TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/GAUSS-NATIVE.txt Original file SHA-256: fe4c32c89f5e14cfea80c4fe55c663d4b64eb428e02b60a01e283ff139a08043 Inclusive original lines: 1-158 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | ACTUAL NATIVE GAUSS SUPPORT AND PROJECTED GIBBS TEST 000002 | L20261102H, 2 November2026. Original v989 callable, no new Hamiltonian. 000003 | Canonical files unchanged. Results: GAUSS-NATIVE.json. 000004 | 000005 | RESULT 000006 | 000007 | The unchanged v989 Hamiltonian is gauge invariant under its two ORIGINAL 000008 | complete interior stars, but its unprojected finite Gibbs state is not 000009 | supported on their G_x=1 sector. At the diagnostic beta=1 its weight 000010 | outside that sector is 000011 | 0.8551616360789673. 000012 | 000013 | The actual common Gauss projector is nonzero, has rank64 in the full 000014 | 1024-dimensional original space, and commutes exactly with H. Projecting 000015 | the SAME Gibbs operator to it produces a positive normalized state that 000016 | satisfies BOTH original interior G_x=1 conditions and is KMS on the 000017 | actual invariant observable algebra. No coupling, hopping, mass, seam 000018 | term, boundary condition or physical temperature was selected by this test. 000019 | 000020 | 1. EXACT ORIGINAL BUILDER AND SCOPE 000021 | 000022 | Freshly hash-read original: 000023 | source:ee2976cb57382871a493:1-70,331-470,670-785, 000024 | verification/v989_lattice_gate_battery.py, 000025 | SHA256 f44d3784698a1fd487024e43effcc4144f6bdc642e4772c56cfa7fabd9e90408. 000026 | Code discovery used the indexed repository graph before the source read. 000027 | 000028 | The native calls executed directly were 000029 | psi,num=_build_ham_ops(), 000030 | H=build_H(psi,num), 000031 | G_x=gauss_unitary(num,x), x=1,2. 000032 | The imported run() and other gate studies were not executed. Python 000033 | bytecode writing was disabled; only the requested work JSON was written. 000034 | 000035 | All original defaults were retained: 000036 | NS_H=4, NLQ_H=3, local links Z4 regular representation, 000037 | LAM_E=1, W_HOP=0.7, MASS=0.5, k_seam=0. 000038 | Thus dimH=4^3*2^4=1024. The original H includes its electric terms, 000039 | all three gauge-dressed link hoppings, staggered masses and the stated 000040 | undressed endpoint seam hopping. Nothing was removed to obtain the result. 000041 | 000042 | The original run674-684 checks precisely the two INTERIOR Gauss stars 000043 | x=1,2. The present test uses that same contract. It does not impose the 000044 | endpoint stars0,3, whose commutators with this native H have maximum 000045 | entry0.9899494936611666. Those are boundary data, not a numerical failure 000046 | of the two original interior constraints. Imposing them as if they also 000047 | commuted would silently change this source problem. 000048 | 000049 | The later v1011 B6 builder was also inspected through its actual callable: 000050 | gauge_matter_hamiltonian uses divergence_occupancy to construct the 000051 | already constrained sector. A comparison with an unconstrained Gibbs 000052 | state would require another full-space construction there. The present 000053 | test instead uses the EXISTING full-space v989 callable, with its original 000054 | complete interior stars, rather than inventing that construction. 000055 | 000056 | 2. ACTUAL PROJECTOR, EXACT NONEMPTY SUPPORT 000057 | 000058 | The native G_x are commuting unitary Z4 generators, G_x^4=I. Hence their 000059 | given G_x=1 constraint has the unique orthogonal group-average projector 000060 | 000061 | P_G=product_(x=1,2) (I+G_x+G_x^2+G_x^3)/4. (1) 000062 | 000063 | This introduces no chosen auxiliary sector or arbitrary projector. 000064 | The execution gives exactly zero residual for 000065 | P_G^2-P_G, P_G*-P_G, [H,P_G], [G1,G2], G_x^4-I, 000066 | and [H,G_x] for both original stars. A basis Q of Ran(P_G) was extracted 000067 | from this actual projector. Q*Q=I, QQ*=P_G and Q*G_xQ=I also have 000068 | zero residual at the matrix-entry level. 000069 | 000070 | Nonemptiness is analytic in this original regular-link case. The interior 000071 | gauge element G1^a G2^b shifts the three original link coordinates by 000072 | (-a,a-b,b) mod4. 000073 | It fixes a link configuration only for a=b=0. The action is therefore 000074 | free on each original link orbit, independently of the finite matter 000075 | phase. Every orbit has16 points and carries one invariant vector. 000076 | Consequently 000077 | rank P_G=1024/16=64>0. 000078 | For an immediate invariant vector, take the original empty matter state 000079 | and the uniform vector on each regular link. No new background charge or 000080 | periodic-lattice neutrality constraint is needed for this interior-star 000081 | statement. The full open-boundary generalization is reviewed separately 000082 | by joint_origin; no periodic counting formula is imported here. 000083 | 000084 | 3. INVARIANCE DOES NOT IMPLY THE GIVEN GAUSS SUPPORT 000085 | 000086 | For finite beta the original full kinematic Gibbs state is 000087 | rho_beta=exp(-beta H)/Tr exp(-beta H). 000088 | Because [H,G_x]=0 it is invariant under each original gauge conjugation. 000089 | But exp(-beta H) is strictly positive on the FULL finite-dimensional 000090 | space. As00. (2) 000093 | 000094 | This proves the distinction without depending on a selected eigenbasis. 000095 | The actual beta=1 evaluation gives 000096 | omega_beta(P_G)=0.14483836392103266, 000097 | omega_beta(I-P_G)=0.8551616360789673. 000098 | Gauge invariance of this state therefore cannot be relabeled support 000099 | on the physical G_x=1 sector. 000100 | 000101 | The diagnostic beta=1 is merely one evaluation of the existing Gibbs 000102 | family, not a derived physical thermal scale. For numerical conditioning 000103 | the SAME scalar energy shift E0=-7.729844052718033 was used in both 000104 | partition traces; it cancels from every ratio and changes no original 000105 | Hamiltonian or vacuum normalization. The shifted traces were 000106 | Z=10.05673621782408, Z_G=1.4566012201750331. 000107 | The independently computed ratio Z_G/Z agrees with omega_beta(P_G) 000108 | within5.56e-17. 000109 | 000110 | 4. PROJECT THE SAME GIBBS STATE; KMS AND GAUSS BOTH HOLD 000111 | 000112 | Use the original constraint, not a new Hamiltonian: 000113 | rho_beta,G=P_G exp(-beta H)/Tr[P_G exp(-beta H)]. (3) 000114 | The denominator is strictly positive because P_G is nonzero and 000115 | exp(-beta H)>0. Since [P_G,H]=0, (3) is a positive normalized state. 000116 | It has P_G support, hence G1=G2=1 exactly. On the physical corner its 000117 | Hamiltonian is simply H_G=Q*HQ, the restriction of the SAME native H. 000118 | 000119 | Every invariant observable A commutes with P_G. The original evolution 000120 | therefore preserves its physical corner and cyclicity of its finite 000121 | trace gives, for all invariant A,B and all real t, 000122 | omega_beta,G(A tau_(t+i beta)(B)) 000123 | =omega_beta,G(tau_t(B) A). (4) 000124 | Here tau_t(B)=exp(itH)B exp(-itH); neither time nor H is rescaled. 000125 | Thus the projected state remains KMS on the actual invariant algebra. 000126 | It need not be KMS on the full algebra of charged sector-changing fields; 000127 | that is not the original gauge-invariant observable claim. 000128 | 000129 | The concrete native test uses actual noncommuting invariant terms: 000130 | A=n1, 000131 | B=psi1* Z_link1 psi2+its original adjoint. 000132 | Both commute exactly with G1 and G2, while max-entry|[A,B]|=1. 000133 | At the diagnostic real time t=0.37 and beta=1, 000134 | left side(4)=-0.0843970939528475-0.014792228485994003i, 000135 | right side(4)=-0.08439709395284752-0.014792228485994022i. 000136 | The KMS residual is2.359479019930958e-17. The physical stationarity 000137 | residual is1.138091725164796e-15. The largest tested residual is the 000138 | latter, below the recorded tolerance2e-10. All native checks passed. 000139 | The analytic proof(4), rather than this one pair of insertions, establishes 000140 | the invariant-observable KMS statement. 000141 | 000142 | 5. CONSEQUENCE AND EXACT LIMIT OF THE RESULT 000143 | 000144 | This test supplies an actual nonempty, Gauss-supported positive KMS 000145 | finite source in the existing native Hamiltonian route, with the entire 000146 | original link/matter Hamiltonian preserved. The physical projection is 000147 | the supplied Gauss condition and introduces no coupling or fitted state 000148 | parameter. It also decisively shows why invariance of an unprojected 000149 | Gibbs state alone was insufficient for that stronger support claim. 000150 | 000151 | The result is a finite native Z4 staggered-source check, within the 000152 | original v9891+1D/reduced-size scope. It is not a3+1D continuum or full 000153 | geometric/Higgs/matter identification. Nor is it a proof that finite 000154 | physical density matrices at different outer volumes are exactly 000155 | projective. The independent H limit-state argument handles consistent 000156 | restrictions after the actual limit, with complete interior stars and 000157 | the original invariant observables. Existing G transport, the whole 000158 | relative source, projective full moments and reference factors remain.