Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Quantum and boundary structure
The published foundation in this area
32 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.
Finite E8, Pauli/Clifford, code, state, covariance, and instrument constructions supply internal quantum models and scoped guarantees. A code or successful channel is not yet a selected, repeatable physical source with the same parent dynamics.
Starting structure in the Atlas orientation
Concrete finite code and operator structures exist, with local results within explicitly specified source and instrument menus.
Which connection is sought
Open items include common physical preparation, source, interaction, repetition, record preservation, and connection to the TFPT parent.
How it connects
Codes and clocks specify internal information and symmetry structures; gravity additionally requires local geometry and dynamics.
Audit of this definition, not a disproof of all other selection theorems. Its electromagnetic alpha is not automatically the quarter-phase parameter in the CAR energy test.
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Result
The definition fixes d*_disc and P_adm first. Its barrier is zero for B≅B_min and infinity otherwise. M therefore optimizes continuous xi on the already selected branch; this definition neither classifies B_min nor derives the least charge-one excitation rule.
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.
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.
Global variational theorem in a given scalar CAR source; no change to B and no finite-search proof. General unitary vacuum symmetries may yield the same minimal state energy. The least-excitation selection rule and origin of fields, state and time are not derived from P1/P2.
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Result
With B_alpha and the reference sea fixed, every unimodular H1 symbol u=exp(i phi) of winding one has E(R_u Omega)=alpha+½⟨(phi prime−1)²⟩. The unique scalar-multiplier minimum is u=exp(i chi)z. Both b_(1/4) and z²/b_(1/4) have excess 1/15. The produced state changes, not the reference sea.
Required assumptions
Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
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.
Complete analytic theorem in the supplied scalar Hardy class. Leakage excludes only the pure-isometry route, not general physical sources. Not a P1 classification.
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Result
Every continuous scalar unimodular winding-one symbol with isometric Hardy compression is a single Blaschke factor modulo phase. With B_alpha fixed, only the centered factor z has constant shift covariance. The strictly oriented control z²/b_(1/4) has leakage 15/4096.
Required assumptions
Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
Conditional analytic reconstruction on a common core; Lean verifies the algebraic identity and mode recurrence, not physical derivation of inputs. One channel is not the five-slot P2 carrier; no event law.
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Result
Under the operator hypotheses, Wold supplies a complete mode basis U^n w and B=N+alpha. With the CAR/sea premise, R c_n R*=c_(n+1), R*QR=Q+1 and R*H_alpha R=H_alpha+Q+alpha. R Omega is a new charge-one state; exp(2pi iH_alpha)=exp(2pi ialpha Q) retains the quarter phase.
Required assumptions
Hypotheses of joint operator reconstruction
For this conditional theorem: orthogonal projection C, unitary U, (1−C)UC=0, index(CUC)=−1, pure isometry and minimal unitary dilation. Actual selfadjoint B satisfies U*BU=B+1 and Bw=alpha w on a common invariant core, alpha=1/4 or 3/4. These conditions are not presumed derived from P1/P2.
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.
Audit of the cited proof only; not a general impossibility theorem for P1/P2 or a revocation of declared unit winding.
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Result
The original argument first states that uniqueness needs classification of the minimal stratum, then asserts it without that classification. Well-ordering N4 yields a minimal value, not a unique operator fiber. Nonzero spectral flow alone also does not prove attainability of one.
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.
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 field-type comparison in the same given source. No universal exclusion of E8 or future source/sector extensions.
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Result
Each of the four candidates acts on the eight original complex fields with two copies of every quarter phase. Their neutral bilinears form A3+4A1 with twenty roots and rank seven, rather than D5+A3 with fifty-two roots. The latter uses the stipulated spinor completion. The new marking changes neither K nor the known ground-state comparison and supplies no event instrument.
Required assumptions
Fully normalized time/charge family in the given E8 completion
The given chiral quarter source, its eightfold pushforward and stipulated even E8 completion supply K_alpha and the full additive U1 generator Q. Only characters in the real span of their torus vectors modulo E8 are searched; no P5 is supplied as a search variable or target projector. Physical identification of this completion and of a composed marking is not assumed.
Stipulated E8 quarter-fold dictionary
The existing Weyl lift M, affine E8 module, Q'=2(H3+H6) and K_alpha=L0+delta_alpha.H0 with alpha=1/4,3/4 are held fixed for comparison. This representation is not derived from P1/P2; an unbroken SM time action is not assumed.
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.
Given chiral source, fold, full normalized charge and stipulated E8 completion. Algebraic construction and Lean lattice theorem; no physical selection of W or source from P1/P2.
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Result
Full lattice saturation reduces the continuous source torus to sixteen four-torsion characters, twelve primitive and exactly four D5+A3 markings. P5 is reconstructed from neutral root moments; a control without M or a target projector finds the same four. W=exp(4πiK)exp(-iπαQ/2) realizes one while preserving quarter time; sheet transport gives W Zq. Compensation was previously known; complete classification and control are added here.
Required assumptions
Fully normalized time/charge family in the given E8 completion
The given chiral quarter source, its eightfold pushforward and stipulated even E8 completion supply K_alpha and the full additive U1 generator Q. Only characters in the real span of their torus vectors modulo E8 are searched; no P5 is supplied as a search variable or target projector. Physical identification of this completion and of a composed marking is not assumed.
Stipulated E8 quarter-fold dictionary
The existing Weyl lift M, affine E8 module, Q'=2(H3+H6) and K_alpha=L0+delta_alpha.H0 with alpha=1/4,3/4 are held fixed for comparison. This representation is not derived from P1/P2; an unbroken SM time action is not assumed.
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.
Global charge transports in the stipulated even completion; not local vertex insertions, an ordinary-local odd CAR/E8 inclusion or event dynamics.
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Result
On the even D8 Fock sector W4=Us Rc holds with the actual ordered CAR/E8 cocycle, Klein signs, all oscillators and current central term; Rc preserves Q,K and the quarter transition. The E8 extension and lift convention are explicitly given.
Required assumptions
Stipulated even E8 completion with actual Klein signs
Assume the E8 extension of the even D8 CAR sector constructed in seam-joint-ward-state, with original ordered cocycle, retained quarter transition and explicit P-lift convention. An ordinary-local inclusion of odd CAR fields is not assumed.
Given source, grade and J>0; the two-time kernel is neither a physical measuring instrument nor a universal three-dimensional memory realization.
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Result
Original grade and transport generate M4(C(S1)) with integer carry. Full H couples grade and carry. The GNS projection has K(t)=B*exp(itLQ)B with rank K(0)=3; its minimal temporal realization may be infinite-dimensional. No external memory is appended.
Required assumptions
Declared native charge/scalar source
Fixed finite periodic spatial lattice source of rounds20–26 with original G, cocycle, neutral Z8 charges, scalar field and H=Hdiag+Hhop; a,m,beta>0 and original couplings are supplied. Nontrivial memory requires J>0. The original grade k is additionally retained, without presuming its selection from P1/P2.
Finite spatial volume and original neutral scalar/charge source; no spatial continuum or P1/P2 selection claim.
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Result
The original H is positive self-adjoint and has a faithful Gibbs state for every prescribed beta. The actual transfer reconstructs Hδ=−log Qδ/δ; removing the temporal regulator makes its unitary dynamics converge to the same H. Spatial lattice, couplings and beta remain declared inputs.
Required assumptions
Declared native charge/scalar source
Fixed finite periodic spatial lattice source of rounds20–26 with original G, cocycle, neutral Z8 charges, scalar field and H=Hdiag+Hhop; a,m,beta>0 and original couplings are supplied. Nontrivial memory requires J>0. The original grade k is additionally retained, without presuming its selection from P1/P2.
Applies to the cited operational seed category and its collar completion; it does not claim that no other original selector exists.
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Result
The original definition already assumes (A_loc, tau_t, Theta, omega, [u_Sigma], D_coll); its collar-completion theorem derives B_Sigma and spectral data from them, but does not select those inputs.
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.
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.
Construction applies only with complete coherent K values and stated domain conditions; K from the original TFPT source has not been derived.
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Result
Given a positive Gram kernel K with normalized reference, compatible composition/adjunction, and null-stable continuation, quotienting by null vectors and completing constructs H_K and the well-defined word action. Cyclic realizations of the same complete K values are unitarily equivalent on the common word domain; for unbounded operators, closures and self-adjointness remain additional requirements.
Required assumptions
Positive word kernel with null stability
The GNS/Gram reconstruction requires a Hermitian positive kernel K, normalized reference, declared composition and adjunction, and stability of null relations under every allowed continuation; these are conditions on a supplied K source, not TFPT axioms proved from bare future answers.