TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_recursion_20261028/PADM-VACUUM-RETENTION-AUDIT.txt Original file SHA-256: 4cbfdb96c2d8e7a75fe8c1423a91f98ff217e6e1fc13814cb86a7c08d3828d1f 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. Conversation metadata and personal execution details are omitted where indicated. 000001 | VACUUM RETENTION BY THE ORIGINAL FULL ADMISSIBILITY PROJECTOR 000002 | 28 October 2026 000003 | 000004 | VERDICT 000005 | 000006 | The positive color-center result does not yet extend to the original full 000007 | projector 000008 | 000009 | P_adm = P_prim P_sing P_Theta. 000010 | 000011 | The obstruction is now precise and source based. 000012 | 000013 | * P_sing keeps the stipulated E8 vacuum Omega and removes the only other H8 000014 | ground state p_*. This was proved in CLUSTERING-ADMISSIBLE-SECTOR.txt. 000015 | * The original P_prim definition does not say that it keeps Omega. Its gap 000016 | factor keeps the spectrum of D_prim^* D_prim above a positive threshold. 000017 | Hence, if an identification sent Omega to a D_prim zero mode, the displayed 000018 | original formula would remove it. 000019 | * The determinant data determine P_Theta as part of the closed branch, but no 000020 | source formula specifies Q_Theta Omega. Thus neither P_Theta Omega=Omega nor 000021 | P_Theta Omega=0 follows in the H8 lattice representation. 000022 | * The sources prove commutation of D_adm with the admissibility factors. They 000023 | do not place H8 on the same Hilbert space or prove [P_adm,H8]=0. 000024 | 000025 | Therefore the one-dimensional kernel and positive gap on Ran(P_sing) are 000026 | exact, while an H8 clustering theorem on Ran(P_adm) remains conditional on one 000027 | typed image and reduction test. No further projector choice is needed or 000028 | allowed; the missing datum is an intertwiner between two already fixed source 000029 | representations. 000030 | 000031 | 000032 | 1. THE DECISIVE P_PRIM TEST 000033 | 000034 | The original primitive Hilbert space is 000035 | 000036 | H_prim = H_Seam tensor H_t, 000037 | 000038 | and its seam selector is P_Seam,+ = (1+iC)/2. The source then defines 000039 | 000040 | K_g = sqrt(lambda_gap) 000041 | 1_[0,epsilon_adm^2/4)( 000042 | P_Seam,+ D_prim^* D_prim P_Seam,+), 000043 | 000044 | P_prim = Pi_ker Delta_prim 000045 | = P_Seam,+ P_prim,gap, 000046 | 000047 | P_prim,gap 000048 | = 1_[epsilon_adm^2/4,infinity)( 000049 | P_Seam,+ D_prim^* D_prim P_Seam,+). 000050 | 000051 | Thus P_prim selects seam-even vectors in the gapped spectral subspace. For a 000052 | seam-even vector xi in the displayed operator domain, 000053 | 000054 | D_prim xi = 0 implies P_prim xi = 0. 000055 | 000056 | This is the opposite of an inference that a geometric zero mode must be the 000057 | retained vacuum. It does not prove that P_prim removes the H8 vacuum, because 000058 | the original source never supplies a map taking the H8 lattice vector Omega 000059 | to a vector xi in H_prim. It proves that retention cannot be inferred from the 000060 | word vacuum or from zero energy. The actual question is the image identity 000061 | 000062 | P_prim W Omega = W Omega 000063 | 000064 | for a source-defined isometry W between the H8 representation and the 000065 | boundary or OS representation. W is absent. 000066 | 000067 | This agrees with the already registered Atlas finding 000068 | 000069 | finding:original-admissibility-requires-image-membership-not-positive-half-or-commutation. 000070 | 000071 | Its source-reviewed contract identifies the direct test P_adm R0=R0 and warns 000072 | that commutation alone is insufficient. The present H8 vacuum is subject to 000073 | the same type requirement. 000074 | 000075 | 000076 | 2. THE DECISIVE P_THETA TEST 000077 | 000078 | The full admissibility source defines 000079 | 000080 | P_Theta = (1+Q_Theta)/2, 000081 | Q_Theta^2=1, 000082 | Q_Theta^*=Q_Theta. 000083 | 000084 | The carrier source fixes the determinant class 000085 | 000086 | (c1(L2),c1(L3))=(1,0) 000087 | 000088 | and says that P_Theta is thereby determined. This determines the later 000089 | determinant-sector selector inside its own Hilbert factor. It does not give a 000090 | representation of Q_Theta on the stipulated E8 lattice module and nowhere 000091 | states 000092 | 000093 | Q_Theta Omega = +Omega. 000094 | 000095 | The retention value is therefore unspecified, not negative. No original 000096 | definition found in the audited sources explicitly excludes Omega through 000097 | P_Theta; equally, none proves that it survives. 000098 | 000099 | 000100 | 3. WHY THE H8 COMMUTATOR DOES NOT FOLLOW 000101 | 000102 | The exact color-center computation gives 000103 | 000104 | [P_sing,H8]=0 000105 | 000106 | because both operators are diagonal Cartan actions in the stipulated E8 000107 | lattice representation. The full source instead states that D_adm commutes 000108 | with iC, P_sing, and Q_Theta. This is a statement about D_adm on 000109 | 000110 | H_Seam tensor H_c tensor H_Theta tensor H_t. 000111 | 000112 | It is not a statement about H8. Neither H8 nor an operator unitarily 000113 | equivalent to H8 occurs in the definition or proof of the full admissibility 000114 | projector. Consequently 000115 | 000116 | [P_adm,H8]=0 000117 | 000118 | is not a well-typed source theorem before an intertwiner W is given. 000119 | 000120 | The later OS/CAR theorem cannot repair this gap. It assumes clustering on 000121 | P_adm and then reconstructs a semigroup exp(-t H_adm) with vacuum Omega. It 000122 | does not identify H_adm with H8, and clustering is an input of that theorem. 000123 | Using it to prove H8 clustering would therefore be circular. 000124 | 000125 | 000126 | 4. STRONGEST CORRECT CONDITIONAL SEMIGROUP STATEMENT 000127 | 000128 | Let W embed the stipulated H8 E8 module into the Hilbert space on which the 000129 | fixed original P_adm acts. Suppose the following three image and reduction 000130 | identities are proved from the source: 000131 | 000132 | P_prim W Omega = W Omega, 000133 | P_Theta W Omega = W Omega, 000134 | P_adm reduces W H8 W^*. 000135 | 000136 | The previous color-center theorem already supplies 000137 | 000138 | P_sing W Omega = W Omega 000139 | 000140 | and removes the second zero state. On Ran(P_adm), the restricted spectrum is 000141 | then a subset of the spectrum on Ran(P_sing). Hence its kernel is the single 000142 | line C W Omega and its gap g obeys 000143 | 000144 | g >= 3/2 for alpha=1/4, 000145 | g >= 2 for alpha=3/4. 000146 | 000147 | For bounded admissible operators A and B preserving this reduced space, the 000148 | connected Euclidean-time matrix element satisfies 000149 | 000150 | | 000151 | - | 000152 | 000153 | <= exp(-g t) 000154 | || A^* Omega || 000155 | || (1-|Omega> H_Seam tensor H_c tensor H_Theta tensor H_t 000170 | 000171 | with the three displayed identities for Omega and H8. A global unitary is 000172 | stronger than necessary. An isometry on the cyclic H8 subspace generated by 000173 | the intended admissible algebra suffices. The source must fix W; inserting a 000174 | new identification would merely choose the desired answer. 000175 | 000176 | Until that diagram is supplied, the exact conclusion remains: 000177 | 000178 | ker(H8 | Ran P_sing) = C Omega, 000179 | 000180 | while retention by P_prim and P_Theta, and therefore clustering on the actual 000181 | full P_adm space, is not proved and is not explicitly ruled out. 000182 | 000183 | 000184 | 6. EXACT SOURCES 000185 | 000186 | Primitive projector and its high-spectrum gap selector: 000187 | 000188 | [OMITTED: non-scientific conversation or execution metadata] 000189 | [OMITTED: non-scientific conversation or execution metadata] 000190 | [OMITTED: non-scientific conversation or execution metadata] 000191 | [OMITTED: non-scientific conversation or execution metadata] 000192 | [OMITTED: non-scientific conversation or execution metadata] 000193 | 000194 | Full P_adm, P_Theta, P_sing, and the stated D_adm commutators: 000195 | 000196 | documents/archive--tfpt-45--source_extracts--04_qft_source/ 000197 | sections/002-section-closure-theorems-from-the-admissibility-complex.tex 000198 | lines 12-110 000199 | SHA-256 61ecee7727bdec47936bbffc028d3dbac5260a025b6dbcce30124a60d635fc9c 000200 |