TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-COLOR-DENSITY-DESCENT-PROOF.txt
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000001 | R073. ACTUAL COLOR SELECTION AND THE ORIGINAL DENSITY SOURCE
000002 | 1 November2026. Exact given-source descent theorem; physical F1 remains open.
000003 | 
000004 | 1. Reuse the complete existing source algebra
000005 | R072 excludes a SINGLE scalar-cover Weyl profile implementing the actual
000006 | nonuniform q Cartan loop. It does not exclude that loop from the algebra
000007 | generated by the original current. The older BILOCAL-RECONSTRUCTION.txt
000008 | already reconstructs neutral interarc CAR bilinears from the ORIGINAL
000009 | current, with full sea contraction, quarter phase and old point splitting.
000010 | On a connected containing interval the gauge-fixed complex CAR net equals
000011 | the current net. Its relative-commutant containment and exact limitations
000012 | are reused, not claimed newly proved. Thus merely constructing Jq needs
000013 | no new elementary interaction or added current algebra. Products/limits
000014 | in the generated algebra are different from a single scalar Weyl group.
000015 | Do not assume dual completion equals the entire ambient physical net.
000016 | 
000017 | The original physical-net definition at boundary_kernel_source/001:319-343
000018 | requires A AND A* to preserve ran(P_adm). For bounded A this is [A,P_adm]=0.
000019 | It does not define locality by arbitrary P_adm A P_adm. Here we execute
000020 | the actual COLOR selector already transported in R070, not a guessed
000021 | representation of the still separately typed seam/determinant/gap factors.
000022 | 
000023 | 2. Actual fixed center on the original NS source
000024 | Keep original a_r, r integer, occupied sea r<0, with relative charges.
000025 | The unchanged rational M gives the actual color weights
000026 |  w=M^T(1,1,-2,0,0,0,0,0)=(1,-1,0,-1,-1,-1,0,-1).
000027 | Z acts on a_r* by exp(2pi i w_(r mod8)/3), and Z Omega=Omega.
000028 | Set P0=(I+Z+Z^2)/3. It selects this GIVEN source's center-neutral STATES.
000029 | This is not an asserted P0=P_adm, P_F or physical96 one-particle projector.
000030 | The original scalar current is
000031 |  J_m=sum_r :a_r* a_(r+m):,
000032 |  J(phi)=phi0 Q+sum_(m>0)(phi_m J_-m+bar(phi_m)J_m),
000033 | for a real smooth profile phi with its original full Fourier tail.
000034 | 
000035 | Question: when does the ENTIRE continuous family exp(it J(phi)), all real
000036 | t, reduce this actual neutral source space? This is a source-variation
000037 | condition, not a classification of single accidental isolated unitaries.
000038 | 
000039 | 3. Exact leakage from the actual source vacuum
000040 | J_-m Omega is the sum of m orthonormal particle-hole configurations
000041 |  a_r* a_(r-m) Omega, r=0,...,m-1, with their original CAR signs.
000042 | Their trialities are w_r-w_(r-m) mod3. Different m have different ORIGINAL
000043 | relative energy, hence no cancellation of their charged parts. Consequently
000044 |  ||(I-P0) J(phi) Omega||^2=sum_(m>0) ell_m |phi_m|^2,        (1)
000045 |  ell_m=#{r=0,...,m-1: w_r!=w_(r-m) mod3}.
000046 | This is an infinite-sea identity, not a finite cutoff inference. Smooth
000047 | phi makes the sum finite since ell_m<=m. Both original quarters give the
000048 | same result: they change the energy by alpha Q but these currents have Q0.
000049 | 
000050 | Write m=8k+s,0<=s<8. Direct periodic counting gives ALL m:
000051 |  D=(0,6,4,6,2,6,4,6), L=(0,1,1,3,1,3,3,5),
000052 |  ell_(8k+s)=k D_s+L_s.                                  (2)
000053 | In particular ell_m=0 iff8 dividesm. A simpler all-mode necessity witness
000054 | is r=0: w0=1 is the UNIQUE such color residue; for m not divisible by8,
000055 | a_0* a_-m Omega is charged and has coefficient of modulus1.
000056 | 
000057 | If all exp(itJ(phi)) preserve P0, differentiating their admitted vacuum at
000058 | t0 forces(1) to vanish. Thus phi_m=0 for every m outside8Z. Conversely
000059 | every J_8n commutes with Z term by term on the actual infinite CAR core,
000060 | including J0. Smooth exponentiation strongly commutes with Z and P0.
000061 | The standard current energy bounds justify the selfadjoint smear and
000062 | derivative on Omega. Equivalently the given CAR implementer automorphism
000063 | commutes with Z for these phases, with the original continuous lift.
000064 | We have the complete equivalence
000065 |  [exp(itJ(phi)),P0]=0 for ALL real t
000066 |     iff Fourier support(phi) subset8Z
000067 |     iff phi(theta+2pi/8)=phi(theta).                     (3)
000068 | No assertion is made that commuting with P0 always implies commuting
000069 | with Z for arbitrary operators. The vacuum argument proves this exact
000070 | connected density-family class, without that false general inference.
000071 | 
000072 | For phi=cos(theta),phi1=1/2 and ell1=1:
000073 |  ||(I-P0)J(cos theta)Omega||^2=1/4.                     (4)
000074 | For the fixed source vacuum, the color-survival probability of this pulse
000075 | is1-t^2/4+o(t^2). This is a diagnostic source response, not a measurement
000076 | or a physical coupling prediction. The vacuum is an analytic vector for
000077 | this finite-mode current; the expansion is justified on the original core.
000078 | 
000079 | 4. Spatial meaning, without inventing physical spacetime
000080 | For the actual Ualpha arc/residue transform the one-particle center matrix
000081 | is Z_arc=Ualpha diag(exp(2pi i w_j/3)) Ualpha*.
000082 | A scalar original phase is arc-diagonal G_phi. Its commutator entries are
000083 |  (g_r-g_s)(Z_arc)_rs. The exact equations have rank7 and constant kernel:
000084 | G_phi commutes with Z_arc iff all eight arc phases coincide. Diagonal
000085 | quarter phases leave this graph and its rank unchanged.
000086 | Analytically, the adjacent circulant entries all have squared modulus
000087 | 3/64, as in(4b), so their connected eight-cycle alone forces this kernel.
000088 | For a smooth real phi, equality of those phases implies differences in
000089 | 2pi Z; continuity and eightfold closure make the difference zero. For a
000090 | CONTINUOUS family it also follows immediately by differentiating in t.
000091 | This recovers(3) independently at the actual source operator action.
000092 | 
000093 | Thus a nontrivial supported admissible scalar density variation repeats
000094 | on all eight preimage arcs. A profile confined to an arc shorter than
000095 | 2pi/8 cannot satisfy(3) unless it vanishes. This is an algebraic support
000096 | constraint from the fixed color mark. It singles out the existing eighth
000097 | periodic scalar-current subfamily. It does NOT derive a physical quotient,
000098 | the identification of internal indices with space,3+1,or a new time law.
000099 | The old J_tot,n=J_cover,8n,level8 and its SAME original K time are reused.
000100 | 
000101 | There is a full coordinate-region version of the same result. A cover
000102 | support projector has arc form P_O(x)=diag_r 1_O((x+2pi r)/8). The given
000103 | color acts as an INTERNAL symmetry on this CAR region only if its linear
000104 | action preserves that support, equivalently [P_O,Z_arc]=0. The rank7
000105 | diagonal commutant calculation forces all eight indicators to agree a.e.:
000106 |  O=F^-1(K), modulo nullsets.                             (4a)
000107 | Conversely every such region is preserved by the same Z. These are the
000108 | finest ordinary coordinate regions FROM THIS GIVEN cover that preserve
000109 | the fixed color action on the full CAR net. This is not uniqueness among
000110 | all possible subnets, noncoordinate localizations or physical realizations.
000111 | 
000112 | An exact exterior test makes the failed finer localization visible. For
000113 | coordinate projections P_r,P_s onto arcs0,1 at the same quotient point,
000114 |  |(Z_arc)_10|^2=3/64,
000115 |  ||[Z_arc P_0 Z_arc^*,P_1]||_HS^2=183/2048>0.             (4b)
000116 | Here P_0 in(4b) denotes the arc0 COORDINATE projection, not the color-neutral
000117 | STATE projector P0. Smear on disjoint original arcs over a common small
000118 | quotient interval. A density originally supported in arc0, after color
000119 | conjugation, fails to commute with an exterior density in arc1. The
000120 | noncentral one-particle commutator detects the nontrivial induced CAR
000121 | automorphism as well: on the common current core, the commutator of the
000122 | two smeared quadratic generators contains this nonzero matrix current,
000123 | not just a central phase. Their bounded current groups cannot all commute.
000124 | Thus this actual color is not an internal symmetry
000125 | of the UNCHANGED fine cover-local net. It is internal on the existing
000126 | eight-channel quotient-local net: its constant arc matrix preserves K
000127 | and commutes with the actual residue-diagonal source time and covariance.
000128 | Quarter row phases leave(4b) invariant. Order3 of Z is not identified
000129 | with an order8 geometric clock; the support relation uses its exact M/DFT
000130 | matrix and no numerical-order analogy.
000131 | 
000132 | 5. Existing composite source and complete words remain essential
000133 | The actual q currents J_q,n transfer modes only within the same residue,
000134 | by8n, with q=M^T beta weights. They commute with this Z and P0 exactly,
000135 | even though their original ARC matrices are non-diagonal. They therefore
000136 | pass the color-only descent test, without a scalar profile replacement.
000137 | No automatic descent under Q_Theta, primitive seam or gap follows.
000138 | 
000139 | For a general original current, the canonical invariant observable part is
000140 |  E_Z(J_m)=sum_r delta_(w_r,w_(r+m)) :a_r* a_(r+m):.         (5)
000141 | It is generally a composite current rather than the original scalar J_m;
000142 | for m outside8Z it need NOT vanish. Group averaging a cover-local density
000143 | can expand its cover support to the eight-preimage union. Therefore(5)
000144 | does not prove membership in the ORIGINAL same-small-region physical net.
000145 | Image membership and original region assignment still have to be shown.
000146 | 
000147 | Also E_Z(A B) is not generally E_Z(A)E_Z(B). The smallest original example
000148 | is J_-1 Omega: it has triality2, so P0 J_-1 P0 Omega=0, whereas
000149 |  <Omega,P0 J1 J_-1 P0 Omega>=1.                         (6)
000150 | Whole neutral responses retain charged intermediate states. Equation(6)
000151 | is the original scalar-current instance of the ALREADY established Ward
000152 | principle in R069/R071; it is not recounted as a new general principle.
000153 | Discarding all m outside8Z from the complete source would lose these words.
000154 | Equation(3) classifies admissible individual continuous density operations,
000155 | not the full observable algebra or all allowed charged field insertions.
000156 | 
000157 | 6. Consequence for the common-source derivation
000158 | Fixed color admissibility and the actual eighth-folded source are linked
000159 | by a necessary AND sufficient all-mode operator law, not number matching.
000160 | The existing full current algebra supplies the composite q source; a new
000161 | elementary coupling is not required merely to define its observable.
000162 | The next primitive step is the ORIGINAL current-net/source image with
000163 | its actual seam, determinant and gap actions, reflection and time, on
000164 | the SAME center-fixed words and transported composite operators.
000165 | Do not insert Gamma(P_S) by definition, equate Q_Theta with CAR parity,
000166 | or replace the primitive source algebra by an abstract dual completion.
000167 | Missing original operator correspondence is not an operator set to zero.
000168 | 
000169 | F1 remains open. Preserve R054/R063/R067/R068/R071/R072,R035/R037/R043,
000170 | full source and Z0/Ref/Eta/H0,all word histories and memory,state/time/RP,
000171 | E8/clocks and existing Yukawa/flavor/alpha. No new Higgs measure,spin
000172 | weight,profile,Y/scale/gap/counterterm or physical3+1/spin2 claim.
000173 | 
000174 | 7. Evidence and sources
000175 | The checker imports the unchanged original M and infinite-Maya add_ribbons
000176 | constructor. Full original current vectors are independently classified
000177 | by relative color charges for the inspected modes; the analytic proof
000178 | (1)-(3) covers ALL modes and profiles. A separate exact cyclotomic arc
000179 | commutant system checks rank7. No finite-matrix CCR surrogate is used.
000180 | Older BILOCAL-RECONSTRUCTION.txt and its prior checked evidence are reused.
000181 | Bischoff/Tanimoto arXiv:1111.1671v3,Proposition3.8 and section3.3 support
000182 | the connected-interval U1 fixed-point/current-net identification;they do
000183 | not identify a physical TFPT source or derive spacetime from E8.
000184 | https://arxiv.org/html/1111.1671
000185 | 
000186 | Original physical-net source: boundary_kernel_source/001:319-343;
000187 | full original selectors:qft_source/002:12-118. These do not give the
000188 | Fock Gamma(P_S) identification proposed by the first model response.
000189 | The actual fixed w and NS center action come from R070 with the original
000190 | fold-results.json;R069/R071 complete-word and trace results stay intact.