TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/GEOMETRIC-R-SPIN-COLLAR-REVIEW.txt Original file SHA-256: f41d9aa53c27d8e167ec82e7692f9367d073b287a157869a2a39a30114dfe164 Inclusive original lines: 1-200 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | THE ACTUAL HEXAGONAL R SPIN PULLBACK AND THE WHOLE TRANSPORTED COLLAR SOURCE 000002 | Independent analytic source review, 30 October 2026. 000003 | 000004 | VERDICT 000005 | The canonical ramified spin pullback supplies a concrete positive map on 000006 | the existing q/p source. Its base inversion has half-form multiplier 000007 | plus/minus i/z, hence constant plus/minus i in the intrinsic cylinder 000008 | frame. On the whole normal Clifford space this is the Spin(2) rotation 000009 | plus/minus i0 Z, with no compulsory finite particle/antiparticle swap. 000010 | Together with the actual coefficient connection and the old unitary W, 000011 | it transports the whole given source, its graph, CAR products, state, 000012 | circle time and weighted corner domain. This closes that IMAGE-source 000013 | transport. It does not select an original physical OS cut or identify an 000014 | independently prescribed maximal Dolbeault/APS source with this image. 000015 | 000016 | 1. Global map and the old coefficient cocycle 000017 | Use exactly C: eta^2=1-w^6, z=eta/(1-w^3), omega=exp(2pi i/3), 000018 | rho=exp(i pi/3)=-omega^2. For c(w)=omega w, s(w)=-w, 000019 | R=s c^2: w -> rho w, pi R=iota pi, iota(z)=1/z, 000020 | R^2=c, R^3=s, R^6=1. 000021 | The marked points are q_a=(-omega^a,0), p_a=(omega^a,0). The actual 000022 | normalized Kummer maps, already proved in the six-line completion, give 000023 | R^*h_a^q=h_(a+1)^p, R^*h_a^p=h_(a+1)^q, 000024 | c^*h_a^(q/p)=h_(a-1)^(q/p). (1) 000025 | Thus the q/p maps commute with the actual c cocycle. They are not a 000026 | constant symmetry asserted for the isolated original q Fuchs equation. 000027 | In its original holomorphic frame the connection really transforms as 000028 | -z^-2 A_q(1/z)=A_q(z)^t-diag(2,1,1)/z. 000029 | The positive flat metric and the G frame must accompany this formula. 000030 | 000031 | [OMITTED: non-scientific conversation or execution metadata] 000032 | [OMITTED: non-scientific conversation or execution metadata] 000033 | [OMITTED: non-scientific conversation or execution metadata] 000034 | [OMITTED: non-scientific conversation or execution metadata] 000035 | [OMITTED: non-scientific conversation or execution metadata] 000036 | [OMITTED: non-scientific conversation or execution metadata] 000037 | [OMITTED: non-scientific conversation or execution metadata] 000038 | [OMITTED: non-scientific conversation or execution metadata] 000039 | [OMITTED: non-scientific conversation or execution metadata] 000040 | [OMITTED: non-scientific conversation or execution metadata] 000041 | [OMITTED: non-scientific conversation or execution metadata] 000042 | [OMITTED: non-scientific conversation or execution metadata] 000043 | [OMITTED: non-scientific conversation or execution metadata] 000044 | [OMITTED: non-scientific conversation or execution metadata] 000045 | [OMITTED: non-scientific conversation or execution metadata] 000046 | 000047 | At x_epsilon=(w=0,eta=epsilon), epsilon=+/-1, 000048 | z-epsilon=epsilon w^3+O(w^6), Rw=rho w. 000049 | Phi is nonzero there, so its true local fiber weight is epsilon lambda. 000050 | The frame e_z vanishes to order one; its coefficient i/epsilon is NOT 000051 | that nonvanishing fiber weight. They agree through the ramification 000052 | factor rho*(epsilon lambda)=i epsilon. At the two infinity points above 000053 | z=zeta=+/-i, set v=1/w. Then z-zeta=zeta v^3+O(v^6), Rv=rho^-1 v, 000054 | and R exchanges the two points. No fiber eigenvalue is assigned to an 000055 | individual exchanged point. Inversion has derivative -1 at +/-1 and +1 000056 | at +/-i. These jets preserve all the relevant vanishing orders. 000057 | 000058 | 3. Intrinsic cylinder and the actual quarter dictionary 000059 | Set z=exp(u), u=r+ix. For a section f(z)sqrt(dz), its cylinder coefficient 000060 | is g(u)=exp(u/2)f(exp u). Equation (2) becomes exactly 000061 | Rhat g(u)=+/- i g(-u). (4) 000062 | The radial and angular inversions are both present. On the NS circle 000063 | g(x+2pi)=-g(x), the spin map is +/-i P0, P0g(x)=g(-x). 000064 | In the periodic NS trivialization g(x)=exp(ix/2)f(x), it is instead 000065 | f(x) -> +/-i exp(-ix)f(-x). (5) 000066 | Plain periodic reflection would omit a connection term. 000067 | 000068 | Using the ORIGINAL W0 f(theta)=exp(i theta/2)f(2theta) and 000069 | W1 f(theta)=exp(3i theta/2)f(2theta), P0 exchanges the two quarter sectors. 000070 | On their periodic base coefficients both off-diagonal entries are 000071 | K0=M_(exp(-ix)) P_(x->-x), K0^2=I, 000072 | K0 A_alpha K0^-1=-A_(1-alpha), alpha=1/4,3/4. (6) 000073 | This includes the integer carry and retains the 1/4 versus 3/4 labels. 000074 | The c lift fixes z and e_z. Its remaining action is the actual coefficient 000075 | deck action, not a freely chosen angular translation on this base circle. 000076 | 000077 | 4. Whole normal Clifford source and the finite coefficient 000078 | Use the existing source, including ALL actual finite slots: 000079 | N=[[0,-1],[1,0]], Z=diag(1,-1), S=NZ, 000080 | D_q=N partial_r+S A_q+Z Phi_q, B_q=Z A_q-S Phi_q, 000081 | gamma=gamma_F, {gamma,Phi_q}=0, [A_q,gamma]=0, 000082 | Gamma=Z gamma, Ical=i0 Gamma, C0=i0 N. 000083 | Here i0 denotes the old complex structure; Ical is the new one. The 000084 | normal tangential Clifford is -i0 S. Its oriented Spin(2) half-turn is 000085 | M=N(-i0 S)=i0 Z, M^2=-I. (7) 000086 | Changing the common spin lift sign changes M to -M. Which chirality is 000087 | called the holomorphic S0 leg follows the original normal orientation; 000088 | the conjugate leg has the opposite half-form phase. This is the usual 000089 | common spin sign, with no fitted 45-degree relative clutch factor. 000090 | 000091 | Let F_R denote the full existing q-to-p coefficient pullback from (1), 000092 | in the original flat metric. Its action on charged slots and their duals 000093 | retains gamma and the given real/dual labels. In raw Fuchs frames its G 000094 | factors and connection continuation are retained. Let T_R=F_R R_x and 000095 | include r->ell-r in the coefficient pullback when Phi depends on r. 000096 | Transport the GIVEN background, rather than selecting a new one: 000097 | A_p=-T_R A_q T_R^-1, 000098 | Phi_p(r)=T_R Phi_q(ell-r) T_R^-1. (8) 000099 | For the normal-independent family this is the old whole finite class. 000100 | For the actual transported normal-dependent family use its original 000101 | Cauchy/Calderon data; no unbounded Cauchy evolution is inferred. 000102 | 000103 | The whole boundary/collar map is 000104 | Theta_spin=P_r M F_R R_x. 000105 | Derivative and matrix multiplication give 000106 | Theta_spin D_q Theta_spin^-1=D_p, 000107 | Theta_spin B_q Theta_spin^-1=-B_p. (9) 000108 | It commutes with Gamma and Ical, hence is NEW-complex-linear. M 000109 | anticommutes with C0; the required endpoint swap compensates this sign. 000110 | Consequently the whole trace map intertwines the oriented 000111 | C_faces=(C0,-C0), 000112 | not two independent copies with the same outward-normal real structure. 000113 | 000114 | 5. Entire charged chart, products, covariance and time 000115 | The already proved entire chart is 000116 | E v=(P_gamma,+ v,i0 P_gamma,- v), 000117 | h_q=A_q gamma-i0 gamma Phi_q, B_q E=E h_q. 000118 | Its exact compatibility with (7) is M E=E(i0 gamma). Thus the charged 000119 | NS map is j_R=i0 gamma F_R R_x and 000120 | j_R h_q j_R^-1=-h_p. (10) 000121 | On T_faces(v0,v1,bar w0,bar w1), the endpoint-swapped map has slots 000122 | (j_R v1,j_R v0,overline(j_R w1),overline(j_R w0)). (11) 000123 | This retains all four slots, including both adjoint slots. Global squares 000124 | retain the actual coefficient deck map in (3). Its descent action is 000125 | evaluated explicitly below; the unquotiented global spin representation 000126 | must not silently be called the descended operator. 000127 | 000128 | For the whole given spectral graph G_q, (9), endpoint swap and C_faces 000129 | give G_q -> G_p. In the normal-independent charged chart this follows 000130 | also from G(-h)=Swap G(h) Swap, including the off-diagonal cosh block. 000131 | The full selfdual CAR map therefore preserves the products and pulls the 000132 | given graph state to the transported graph state, by the old uniqueness 000133 | theorem. No new Gaussian assumption or chosen state has entered. 000134 | Since j_R is linear, its actual circle-time relation is 000135 | j_R exp(it h_q)j_R^-1=exp(-it h_p). (12) 000136 | Time reversal is part of this source-family map. It is not an assertion 000137 | that the circle generator is a physical radial OS Hamiltonian. 000138 | 000139 | 6. Concrete W, genuine pullback image, and CLOSED transported domain 000140 | Use the already explicit G of ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt with 000141 | the flat-unitary coefficient metric H=G^-dagger G^-1. The old map is 000142 | W=3^-1/2 sqrt(pi') G^-1, 000143 | with the fixed original analytic continuation and spin frame. It maps 000144 | the original source boundary space unitarily onto its C3-EQUIVARIANT 000145 | pullback image. Equations (1)-(2) and pi R=iota pi show that R takes the 000146 | q image ONTO the corresponding p image. All actual deck cocycles commute 000147 | with this open map. In horizontal NS charts the resulting 000148 | U_geom=W_p R_spin W_q^-1 000149 | is precisely the map i0 gamma F_R R_x above, with the normal half-turn 000150 | and endpoint swap on the full collar. The raw chart includes the known 000151 | G, spin-connection and Jacobian terms; it is not a plain rowwise pullback. 000152 | 000153 | Here is the decisive TOTAL deck test. The actual G satisfies 000154 | G(cy)=G(y)Pc, Pc=[[0,0,1],[1,0,0],[0,1,0]]. 000155 | In its horizontal coefficient frame, with the canonical e_z spin frame, 000156 | c* (G^-1 f(pi(y)))=Pc^-1 G^-1 f(pi(y)). 000157 | Consequently the descent operator is delta_c=Pc c*, and 000158 | delta_c W=W. (14) 000159 | It is delta_c, NOT plain c* on scalar meromorphic functions, that defines 000160 | the genuine image. The two applications of (1) give precisely the same 000161 | coefficient deck identification. With that identification the descended 000162 | inversion lift has square -delta_c=-I and order four on this image. 000163 | Its sixth power is still -I. This does NOT identify its representation 000164 | with the faithful order-twelve global S0 lift on unquotiented sections. 000165 | 000166 | Equation (14) retains all three Pc coefficient eigenspaces with their 000167 | required inverse geometric automorphy. For a geometric c eigenvalue chi, 000168 | an invariant tensor requires coefficient eigenvalue chi^-1. In particular 000169 | the naked scalar hexagonal Hodge line, and the rho line of the old Voff 000170 | representation, have c eigenvalue rho^2=omega. A plain invariants 000171 | projection removes them. A true coefficient leg of character omega^2 000172 | would retain their tensor product. The W construction alone does not 000173 | supply an identification of that leg with the physical CP reader. Thus 000174 | this whole domain map proves neither an unchanged faithful U6/C12 clock 000175 | on the descended source nor a general physical exclusion of the rho 000176 | channel. The spaces and their total deck characters must be compared. 000177 | 000178 | For the GIVEN base domain D_q, including its original interior/exterior 000179 | continuation and transmission, the image domain is W_q D_q. R carries 000180 | it to W_p D_p with the same graph norm. This is an actual unitary domain 000181 | map of this source, not a separately fitted selfadjoint extension. 000182 | At a branch point the old raw Fuchs powers 0,1/3,2/3 become integer 000183 | coefficient powers 0,1,2; with the spin factor they are 1,2,3. The first 000184 | jets above preserve these powers and the original gluing. The known 000185 | weighted Sobolev domain and its corner transmission are consequently 000186 | transported. No extra curve Dirac or independently chosen Phi_C is needed 000187 | for the image of the GIVEN D_q and Phi_q under W. 000188 | This does not equate that image with the entire rank-nine pushforward, 000189 | all holomorphic spin sections, or a separately specified unweighted/maximal 000190 | Dolbeault/APS source. Such a different identification is not the present 000191 | source-image transport question. 000192 | 000193 | [OMITTED: non-scientific conversation or execution metadata] 000194 | [OMITTED: non-scientific conversation or execution metadata] 000195 | [OMITTED: non-scientific conversation or execution metadata] 000196 | [OMITTED: non-scientific conversation or execution metadata] 000197 | [OMITTED: non-scientific conversation or execution metadata] 000198 | [OMITTED: non-scientific conversation or execution metadata] 000199 | [OMITTED: non-scientific conversation or execution metadata] 000200 | [OMITTED: non-scientific conversation or execution metadata]