TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ERASURE-AND-JOINT-25-MULTILOCAL-SUPPORT.txt
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000001 | UR.NATIVE.JOINT_OPERATOR.01 -- Original erasure20 and joint25 field support
000002 | 19 October 2026. Scoped addendum; no physical P1 or original-ledger promotion.
000003 | 
000004 | 1. Same objects, no freely normalized twenty
000005 | 
000006 | Retain the original eightfold CAR, V=M^T O U_+, its correlated source sea,
000007 | alpha=1/4,3/4, the actual five raw quartics P_i and norms N=(96,8,8,8,1).
000008 | The normalized quartics are f_i=P_i/sqrt(N_i). The twenty fields tested here
000009 | are PRECISELY g_(i,a)=partial_a P_i/sqrt(N_i), i=0,...,4, a=0,...,3.
000010 | The original Fock Gram of the raw derivatives is diag(N_i repeated four
000011 | times); the checker verifies this from the existing polynomial inner product.
000012 | Thus Gram(g)=I20, with no fitted rescaling or arbitrary twenty-dimensional
000013 | basis. The original erasure/native intertwiner is unchanged.
000014 | 
000015 | Source anchors: LOCAL-CURRENT-ERASURE-AND-ORDERING.txt39--67;
000016 | FOUR-STARTING-POINTS-STATE-RELATIVE-EQUIVALENCE.txt140--155;
000017 | FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt117--125,182--185.
000018 | This is the same erasure20 in the existing joint25 fields, not a charged E8
000019 | twenty, a new field species, or a new source/preparation hypothesis.
000020 | 
000021 | 2. Small exact cubic principal map
000022 | 
000023 | Use the CAR filtration and exact current/arc dictionary of
000024 | NATIVE-QUARTIC-MULTILOCAL-SUPPORT.txt sections1--3. For each three-subset
000025 | K of{0,...,7}, define the RAW coefficient matrix
000026 |  C3_(K,(i,a))=[product_(k in K)t_k] (partial_a P_i)(V^T t).
000027 | Here t_k=:chi_k* chi_k: is even and t_k^2=0 in the highest CAR-degree
000028 | exterior algebra. The matrix has56 rows and20 columns; columns are ordered
000029 | (i,a) with i first, a second. The actual original unit-field matrix is
000030 | C3 diag(1/sqrt(N_i) repeated four times).
000031 | 
000032 | check_erasure_multilocal_support.py computes these coefficients directly
000033 | with the existing derivative/code/M/O functions. The original M-file and
000034 | O-matrix pins agree with the independently reviewed quartic test. Results:
000035 |  rank(C3)=20, kernel(C3)={0},
000036 |  det(C3* C3)=1032386052096,
000037 |  det(G3_unit)=194481/1099511627776.
000038 | The complete matrices and original norms are saved in
000039 | erasure_multilocal_support_results.json. Since G3=C3* C3 and rank(C3)=20,
000040 | G3 is strictly positive; this is not merely a positive determinant assertion.
000041 | 
000042 | 3. Every original sheet has a nonzero CAR commutator
000043 | 
000044 | The degree-six principal part has three creators and three annihilators
000045 | with matching K in the eigenchannel basis. Commuting with eta_r contracts
000046 | one creator. Distinct pairs(K,k in K) have disjoint exterior basis labels
000047 | (annihilator K, creator K without k), so their Gram has no cross-K
000048 | interference. The actual Fourier matrix has |U_alpha(r,k)|^2=1/8. Hence
000049 |  G3_r=sum_(K,k in K) conjugate(C3_Kj) C3_Kl /8=(3/8)G3.
000050 | It is strictly positive and has rank20 at EVERY r=0,...,7, in BOTH original
000051 | quarter cases. The same formula holds after the original unit normalization.
000052 | The checker constructs the contraction Gram explicitly, preserving these
000053 | basis multiplicities; no source-independent dimension matching is used.
000054 | 
000055 | Both original affine current shifts are checked independently. The
000056 | degree-three part of (partial_a P_i)(H+b/z) remains (partial_a P_i)(H).
000057 | Shift terms and point-split normal-ordering/derivative corrections have
000058 | lower CAR degree and cannot cancel the degree-six field or degree-five
000059 | commutator witness. The full fields retain those lower terms.
000060 | 
000061 | Therefore every nonzero original erasure combination has operator-valued
000062 | support at all eight original preimages. A properly localized quotient
000063 | profile pulls back to the union of its eight source arcs; it does not
000064 | become a single original point or a single selected preimage arc.
000065 | 
000066 | 4. All twenty-five actual joint fields
000067 | 
000068 | A general joint field is a sum of a quartic combination and an erasure
000069 | combination, with their fixed original normalizations. If its quartic
000070 | coefficient is nonzero, the reviewed degree-eight principal field term
000071 | (degree-seven commutator) survives; cubic erasure terms cannot cancel it.
000072 | If the quartic coefficient vanishes, injectivity of C3 leaves a nonzero
000073 | degree-six erasure term (degree-five commutator). Consequently the graded
000074 | joint map has rank5+20=25, and every nonzero joint combination retains
000075 | all eight source preimages. Quartic lower-degree terms may overlap cubic
000076 | terms, but cannot cancel a surviving higher quartic principal degree.
000077 | 
000078 | This concerns the complete ALREADY GIVEN joint25 field space under the
000079 | specified conditional original-fold/current dictionary. It is no claim
000080 | about all source fields, all E8, arbitrary nonlinear products, P1 physical
000081 | locality, or physical q/plane selection. It adds no closed-code requirement.
000082 | The existing quartic files and their results are not changed by this addendum.
000083 | 
000084 | No separate source-energy24 or exact local-vacuum preparation result is
000085 | claimed here: this calculation tests the twenty field identities and their
000086 | support, not a newly reconstructed source-Schur state packet. The earlier
000087 | bounded-local eigenstate-preparation lemma retains its own explicit full-
000088 | Hamiltonian, reference, covariance, and separating-vacuum premises.
000089 | 
000090 | Reproduce with python3 -B check_erasure_multilocal_support.py; use -OO and
000091 | an alternate --output for an optimized independent result-file comparison.
000092 | The falsifiable cancellation test is the actual kernel of C3 or of any
000093 | original-sheet principal commutator. Their observed nullities are all0.