TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/PROVENANCE-AND-ORIGIN.txt
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Inclusive original lines: 15-108
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000015 | 1. The doily POVM is an additional logical measurement channel
000016 | ---------------------------------------------------------------
000017 | 
000018 | Let
000019 | 
000020 |     V : H5 -> (C^4)^(tensor 4)
000021 | 
000022 | be the original normalized quartic-code isometry and let
000023 | 
000024 |     A_a = (<a| tensor I_64) V,       a=0,1,2,3,
000025 | 
000026 | be the four Kraus operators obtained by losing the first register in the
000027 | stored computational basis.  Direct reconstruction from the five original
000028 | word supports gives the exact Knill-Laflamme identity
000029 | 
000030 |     A_a^* A_b = delta_ab I5/4.                              (1.1)
000031 | 
000032 | The same identity holds for any of the four tensor slots.  Consequently the
000033 | four effects of the original labelled erasure instrument are
000034 | 
000035 |     F_a = A_a^* A_a = I5/4.                                (1.2)
000036 | 
000037 | An exact trace-preserving recovery with branch Kraus operators R_(a,mu) does
000038 | not change branch probabilities, because its composed logical effect is
000039 | 
000040 |     sum_mu A_a^* R_(a,mu)^* R_(a,mu) A_a = A_a^* A_a = I5/4.
000041 | 
000042 | Thus the recovered effects are still F_a.  Equivalently, in the existing
000043 | decoded coordinates the erased state is
000044 | 
000045 |     E(rho) = rho tensor I4/4,
000046 | 
000047 | and every measurement {M_b} confined to the erased/gauge factor induces the
000048 | logical effect Tr(M_b) I5/4.  Thus even changing the four-outcome basis or
000049 | coarse-graining the outcomes keeps the whole logical effect space scalar.
000050 | 
000051 | Exact signature of the original effects:
000052 | 
000053 |     number of displayed branches             4
000054 |     rank(F_a)                                 5
000055 |     trace(F_a)                                5/4
000056 |     linear span dimension                     1
000057 |     common commutant dimension in End(H5)    25
000058 |     action of all native G31 words             each F_a is fixed
000059 | 
000060 | The doily-quartic theorem instead gives, in the original unnormalised quartic
000061 | word basis,
000062 | 
000063 |     G = diag(1,3,3,3,6),
000064 |     E_r = q_r q_r^T G / 12,                  r=1,...,6,       (1.3)
000065 | 
000066 | for the six displayed primitive rays q_r.  Reconstructing these exact
000067 | rational matrices from the current checker gives
000068 | 
000069 |     sum_r E_r = I5,
000070 |     rank(E_r) = 1,
000071 |     trace(E_r) = 5/6,
000072 |     E_r^2 = (5/6) E_r,
000073 |     trace(E_r E_s) = 1/36                    for r != s.      (1.4)
000074 | 
000075 | The six effects are linearly independent, so their span has dimension six.
000076 | No distinct pair commutes.  Solving X E_r = E_r X for all six r gives the
000077 | one-dimensional commutant C I5.  All sixty native quartic reflections permute
000078 | the six effects by the standard six-point action.
000079 | 
000080 | Therefore (1.2) and (1.3) are not two presentations of one instrument.  Their
000081 | ranks, effect-space dimensions, commutants and marked group actions differ.
000082 | The new doily POVM is a concrete covariant measurement on the already existing
000083 | H5.  To realize it after erasure one must additionally measure or couple to
000084 | the logical H5 factor; the original erasure/recovery map does not contain that
000085 | measurement.
000086 | 
000087 | Original paths for this comparison:
000088 | 
000089 | * tfpt_explorer/process.py:242-258 constructs the normalized five codewords.
000090 | * _newest2/TFPT_Gesamtdokumentation2_20260927.md:8506-8568 reconstructs the
000091 |   code and checks (1.1), including observation ranks 1,10,25.
000092 | * experiments/theory-contracts/joint-native-algebra-selection-20261016/
000093 |   PROOF.md:42-54 states the actual erasure intertwiner and
000094 |   E(rho)=rho tensor I4/4, R(X)=Tr_4 X.
000095 | * experiments/theory-contracts/joint-native-algebra-selection-20261016/
000096 |   operator_audit.md:72-90 states W^*W=I20 and the physical preparation
000097 |   W(rho tensor I4/4)W^*.
000098 | * experiments/theory-contracts/atlas-defect-synthesis-20261024/
000099 |   doily_quartic_bridge.py:638-669 constructs G and the six native effects;
000100 |   lines 671-693 check all sixty covariances.
000101 | * experiments/theory-contracts/atlas-defect-synthesis-20261024/
000102 |   DOILY-QUARTIC.txt:141-180 states the representation-level POVM verdict.
000103 | 
000104 | 
000105 | 2. The P1/P2 loop is a closure check, not a P1-only selection of five
000106 | ---------------------------------------------------------------------
000107 | 
000108 | The current split original source says explicitly that