TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-observation-arity-20261025/COHERENT-QUARTIC-RESPONSE.txt Original file SHA-256: 012879c9ab78413ce229cb1dbc99eac4e2f4c229550d5853b74626385d389c83 Inclusive original lines: 1-134 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 | ORIGINAL QUARTIC SOURCE, CLOCK OVOIDS AND THREE-REGISTER RESPONSE 000002 | ================================================================ 000003 | 000004 | Claim MATH.NATIVE.COHERENT-QUARTIC-DERIVATIVE.01 000005 | Status: exact in the declared native tensor category; experiment only. 000006 | 000007 | Question and decisive criterion 000008 | ------------------------------ 000009 | Do the six directions obtained from the actual E8/J quotient also occur 000010 | in the original source product, and does the existing erasure operator 000011 | give their coherent response without an independently fitted coupling? 000012 | Success requires equality of the marked rays and actual operator maps, 000013 | including conjugation and normalization. A dimension or count match would 000014 | not suffice. Physical preparation is a separate question. 000015 | 000016 | 1. The original data and the three views of the same six directions 000017 | ----------------------------------------------------------------- 000018 | Use precisely process._code_basis, with its five symmetric word supports 000019 | S_i of sizes s_i=(4,12,12,12,24), and the original October 16 erasure map. 000020 | The normalized code columns are V_i=sum_{w in S_i}|w>/sqrt(s_i). 000021 | The October 24 doily calculation gives the following oriented rays in 000022 | the common unnormalized chart with metric G=diag(1,3,3,3,6): 000023 | 000024 | (1,-1,-1,-1,0), (1,1,-1,1,0), (1,1,1,-1,0), 000025 | (-2,0,0,0,1), (1,-1,1,1,0), (-2,0,0,0,-1). 000026 | 000027 | Their squared G norms are10 and their distinct pairings are-2. 000028 | 000029 | Independently, the October 16 source multiplication M:Sym²H5 -> E8[8] 000030 | has Gram spectrum28^1,10^5,18^9. Its Pi9(v tensor v)=0 locus, writing 000031 | v=(a,sqrt3 b,sqrt3 c,sqrt3 d,sqrt6 e), is defined by 000032 | 000033 | a²-d²-4e²=b²-d²=c²-d²=0, 000034 | ab+cd=ac+bd=ad+bc=0, 000035 | be=ce=de=0. 000036 | 000037 | If e is nonzero, b=c=d=0 and a=+/-2e: exactly two projective rays. 000038 | If e=0, a cannot vanish and normalization a=1 gives b,c,d=+/-1 with 000039 | bcd=-1: exactly four rays. This argument is over C, not only over R. 000040 | These are literally the six displayed doily rays, with permutation only. 000041 | The checker substitutes every marked ray into all nine source quadrics. 000042 | Thus the source-product configuration, the old quartic W6 configuration, 000043 | and the actual clock doily ovoid configuration are the same marked object. 000044 | The old source-product and Igusa results are not claimed as new results. 000045 | 000046 | 2. The decoder is the derivative, with no fitted source map 000047 | --------------------------------------------------------- 000048 | For x in C4 define the original coherent quartic response 000049 | 000050 | phi_i(x) = 000051 | = sum_{w in S_i} product_{k=0}^3 x_{w_k}/sqrt(s_i). 000052 | 000053 | The first-slot erasure decoder in gauge-first order has entries 000054 | 000055 | W[row(w1,w2,w3),5a+i] = 2/sqrt(s_i), if (a,w1,w2,w3) lies in S_i, 000056 | 000057 | with zero entries otherwise. The checker compares all20 columns directly 000058 | with erasure_isometry from the original October 16 operator_checker.py; 000059 | only its old logical-first column order4i+a is changed to5a+i. 000060 | 000061 | Because each S_i is invariant under all four slot permutations, 000062 | 000063 | (W* x^tensor3)_{5a+i} = (1/2) partial_a phi_i(x). 000064 | 000065 | This is an identity of cubic polynomials. In the usual convention Jac(phi) 000066 | is5x4; the gauge-first4x5 reshape is (1/2)Jac(phi)^T. No untransposed 000067 | matrix identification is intended. The bar V4 gauge action in 000068 | Sym³V4 = bar V4 tensor H5 is the contragredient derivative action: 000069 | for unitary g, inverse transpose is bar g. It is not an arbitrary 000070 | assignment made from dimensions20=4*5. 000071 | 000072 | 3. Six marked quartics and their exact Born response 000073 | -------------------------------------------------- 000074 | Put D=diag(1,sqrt3,sqrt3,sqrt3,sqrt6), and let q_A be the displayed rays. 000075 | The actual normalized H5 effect is E_A=(Dq_A)(Dq_A)^T/12. Define 000076 | 000077 | R_A(x) = q_A^T D phi(x) 000078 | = (1/2) sum_i q_Ai sum_{w in S_i} product_k x_{w_k}. 000079 | 000080 | Then the already known Igusa relation holds in these precise marked 000081 | coordinates, rather than after a fitted rotation: 000082 | 000083 | sum_A R_A=0, 000084 | (sum_A R_A²)²=4 sum_A R_A^4. 000085 | 000086 | For the original three-register input x^tensor3 the erasure readout is 000087 | 000088 | p_A(x) = 000089 | = (1/48) sum_{a=0}^3 |partial_a R_A(x)|². 000090 | 000091 | The factor48 is4 from the two half derivatives times12 from the effect. 000092 | The six probabilities satisfy the complete norm identity 000093 | 000094 | sum_A p_A(x) = (sum_a |x_a|²)^3. 000095 | 000096 | For normalized x this is a POVM distribution. The checker proves every 000097 | equality coefficient by coefficient, first with independent conjugate 000098 | variables y and then on the physical slice y=bar x. This treats arbitrary 000099 | complex inputs and does not replace conjugation by transposition. 000100 | 000101 | For x=(1,0,0,0), the six probabilities are 000102 | 000103 | (1/12,1/12,1/12,1/3,1/12,1/3). 000104 | 000105 | For x=(1,i,-1,i)/2 they are 000106 | 000107 | (1/12,1/3,1/12,1/12,1/12,1/3). 000108 | 000109 | The independent review checked that the erroneous substitution y=x in 000110 | the latter example instead produces six zeros. This is a useful specific 000111 | control on the operator adjoint and phase convention. 000112 | 000113 | 4. What this removes, and the remaining source obligation 000114 | -------------------------------------------------------- 000115 | There is no need to postulate another five-dimensional carrier or a 000116 | separately chosen coupling to compare these responses: the original 000117 | quartic source, its actual derivative/erasure, and the actual clock 000118 | incidence geometry already give the same six directions and normalization. 000119 | The companion arity theorem determines why an informative readout with 000120 | these labels first occurs on three native registers. 000121 | 000122 | The coherent input x^tensor3 is a specified family of inputs. Its existence 000123 | does not construct copies of an unknown state or explain their physical 000124 | preparation. The source-product Pi9 null condition reconstructs a geometric 000125 | configuration; it does not select one of its rays as the physical state. 000126 | Neither the Igusa relation nor differentiation chooses the instrument, 000127 | physical clock rate, P1 insertion rule or spacetime dimensions. The entire 000128 | native construction still starts from the retained E8/J representation. 000129 | 000130 | Primary literature context only: Krasnov, Real Kummer quartics and their 000131 | Heisenberg invariance, Izvestiya Mathematics84(2020),95-145, section4.1, 000132 | describes the Heisenberg invariant quartic quotient as the Igusa quartic. 000133 | https://doi.org/10.1070/IM8734 000134 | This conventional identification is not used in the polynomial proof.