TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DOILY-RECORD-ORIENTATION-PROOF.txt
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000001 | R064. ORIGINAL E8 RECORD ACTION ON DOILY15, ITS SIGNED LIFT AND ORDERED RESPONSE
000002 | 2026-11-01 | scoped exact finite mathematics | F1 remains in progress
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000012 | 
000013 | ORIGINAL INPUT, NOT A NEW MODEL
000014 | compiler-correlated-event-clock-20260919/source_channel.py, unchanged SHA256
000015 | 3852fabaf10c143fe20238b67406df6396f85c931dfd4834fe91a2d7ef900593,
000016 | source_rays49-68, Pauli dictionary70-106, reflection_actions112-149.
000017 | Its PROOF.txt already supplies the sixty-event source and explicitly states
000018 | its physical origin limits. The source consists of norm-square4 rays z in
000019 | C4, P_z=zz*/4, r_z=I-2P_z. Its actual memory is H10=Sym2(C4), not C4:
000020 | U_z=Sym2(r_z), B_z=U_z/sqrt60 for the uniform unrefined event-record
000021 | realization of the given source. This is not the selected stay/jump
000022 | refinement, stopped Petersen reader or physical deployed cusp clock.
000023 | Those instruments retain their own actual record and frame contracts.
000024 | The quarter clock, deployed cusp clock and this finite source must not be
000025 | identified merely by matching rates or dimensions. The original proof
000026 | already retains phase, primitive counts and the tracked history frame.
000027 | The new content here is the explicit incidence/action comparison, the typed
000028 | 15-direction lift into this actual memory, and exact ordered witnesses.
000029 | 
000030 | 1. POINTS ARE PAULI INVOLUTIONS MODULO SIGN, NOT FIFTEEN ORTHOGONAL SECTORS
000031 | Let W_p be the fifteen nonidentity Hermitian two-qubit Paulis, p in F2^4 minus {0}.
000032 | Commutation is the symplectic pairing <p,q>=0. Each line L is a maximal
000033 | commuting triple. The fifteen lines and their unsigned adjacency are known
000034 | Doily mathematics, not a new derivation of physical space. In particular,
000035 | Pi_p^+=(I+W_p)/2 is rank2 and Pi_p^- is its complementary rank2 sector.
000036 | Different p need not give orthogonal projections. Retaining only fifteen
000037 | plus sectors loses information under the original action.
000038 | 
000039 | For each of the four native rays with stabilizer L, exact conjugation is
000040 | r_z W_p r_z*=epsilon_z,p W_fL(p), epsilon_z,p in {+1,-1}.
000041 | The common unsigned involution fixes L and interchanges six external pairs.
000042 | For p not in L, let q_L(p) be the unique member of L commuting with p. Then
000043 | 
000044 |     f_L(p)=p+q_L(p),             q_L(p)=p+f_L(p)  in F2^4.
000045 | 
000046 | The original unitary does NOT send p to q_L(p): its output is outside L.
000047 | All720 external ray/point cases, covering180 incidence cases with four
000048 | native phase lifts each, satisfy this identity. The 12-to3 map p->q has four
000049 | preimages per q and cannot equal invertible conjugation on15 labels.
000050 | This corrects the first proposed identification before building on it.
000051 | The original program already checks Pauli conjugation; that existing fact
000052 | is reused, not counted as a new sixty-event discovery.
000053 | 
000054 | 2. ERASING SIGNS IS A DIFFERENT READER
000055 | Let A be the known Doily adjacency, A_pq=1 for distinct commuting points.
000056 | The unsigned label accounting matrix for uniform original events is
000057 | 
000058 |     T=(12I+8A)/60=(3I+2A)/15,
000059 |     spec T={1^1,(1/3)^9,(-1/5)^5}.
000060 | 
000061 | It is symmetric. For each single native event the unsigned map is itself an
000062 | involution, so every move has a reverse under the same allowed event.
000063 | Any positive reweighting of these events also gives a symmetric label
000064 | matrix. Therefore no strict irreversible order on the unchanged15 labels
000065 | can accommodate all these moves as increasing steps. This is a restricted
000066 | exclusion of that proposed event identification, not a no-go for clocks.
000067 | A chosen ordered word can still have nontrivial direction and response.
000068 | 
000069 | In the fundamental Pauli operator reader, signs cancel differently:
000070 | 
000071 |     (1/60) sum_z r_z W_p r_z* = W_p/5.
000072 | 
000073 | The signed coefficient matrix is12I/60, not T. Independently, the already
000074 | established original projective2design has meanP=I/4 and
000075 | mean(P X P)=(X+Tr(X)I)/20. Expanding (I-2P)X(I-2P) therefore
000076 | gives (X+Tr(X)I)/5, hence W_p/5. This C4 map is CPTP. The
000077 | unsigned T is a valid classical walk, but not its Pauli transfer matrix. Neither this C4 restriction
000078 | nor T is the full original channel on End(H10), whose already established
000079 | spectrum is {1^1,(1/3)^9,(1/5)^45,(1/15)^45}. In particular the negative
000080 | label eigenvalue is not an eigenvalue of that full CP source channel.
000081 | No transition probabilities between orthogonal physical states follow
000082 | from counting these15 nonorthogonal operator directions.
000083 | 
000084 | 3. ACTUAL TEN DIMENSIONAL RECORD ACTION
000085 | Let F:C10->C4 tensor C4 have columns vec(S_a)/2, where the S_a are the
000086 | original ten symmetric Hermitian Pauli matrices. F*F=I10 and FF* is the
000087 | symmetric tensor projector. These are actual original source conventions.
000088 | Define the type-correct observable and sector lifts
000089 | 
000090 |  J_p=F*(W_p tensor I+I tensor W_p)F,
000091 |  Q_p^+=F*(Pi_p^+ tensor Pi_p^+)F.
000092 | 
000093 | J_p has eigenvalues +2 with multiplicity3,0 with multiplicity4,-2 with
000094 | multiplicity3. Q_p^+ is rank3, not rank2 onH10. Define Q_p^- similarly;
000095 | the mixed symmetric sector has rank4. Functorial conjugation gives
000096 | 
000097 |  U_z J_p U_z*=epsilon_z,p J_fL(p),
000098 |  U_z Q_p^+ U_z*=Q_fL(p)^(epsilon_z,p).
000099 | 
000100 | Both identities are executed for all60x15 original matrix pairs.
000101 | This lift is an explicit mathematical observable construction on the
000102 | existing carrier. It does not identify these probes as physical events,
000103 | select a physical state, or derive spacetime dimension from their ranks.
000104 | The familiar Herm2 positive cone for a rank2 C4 sector remains an internal
000105 | matrix cone; it is not a derived common Lorentz cone of spacetime events.
000106 | 
000107 | 4. SAME LABEL CAN MEAN OPPOSITE SECTOR AFTER A WHOLE WORD
000108 | In the unchanged deterministic source ordering, take ray indices0 and54
000109 | and p=(0,0,1,0), namely I tensor X. Exact Gaussian rational multiplication
000110 | of r_54 r_0 gives
000111 | 
000112 |  r_54 r_0 W_p r_0* r_54* = -W_p,
000113 |  f_L54(f_L0(p))=p.
000114 | 
000115 | Here z54=(0,0,-2,0), so r54 r0=diag(-1,1,-1,1)=-I tensor Z.
000116 | This directly proves the sign reversal, without a group-extension claim.
000117 | Thus Pi_p^+ ends at Pi_p^-, and likewise Q_p^+ ends at Q_p^- in the actual
000118 | memory. Returning to a15-label endpoint is not returning to the same
000119 | sector or full history. This is a concrete source-specific connection to
000120 | the existing requirement to keep complete words, phases and Q memory.
000121 | It does not newly prove that general compression loses information.
000122 | 
000123 | 5. EXACT ORDERED RESPONSE IN THE ORIGINAL H10, NOT A TOY QUBIT MODEL
000124 | Use actual source rays z_r=(-2,0,0,0), z_s=(-1,-1,-1,-1), indices0 and13.
000125 | Then r=diag(-1,1,1,1), s=I-J4/2, where J4 is the allones matrix.
000126 | Choose the explicitly specified mathematical preparation and final effect
000127 | 
000128 |  rho=Q_(I tensor Z)^+/3,         E=Q_(X tensor I)^+.
000129 | 
000130 | These are normalized positive rho and a projection E on the ACTUAL H10.
000131 | Write U_r=Sym2(r), U_s=Sym2(s). Exact symbolic rational traces give
000132 | 
000133 |  Tr E U_s U_r rho U_r* U_s* =1/3,
000134 |  Tr E U_r U_s rho U_s* U_r* =1.
000135 | 
000136 | For an elementary verification use computational indices0,1,2,3.
000137 | The input plus subspace of I tensor Z is span(e0,e2), and the output plus
000138 | subspace of X tensor I has basis (e0+e2)/sqrt2,(e1+e3)/sqrt2.
000139 | The compressed fundamental amplitudes M=B*(sr)A and B*(rs)A are
000140 |  M_sr=[[0,0],[1/sqrt2,-1/sqrt2]],
000141 |  M_rs=[[-1/sqrt2,1/sqrt2],[-1/sqrt2,-1/sqrt2]],
000142 | where A=(e0,e2), B=((e0+e2)/sqrt2,(e1+e3)/sqrt2).
000143 | Their squared singular values are (1,0) and (1,1), respectively. For K=M*M,
000144 | Tr Sym2(K)=((Tr K)^2+Tr K^2)/2, divided by the input rank3.
000145 | This yields1/3 and1 independently of the exact10d matrix calculation.
000146 | 
000147 | Each two-record word has original record probability1/3600. Including the
000148 | final effect yields1/10800 versus1/3600, difference1/5400. These numbers
000149 | refer to those chosen probes and recorded words, not an experimental
000150 | prediction, derived vacuum, or physical temporal ordering.
000151 | The result disproves the inference 'symmetric one-step incidence means
000152 | no order-sensitive dynamics'. The word already has an order. It does not
000153 | derive which records represent local physical events or their causal net.
000154 | 
000155 | 6. ACTUAL RECORD BASIS VERSUS THE UNREAD CHANNEL
000156 | For each line L, its four r_j commute and product_j r_j=-I4. Consequently
000157 | the actual four U_j commute with productI10. The relatives H_j=U_j U_0
000158 | form Klein four with H_0=I, H_1H_2=H_3. For its four real Walsh characters
000159 | 
000160 |  Q_chi=(1/4) sum_j chi(j)H_j
000161 | 
000162 | are orthogonal projections summing toI10, of ranks4,2,2,2. These Q_chi are
000163 | context-character projections, distinct from the rank3 Q_p^+ above.
000164 | Rotate the four original record coordinates by the unitary Walsh/2.
000165 | The transformed GLOBAL Kraus operators and effects are
000166 | 
000167 |  B'_chi=(1/(2sqrt60))sum_j chi(j)U_j=U_0Q_chi/sqrt15,
000168 |  (B'_chi)*B'_chi=Q_chi/15,
000169 |  B_j*B_j=I10/60.
000170 | 
000171 | All fifteen context groups satisfy the exact Kraus identity. The unread
000172 | context channel equals Ad(U_0) composed with character pinching,
000173 | (1/4)sum_j Ad(U_j)=Ad(U_0) o [sum_chi Q_chi (.)Q_chi].
000174 | It is not asserted to be the plain rank1 dephasing channel on C4.
000175 | For rho supported in one Q_chi, the alternative global record probability
000176 | is1/15 for that character and0 for the other three. Original four records
000177 | each have probability1/60. Their coarse context probability is1/15 in
000178 | both readouts. Their memory channel agrees; the record statistics do not.
000179 | 
000180 | This is a conditional alternative coherent measurement on a record carrier,
000181 | not an assertion that an already classical record can be rotated without
000182 | extra access. The original specified instrument is not declared ambiguous
000183 | by definition. Its actual physical realization and readout selection must
000184 | be derived at the original seam if it is to identify physical event data.
000185 | The general record-basis observation was already known in the original
000186 | record and Hodge audits; only its explicit actual H10 realization is added.
000187 | The old twenty-event/five-Hodge-block instrument is a DIFFERENT grouping;
000188 | its ranks and Choi multiplicities must not be substituted for these15x4.
000189 | 
000190 | 7. WHAT THIS CHANGES IN F1
000191 | The proposed15-label literal routing and a causal arrow inferred solely
000192 | from its unsigned adjacency fail at their first structural identification.
000193 | The original source itself nevertheless carries order-sensitive responses
000194 | and sector distinctions invisible to those labels. Thus the next origin
000195 | step is to derive the actual physical local insertions, retained record
000196 | algebra and state/clock map from the SAME original full process, preserving
000197 | its phases, event counts, reference and whole-word memory. Then compare its
000198 | ordered responses with the already established seam/contact and causal
000199 | response constructions. No further static Doily census, free detector,
000200 | physical state, clock fit or new Herm2-to-spacetime identification is chosen.