TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_recursion_20261028/SOURCE-RECORD-HODGE.txt
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000001 | THE ACTUAL TWO EVENT RECORD CAN HERALD THE HODGE CHANNEL
000002 | 
000003 | 2026-10-28 | exact finite source test | read only upstream analysis
000004 | 
000005 | RESULT
000006 | 
000007 | One recorded source event is insufficient.  Two actual unmarked source events
000008 | are sufficient, provided their 3600 coherent history amplitudes remain
000009 | accessible and a particular record filter is applied.
000010 | 
000011 | For one event, every one of the six marked twenty jump records and the actual
000012 | unmarked sixty record share only two of the five Hodge Kraus directions.  No
000013 | positive record effect can produce p Phi for any p greater than zero.
000014 | 
000015 | For two consecutive events, the actual products U_second U_first span all of
000016 | M10.  The exact source Choi matrix then contains the Hodge Choi matrix as a CP
000017 | subchannel.  The largest constant success probability is
000018 | 
000019 |   p_max = 1/25.
000020 | 
000021 | At p=1/25 the success map is exactly (1/25) Phi, so its success probability is
000022 | input independent.  This uses the actual sixty source matrices and their actual
000023 | phases.  It does not introduce event matrices or combine the six charts.
000024 | 
000025 | The source does not select the required coherent record filter.  That filter is
000026 | constructed after choosing a duad chart and the Hodge target.  Thus the result
000027 | is a real source capability under the new coherent access premise, not an
000028 | original P1 selection law or a predicted physical event rate.
000029 | 
000030 | ONE EVENT OBSTRUCTION
000031 | 
000032 | For every mark, the actual twenty jump matrices span a 15 dimensional subspace
000033 | of M10.  The five Hodge matrices span a 5 dimensional subspace.  Their joint
000034 | span has dimension 18, so their intersection has dimension 2.
000035 | 
000036 | For the actual unmarked sixty matrices, the Kraus span has dimension 55.  In
000037 | every source derived duad chart its joint span with the Hodge matrices has
000038 | dimension 58.  The intersection again has dimension 2.
000039 | 
000040 | The exact intersection planes, in the C_0,...,C_4 basis of the corresponding
000041 | duad chart, are
000042 | 
000043 |   mark 0:       span(C_3,C_4)
000044 |   marks 1,2,3:  span(C_1,C_2)
000045 |   marks 4,5:    span(C_0,C_4).
000046 | 
000047 | These are six coordinate descriptions of the same sixty matrices.  They were
000048 | not stacked into a new 360 event source.  The two dimensional overlap is an
000049 | exact algebraic relationship after a chart is fixed.  It is not a physical
000050 | identification of the old Bell10 module with Lambda^2 C5.
000051 | 
000052 | Write a recorded source isometry as
000053 | 
000054 |   V_source = sum_l |l> tensor K_l.
000055 | 
000056 | For any positive effect E on the coherent record, the heralded Choi matrix has
000057 | the form
000058 | 
000059 |   J_E = W E^T W^dagger,
000060 | 
000061 | where the columns of W are vec(K_l).  Hence range(J_E) is contained in
000062 | range(W).  The transpose convention has no effect on this statement.
000063 | 
000064 | For the Hodge channel
000065 | 
000066 |   Phi = (1/3) sum_m Ad(C_m),
000067 |   J_Phi = (1/3) V V^dagger,
000068 | 
000069 | and range(V) has dimension 5.  Positivity of J_source minus p J_Phi for a
000070 | positive p would require range(J_Phi) to be contained in range(J_source).  The
000071 | exact one event ranks violate this inclusion.  Therefore the largest p is zero
000072 | for every marked twenty jump source and for the one event whole sixty source.
000073 | 
000074 | The accessible two dimensional slice cannot hide a constant probability Hodge
000075 | subchannel.  For each intersection plane the checker solves the stronger
000076 | unconstrained equation
000077 | 
000078 |   sum_mn k_mn C_n^dagger C_m = p I.
000079 | 
000080 | The four coefficient columns and the scalar column are linearly independent.
000081 | The only solution has all k_mn and p equal to zero.  Positivity cannot improve
000082 | that result.
000083 | 
000084 | TWO EVENT CONSTRUCTION AND OPTIMALITY
000085 | 
000086 | Fix one of the six existing duad charts.  Let A be the 100 by 3600 matrix whose
000087 | columns are
000088 | 
000089 |   vec(U_second U_first)
000090 | 
000091 | for the two actual uniform source events.  Let V have columns vec(C_m), and set
000092 | 
000093 |   S = A A^dagger.
000094 | 
000095 | The exact rank of S is 100.  This reuses and sharpens the original source fact
000096 | that two event products span M10.
000097 | 
000098 | The minimum norm record coefficients for the five Hodge amplitudes are
000099 | 
000100 |   X = A^dagger S^(-1) V,
000101 |   A X = V.
000102 | 
000103 | Their exact Gram matrix is diagonal.  Its entries are 1/64 on the two Hodge
000104 | directions already present after one event, and 1/48 on the other three.  For
000105 | example, in chart 0,
000106 | 
000107 |   X^dagger X = diag(1/48,1/48,1/48,1/64,1/64).
000108 | 
000109 | The two event source has Choi matrix S/3600.  Therefore
000110 | 
000111 |   S/3600 - p V V^dagger/3 is positive
000112 | 
000113 | exactly when
000114 | 
000115 |   I - 1200 p S^(-1/2) V V^dagger S^(-1/2) is positive.
000116 | 
000117 | The nonzero eigenvalues in the second term are the eigenvalues of
000118 | V^dagger S^(-1)V=X^dagger X.  Its largest eigenvalue is 1/48.  Hence
000119 | 
000120 |   p <= 1/(1200/48) = 1/25.
000121 | 
000122 | At p=1/25 the inequality is saturated in the three previously missing
000123 | directions and remains positive by 1/4 in the two one event directions.  This
000124 | proves both existence and optimality without a numerical SDP.
000125 | 
000126 | An explicit coherent record contraction is obtained from
000127 | 
000128 |   R_mw = 60 sqrt((1/25)/3) X_wm.
000129 | 
000130 | Its row Gram matrix has diagonal 1 on the three previously missing directions
000131 | and 3/4 on the two old intersection directions.  Thus R is a contraction and
000132 | E=R^dagger R is a valid positive record effect.  On success, retaining five
000133 | orthogonal output record modes and then forgetting that residual label yields
000134 | exactly (1/25)Phi.  No system side correction is required for this conditional
000135 | mathematical construction.
000136 | 
000137 | ACTUAL SOURCE MATRICES AND SCOPE
000138 | 
000139 | For a fixed mark q, the one event checker uses exactly the twenty jump matrices
000140 | 
000141 |   K_l = U_l D_jump,l
000142 | 
000143 | from source_channel.py.  It preserves their phases.  The normalized conditional
000144 | jump channel is (1/12) sum_l Ad(K_l).
000145 | 
000146 | The unmarked one event test uses exactly (1/60) sum_l Ad(U_l).  The two event
000147 | test uses its actual composition, with amplitude 1/60 on each ordered history.
000148 | It does not use a freely chosen unitary gauge, a classical merge, an averaged
000149 | phase twirl, or six independent copies of the source.
000150 | 
000151 | The additional operation is now precise.  It is the dense target specific
000152 | record contraction R over the 3600 coherent histories, together with a success
000153 | herald.  The source fixes A and therefore fixes the minimum norm R once the
000154 | chart and Hodge target are given.  It does not choose that chart, target,
000155 | effect, apparatus or successful branch.  Classical access to the two event
000156 | labels is insufficient because R uses coherent superpositions of histories.
000157 | 
000158 | HOLOMORPHIC PRODUCT TYPING BOUNDARY
000159 | 
000160 | The five epsilon tensors have two complementary algebraic roles.  The wedge
000161 | map
000162 | 
000163 |   w: Sym^2(Lambda^2 C5) -> Lambda^4 C5
000164 | 
000165 | has rank 5 and kernel dimension 50.  Its five output components are represented
000166 | by the symmetric matrices C_m.  In this sense their span is the image side of
000167 | the wedge map.
000168 | 
000169 | For the holomorphic Pluecker coordinate ring of Gr(2,5), the multiplication map
000170 | 
000171 |   mu: Sym^2(10) -> H^0(Gr(2,5),O(2))
000172 | 
000173 | has image dimension 50 and kernel dimension 5.  That kernel is generated by
000174 | the five three term Pluecker quadrics.  The original
000175 | s6_plucker_hadamard_probe.py explicitly calls Q_i the Gr(2,5) relations and
000176 | checks that their signs equal the Hodge wedge signs on all 15 monomials.
000177 | 
000178 | Thus the same five tensors are the wedge output components and the holomorphic
000179 | multiplication relations.  The two statements use complementary maps.  If the
000180 | physical source is specifically the holomorphic coordinate multiplication,
000181 | the five dimensional space is killed rather than emitted as a degree two
000182 | product.  The coherent record theorem remains a valid CP map statement because
000183 | there the C_m are nonzero Kraus operators on the ten dimensional carrier.  It
000184 | does not by itself turn the Pluecker relation space into a physical product
000185 | output.  A source interpretation must specify which of these two maps is
000186 | actually implemented.
000187 | 
000188 | RELATION TO THE OLDER RECORD CONTRACTS
000189 | 
000190 | COHERENT_COMPILER_RECORD.md, COHERENT_RECORD_SELECTION.md and
000191 | PRIMITIVE_RECORD_RECOVERY.md establish conditional record constructions for a
000192 | different sixteen label Pauli word process.  They explicitly leave the record
000193 | carrier, access protocol and physical dynamics underived.  They do not supply
000194 | an unnoticed Bell10 to Hodge identification.
000195 | 
000196 | The actual sixty event PROOF.txt supplies a recorded finite frame protocol and
000197 | proves the full two event product span.  It also distinguishes the measured
000198 | instrument from an unmeasured coherent history.  The current result grants the
000199 | stronger premise requested here: a coherent two event environment record and
000200 | active heralding.  It derives what that resource can do and identifies the