Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Complete physical Hamiltonian source with Gauss sector and thermal word limit
Which whole source is constructed on the original local Hamiltonian route, how do physical constraints and thermal words follow, and how is its actual IR response distinguished from the original covariant geometric source branch?
Complete bounded physical word source of the supplied class, not a positive Grassmann/ghost configuration space. Thermal reflection is not automatically a separate P1 seam reflection. Ambient and physical subnet states remain distinct. q[0]=1 normalizes only this local source; original unnormalized Z_rel, reference and energy offsets are not replaced. No unbounded field limit, IR universality, general4D covariance or fundamental total promotion.
Read the exact result, assumptions and evidence
Result
All ordered bounded real multitime words and Dyson sources converge on the same original Hamiltonian source. Whole Euclidean thermal words obey volume-independent Schatten Holder |S_k|<=product||A_i||. Fixed Gaussian smearing gives complex-time norm convergence; uniform local derivation bound and multilinear Holder continuity remove smearing. Thus all unsmeared ordered words converge in the same subnet, without assuming unsmeared imaginary-time norm convergence. Whole source series have majorant exp(sum||J_i||_1||A_i||). Reflected half-circle word kernel Tr(B_F*B_G)/Z is a Hilbert-Schmidt Gram matrix, so positivity survives the limit. Canonical beta-s reflection and -is for latest-left ordering are explicit.
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.
Original local Hamiltonian class and common limiting caller
One fixed original selfadjoint interaction Phi(X) on the existing finite local tensor factors, finite range and finite uniform interaction moment. Original quasilocal algebra, inclusions and free-boundary H_Lambda=sum_X Phi(X). Use the scoped v1013 real-time norm dynamics and common weak-* ground/KMS subnet construction. Graph, frozen model couplings and actual field factors are supplied class data, not newly derived from P1.
Original regular finite gauge links and complete interior Gauss stars
Actual regular finite-G links with original matter representation. Impose the original Gauss=1 projectors only for complete contained stars preserved by actual H. Open boundary with a path from every projected link component to an unprojected boundary vertex. Local gauge actions commute and preserve each interaction term. Admissible observables belong to A^G; do not add broken seam/boundary stars. No periodic neutrality or alternate sector choice.
Given thermal parameter and bounded admissible source legs
Given beta>0 and actual bounded local physical insertions, in particular the even admissible algebra when parity requires it. Insertions preserve the Gauss projectors used. Retain original real time tau_t=exp(itH)·exp(-itH), word order and subnet. Thermal reflection is beta-s with adjoint/word reversal. Do not identify another P1 reflection or whole master caller without its original map. Keep original unnormalized reference factors and energy offsets separately.
New constructive application theorem on the same given Hamiltonian class and actual constraints. Projected state can differ from earlier ambient state. KMS/ground energy only on admissible algebra; whole stars, regular finite links and open topology are premises. Native1024D v989 is a separate original control, not execution of16777216D v1013 DET16 cell. No P1/whole-master identity, new coupling or total physical closure.
Read the exact result, assumptions and evidence
Result
Gauge invariance of Gibbs states does not impose Gauss=1 support. For original regular finite links with open boundary, the interior gauge group acts freely on link configurations; ||P(delta_u tensor v)||²=||v||²/|G|^|X| proves nonemptiness without choosing a vacuum. The same H gives positive rho_phys=P exp(-beta H)/Tr(P exp(-beta H)), KMS on A^G. The original norm/weak-* argument yields a limit with omega(P_F)=1. Centrality forces pi(P_F)=I throughout GNS; state and dynamics descend to the physical quotient. Unchanged v989 control has dim1024/rank64; diagnostic beta1 ambient invariant Gibbs has0.8551616360789673 outside, projected0. Actual noncommuting invariant-operator KMS residual2.36e-17.
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.
Original local Hamiltonian class and common limiting caller
One fixed original selfadjoint interaction Phi(X) on the existing finite local tensor factors, finite range and finite uniform interaction moment. Original quasilocal algebra, inclusions and free-boundary H_Lambda=sum_X Phi(X). Use the scoped v1013 real-time norm dynamics and common weak-* ground/KMS subnet construction. Graph, frozen model couplings and actual field factors are supplied class data, not newly derived from P1.
Original regular finite gauge links and complete interior Gauss stars
Actual regular finite-G links with original matter representation. Impose the original Gauss=1 projectors only for complete contained stars preserved by actual H. Open boundary with a path from every projected link component to an unprojected boundary vertex. Local gauge actions commute and preserve each interaction term. Admissible observables belong to A^G; do not add broken seam/boundary stars. No periodic neutrality or alternate sector choice.
Reuse later original proofs and correct excessive source requirements. v330 transfer labels are not all geometry/Higgs/matter fields. A whole S_adm field dictionary does not follow from a finite example. No phase uniqueness, limit gap or IR universality claim; no new continuum gate or total physical closure.
Read the exact result, assumptions and evidence
Result
The fully read v1013 original proof constructs common norm dynamics and at least one ground/beta-KMS state on the same A/A^G. Its whole local densities are exactly compatible by restriction; testing all matrix units gives Tr_(G-F)rho_G=rho_F. Free finite Gibbs weights themselves need not be projective. Existing whole fiber integration carries effective weights, exterior determinants and all source legs; unchanged bare spectral form is not required. A single global reference density is no general existence gate. Ledger760 makes a→0 optional and thermodynamics plus controlled IR universality binding.
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.
Original local Hamiltonian class and common limiting caller
One fixed original selfadjoint interaction Phi(X) on the existing finite local tensor factors, finite range and finite uniform interaction moment. Original quasilocal algebra, inclusions and free-boundary H_Lambda=sum_X Phi(X). Use the scoped v1013 real-time norm dynamics and common weak-* ground/KMS subnet construction. Graph, frozen model couplings and actual field factors are supplied class data, not newly derived from P1.
Construct physical Gauss source and whole thermal words
Apply existing common limit theorem to actual Gauss constraints and all bounded ordered source words with RP; choose no new source or boundary contract.
Derive whole kinetic-metric test and evaluate actual IR callers
Prove actual shared symbol/Ward implication with its hypotheses and full mixed tensors; execute native input actions/consumers instead of identifying unrelated speed labels.
Concrete common Hamiltonian-source completion and actual Gauss constraints replace blanket source-absence or new-continuum requirements. Preserve G reference and geometric reorganization; do not preempt whole-master caller identity.