inputs and do not select those inputs.
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.
Same original circle with trace-class changed polarization
For the mathematical continuation retain H,B_a psi_n=(n+a)psi_n,0<a<1,u psi_n=psi_(n+1),P=1_(B_a<0),uPu*=P+Pi. C is a positive contraction with trace-class C-P; actual functional-calculus C_beta=(I+exp(beta B_a))^-1,beta>0 belongs to this class. Physical KMS/OS/EM source identification is not presumed.
Test a positive joint CAR source against the full geometric projection
Consider an arbitrary normalized positive state on the specified joint charged two-face CAR algebra,without assuming quasifreeness or U1 invariance. The geometric target is the existing closed chiral annulus graph P_A with both marked faces,W=1/2 and original NS B_Sigma. Actual identity C_source=P_A is an explicit condition of the uniqueness theorem and an unresolved P1 source equation,not a fact inferred from geometry or RP. A finite error test applies only on a genuinely P_A reducing CAR block. Complete Dirac vector source,OS Padm and physical time are not identified with this chiral scalar target. No freely chosen purification supplies missing physical cross correlations.
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 response in the specified existing source, not a new field or fitted exponent. Does not derive physical preparation, u=48c3^4, q=exp(-2alpha), outer uq, baseline1/(6pi), or original EM instrument. Common time is not sector-resolved D5 readout. Replay26October2026:21/21; finite coefficient checks through degree12 do not replace the general series proof.
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Result
Exactly G_u(q5,q3)=[(1-u)/(1-uq5)]^(5/4)[(1-u)/(1-uq3)]^(3/4). Hence G_u(q,1)/G_u(0,1)=(1-uq)^(-5/4); the common-time diagonal G_u(q,q) instead has exponent2. Powers follow from actual charge norms and the convergent oscillator series. Normalized local T5 insertion has Ward coefficient5/8. Fixed C/J do not generally preserve P5 but jointly transport charge and marking covariantly.
Required assumptions
Marked regularized E8 charged state and sector-grade response
Existing positive E8_1 lattice vacuum source with original D5/A3 marking and 48 retained charges beta: beta^t P5 beta=5/4, beta^t P3 beta=3/4, beta²=2. The regularized charged vertex state has free preparation parameter u=r² in (0,1). N5,N3 are strongly commuting nonnegative oscillator grades; q5,q3 in [0,1] parametrize spectral readout. Keep original C/J matrices. u=48c3^4, q=exp(-2alpha), physical preparation, instrument and common time do not follow.
Model results remain conditional
Calculations in a fixed E8 net, quotient, family packet, or readout apply only under their stated inputs and do not select those inputs.
Mathematical theorem for the fixed function; not a cubic polynomial because the seam response depends on x. No parameter or physical Thomson-transfer derivation; ledger Numerical unchanged.
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Result
Log-convexity of the stated seam response and the sign before x=2c³ prove exactly one positive simple root globally. For M=41 its reciprocal is approximately 137.0359992168407. The analytic proof is distinct from the 10 rerun symbolic/interval checks. It holds for every M>0 and therefore does not select M=41.
Required assumptions
Specified EM function with positive seam response
F(x)=x³−2c³x²+(4/5)c⁶M log(p+q(1−q)^(−5/4)), c=1/(8π), p=1/(6π), q=48c⁴exp(−2x), M>0; original budget M=41. The function and inputs are premises here.
Historical minimal-transfer definition with fixed readout; the single line has since been canonically reconstructed on the old circle. Neither this reconstruction nor independently proved root uniqueness is retracted. Original physical P1/EM mapping and renormalized Thomson observable remain separate.
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Result
For the defined rank-one projector U_Sigma^min and stated parameters, T_Sigma=(1-delta_top exp(-2 alpha))^(-5/4) and the displayed scalar closure equation follow algebraically. This calculation does not independently prove physical selection of that projector, parameter set, or transfer to the Thomson readout.
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.
Model results remain conditional
Calculations in a fixed E8 net, quotient, family packet, or readout apply only under their stated inputs and do not select those inputs.
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.
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.
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.
Applies on a faithful finite KMS sector or the same gapped transfer sector; physical TFPT transfer remains unidentified.
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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.
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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.