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Original repository path: experiments/theory-contracts/clock-source-twist-transport-20261015/ADDENDUM.md
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000015 | ## 1. Neutral here has a precise meaning
000016 | 
000017 | Use the already stipulated lattice Fock representation from `PROOF.md`:
000018 | 
000019 |     H = direct_sum_(p in Ls) M(1) tensor |p>,
000020 |     K_delta=N+|p|²/2+delta.p,
000021 |     Q_q=H(q)_0,  t_tau=exp(2 pi i H(tau)_0).
000022 | 
000023 | The vacuum is Omega=|oscillator vacuum> tensor |0>. Every individual allowed
000024 | insertion is generated by Heisenberg modes h_i(n), their normal-ordered
000025 | quadratics and the resulting Jordan/OPE coefficients, with the *same*
000026 | normal ordering and operator labels in each candidate. Such insertions
000027 | commute with **all eight** Cartan zero modes h_i(0).
000028 | 
000029 | This is stronger than preserving the one scalar Q_q. A vertex of lattice
000030 | charge p with q.p=0 can still change other Cartan charges and is not included
000031 | in this insertion class. Nor does overall charge cancellation in a word
000032 | imply that its individual intermediate insertions are neutral.
000033 | 
000034 | ## 2. All-word and all-time blindness, without a cutoff
000035 | 
000036 | The Heisenberg relations give [h_i(0),h_j(n)]=0. Hence every Heisenberg mode
000037 | preserves each full charge sector p. Normal-ordered products, derivatives,
000038 | linear combinations and OPE coefficients of these modes do so as well.
000039 | For fixed current modes acting on a finite-energy vector, the relevant
000040 | normal-ordered sums are well defined on the common finite-energy core.
000041 | 
000042 | The entire vacuum-generated representation of this insertion algebra is
000043 | 
000044 |     H_neutral = M(1) tensor |0>.
000045 | 
000046 | It is exactly the oscillator Fock module: creation modes generate it from
000047 | Omega. The intervening K_delta evolution preserves it, and on this module
000048 | 
000049 |     K_delta=N,   Q_q=0,   t_tau=I
000050 | 
000051 | for every delta, q and tau. No subtraction of a charged phase is involved;
000052 | the phase character is evaluated at the actual zero charge.
000053 | 
000054 | Consider any finite ordered word of permitted insertions, their adjoints,
000055 | and real-time unitary or positive-time Euclidean propagators, beginning and
000056 | ending at Omega. Every intermediate vector remains in H_neutral. Replacing
000057 | each propagator by its restriction therefore replaces K_delta by N at every
000058 | step. The resulting matrix element is independent of delta, q and tau.
000059 | The same argument applies to Gram/decoherence entries between two such
000060 | words. It covers arbitrary word length and arbitrary allowed times, not
000061 | merely a finite collection of moments.
000062 | 
000063 | In particular, **if** a complete 36-dimensional symmetric-matrix
000064 | tomography uses only T_A=:h^T A h: and their Heisenberg/Jordan/OPE vacuum
000065 | responses, even perfect reconstruction of every A does not identify delta
000066 | or its charged time law. This does not negate that positive matrix-algebra
000067 | reconstruction. It identifies a blind direction of the response experiment
000068 | from which it is inferred. Operators with a delta inserted into their own
000069 | definition, a delta-dependent initial state, Gibbs traces over charged
000070 | sectors, and charged intermediate fields change this question.