Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Fine-structure constant
The published foundation in this area
7 authored claim references lead into the original ledger and paper sections. This source record remains available even where no separate reviewed research packet is attached.
The α program contains an explicit fixed-point equation, determinant-line reformulations, and precise finite checks. A positive root is exhibited for the specified equation; finite checks do not establish global uniqueness. make_F ignores c3 and Nfac=8 is selected, so physical origin and transfer to Thomson α remain open.
Starting structure in the Atlas orientation
The specified equation has a positive solution. v3 finite tests support only their checked cases; make_F ignores c3 and Nfac=8 is selected. They do not prove global uniqueness.
Which connection is sought
The open work is a source derivation of the function and its parameters, determinant/Ward origin, and a justified transfer to Thomson α.
How it connects
φ0 and other architectural inputs connect the α branch to flavor and other outputs; this shared dependence limits the independence of matches.
Complete matrix reconstruction in the specified neutral quadratic vacuum sector; neither four measurements nor all charged, state or multitime responses of a physical source.
Read the exact result, assumptions and evidence
Result
P5, CPCᵀ, JPJᵀ and C²P(C²)ᵀ generate all36 real-symmetric8x8 matrices through Jordan words up to degree4. The double-pole OPE realizes these products. An explicit rational inverse reconstructs A from36 mixed responses Tr(BℓA)/2; prescribed trace leaves35 directions.
Required assumptions
Native clock fields and mixed OPE access
Fixed free eight-current vacuum, original C/J/P5 in the same E8 index basis and real-symmetric quadratic fields; access to mixed two-point responses of iterated weight-two double-pole OPE coefficients is assumed. Four individual measurements or a physical detector from P1/P2 are not thereby supplied.
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.
Read the exact result, assumptions and evidence
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.
Read the exact result, assumptions and evidence
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.
Read the exact result, assumptions and evidence
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.
Read the exact result, assumptions and evidence
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.