TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-observation-arity-20261025/NATIVE-SOURCE-INSERTION.txt
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000001 | ONE ORIGINAL OCTIC AND THE ACTUAL INSERTION OBLIGATION
000002 | =====================================================
000003 | 
000004 | Scope: exact native algebra and a conditional local-field realization.
000005 | This is a cross-source dictionary, not a physical source-selection theorem.
000006 | 
000007 | 1. Compression to one polynomial
000008 | -------------------------------
000009 | Let phi_i be the normalized original tensor quartics in the companion
000010 | COHERENT-QUARTIC-RESPONSE proof. The original positive Fock polynomials
000011 | are f_i=phi_i/sqrt(4!). In orthonormal isotropic current variables x_a,
000012 | 
000013 |   f0=(sum_a x_a^4)/sqrt96,
000014 |   f1=(x0²x1²+x2²x3²)/sqrt8,
000015 |   f2=(x0²x2²+x1²x3²)/sqrt8,
000016 |   f3=(x0²x3²+x1²x2²)/sqrt8,
000017 |   f4=x0x1x2x3.
000018 | 
000019 | The original Fock norm of a monomial x^alpha is alpha!. All five f_i
000020 | are orthonormal. In these exact, unfitted coordinates put
000021 | 
000022 |   I8(x)=sum_a x_a^8
000023 |         +14 sum_{a<b} x_a^4 x_b^4
000024 |         +168 product_a x_a².
000025 | 
000026 | For the six marked quartics R_A from the actual clock ovoids,
000027 | 
000028 |   I8 = 96 sum_i f_i² = (1/3) sum_A R_A².                    (1)
000029 | 
000030 | These are holomorphic squares, without complex conjugation. Equation(1)
000031 | is not a positive energy or a probability formula for complex x. The
000032 | companion Born formula is instead p_A=||grad R_A||²/48.
000033 | 
000034 | There is an independent original-code definition of the SAME polynomial.
000035 | For the actual binary extended Hamming code C of length8 from
000036 | process._hamming_words, enumerate all256 ordered codeword pairs:
000037 | 
000038 |   W_C^(2)(x00,x01,x10,x11)
000039 |     = sum_{c,d in C} product_{k=1}^8 x_(c_k,d_k) = I8.       (2)
000040 | 
000041 | The checker restores all240 real roots from the60 Gaussian mu4 lines, then
000042 | compares their fourteen tetrads with the fourteen weight4 supports of C.
000043 | The60 canonical line representatives alone carry only10 different supports.
000044 | After restoring the actual phase orbits, the two sets are equal
000045 | as marked subsets, not merely equivalent abstract codes. It then proves
000046 | equations(1)-(2) as full polynomial identities. Thus the original E8 code,
000047 | the source quartics, and the clock ovoid directions meet in one octic.
000048 | 
000049 | The polynomial is classical: it is the degree8 Maschke invariant and the
000050 | genus-two complete weight enumerator. This name does not imply physical
000051 | two-dimensional genus or eight-dimensional spacetime. Primary references:
000052 |   Planat, On the geometry and invariants of qubits, quartits and octits,
000053 |   arXiv:1005.1997, section3, equation(30).
000054 |   https://arxiv.org/abs/1005.1997
000055 |   Edge, The geometrical construction of Maschke's quartic surfaces,
000056 |   section6, Maschke's Invariant.
000057 |   https://webhomes.maths.ed.ac.uk/cheltsov/edge2020/pdf/1945a.pdf
000058 | Neither the classical polynomial nor its code interpretation is claimed
000059 | as a new theorem of invariant theory. The result here is its exact common
000060 | normalization and marking in the retained TFPT operator constructions.
000061 | 
000062 | 2. The actual field and a possible canonical scalar input
000063 | --------------------------------------------------------
000064 | In the already supplied local E8 field realization, actual multiplication
000065 | M:Sym²H5 -> E8[8] has M*M spectrum28^1,10^5,18^9. For