TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/FOUR-POLAR-CONTEXT-ALGEBRA.txt
Original file SHA-256: f9bb309bae0d65b31df6a38899c526547f027b69eb89b34800497e72b4ea9bdb
Inclusive original lines: 27-118
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000027 | Put a=(sqrt6-1)/5, n_j=N[:,j], q=2,3,4,5 and
000028 |  P_q=sum_(j=0,1) (n_j-a n_q)(n_j-a n_q)*.
000029 | The actual rooted polar columns are orthonormal. Define
000030 |  e=(n0-n1)/sqrt2,
000031 |  v_q=(n0+n1-2a n_q)/sqrt2,
000032 |  Z=|e><e|, Q_q=|v_q><v_q|.
000033 | Direct use of the actual simplex Gram gives
000034 |  P_q=Z+Q_q, <e,v_q>=0, ||e||=||v_q||=1,
000035 |  <v_q,v_r>=t=(11+4sqrt6)/25 for q!=r.
000036 | Here 0<t<1. The four-v Gram is (1-t)I4+t J4, strictly positive;
000037 | therefore the four v_q span the whole e-perpendicular four-space.
000038 | 
000039 | This common line is genuinely in the algebra GENERATED by the four P_q.
000040 | Let S=sum_q P_q. Its eigenvalues are 4 on e, lambda+=1+3t on the common
000041 | sum direction of the v_q, and lambda-=1-t on the remaining three-space.
000042 | All three are distinct, so exactly
000043 |  Z=((S-lambda+ I)(S-lambda- I))/((4-lambda+)(4-lambda-)).
000044 | Consequently all Q_q=P_q-Z belong to the generated algebra, and
000045 |  Q_q Q_r=t |v_q><v_r| for q!=r.
000046 | The sixteen outer products are linearly independent and span every operator
000047 | on e-perpendicular. Hence the COMPLETE unital complex *-algebra is
000048 |  A_marks = C Z direct_sum M4(C)|_(e-perp), dimension17,
000049 |  A_marks' = C Z + C(I-Z), dimension2.
000050 | This is a classification, not merely a dimension coincidence or a numerical
000051 | commutant search. The explicit spectral polynomial and products construct it.
000052 | 
000053 | 3. WHAT THE ORIGINAL SHARP POINTER TEMPLATE DOES AND DOES NOT PERMIT
000054 | 
000055 | For every distinct q,r,
000056 |  ||[P_q,P_r]||_HS^2 = 2 t^2(1-t^2)>0.
000057 | Thus the four P_q cannot be four orthogonal outcomes of ONE original sharp
000058 | pointer family. More strongly they cannot be sharp binary marginals of one
000059 | joint POVM: a positive joint effect below a projection is supported in its
000060 | range; summing the joint effects then forces the two sharp marginals to
000061 | commute. The explicit nonzero commutators contradict this condition.
000062 | Each pair {P_q,I-P_q} separately defines a mathematical binary Lueders
000063 | instrument under the original conditional template. Existence of its physical
000064 | implementation in the original full source is NOT inferred from this formula.
000065 | Sequential and unsharp measurements are not excluded.
000066 | 
000067 | The four unit quartic rays IN THE NATIVE COEFFICIENT METRIC w_q=sqrt(6/5)n_q satisfy
000068 |  <w_q,w_r>=-1/5 for q!=r. They are distinct but nonorthogonal. No deterministic
000069 | measurement can perfectly identify which of these pure states was given:
000070 | if an effect E_q has expectation1 in w_q and0 in w_r, positivity of E_q and
000071 | I-E_q implies E_q w_q=w_q and E_q w_r=0, forcing <w_q,w_r>=0.
000072 | This statement concerns preparations IF these rays were prepared in that
000073 | metric. It is not an assertion about physical R_b,4 w_q preparations, whose
000074 | actual Gram must be carried separately. Correlation distinguishability in
000075 | the fixed fold state is not itself such a preparation claim.
000076 | 
000077 | The actual result is therefore a menu of incompatible potential contexts,
000078 | not four physical theories, not a fundamental obligation to choose one q,
000079 | and not a derivation of a detector or a 3+2 physical decomposition.
000080 | 
000081 | 4. THE ORIGINAL STATE RESPONSE BREAKS THE APPARENT CENTRAL 1+4 SPLIT
000082 | 
000083 | Use the FULL original D1, defined by the derivative Wick contraction of the
000084 | same shifted quartics and stored for both quarter phases. D1 is Hermitian.
000085 | For the SAME e and Z above, the exact squared cross-block norm is
000086 |  ||(I-Z)D1 e||^2 = 7882461/4398046511104       for alpha=1/4,
000087 |                     3201175/13194139533312    for alpha=3/4.
000088 | Both are strictly positive. A member xZ+y(I-Z) of A_marks' commutes with D1
000089 | only if x=y. Hence
000090 |  alg_*(P_2,P_3,P_4,P_5,D1)' = C I5,
000091 |  alg_*(P_2,P_3,P_4,P_5,D1) = M5(C).
000092 | The last equality follows from the finite-dimensional double-commutant
000093 | theorem; equivalently M4 acts on the nonzero cross-block to connect e to
000094 | every vector in its orthogonal complement.
000095 | 
000096 | D1 here is a SOURCE-RELATIVE CORRELATION COEFFICIENT on the observation space,
000097 | not an assumed physical measuring instrument or freely executable operation.
000098 | The conclusion is a response-algebra statement. The apparent central 1+4
000099 | split belongs only to the restricted four-mark menu at fixed S01; it is
000100 | already coupled by the given source. Neither it nor a 3+2 split is a proven
000101 | full-source superselection rule. Full E8, erasure, time and carry are retained
000102 | by referring to the original source, not replaced by this finite algebra.
000103 | 
000104 | 5. CHECKER, REVIEW AND DECISIVE REMAINING STEP
000105 | 
000106 | check_four_polar_context_algebra.py checks actual simplex normalization, all
000107 | four exact projector identities, the spectral construction of Z, all16
000108 | generated independent outer products, all6 nonzero commutators, the complete
000109 | four-ray Gram, an orthogonal compatible control, and BOTH original D1 mixing
000110 | norms. four_polar_context_algebra_results.json records source tensor SHA256.
000111 | This is exact SymPy arithmetic; no fits, scan, numerical threshold or cutoff.
000112 | 
000113 | To connect the existing original instrument to physics, identify an actual
000114 | P1-admissible preparation and stable record family in the SAME source net,
000115 | with its original support, reference, time and A/A*-invariant restriction.
000116 | Then compare the resulting pointer/context map with these already available
000117 | observables. Their algebra does not supply this missing identification.
000118 | No ledger, paper, website or full-TFPT proof status is promoted here.