TFPT scientific source excerpt Original repository path: experiments/theory-contracts/compiler-correlated-event-clock-20260919/PROOF.txt Original file SHA-256: bf1f85c99f5f89a0a15da1af5e3ff321c2756fda132a9f8d644f9ce0e08c2f5c 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 | UR.COMPILER.CORRELATED_EVENT_CLOCK.13 000002 | 2026-09-19 | PARTIAL | exact finite reconstruction and extension 000003 | 000004 | MAIN RESULT 000005 | 000006 | The full sixty-event correlated source and the marked Petersen quantum jump 000007 | readout can be realized by one recorded finite process, without silently 000008 | removing the forty events outside the marked branch. The necessary extra 000009 | object is a reference frame computed from those forty executed events. Their 000010 | native group action conjugates the marked twenty-event alphabet into itself, 000011 | including all Bell phases and the complete stay/jump instrument. 000012 | 000013 | This closes a finite source/readout composition problem. It does not derive 000014 | the physical event distribution, record, observer/frame, clock, local charged 000015 | fields, vacuum or spacetime from TFPT. No T1-T8 gate is promoted. 000016 | 000017 | INPUT AND RECONSTRUCTION 000018 | 000019 | [OMITTED: non-scientific conversation or execution metadata] 000020 | [OMITTED: non-scientific conversation or execution metadata] 000021 | [OMITTED: non-scientific conversation or execution metadata] 000022 | [OMITTED: non-scientific conversation or execution metadata] 000023 | [OMITTED: non-scientific conversation or execution metadata] 000024 | [OMITTED: non-scientific conversation or execution metadata] 000025 | 000026 | Let z_l be the sixty norm-squared-four Gaussian ray representatives, 000027 | P_l=z_l z_l^dagger/4 and r_l=I-2P_l. On Sym^2(C^4), use the orthonormal Bell 000028 | basis vec(P_a)/2 for the ten symmetric Hermitian two-qubit Pauli matrices. 000029 | U_l=Sym^2(r_l) is a monomial Hermitian involution in this basis. 000030 | 000031 | 1. VERIFIED QUANTUM RESPONSE OF THE FULL SOURCE 000032 | 000033 | C60(X)=(1/60) sum_l U_l X U_l^dagger. 000034 | The first three projective moments equal Psym,t / dim Sym^t(C4), exactly. 000035 | Writing D4=M4^60-Psym,4/35, direct expansion of the four reflection factors 000036 | gives, even before restriction to Sym^2, 000037 | 000038 | C60-C_Haar = 16 R(D4). 000039 | 000040 | The reshuffle R takes an eight-index tensor with order 000041 | (b,d,e,g,a,c,f,h) to (a,c,e,g,b,d,f,h). In zero-based axis notation its 000042 | permutation is (4,5,2,3,0,1,6,7). The checker independently expands the Haar 000043 | integral using E[P^tensor t]=sum_{permutations} V_perm/(4)_t, so this identity 000044 | is not verified by defining the Haar channel as the desired difference. 000045 | 000046 | On End(Sym^2 C4): 000047 | spec C60 = {1^1, (1/3)^9, (1/5)^45, (1/15)^45}; 000048 | spec C_Haar = {1^1, (1/5)^15, (1/7)^84}. 000049 | For the same |e0,e0> preparation and return measurement: 000050 | p60=3/10; pHaar=9/35; difference=3/70. 000051 | This is a finite channel response, not an experimental particle prediction. 000052 | The marked twenty return is 1/4 for two of the six marks in this fixed basis; 000053 | the other four give 13/40. The submitted 1/4 comparison is a particular mark. 000054 | 000055 | The Kraus span has rank55. Two-event products span all M10(C), witnessed 000056 | by rank100 modulo5 with i mapped to2. A nonzero full minor over this quotient 000057 | proves a nonzero Gaussian-integer minor. This supports injectivity after 000058 | blocking two sites and the stated transfer trace 000059 | Z_n=1+9/3^n+45/5^n+45/15^n. It does not supply a physical parent Hamiltonian 000060 | or a new uniform energy-gap value. No such value is claimed here. 000061 | 000062 | The design/Clifford moment background is established mathematics: 000063 | Zhu, Multiqubit Clifford groups are unitary 3-designs, 000064 | https://arxiv.org/abs/1510.02619 000065 | Zhu et al., The Clifford group fails gracefully to be a unitary 4-design, 000066 | https://arxiv.org/abs/1609.08172 000067 | The source-specific finite identities above are independently checked here; 000068 | no worldwide novelty priority is asserted. 000069 | 000070 | 2. THE MARK AND THE COMPLETE QUANTUM INSTRUMENT 000071 | 000072 | The native action on the six odd quadratic forms sends each reflection to a 000073 | triple transposition, not a single transposition. Sylvester's six pentads 000074 | provide the outer S6 action in which these become ordinary transpositions. 000075 | The checker constructs this duality and an equivariant bijection between 000076 | Bell10 rays and unordered 3+3 partitions, for all sixty source events. 000077 | 000078 | Fix an outer mark q. A partition becomes the duad e obtained by deleting q 000079 | from its containing triple. The twenty events moving q give eight self 000080 | transitions and four transitions to each of the three disjoint duads. Thus 000081 | 000082 | K20=(2I+A)/5, 000083 | A^2=2I-A+allones, 000084 | 000085 | where A is Petersen adjacency, checked as a full matrix for every mark. 000086 | 000087 | Let D_l project onto the four Bell labels fixed by U_l. All four fixed 000088 | amplitudes are +1, rather than label-dependent idle phases. Define 000089 | 000090 | S20(X)=(1/20) sum_{l in20} U_l D_l X D_l U_l^dagger, 000091 | J20(X)=(1/20) sum_{l in20} U_l (I-D_l) X (I-D_l) U_l^dagger. 000092 | 000093 | Exact cancellation of cross terms among the four native phase lifts gives 000094 | S20+J20=C20; S20^*(I)=2I/5; J20^*(I)=3I/5. 000095 | This binary Kraus refinement preserves the averaged channel. The existence 000096 | of this refinement does not make a particular physical apparatus automatic. 000097 | 000098 | For E_ef=|e>=0} J20 S20^n=J20(I-S20)^(-1). 000110 | Convergence follows from the state-independent 2/5 idle probability. 000111 | 000112 | spec Phi20 = {1^1,(1/3)^5,(-2/3)^4,0^60,(1/5)^15,(-1/5)^15}. 000113 | spec Phi20^6 = {1^1,(1/729)^5,(64/729)^4,0^60,(1/15625)^30}. 000114 | 000115 | The extra thirty coherence modes distinguish the complete quantum clock 000116 | from its classical population readout. Normalizing J20 by3/5 alone would 000117 | incorrectly give +/-1/3 on these coherences. 000118 | 000119 | The primitive-event counting generator is 000120 | Phi20(z)=z J20(I-z S20)^(-1). 000121 | On populations it is z A/(5-2z); on disjoint coherences it is +/-z/5, 000122 | and on intersecting coherences it is zero. Consequently the number N of 000123 | selected-branch events consumed by six accepts satisfies 000124 | 000125 | Pr(N=n)=binom(n-1,5)(3/5)^6(2/5)^(n-6), n>=6; 000126 | E[N]=10, Var[N]=20/3. 000127 | 000128 | Although count probabilities are state independent, the count-resolved 000129 | endpoint maps do not factor: coherence and population count factors differ. 000130 | 000131 | 4. THE FORTY OTHER SOURCE EVENTS CANNOT SIMPLY BE FORGOTTEN 000132 | 000133 | The original full source contains forty mark-fixing events in addition to 000134 | the selected twenty. If they physically execute and are then ignored in a 000135 | fixed laboratory frame, the accept instrument is J20/3 and the idle map is 000136 | (2/3)C40+S20/3. Exact summation gives a different stopped quantum channel: 000137 | 000138 | spec Phi_fixed60 = {1^1,(1/11)^5,(-1/4)^4,0^60,(1/13)^15,(-1/13)^15}. 000139 | 000140 | More generally let the selected branch have prior weight t in (0,1]. The 000141 | population rates are t/(4-t) and -4t/(5+t), and the surviving coherence 000142 | rates are +/-t/(4+t). The full uniform source is t=1/3. Petersen's rates 000143 | require t=1 within this uncorrected fixed-readout family. The condition is 000144 | not a no-go for recorded or other native readouts. 000145 | 000146 | 5. CONSTRUCTIVE RESOLUTION: KEEP THE NATIVE HISTORY FRAME 000147 | 000148 | The forty background events are not arbitrary noise: they fix the same 000149 | outer mark. For each background U_h, conjugation 000150 | U_l -> U_h^dagger U_l U_h 000151 | permutes the twenty selected source matrices exactly, with all their 000152 | complex phases. Their stay/jump projectors and Kraus operators covary too. 000153 | These identities are checked for all 6*40*20 combinations. Closure for any 000154 | background word follows by composition. 000155 | 000156 | Maintain a reference word F, initially I. At a background event U_h update 000157 | F <- U_h F. 000158 | At a selected-branch event leave F unchanged. Use the relational state 000159 | rho_rel=F^dagger rho F, 000160 | or equivalently the final observable F O F^dagger on the physical memory. 000161 | 000162 | For a background event: 000163 | (U_h F)^dagger (U_h rho U_h^dagger)(U_h F)=F^dagger rho F. 000164 | It is identity in this tracked readout. For a selected event the relative 000165 | Kraus map is the covariantly relabeled U'_l=F^dagger U_l F. Conditional on 000166 | any previous record the twenty-event distribution remains uniform. Hence 000167 | the same averaged relative instrument applies at every tick. 000168 | 000169 | All sixty physical events are retained. The effective maps are 000170 | S_recorded=(2/3)Id+(1/3)S20, 000171 | J_recorded=(1/3)J20. 000172 | Therefore 000173 | 000174 | Phi_recorded=J_recorded(I-S_recorded)^(-1) 000175 | =J20(I-S20)^(-1)=Phi20. 000176 | 000177 | THIS IS THE COMMON-PROCESS RESULT: the full C60 fourth-moment response and 000178 | the complete Petersen quantum jump readout are compatible marginals/readouts 000179 | of one recorded instrument. Explicitly, use K_l=U_l/sqrt60 for background 000180 | events and K_l,s=U_l D_l/sqrt60, K_l,j=U_l(I-D_l)/sqrt60 for selected events. 000181 | Ignoring the records gives exactly C60; retaining and using them yields the 000182 | tracked clock above. The two readouts are not asserted to be the same 000183 | unobserved marginal, nor simultaneously identical observables. 000184 | 000185 | The selected stay/jump refinement preserves C20 only after averaging over 000186 | the source labels. For a fixed selected l, deleting the cross terms between 000187 | D_l and I-D_l generally changes Ad(U_l). Thus the theorem specifies an 000188 | instrument; it is not permission to measure an already fixed coherent 000189 | source history without disturbance. Nor does equality of the reduced memory 000190 | channel identify this measured record process with the submitted unmeasured 000191 | coherent MPS or preserve interference between different recorded histories. 000192 | The audited phase covariance concerns the actual Kraus matrices, not an 000193 | unproved preservation of every global history amplitude. 000194 | 000195 | No active reversal of the physical background dynamics is necessary for 000196 | this mathematical final-readout protocol. The frame depends on actual event 000197 | records; throwing them away does not implement the protocol. Arbitrary 000198 | laboratory interventions need their own covariance analysis. 000199 | 000200 | With all sixty events counted, the generator is