TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/NATIVE-QUARTIC-MULTILOCAL-SUPPORT.txt
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000001 | UR.NATIVE.JOINT_OPERATOR.01 -- Native quartic support under the original fold
000002 | 18 October 2026. Scoped mathematical continuation; no original-ledger promotion.
000003 | 
000004 | 1. Question and fixed source
000005 | 
000006 | Which of the ALREADY GIVEN native quartic rays or polar planes becomes a
000007 | single-point local original CAR field under the exact eightfold dictionary?
000008 | Does proper-interval locality supply an exact preparation of their finite
000009 | energy states from the SAME source vacuum? These are different questions.
000010 | No unique-q condition, closed-C5 axiom, fitted profile, or physical measurement
000011 | rule is added. Physical P1 selection of the source/reference remains open.
000012 | 
000013 | The actual conditional conformal-NS/current dictionary is retained:
000014 |  eta_r(x)=psi((x+2pi r)/8)/sqrt8,
000015 |  U_alpha(r,k)=exp(2pi i(k+alpha)r/8)/sqrt8, chi=U_alpha* eta,
000016 |  V=M^T O U_+, H_a=sum_k V_ka :chi_k* chi_k:_sea,
000017 |  F_i=:f_i(H+b/z):, alpha=1/4 or3/4.
000018 | M is the original Weyl matrix; O=T S^-1/sqrt2 is the original J-compatible
000019 | isometry. The code polynomials, in ORIGINAL order, are
000020 |  f0=sum_a h_a^4,
000021 |  f1=h0^2 h1^2+h2^2 h3^2,
000022 |  f2=h0^2 h2^2+h1^2 h3^2,
000023 |  f3=h0^2 h3^2+h1^2 h2^2,
000024 |  f4=h0 h1 h2 h3,
000025 | with original Fock norms N=(96,8,8,8,1). Unit fields are f_i/sqrt(N_i).
000026 | 
000027 | Original exact field prescription: ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-
000028 | GRADING.txt108--160. Its lines64--68 ALREADY warn that Hall is a state/current
000029 | map, NOT a point-local source weight32 field map. We do not relabel that warning
000030 | as a new theorem. The new result below excludes cancellations in EVERY nonzero
000031 | linear combination of the actual five fields.
000032 | 
000033 | 2. Small exact CAR-filtration test
000034 | 
000035 | Filter point-split CAR expressions by number of fermionic factors. Each H_a
000036 | has degree2. In degree8 the current polynomial is obtained by replacing H_a
000037 | by sum_k V_ka t_k, where t_k=:chi_k* chi_k: and t_k^2=0 in the highest-degree
000038 | exterior algebra. Define, for each four-subset K of{0,...,7},
000039 |  C_(K,i)=[product_(k in K)t_k] f_i(V^T t).
000040 | There are70 rows, not a matched physical70-dimensional carrier. A common
000041 | creator-order sign does not change this coefficient map's kernel or Gram.
000042 | 
000043 | The checker uses the ORIGINAL code/fold constructors and Gaussian-rational
000044 | coefficients. Its exact raw Gram G=C* C is
000045 |  [[243/8, -51/16, 0, 0, 0],
000046 |   [-51/16,115/32, 0, 0, 0],
000047 |   [0,0,13/8,-3/8,0],
000048 |   [0,0,-3/8,13/8,0],
000049 |   [0,0,0,0,1/4]].
000050 | Its leading principal minors are243/8,99,1287/8,495/2,495/8, all positive.
000051 | Thus rank(C)=5. For UNIT native polynomials the Gram is N^-1/2 G N^-1/2,
000052 | with determinant165/131072>0. No nonzero ray in this five-space cancels its
000053 | highest degree-eight CAR term. This includes each q=2,3,4,5 ray and every
000054 | nonzero field in any of the original polar planes, without choosing q.
000055 | 
000056 | Both original b shifts are separately evaluated. In f_i(H+b/z), their terms
000057 | of current-degree4 remain precisely f_i(H); all shifted terms have lower CAR
000058 | degree. Current normal-ordering subtraction/point-split derivative corrections
000059 | also have lower fermion degree. They are retained in the FULL field, but cannot
000060 | cancel the degree-eight witness. No replacement of bosonic normal order by a
000061 | bare fermion monomial is made for lower-degree terms.
000062 | 
000063 | 3. Why this proves support on ALL eight original preimages
000064 | 
000065 | Let T_K=product_(k in K):chi_k* chi_k: in degree8. In a creator/annihilator
000066 | exterior basis, T_K has four creators and four annihilators with matching K.
000067 | Commuting an even degree-eight expression with the original odd eta_r field
000068 | contracts one creator. In eigenchannels, contraction by eta_r has coefficients
000069 | of modulus1/sqrt8. Its basis labels (annihilator K, creator K without k) are
000070 | distinct for all pairs(K,k in K). Consequently the EXACT Gram of this degree-
000071 | seven commutator map on the five original polynomials is
000072 |  G_r=sum_(K,k in K) conjugate(C_Ki) C_Kj |U_alpha(r,k)|^2=G/2.
000073 | It has rank5 for EVERY r=0,...,7 and BOTH alpha. No interference between
000074 | different K can hide one original arc. Lower CAR degrees produce commutator
000075 | degrees at most5 and cannot remove this degree-seven witness.
000076 | 
000077 | The holonomy-aware cyclic deck recurrence is also checked exactly. With
000078 | zeta=exp(i*pi/16), U_alpha(r,k)=zeta^((4k+4alpha)r)/sqrt8 and wrap phase
000079 | zeta^(32alpha), every eigenchannel density is deck invariant. This agrees
000080 | with the explicit eight separate commutator tests; support is not inferred
000081 | from dimensional equality or from an arbitrary eight-vacuum construction.
000082 | 
000083 | For a quotient test function supported in a nonempty proper interval J,
000084 | the field pulls back to the union of the eight preimage arcs
000085 |  I_r=(J+2pi r)/8.
000086 | For any nonzero original quartic combination the operator-valued distribution
000087 | has a nonzero CAR commutator witness at EACH corresponding preimage. Smearing
000088 | and separated probe tests can therefore detect every arc. It cannot be an
000089 | operator-valued distribution localized at only one source point or one selected
000090 | preimage arc. Localizing a quotient field means a MULTI-INTERVAL original
000091 | localization, not an absence of localization. This does not classify arbitrary
000092 | other original composites, state-dependent matrix-element cancellations, or
000093 | the still unproved physical P1 net/current identification.
000094 | 
000095 | 4. Distinct exact bounded-local preparation statement
000096 | 
000097 | The existing full CAR source construction has charge0 in EACH residue channel
000098 | and original excitation energy32 on every one of the five inputs, equivalently
000099 | K_alpha=4. A small direct call of its original source_packets() constructor
000100 | checks both quarters: supports(694,710,711,713,688), only energy32, residue
000101 | charges all0, actual Gram rank5. Thus all nonzero q targets/plane vectors remain
000102 | genuine nonzero positive-energy eigenvectors in the full source Hilbert space.
000103 | This is NOT merely the compression of a larger H to a finite code.
000104 | Source anchors: check_original_current_schur_kernel.py96--119;
000105 | check_original_density_current.py48--52,81--85; ORIGINAL-ORDERED-CURRENT-