TFPT scientific source excerpt Original repository path: experiments/theory-contracts/clock-source-twist-transport-20261015/ADDENDUM.md Original file SHA-256: 54adf5088665c2eb9c7ae1f280f5e641ea30df39b06d53c4576e141f0458158d Inclusive original lines: 72-150 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. 000072 | ## 3. A neutral total word can nevertheless distinguish twists 000073 | 000074 | In the *given* lattice group algebra the original charge displacement E_p 000075 | is unitary and E_p Omega is, up to a harmless unit phase, the normalized 000076 | primary |p>. It is a global zero-mode charge displacement, not a derived 000077 | local physical vertex instrument. Define the return amplitude 000078 | 000079 | C_delta(p,t)= 000080 | = 000081 | =exp[-t(|p|²/2+delta.p)], t>0. 000082 | 000083 | The total word begins and ends at zero charge. It is not blind, because the 000084 | intermediate state has charge p. This is the concrete reason not to extend 000085 | the all-word result to every overall-neutral observable. Even a Q_q-neutral 000086 | root insertion can discriminate part of delta while violating the stronger 000087 | eight-Cartan-neutral premise. 000088 | 000089 | The source cocycle phases in E_p cancel from this normalized diagonal 000090 | amplitude; no replacement of the original cocycle is needed. The full local 000091 | vertex field contains additional oscillator factors and locality/domain 000092 | requirements. Identifying a physical preparation or detector with E_p is 000093 | an extra operator/instrument question, not supplied by this formula. 000094 | 000095 | ## 4. Eight original lattice charges reconstruct delta 000096 | 000097 | The odd energy difference removes the known quadratic term: 000098 | 000099 | C_delta(p,t)/C_delta(-p,t)=exp[-2t delta.p]. 000100 | 000101 | For the eight columns b_i of the original integral E8 basis B, define 000102 | 000103 | y_i=-(1/(2t)) log[C_delta(b_i,t)/C_delta(-b_i,t)]. 000104 | 000105 | For positive Euclidean amplitudes the real logarithm has no branch 000106 | ambiguity. Then y=B^T delta and 000107 | 000108 | delta=(B^T)^(-1) y. 000109 | 000110 | Equivalently, the exact spectral energy differences give 000111 | `y_i=(E_delta(b_i)-E_delta(-b_i))/2`. B is the existing original basis, 000112 | not a newly chosen geometry; |det B|=1. Eight independent linear probes 000113 | therefore determine an unrestricted eight-component delta. Fewer than eight 000114 | independent such linear functionals leave a nontrivial kernel unless a 000115 | smaller admissible delta class is supplied independently. This minimality 000116 | is about the stated linear data class, not every possible experiment. 000117 | 000118 | Known nonzero t means the same fixed K/time normalization in all candidates. 000119 | If only ratios with an unknown time scale are available, the result 000120 | determines t delta. The symmetric product 000121 | `C_delta(p,t) C_delta(-p,t)=exp[-t |p|²]` can calibrate this dimensionless 000122 | time within the already fixed quadratic K normalization when those absolute 000123 | normalized amplitudes are also given. None of this derives a physical 000124 | second/GeV calibration from P1/P2. 000125 | 000126 | These amplitudes identify delta, not q and tau by themselves. Reconstructing 000127 | the latter would require additional charged Q matrix elements and full 000128 | transition characters with their own operator and phase conventions. The 000129 | neutral vacuum experiment supplies neither. 000130 | 000131 | ## 5. Consequence for automated candidate discrimination 000132 | 000133 | Candidates differing only in delta must be treated as observationally 000134 | equivalent when the retained data consist exclusively of the specified 000135 | neutral vacuum algebra, even if its entire matrix/Jordan closure is known. 000136 | More such neutral moments cannot discriminate the candidates. A search 000137 | should keep that equivalence and the absent charged probe explicit. 000138 | 000139 | If the original source supplies normalized charged primary responses with 000140 | known common time, the eight-basis reconstruction is an exact discriminator 000141 | requiring only linear algebra. The checker confirms that both original 000142 | quarter cases have four distinct J-twist response vectors on these charges. 000143 | Thus the covariant frame family in the parent proof is mathematically 000144 | equivalent under simultaneous transport, while a comparison using *fixed 000145 | marked charged probes* can distinguish its members. Transporting the probes 000146 | as well would simply restore the covariance equality. 000147 | 000148 | Neither result selects a physical twist, provides a charged preparation, 000149 | licenses a local vertex instrument, or makes the covariance orbit into 000150 | four physical sectors. No P1/P2 or physical T1–T8 closure follows.