TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/LOCAL-WEYL-COMPRESSION-PROOF.txt
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000001 | R059. Proper-local operations and the actual native-code compression
000002 | 31 October 2026. Source-specific exact result. No physical TFPT promotion.
000003 | 
000004 | 1. Decisive question and changed identification
000005 | 
000006 | R058 asks whether the whole process determines spatial relations. Before
000007 | reading an influence graph from reduced operators, test preservation of
000008 | the locality that the given original source ALREADY has. The candidate
000009 | under test is A -> P A P, followed by ordinary reduced-matrix composition
000010 | and its commutator as a local relation. P is the actual five-code projector,
000011 | NOT the original physical H_adm or an executed measurement.
000012 | 
000013 | Earlier source-memory results tested return amplitudes of the single-beta
000014 | L/J alphabet. The present test uses bounded Weyl operators of genuinely
000015 | disjoint smooth local currents in the given whole E8 source. It is not a
000016 | new source/detector, a free dynamics fit or a repeat of the finite-fiber
000017 | Weyl determinant obstruction. Success: exhibit the original local relation
000018 | and its exact compressed defect. Stop if no such defect is obtained.
000019 | 
000020 | 2. Same full source and genuinely local operations
000021 | 
000022 | Use the actual J-positive Cartan polarization u_a,bar(u_a), a0..3,
000023 | bar(u_a).u_b=delta_ab, and the full positive neutral oscillator Fock sector
000024 | inside the given E8_1 source. The actual five homogeneous mode1 polynomials
000025 | are f0=sum z_a^4; f1=z0²z1²+z2²z3²; f2=z0²z2²+z1²z3²;
000026 | f3=z0²z3²+z1²z2²; f4=z0z1z2z3. Their raw Gram is G=diag(96,8,8,8,1).
000027 | The unchanged original code() constructor checks these against the native
000028 | tensor supports. All source charges and other fields remain available.
000029 | 
000030 | For any two disjoint nonempty proper intervals I,J of the existing source
000031 | circle choose real smooth vector profiles V,W supported respectively there.
000032 | The original level1 current relation and its local derivative symplectic
000033 | form give sigma(V,W)=0. Hence the bounded genuine Weyls
000034 |  A=exp(iJ(V)), B=exp(iJ(W)) obey AB=BA on the whole represented source.
000035 | This uses actual supported fields, not truncated Fourier profiles. No
000036 | physical4D support, physical history or P1 intervention has been inferred.
000037 | 
000038 | 3. Exact compact-support construction of the required first moments
000039 | 
000040 | For ANY nonempty interval I choose w in C_c^infinity(I), w>=0 and positive
000041 | on a smaller open interval. Set v_a(theta)=w(theta)sum_(m=-1,+1)c_(a,m)
000042 | exp(i m theta), and V=sum_a(v_a u_a+bar(v_a)bar(u_a)), a real vector field.
000043 | The two moments vhat_(a,n), n=-1,+1, are G_I c_a, where
000044 |  (G_I)_(n,m)=int w(theta) exp(i(m-n)theta) dtheta/(2pi).
000045 | This Hermitian Gram is strictly positive: its quadratic form is the
000046 | integral of w times the squared modulus of c_- exp(-i theta)+c_+exp(i theta).
000047 | Vanishing would force this analytic trigonometric polynomial to vanish on
000048 | an open interval and hence both coefficients to be zero. Thus invert G_I.
000049 | 
000050 | For any desired alpha in C4 prescribe vhat_-1=-i alpha, vhat_+1=0. Reality
000051 | then gives (Vhat_-1)_u=-i alpha and (Vhat_-1)_bar(u)=0. Thus exp(iJ(V)) has
000052 | exactly alpha as its first-mode displacement in the four code oscillators
000053 | and zero first-mode displacement in their four orthogonal companions.
000054 | The identical construction works independently in J for any beta. There
000055 | is NO requirement to prescribe or cancel all higher Fourier moments.
000056 | 
000057 | 4. Full higher modes retained; compression is exact
000058 | 
000059 | For n>0 the normalized creation displacement is i sqrt(n) Vhat_-n in the
000060 | eight complex oscillator coordinates. Smoothness implies finite total
000061 | S_V=sum_(n>0)n||Vhat_-n||². Current zero modes act0 on this neutral sector.
000062 | The whole oscillator displacement has the usual Bargmann action
000063 |  D(alpha_full)f=exp(-S_V/2) exp(alpha_full.z) f(z-bar(alpha_full)).
000064 | The code is vacuum in all unused oscillators. Consequently
000065 |  P A P = lambda_A M(alpha), lambda_A=exp(-S_V/2)>0,
000066 | where M(alpha) is EXACTLY the degree4 part of
000067 |  exp(alpha.z) f(z-bar(alpha)) on the four code coordinates. Degree4
000068 | extraction terminates because the input degree is4. It does NOT truncate
000069 | the Weyl exponential or the supported profile: every excluded oscillator
000070 | contributes to the retained nonzero scalar lambda_A. Likewise for B.
000071 | The scalars are not probability normalization or a changed physical measure.
000072 | These compressions ARE Hermitian in the actual code metric: total oscillator
000073 | parity acts as+1 on the homogeneous degree4 code and conjugates D(alpha)
000074 | to D(-alpha)=D(alpha)*. They are not unitary (at a=b1 they kill f4).
000075 | The Bargmann vacuum used for matrix elements does not replace the actual
000076 | physical state by a vacuum. This operator compression test does not set or
000077 | reconstruct physical Z0; the original whole source normalization is retained.
000078 | 
000079 | 5. Entire two-strength class and explicit original witness
000080 | 
000081 | Take alpha=(a,0,0,0), beta=(b,b,0,0), a,b real. Raw matrix coordinates
000082 | divide <f_i,Mf_j> by G_ii. They must not be treated as orthonormal.
000083 | The exact commutator has only entries
000084 |  [M(alpha),M(beta)]_01=c(a,b), entry10=-12c(a,b),
000085 |  c=a^4 b^4(a²-4)(a²-12)(b²-2)(b²-6)/4608.
000086 | It is nonzero of rank2 except a0,b0,a²4/12 or b²2/6. This is the whole
000087 | declared two-real-strength family, not an assertion about all local Weyls.
000088 | For profiles chosen linearly in a,b, the first joint nonzero Taylor order
000089 | is a^4 b^4/8, totalorder8; nonzero vacuum scalars do not change that order.
000090 | 
000091 | At a=b=1, M(alpha)=diag(19/32,1/4,1/4,1/4,0), while
000092 |  M(beta)=[[5/24,5/48,0,0,0],[5/4,5/8,0,0,0],
000093 |           [0,0,-1/2,1/2,0],[0,0,1/2,-1/2,0],[0,0,0,0,0]].
000094 | The raw01 commutator is55/1536. In the actual orthonormal code basis it is
000095 | 55sqrt(3)/768, with negative transpose10. Squared Hilbert-Schmidt norm is
000096 | 3025/98304 before the strictly positive factor exp[-(S_V+S_W)].
000097 | Thus for EVERY disjoint nonempty proper pair I,J the construction supplies
000098 | commuting full local operators whose bare code compressions do not commute.
000099 | This is an existential pair in each pair of intervals, not every profile.
000100 | 
000101 | 6. Where the missing cancellation is and how the existing full reader works
000102 | 
000103 | For Q=1-P, bounded A,B with AB=BA give exactly
000104 |  [PAP,PBP]=PBQAP-PAQBP.
000105 | The whole word readouts still satisfy PABP=PBAP. Including the actual
000106 | intermediate Q sector restores that relation without adding any coupling.
000107 | The previously established all-word Schur source reader retains precisely
000108 | these intermediate contributions. The68D realization was for L,J,Jdagger;
000109 | these new proper-local Weyl letters have infinite oscillator tails and its
000110 | sufficiency must not be asserted. Earlier displacement results already
000111 | showed that nonzero displacements do not preserve a finite polynomial space.
000112 | 
000113 | If the original H_adm is reducing for the actual local field algebra then
000114 | its restriction preserves locality. No equality with the observed code is
000115 | given here. This result does not forbid other locality-preserving effective
000116 | descriptions, quantum channels, reconstructed subalgebras or infinite fields.
000117 | 
000118 | 7. Meaning for the requested reconstruction and prior positive results
000119 | 
000120 | The bare-five-code commutator graph can invent connections between disjoint
000121 | original local operations. It is not by itself the original causal graph.
000122 | There is no physical signal, superluminal effect, measurement prescription
000123 | or new interaction in this compression calculation. It is a failed proposed
000124 | readout, not a failure of the local source.
000125 | 
000126 | The ACTUAL constructive neutral Moebius standard pair is already present in
000127 | elliptic-moebius-source-bridge-20261026/neutral_moebius_source_20261029/PROOF.txt.
000128 | Do not reopen its infinite realization or infer absence from the older
000129 | abstract Lean input. Its vacuum source/state boundary remains: it does not
000130 | replace the original elliptic Gibbs state, physical P1 reflection/time or
000131 | charged whole-source localization map. R058's Lean statement is limited to
000132 | the literal cited formal composition, not a denial of that positive result.
000133 | 
000134 | Next determine the actual original local word map into the full operational
000135 | source, preserving real support,inclusions,state,time and reflection. Use
000136 | the full mixed word response; do not infer space from bare-code commutators,
000137 | forget Q intermediates, choose a new graph/metric or replace the given state.
000138 | Keep Yukawa/Flavor/alpha,E8/clocks,Z0/reference/eta/H0 and prior source results.
000139 | 
000140 | 8. Reproducibility and literature
000141 | 
000142 | check_local_weyl_compression.py imports unchanged original code() and checks
000143 | the Fock Gram. Exact polynomial degree extraction is independently compared
000144 | with products of four one-oscillator coefficient formulas, all25 entries
000145 | for each generator. It verifies the whole polynomial commutator, actual
000146 | metric adjoint, rank, norm and totalorder8.60 controls pass. Local support
000147 | and all-mode scalar statements are analytic proofs above,not conclusions
000148 | from finitely many Fourier modes. LOCAL-WEYL-COMPRESSION.json stores output.
000149 | No original paper/verification/ledger changed; no new Lean formalization.
000150 | 
000151 | Primary context: Carpi,Kawahigashi,Longo,Weiner,arXiv1503.01260v4,
000152 | section6 supported smeared fields and conformal nets. Local source
000153 | conventions are LOCAL-CURRENT-ERASURE-AND-ORDERING.txt sections2-5.
000154 | The compact-moment construction,actual-code witness and correction identity
000155 | above supply the source-specific test; a literature analogy is not its proof.