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.
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.
Supplied finite representations. Clifford monomial grammar and generic equal matrix dimensions up to16; not a complete operator-program, kernel or continuum search. Twelve generic Lean transport lemmas; concrete matrices checked separately and exactly.
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Result
A CEGIS word search generates 2048 Majorana word/linearity classes and recovers the known five-mode particle-hole map. An automatically found imaginary mixing component eliminates the linear candidate. A second solver constructs simultaneous linear/antilinear intertwiners from exact operator pairs; evaluation identities include constraints, code and search configuration. These are software and recovery results, not new source selection.
Required assumptions
Explicit finite operator-synthesis inputs
The recovery control assumes the given five-mode CAR representation. The inverse solver assumes explicitly supplied finite operator pairs and adjoints. The map is unknown; physical source selection is not assumed derived.
Representation types of the supplied free realization; no selection from P1/P2 and no new spinor-CAR replacement.
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Result
D5_1 has ten real vector fermions; S± each contain 16 complex Ramond/spin-field states. Using S+ as an ordinary CAR coefficient would give c=16 and Spin(10) level 4. The documented c=8 connection uses 10+6 real fermions and mixed spin fields of weight 5/8+3/8=1.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
Supplied free D5+A3 realization
Compare the documented free D5_1+A3_1 realization with five plus three complex fermions and spin-field sectors; realization by the raw P1 source is not presumed.
Only fixed fold time and fixed 5+3 marking. Covariant marking transport is positively constructed; other sources or dynamical dictionaries are not excluded.
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Result
After actual quarter conjugation the D5 projections of the time axis have distinct squared norms 129/256 and 137/256. Hence no orthogonal T satisfies T d_beta=d_beta and T O(P5)T^-1=P5. This excludes every Weyl correction in this fixed comparison.
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.
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.
First 5+3 marking and Z4 character supplied
In the supplied E8 dictionary, D5 on the first five coordinates, D3 on the last three and k=2sum(p1..p5) mod4 are given for the first quarter domain. The other domain may transport these data covariantly; their original physical selection is not presumed.
Exact map of the supplied source and conditional E8 lift; first marking, source realization and physical event law are not derived from P1/P2. Both quarter domains remain explicit; no switch to another branch to solve a fixed-sector question.
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Result
n→−n−1 forces j→7−j and O=−MRM^T with O(x)=(x4,−x5,−x6,x1,−x2,−x3,x8,x7). O d_alpha=d_(1−alpha); P5_beta=O P5_alpha O^-1 and k_beta=k_alpha∘O^-1 need no new marking choice. Sea state and positive time are transported; the E8 current lift assumes the existing extension.
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.
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.
First 5+3 marking and Z4 character supplied
In the supplied E8 dictionary, D5 on the first five coordinates, D3 on the last three and k=2sum(p1..p5) mod4 are given for the first quarter domain. The other domain may transport these data covariantly; their original physical selection is not presumed.
Correction of the finite selection inference; no identification of I with the actual Calderon operator and no P1 countermodel.
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Result
CAR.PAIR.01 counts complex-bilinear singlets. The Hermitian scalar <f,g> already exists on S+; CAR.PAIR.02's 'scalar iff sheet-odd' inference is false in this reading. The previous J_C includes reversion signs and is fully Spin(10)-equivariant.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
Given chiral source. No selection from P1/P2, no equal responses within one fixed twist, and no identification with affine r0.
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Result
n→−n−1, alpha→1−alpha gives C D_alpha C^-1=−D_(1−alpha). Particles become holes and conversely; normal charge reverses and positive excitation energy is preserved. The unique sea state maps to the other sea state without adding a zero vector within one domain.
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.
Representation with metric/orientation; J_C is not yet the actual boundary/OS conjugation. Two carrier labels are not two vacua.
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Result
J_C=∏(epsilon_i+iota_i)K is antiunitary, exchanges S± and reverses Y. Exactly Lambda0 and Lambda5 have Y=0. J_C dGamma(h) J_C^-1=tr(h)I−dGamma(h). It is fully Spin(10)-equivariant. Reuse CAR.PAIR.01's bilinear pairing; CAR.PAIR.02's Hermitian selection inference has been corrected and is not evidence.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
Only the supplied affine dictionary. Backgrounds or time-dependent covariant symmetries have different premises; no general gauge-physics no-go.
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Result
Mdelta=(-1/2,0,alpha/4,-1/2,0,(alpha-1)/4,-1/4,-1/4). The six (3,2) root energies are (0,1/2,1/2,1,1/2+alpha/4,1+alpha/4). K therefore does not commute with the fixed full SM action. The quarter phase is retained.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
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.
Original carrier plus supplied E8 charge dictionary; no physical field identification.
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Result
epsilon1 epsilon4 produces u1∧u4 in the carrier: Y=1/6 and source charge change0. Root operator, spinor carrier vector and affine root-current state have different types. The exterior unit is the internal nu^c label, not the affine vacuum.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
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.
Only the named unchanged CAR/QWZ representation with the same time. No TFPT no-go; choosing a state removes no added zero vector.
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Result
All four hits add the same half-integer Cartan sectors and second K-zero vector. The already known integer-Cartan/ground-space obstruction therefore excludes equivalence to the unchanged source; positive algebraic extension remains possible.
Required assumptions
Fixed CAR source and adjoined half sector
Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2.
Unchanged specified CAR/QWZ source
Compare the known quarter-CAR source or full mass=1, ny=8 QWZ operator with the identical Hilbert representation, Cartan action and physical time. Preserve its previously proved unique ground state and spectral gap. This does not compare all P1/P2 realizations.
Normal ground densities in the supplied K space; no state selection from P1/P2. The ground space was already known.
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Result
All dual-invariant ground mixtures diag(a,1-a) on {0,r0} give the same shift functional. Original Cartan vacuum values select |0><0|; additional Rc invariance requires I2/2. These two conditions conflict.
Required assumptions
Fixed CAR source and adjoined half sector
Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2.
Positive base functional and explicit observable algebra
For finite crossed words omega0 is positive; dual Z2 invariance is additionally required for unique algebraic extension. Comparing physical densities must retain the actual Cartan observables and the common normal ground space of supplied K.