Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Seam, origin and declared inputs
The published foundation in this area
25 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.
TFPT starts from the oriented seam normalization P1 (c3=1/(8π)) and the marked five-slot carrier P2. Integer anchors and later compiler readouts connect these inputs, but do not derive their physical origin or selection.
Starting structure in the Atlas orientation
P1 declares c3=1/(8π); P2 declares five carrier slots with a 3+2 marking. The (1,1,2) anchor organizes internal numerical relations.
Which connection is sought
The open work is a non-circular source for these inputs, the physical seam, and derivations of their markings and readout rules.
How it connects
The declared seeds feed E8, clock, flavor, α, and scale readouts. Shared seeds make those outputs correlated.
Exact finite original case. No nonexistence of B_N, selected continuum state or E8 refutation. Previous energy test uses twist1 without zero modes and is unchanged. Numerical norm diagnostics normal and -OO byte-identical; independent proof review and rerun accepted.
Read the exact result, assumptions and evidence
Result
On unchanged v988 Nx16 Ny6 M1, explicit integral boundary zero vectors, modular ranks over F65537 and exact sign pairing prove ranks190/192/192/192. Periodic counts are95 negative,95 positive,2 zero; other twists96/96/0. Original covariance C=Pminus+Pi0/2 differs from pure occ_frame projector P=Pminus+Q with rankQ1 in the nullspace: norm(C-P)HS²=1/2 and norm(C-C²)HS²=1/8. Both original functions are valid and deliberately distinct. A state-preserving microscopic dictionary cannot silently exchange these periodic states.
Required assumptions
Actual periodic v988 state and occupied frame
Keep unchanged qwz_cylinder, occ_frame and fermi_cov at Nx16 Ny6 M1 and original twists0,1,2,3. Compare their actual finite CAR covariances. Physical selection of either state or equality with the E8 vacuum is not assumed.
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.
Only this proxy-fitted constant in this actual finite original case is excluded. The formula is exact; numerical values are checked to1e-9 tolerance. Larger constants, separate historical value2.86 and general continuum estimates are not excluded.
Read the exact result, assumptions and evidence
Result
On unchanged v988 data Nx16,Ny6,M1 the twist1 Slater state against h0 has exact variance formula Tr(P h0(1-P)h0P)=||(1-P)h0P||HS^2. Numerically it gives7.0875042817 and first excitation norm2.6622367065. The original first central sum proxy is identically0; its fit C=1.2040009128 gives C^2=1.4496181980 and already fails to bound the actual norm here.
Required assumptions
Literal charged scaling contract and original Slater case
Audit actual H_N, Psi_lambda,N and bosonic refinement definitions in the original charged scaling contract and unchanged v988 qwz_cylinder/occ_frame functions at Nx16,Ny6,M1, h0 at twist0 and Slater state at twist1. This audits that source path and finite energy case, without assuming microscopic bosonization or a uniform continuum bound.
Source audit of this particular original path. No nonexistence claim for B_N and no rejection of the established P1/P2 compiler or mathematical E8 field extension.
Read the exact result, assumptions and evidence
Result
The original charged scaling contract starts with H_N as the actual QWZ/CAR GNS space and declares Psi_lambda,N by a normal boson exponential. Its concrete refinements I_charge tensor Gamma_s(j) act instead on a separately chosen bosonic lattice Fock space. No microscopic B_N between these systems with B_2N J_N,2N=V_N,2N B_N and state, energy, parity, sector and localization compatibility is constructed there. The exact charge-lattice isometry F(2q)=lambda does not replace this field morphism.
Required assumptions
Literal charged scaling contract and original Slater case
Audit actual H_N, Psi_lambda,N and bosonic refinement definitions in the original charged scaling contract and unchanged v988 qwz_cylinder/occ_frame functions at Nx16,Ny6,M1, h0 at twist0 and Slater state at twist1. This audits that source path and finite energy case, without assuming microscopic bosonization or a uniform continuum bound.
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.
Stipulated E8_1 lattice vacuum representation. Bosonic currents are not identified with elementary4D fermions or native CCR mediators. Internal quarter grading is not physical quarter time. Contact terms, Z[0], absolute phase and original P1 Phi-dependence do not follow from these normalized coefficients.
Read the exact result, assumptions and evidence
Result
The executable E8 dictionary contains240 root vertices and8 Cartan currents with cocycle, adjoint signs and quarter grades60/64/60/64. The vertex formula and Cartan Ward recursion determine all separated current coefficients of the normalized source series. Mixed spinor currents have5/8+3/8=1; the checked connected four-point response is nonzero. Standard lattice theory implemented and independently checked here, not a new physical interaction.
Required assumptions
Reverse selection in the discrete closure framework
The origin paper discrete E8 closure framework with rank, family, spinor and simple-current conditions is given. Its numerical fixed-point identities are not identified with a physical operator realization.
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
Quarter phase is retained
Comparisons retain the original charged quarter phase; switching to another phase branch is not an admissible solution to the selection problem.
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.
Source audit of the specified reverse chain. No retraction of numerical self-consistency and no independent derivation of the framework.
Read the exact result, assumptions and evidence
Result
Origin954–981 and v6/v15/v350–v352 recover g_car=5 and eight within the chosen closure framework; v154 supplies the conditional local extension. The checked v350–v352 functions do not construct Phi-dependent D_rel/D_ref or physical insertion operators. This scopes those calculations, rather than asserting general impossibility.
Required assumptions
Reverse selection in the discrete closure framework
The origin paper discrete E8 closure framework with rank, family, spinor and simple-current conditions is given. Its numerical fixed-point identities are not identified with a physical operator realization.
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.
Two canonical invariants of the given four-mark object. Not a new proof task; evaluate original field/state maps separately. This records an existing result to repair retrieval and continuity, without claiming a new scientific finding.
Read the exact result, assumptions and evidence
Result
Existing v189 and the Origin paper obtain rank H1(P1\D)=3 and h0(P1,O(D))=5 from ONE four-point divisor D=mu4 on P1. This is an established common geometrical/topological origin of both numbers, not a new hypothesis or reopened task. Retain known anchor identities and reverse selection. Every genesis continuation starts from this theorem; open further source maps do not downgrade it.
Required assumptions
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.
Only the specified R096 branch under its jointly required premises. No derived physical Higgs measure, general H(x) field theory or P1 source completion.
Read the exact result, assumptions and evidence
Result
For actual X_H=(2cosh(ell h_H))^-1, g(u)=Tr_Q X_H=6sum_j[2cosh(ell sqrt(p²+y_j²u))]^-1. Under the specified real face frame and positive nonconstant radial law g strictly decreases, f(u)=u/(1+u) strictly increases, and Cov_mu(f,B_Q)<0. The face identifications cannot be independently rotated without the actual transport.
Required assumptions
Same constant Higgs and two-face carrier
R096 compares the same original constant Higgs parameter on the same actual two-face CAR carrier and fixed source caller. Face labels are not internal chiral L/R labels. This is a conditional source comparison, not derivation of the primitive P1 identification.
Real face frame and positive nonconstant radial law
The negative mixed covariance witness uses the original 36-dimensional quark particle module with one common real face frame, ell>0, p>0 and at least one nonzero original Yukawa singular value. Mu is positive normalized and u=H†H is not almost surely constant. Independent face gauge transport is not supplied.
Only the specified R096 branch under its jointly required premises. No derived physical Higgs measure, general H(x) field theory or P1 source completion.
Read the exact result, assumptions and evidence
Result
The conditional identities <O>_H=24m(H)^2 and <N>_H=3m(H) give Z[O]=8Z[mN] with the same integrable unnormalized weights. Positivity is unnecessary for this branch. Actual origin of the mixed insertion in the original solver remains open.
Required assumptions
Same constant Higgs and two-face carrier
R096 compares the same original constant Higgs parameter on the same actual two-face CAR carrier and fixed source caller. Face labels are not internal chiral L/R labels. This is a conditional source comparison, not derivation of the primitive P1 identification.
Actual R054 color-symmetric quasifree fibers
The four-word connection uses the actual constant color-symmetric conditionally quasifree R054 fermion fibers on the same carrier: <O>_H=24m(H)^2 and <N>_H=3m(H). The averaged source is not assumed quasifree.
Identical integrable unnormalized weights and insertions
The linear identity Z[O]=8Z[mN] uses identical integrable unnormalized weights and conditional fiber values on both sides. Positivity or finite complex normalization is not assumed for this linear statement. Physical use still requires deriving the mixed insertion from the original alphabet.
Only the specified R096 branch under its jointly required premises. No derived physical Higgs measure, general H(x) field theory or P1 source completion.
Read the exact result, assumptions and evidence
Result
For Pbar=integral P_H dmu and m=Tr(L P_H L), Tr L(Pbar-Pbar²)L>=Var_mu(m)/6. Under the additional named R054 fiber premises Delta_O=24Var_mu(m), giving the same lower bound Delta_O/144. Var(m)>0 excludes projection of Pbar; a pure primitive state before H integration remains possible.
Required assumptions
Same constant Higgs and two-face carrier
R096 compares the same original constant Higgs parameter on the same actual two-face CAR carrier and fixed source caller. Face labels are not internal chiral L/R labels. This is a conditional source comparison, not derivation of the primitive P1 identification.
Positive normalized mixture of covariance contractions
For the quantitative two-face defect mu is positive normalized, each conditional covariance P_H is self-adjoint with 0<=P_H<=1 on the same CAR space, and L is the fixed rank-six projector. Pbar uses precisely these weights.
Actual R054 color-symmetric quasifree fibers
The four-word connection uses the actual constant color-symmetric conditionally quasifree R054 fermion fibers on the same carrier: <O>_H=24m(H)^2 and <N>_H=3m(H). The averaged source is not assumed quasifree.
Constant Higgs and same actual two-face carrier, color symmetry and R054 conditional quasifreeness for the four-word part. No derived physical Higgs measure, general H(x), P1 seam, or independent gauge transport. The mixed source insertion is not assumed present in search alphabets without an actual source map.
Read the exact result, assumptions and evidence
Result
For one positive Higgs mixture and full two-face covariance, Tr L(Pbar-Pbar²)L >= Var(m)/6 = DeltaO/144 under R054 conditional quasifree premises, strengthening the earlier qualitative nonprojection result. The unnormalized linear identity Z[O]=8 Z[mN] requires matching conditional fibers and weights, not positivity. On the original 36-dimensional quark carrier the two-face trace g(u)=6 sum_j[2 cosh(ell sqrt(p²+y_j²u))]^-1 is strictly decreasing, implying Cov(u/(1+u),B_Q)<0 for a positive nonpoint radial law. Original 96 constructor and exact abstract rank-six control checked.
Required assumptions
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.
Same constant Higgs and two-face carrier
R096 compares the same original constant Higgs parameter on the same actual two-face CAR carrier and fixed source caller. Face labels are not internal chiral L/R labels. This is a conditional source comparison, not derivation of the primitive P1 identification.
Fixed original CAR time and admissible eight-arc word; the P1 seed does not specify the clustering direction. No exclusion of other P1 times or states, finite KMS interval, CAR vacuum, or excited insertions; no derivation of P1 localization, reflection or physics. The input basis is not orthonormal.
Read the exact result, assumptions and evidence
Result
All five original charge-zero inputs have K_alpha energy 4. The original J8 Gram is positive definite for both quarters with the stated exact determinant. A smooth quotient profile with nonzero first coefficient gives a bounded eight-arc Weyl word that reaches energy 3 from every nonzero vector in the input span. The connected Euclidean response based on K_alpha-4 grows on the infinite half-line; the same applies to positive states supported entirely in this span. The P0 vacuum channel J32 has the two exact R32 weights in the source. The R093 response remains an original vacuum word with the finite raising-current polynomials B_i.
Required assumptions
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.
Given CAR source time, smooth profiles and direct cover pullback; the one-mode Fock case tests an identity and is not a TFPT source state. Other local P1 embeddings or compressions are not excluded. No identification with OS time, all E8 Cartan currents, Higgs measure or F1.
Read the exact result, assumptions and evidence
Result
In the given folded CAR source contract the derivative of the same ordered Weyl source computes the left-inserted smeared total current response J(h)T(t)T*, including its noncommutative phase and alpha-dependent state kernels. An independent one-mode Fock test confirms the operator order and sign. For a nonconstant smooth phase profile the direct pullback T=W(-b∘F), F(theta)=8theta, acts nontrivially on a bounded even CAR bilinear in each of the eight disjoint cover arcs; hence T is not in the even local von Neumann algebra of any single such arc.
Required assumptions
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.