TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/GAUSS-SOURCE-PROOF.txt
Original file SHA-256: 98c2394d9e60ca195dd12d89f54d144c6bf4ad83a4d08b4c31da9958db53b76e
Inclusive original lines: 1-200
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | PHYSICAL GAUSS-SECTOR SOURCE FROM THE EXISTING v1013 FAMILY
000002 | joint_origin, 2 November2026. Analytic continuation of L20261102H.
000003 | No canonical edit, new Hamiltonian, coupling or Hilbert-space embedding.
000004 | 
000005 | RESULT
000006 | 
000007 | The original invariant-algebra theorem is valid but gauge invariance of
000008 | a state does not by itself impose its Gauss sector. A finite full Gibbs
000009 | density is full rank, so it has positive weight outside any nontrivial
000010 | proper Gauss projection. This does not invalidate v1013's dynamics or
000011 | state-existence theorem. The stronger physical source can be constructed
000012 | with the SAME original Hamiltonians and original Gauss constraints:
000013 | 
000014 |  rho_phys,Lambda=P_Lambda exp(-beta H_Lambda)/Z_phys,Lambda,
000015 |  Z_phys,Lambda=Tr[P_Lambda exp(-beta H_Lambda)]>0.           (1)
000016 | 
000017 | Here P_Lambda projects only at vertices whose ENTIRE original gauge
000018 | stars lie in the available finite region. This uses the original free
000019 | boundary, leaving boundary flux unrestricted. For the original regular
000020 | finite-group links its range is nonempty, by the proof below.
000021 | 
000022 | The original common-algebra compactness argument then gives physical
000023 | ground/KMS states omega_phys on A^G, with omega_phys(P_F)=1 for every
000024 | fixed finite Gauss projector. Their GNS representations factor through
000025 | the actual Gauss quotient; the given dynamics descends. No naive tensor
000026 | embedding of P_Lambda H_Lambda into P_Lambda' H_Lambda' is used.
000027 | 
000028 | 1. USE THE ACTUAL LOCAL CONSTRAINTS AND THE ORIGINAL FREE BOUNDARY
000029 | 
000030 | Use v1013's one fixed finite-range interaction H_Lambda=SUM_X Phi(X),
000031 | its common ambient finite tensor factors and inclusions, and its actual
000032 | local gauge representation U_x(g). For the finite gauge group G put
000033 | 
000034 |  P_x=|G|^-1 SUM_(g in G) U_x(g),
000035 |  X_Lambda={x: the whole gauge-star support of U_x lies in Lambda},
000036 |  P_Lambda=PRODUCT_(x in X_Lambda) P_x.                    (2)
000037 | 
000038 | These are the original Gauss=1 projectors, including any originally
000039 | specified matter/background charge in U_x. They are not new choices of
000040 | sector labels. Distinct vertex gauge actions commute: on a shared regular
000041 | link they act by commuting left/right translations; on matter they have
000042 | disjoint vertex support. Hence the P_x are commuting orthogonal
000043 | projections. Original termwise gauge invariance gives
000044 | 
000045 |  [P_Lambda,H_Lambda]=0.                                  (3)
000046 | 
000047 | The original free-boundary Hamiltonian is explicitly equation1.4 of the
000048 | v1013 theorem. It is not replaced by a periodic box with a different
000049 | global flux constraint. The regulator retains only fully contained gauge
000050 | stars so that no partial Gauss operator is mistaken for the full one.
000051 | 
000052 | 2. NONEMPTINESS: A COMPLETE FREE-LINK-ORBIT PROOF
000053 | 
000054 | Write the finite regular-link factor in its group-configuration basis
000055 | delta_u, u in G^E. Let K=G^(X_Lambda) be the gauge group imposed in(2).
000056 | Every connected component of the acted vertices has a path along included
000057 | links to an unacted boundary vertex. This follows for finite subsets of
000058 | the connected infinite square lattice with all incident links retained
000059 | at each acted vertex. Finite components cannot be closed off with every
000060 | vertex and all its neighbours acted indefinitely.
000061 | 
000062 | K acts FREELY on these link configurations. If g fixes u, then on an
000063 | oriented link x->y,
000064 |  g_x u_xy g_y^-1=u_xy.
000065 | At an unacted boundary endpoint g_y=1. Following a path inward forces
000066 | g_x=1 at every acted vertex. This proof also works for nonabelian finite
000067 | groups; no character diagonalization or abelian charge assumption is used.
000068 | 
000069 | For any nonzero vector v in the ACTUAL remaining matter/clock factors,
000070 | including their original gauge representation V(g),
000071 | 
000072 |  P_Lambda(delta_u tensor v)
000073 |    =|K|^-1 SUM_(g in K) delta_(g.u) tensor V(g)v,
000074 |  norm(P_Lambda(delta_u tensor v))^2=norm(v)^2/|K|>0.        (4)
000075 | 
000076 | The equality follows because the different link configurations are
000077 | orthogonal and V(g) is unitary. Thus P_Lambda is nonzero for every such
000078 | finite region. No gauge-invariant matter vacuum was assumed, no matter
000079 | charge sector was changed, and no coupling to boundary matter was added.
000080 | This closes nonemptiness for the original regular finite-group/free-
000081 | boundary contract. Arbitrary truncated nonregular link representations
000082 | or a fixed periodic charge sector require their own supplied check;
000083 | they are not silently substituted for the actual regular-link contract.
000084 | 
000085 | Because H_Lambda is finite-dimensional selfadjoint, exp(-beta H_Lambda)
000086 | is strictly positive. Equations(3)-(4) imply Z_phys,Lambda>0 and that(1)
000087 | is a positive normalized density. Every fixed interior projector obeys
000088 | Tr(rho_phys,Lambda P_F)=1 once all its vertices belong to X_Lambda.
000089 | 
000090 | 3. FINITE KMS AND GROUND CONDITIONS ON THE ACTUAL ADMISSIBLE ALGEBRA
000091 | 
000092 | For every allowed A in A^G, [A,P_Lambda]=0 whenever its finite support
000093 | is present. Thus P_Lambda is central relative to the finite invariant
000094 | observable algebra. Cyclicity of the finite trace, with (3), gives
000095 | 
000096 |  omega_phys,Lambda(A B)
000097 |    =omega_phys,Lambda(B tau_(i beta)^Lambda(A)).          (5)
000098 | 
000099 | One can also see this as the ordinary Gibbs trace on the actual nonzero
000100 | block P_Lambda H_Lambda, with the restrictions of A and H. It imposes no
000101 | identification of this Hilbert block with a tensor factor in another box.
000102 | 
000103 | For zero temperature choose a normalized lowest-energy eigenvector of
000104 | H_Lambda RESTRICTED to Ran(P_Lambda). Such an eigenvector exists by(4).
000105 | For every local invariant A, A preserves this block, so
000106 | 
000107 |  omega_phys,Lambda(A*[H_Lambda,A])>=0.                    (6)
000108 | 
000109 | The energy in(6) is the minimum IN THE ORIGINAL PHYSICAL SECTOR. It need
000110 | not be the minimum of the unconstrained ambient field Hamiltonian.
000111 | 
000112 | Consequently(5) and(6) are KMS/ground conditions on A^G. The projected
000113 | state is also a positive state on the ambient full field algebra, but
000114 | it is generally NOT KMS there: isolated gauge-changing fields do not
000115 | commute with P_Lambda. No ambient charged-field positive-energy or
000116 | thermal-cyclicity assertion is inferred. Physical histories use the
000117 | actual admissible operations and originally supplied gauge carriers.
000118 | 
000119 | 4. COMMON-ALGEBRA LIMIT, INCLUDING EXACT GAUSS SUPPORT
000120 | 
000121 | Extend each finite density(1) to a state on v1013's common ambient A,
000122 | then take a weak-* convergent subnet. Such extensions and compactness
000123 | are precisely the original v1013 construction. No embedding of physical
000124 | Hilbert spaces is required. Restrict the limit to A^G.
000125 | 
000126 | For every fixed finite F, eventually P_Lambda P_F=P_Lambda, so
000127 | 
000128 |  omega_phys(P_F)=lim_Lambda omega_phys,Lambda(P_F)=1.     (7)
000129 | 
000130 | The original norm limit tau_t^Lambda on local invariant observables is
000131 | unchanged: projecting the statistical state has not changed H or tau.
000132 | For positive temperature use exactly v1013's Gaussian-smeared local
000133 | analytic elements. Every smear remains invariant and commutes with the
000134 | relevant P_Lambda. Thus the finite identity(5) passes to the common
000135 | subnet by the already proved norm dynamics and Gaussian tail bound.
000136 | The limit is beta-KMS for tau restricted to A^G.
000137 | 
000138 | For ground states, the local commutator derivation stabilizes as in the
000139 | original proof, and(6) passes to the weak-* limit. The limit is a ground
000140 | state on A^G. At least one such state exists; neither phase uniqueness
000141 | nor a thermodynamic gap is asserted.
000142 | 
000143 | All local restrictions of this SAME physical state are exactly compatible
000144 | as in LATER-SOURCE.txt. The local projected free-boundary Gibbs densities
000145 | themselves need not be exactly projective before taking the subnet.
000146 | 
000147 | 5. THE PHYSICAL QUOTIENT IS CARRIED BY THE STATE AND DYNAMICS
000148 | 
000149 | Each local P_x belongs to A^G and commutes with every element of A^G.
000150 | Let I_G be the closed ideal in A^G generated by {I-P_x}. Positivity and
000151 | (7) give pi_omega(I-P_x)Omega=0. Centrality then gives
000152 | 
000153 |  pi_omega(I-P_x)pi_omega(A)Omega=0 for every A in A^G.
000154 | 
000155 | These vectors are dense in the GNS space, hence pi_omega(P_x)=I.
000156 | The representation annihilates I_G and therefore factors through
000157 | 
000158 |  A_phys=A^G/I_G.                                         (8)
000159 | 
000160 | For sufficiently large Lambda, tau_t^Lambda(P_x)=P_x by(3); the norm
000161 | limit gives tau_t(P_x)=P_x. Thus I_G is invariant and the same dynamics
000162 | descends to A_phys. The ground/KMS states and their full admissible
000163 | word/source functionals descend as well. This proves the actual sector
000164 | constraint throughout the physical representation, not just its value
000165 | in one chosen vector.
000166 | 
000167 | 6. SOURCES AND EXACT SCOPE
000168 | 
000169 | Original same interaction, finite factors and free boundary:
000170 | paper-source:4cc6102a923009e972617edc:31-139,225-363,
000171 | SHA256 9f53a07911c9e2e8f6b2f594b7c154f5a8dc0faeb2e9a429cc04a229cc6f4294.
000172 | Original local Gauss projections and invariant algebra:
000173 | paper-source:fcd23741dc02fbd857432770:301-363 of the same file/hash.
000174 | Original state existence and Gaussian KMS passage:
000175 | paper-source:6ffa2ad8a0ae00e8cc167c8e:1-61,
000176 | SHA256 32dea10dc2686956bf07afc299b9ebd65c7dd25b39783a1bf87579a87d9056c0.
000177 | 
000178 | Native original link constructor independently read graph-first:
000179 | verification/v989_lattice_gate_battery.py:_build_ham_ops375-378,
000180 | build_H381-394, gauss_unitary397-404; hash-validated complete coefficient
000181 | context at source:ee2976cb57382871a493:330-404, SHA256
000182 | f44d3784698a1fd487024e43effcc4144f6bdc642e4772c56cfa7fabd9e90408.
000183 | Its actual link shifts are regular Z4 translations. The two COMPLETE
000184 | interior stars1,2 act on its three link configurations by
000185 |  (a,b).(u0,u1,u2)=(u0-a,u1+a-b,u2+b) mod4.
000186 | Only a=b=0 fixes any configuration, so each orbit has16 points. With
000187 | four original fermion modes and three regular Z4 links the full dimension
000188 | is1024 and rank(P_G)=1024/16=64, an exact application of(4).
000189 | 
000190 | The sibling origin_topology native execution GAUSS-NATIVE.json uses the
000191 | UNCHANGED entire source at all original defaults. Its full Gibbs state
000192 | at beta1 is gauge invariant but has0.8551616360789673 weight outside P_G;
000193 | the projected state has zero outside weight, and its KMS test on invariant
000194 | number/hopping operators has residual2.36e-17. The actual seam term breaks
000195 | the endpoint stars0,3; those are correctly NOT imposed. This supplies a
000196 | concrete original verification of the precise inner-star scope, not a
000197 | replacement open-boundary Hamiltonian. The all-volume argument remains
000198 | the analytic proof(2)-(8); no second native suite was run here.
000199 | 
000200 | The algebra/constraint distinction is also explicit in the original cited