No complete concrete Finsupp kernel instantiation, physical state selection or automatic K-invariance of the bare shift algebra.
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Result
E0(y*y)=a0*a0+phi^-1(a1*a1) proves positivity of omega0 o E0. General algebra/star/positivity and dual uniqueness theorems are Lean-checked; concrete finite-lattice transfer is analytically proved.
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.
Global crossed operators; no local field equivalence or physical selection.
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Result
262144 candidates on two H200: four universally Lean-checked presentations. X_plus=Pi X_minus preserves source,Q,K,quarter marker and canonical state; two marked P classes remain. Full T1 and Ward are checked. Graded C lifts realize both P signs; scalar X covariance does not follow from the integer endpoint.
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.
Fixed phases and corresponding invariant starts; real time retains conjugation/time reversal. No assertion about all P1/P2 sources.
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Result
C and K may separately break; A=CK preserves H for all fixed link phases and T_p→T_−p. Invariant starts retain equal Euclidean opposite-charge returns, including quarter phases/flux.
Required assumptions
Full original charge-field source
Finite Round20 lattice/scalar space with the original bilinear E8 cocycle, complete integer charges, onsite T and balanced W, coercive quadratic Hdiag and supplied hopping. No P1/P2 selection or local chiral field identification is presumed.
Fixed link phases and A-invariant start
Arbitrary fixed unitary link phases in the same hopping, real original Hdiag/cocycle and initial density invariant under A=C times complex conjugation. Positive Euclidean T return responses are considered; no general real-time equality is assumed.
Fixed nonzero chiral quarter twist
The comparison target is the stipulated Kδ Fock space with original nonzero quarter twist, normalized ±Bp primary states, fixed charge marking and the same positive time. Equality of all these responses is required.
Balanced W remain within the sector; T insertion/removal responses require Cp. No derived physical measuring instrument.
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Result
η·Qtot is zero on C0. T_p enters Cp; full propagation yields ratio exp(−2tη·p). For v=Bp, conditionally η=Bᵀδ. Only the odd factor is matched.
Required assumptions
Full original charge-field source
Finite Round20 lattice/scalar space with the original bilinear E8 cocycle, complete integer charges, onsite T and balanced W, coercive quadratic Hdiag and supplied hopping. No P1/P2 selection or local chiral field identification is presumed.
Chemical comparison with charged intermediate sectors
Additionally supplied Hη=H+η·Qtot, C-invariant initial density in C0 and full propagation after onsite T moves C0 into Cp. η is comparison input, not a derived seam parameter.
Unchanged finite source and specified invariant start; different states or operator dictionaries are not excluded.
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Result
Original C T_p C=T_−p, T_p*=(-1)^e(p)T_−p and C H C=H. At an invariant start all C-conjugate bounded words agree.
Required assumptions
Full original charge-field source
Finite Round20 lattice/scalar space with the original bilinear E8 cocycle, complete integer charges, onsite T and balanced W, coercive quadratic Hdiag and supplied hopping. No P1/P2 selection or local chiral field identification is presumed.
Charge-conjugation-invariant start
The initial density is invariant under linear C:n→−n, including full and C0 Gibbs starts of unchanged H. Compare actual adjoint-paired, opposite marked T responses at the same time.
Only the fixed marked response identification at these starts and the same time.
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Result
The native ratio is one; the fixed quarter twist gives exp(−2tδ·Bp), which differs from one on at least one original basis direction.
Required assumptions
Full original charge-field source
Finite Round20 lattice/scalar space with the original bilinear E8 cocycle, complete integer charges, onsite T and balanced W, coercive quadratic Hdiag and supplied hopping. No P1/P2 selection or local chiral field identification is presumed.
Charge-conjugation-invariant start
The initial density is invariant under linear C:n→−n, including full and C0 Gibbs starts of unchanged H. Compare actual adjoint-paired, opposite marked T responses at the same time.
Fixed nonzero chiral quarter twist
The comparison target is the stipulated Kδ Fock space with original nonzero quarter twist, normalized ±Bp primary states, fixed charge marking and the same positive time. Equality of all these responses is required.
Known preparation/time normalization in the given E8 Fock space; identifies δ, not q,τ, physical instruments or P1/P2 source selection. Eight directions require their respective ± responses.
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Result
Cδ(p,t)=exp[−t(|p|²/2+δ·p)]. With yi=−log[Cδ(bi,t)/Cδ(−bi,t)]/(2t), y=Bᵀδ and δ=(Bᵀ)⁻¹y. Eight independent directions suffice and are necessary for unrestricted δ in this linear data class.
Required assumptions
Normalized charged primary responses with known time scale
For the given Kδ, normalized primary charge states ±bi for eight independent original E8 basis charges and their positive e^(−tKδ) return amplitudes at known common t>0 are supplied. Global charge shifts are not thereby derived local physical instruments.
Individually all-Cartan-neutral insertions and fixed vacuum; not charged intermediate fields, charged Gibbs traces or δ-dependent probe operators.
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Result
Each admissible intermediate insertion stays in M(1)⊗|0>. There Kδ=N,Qq=0,tτ=I. All ordered vacuum multitime and Gram responses in this class are therefore identical at arbitrary word length, even with complete36-dimensional quadratic tomography.
Required assumptions
Every insertion preserves all eight Cartan charges
In the same given E8 Fock space every word starts in the vacuum. Each individual insertion belongs to the fixed Heisenberg/quadratic Jordan OPE algebra and commutes with all eight Cartan zero modes; normal ordering, operator labels and initial state do not change with δ. Overall neutrality or Qq neutrality alone is insufficient.
Exclusion only of the affine lift model at fixed K,Q and oscillator level; covariant tuple transport remains possible.
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Result
An isometric affine lattice lift Ap+r with A²=−I and unchanged oscillator level would require r=Aδ−δ and hence −2δ=(I+A)r in the E8 lattice. Adjacent coordinate differences of2δ are1/4; this fails exactly. Also Jq≠q.
Required assumptions
Stipulated E8 Fock space with marked clock tuple
Given the original ordered CAR/E8 cocycle, E8 Fock representation, Kδ(p,N)=|p|²/2+δ·p+N, original q and full uncompensated quarter transition. J is the actual Gaussian lattice operator, not the cyclotomic Galois operator. A covariant comparison transports δ,q,τ together; a fixed marked source holds them fixed.
Given E8 completion; no physical sectors, switching rates or local instruments selected from P1/P2. As a torus operator τ is identifiable only modulo the dual lattice.
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Result
The original cocycle admits a unitary lift UJ with UJ⁴=I. It intertwines Kδ,Qq and the full quarter transition with KJδ,QJq,tJτ on the entire stipulated Fock space. The four frames are a covariant family; q must be transported.
Required assumptions
Stipulated E8 Fock space with marked clock tuple
Given the original ordered CAR/E8 cocycle, E8 Fock representation, Kδ(p,N)=|p|²/2+δ·p+N, original q and full uncompensated quarter transition. J is the actual Gaussian lattice operator, not the cyclotomic Galois operator. A covariant comparison transports δ,q,τ together; a fixed marked source holds them fixed.
Complete matrix reconstruction in the specified neutral quadratic vacuum sector; neither four measurements nor all charged, state or multitime responses of a physical source.
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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.
Applies as an identifiability limit for matter-only observations and an open bridge to the fixed original functional; it is not an alternative TFPT source or universal gravity no-go.
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Result
In the matter-only observation-class audit, geometric replica responses remain indistinguishable; the scope correction explicitly states that W→W+lambda Q[g] is therefore not automatically an admissible change to the original profile-free Gamma_grav. The original functional is fixed once the full operator, reference, domain, and continuation are fixed; its common off-shell geometric family and normalization must be derived from the source.
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.
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.