TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/HEXAGONAL-OPEN-SOURCE-PHASE-REVIEW.txt Original file SHA-256: 34e897a08c91ad4110f56baaceac82745640a17b675e5a70a1619a105f5f8b8c Inclusive original lines: 1-160 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 | HEXAGONAL OPEN SOURCE TRANSPORT AND ITS ACTUAL PHASE LINE 000002 | Independent analytic review, 2026-10-30 000003 | 000004 | VERDICT 000005 | 000006 | There is a concrete positive source connection on the ALREADY GIVEN genus 000007 | two curve. Its positive hexagonal rotation exchanges the actual q and p 000008 | packets, covers z -> 1/z, and induces the exact rho character on a canonical 000009 | holomorphic differential line. This is a geometric operator producing rho, 000010 | not multiplication of a flavor matrix by a freely inserted scalar. 000011 | 000012 | The spin lift and ordinary marked point norm are different representations: 000013 | the spin lift has order twelve, while the paired point norm has eigenvalues 000014 | +i and -i. Neither is itself the rho Hodge line. The existing joint q+p 000015 | construction supplies the required open coefficient transport. This result 000016 | does not identify the Hodge line with a physical CP insertion, an OS/Pin 000017 | reflection, or the extra scalar reader in a closed flavor observable. 000018 | 000019 | 1. REUSE OF THE ACTUAL SOURCE AND ITS ORIENTATION 000020 | 000021 | Put omega=exp(2 pi i/3), rho=exp(i pi/3)=-omega^2, and retain 000022 | 000023 | C: eta^2=1-w^6, z=eta/(1-w^3), 000024 | q_a=(-omega^a,0), p_a=(omega^a,0), a mod 3. 000025 | 000026 | The old source already gives c:(w,eta)->(omega w,eta) and 000027 | s:(w,eta)->(-w,eta). Define the positive rotation 000028 | 000029 | R=s c^2: (w,eta)->(rho w,eta). 000030 | 000031 | It is the INVERSE of the old T=s c, whose w multiplier is -omega=rho^-1. 000032 | Thus using T rather than R reverses the phase orientation. Directly, 000033 | 000034 | R^2=c, R^3=s, R^6=1, 000035 | z(RP)=eta/(1+w^3)=1/z(P), 000036 | R(q_a)=p_(a+2), R(p_a)=q_(a+2). 000037 | 000038 | The equality for z uses eta^2=(1-w^3)(1+w^3) and extends meromorphically. 000039 | For the existing normalized Kummer roots one may reuse the old c,s lifts: 000040 | 000041 | R^*h_a^q=h_(a+1)^p, R^*h_a^p=h_(a+1)^q. 000042 | 000043 | These substitutions preserve both square relations and both product-one 000044 | relations. No new cover is required. As in the old six-line completion, 000045 | the isolated q packet is not preserved; the paired q+p packet is preserved. 000046 | Different Kummer lifts differ by the already recorded deck torsor. 000047 | 000048 | For the original differential equation y'=A(z)y the pulled-back coefficient 000049 | is exactly 000050 | 000051 | -z^-2 A(1/z)=A(z)^t-diag(2,1,1)/z. 000052 | 000053 | Accordingly this is an open map to the inversion-pulled source. It is not 000054 | an assertion that R is a constant symmetry of the original isolated q ODE. 000055 | The map is holomorphic; identifying it with an antilinear physical reflected 000056 | source requires the actual real/Pin/domain map and cannot follow from its 000057 | action on the six points alone. 000058 | 000059 | 2. THE SOURCE SELECTED SPIN AND THE POINT NORM 000060 | 000061 | Retain the existing S0=O(Dq-Kinf), S0^2=K_C, and put 000062 | 000063 | u=(w^3+1)/eta, Phi^2=u dw, Chi=Phi/u. 000064 | 000065 | Here Phi and Chi are meromorphic sections of S0; their divisors are 000066 | Dq-Kinf and Dp-Kinf, respectively. Since R^*u=1/u and R^*dw=rho dw, 000067 | a lift compatible with the FIXED spin square has the form 000068 | 000069 | Rhat^*Phi=lambda Chi, Rhat^*Chi=lambda Phi, lambda^2=rho. 000070 | 000071 | There are the usual two choices lambda=+/-exp(i pi/6). In particular 000072 | 000073 | Rhat^6=-I, Rhat^12=I. 000074 | 000075 | The square is the negative of the normalized order-three c spin lift: 000076 | that lift acts on Phi and Chi by omega^2, whereas lambda^2=rho=-omega^2. 000077 | This also shows why the spin action cannot simply be called multiplication 000078 | by rho on one fixed q spin fiber: R exchanges q and p fibers. 000079 | 000080 | Let Nq= tensor_a S0|q_a and Np= tensor_a S0|p_a. Use Chi at each q_a and 000081 | Phi at each p_a as nonzero evaluation frames. The induced map on Nq+Np 000082 | has the off-diagonal multiplier lambda^3. The index shift a->a+2 is an 000083 | even three-cycle, so the same assertion holds for the evaluation determinant 000084 | Bq+Bp in the corresponding ordered frames. Therefore 000085 | 000086 | Rhat_norm^2=-I, spectrum(Rhat_norm)={+i,-i}. 000087 | 000088 | The inverse norms have the same unordered eigenvalue set. A change of the 000089 | overall spin lift sign exchanges the signs; it does not produce rho. This 000090 | uses the existing ordinary point norm, not a newly identified determinant 000091 | of cohomology or theta representation. 000092 | 000093 | 3. THE EXACT RHO READER ALREADY IN THE CURVE 000094 | 000095 | The quotient by (w,eta)->(-w,-eta) has invariant coordinates 000096 | 000097 | X=1/w^2, Y=eta/w^3, E_hex: Y^2=X^3-1. 000098 | 000099 | It is a degree-two elliptic quotient, with j=0. The genus-two operator R 000100 | descends to the actual order-six elliptic automorphism 000101 | 000102 | (X,Y)->(omega^2 X,-Y). 000103 | 000104 | On its invariant differential this gives 000105 | 000106 | R^*(dX/Y)=(-omega^2)dX/Y=rho dX/Y, 000107 | pullback(dX/Y)=-2 dw/eta. 000108 | 000109 | Thus the canonical eigenspace ell_hex=C.(dw/eta) in H^0(C,K_C) carries the 000110 | exact rho character. The other Hodge eigenline C.(w dw/eta) carries omega; 000111 | det H^0(C,K_C) carries rho*omega=-1. The rho line must therefore be retained, 000112 | rather than replaced by the determinant of the entire Hodge space. 000113 | 000114 | This is a constructive realization of the old hexagonal j=0 phase fiber 000115 | inside the existing curve and its actual automorphisms. Its inverse R^-1 000116 | carries rho^-1. The old positive/negative orientation must be specified 000117 | when comparing the two; no geometric operator fixes that choice by its 000118 | unordered eigenvalue set alone. 000119 | 000120 | 4. WHAT THIS ADDS TO THE OPEN SOURCE MAP 000121 | 000122 | The source geometry supplies together (i) an actual q-to-p coefficient 000123 | transport, (ii) transport of the already selected spin line, and (iii) a 000124 | distinguished Hodge line with rho action. Hence rho now has an explicit 000125 | operator carrier on the original C, beyond the older arithmetic check. 000126 | All existing clock and common-cover results can be transported along this 000127 | map with their full marked data. No new 48-cover construction is needed. 000128 | 000129 | The next identification is precise: show that the ORIGINAL source/output 000130 | reader uses ell_hex in the relevant open Hom map or evaluation functional. 000131 | Tracing an End(F)-valued expression tensored with ell_hex produces a value 000132 | IN ell_hex, not a scalar, until its actual closing reader is specified. 000133 | Transport of a line and its dual alone does not insert a net rho character. 000134 | The scalar v117 closed trace therefore cannot acquire rho merely by calling 000135 | its existing flat coefficient frame hexagonal. The present construction 000136 | provides the candidate geometric phase leg; it does not manufacture that 000137 | missing physical contraction or identify it with the frozen CP readout. 000138 | 000139 | 5. BOUNDED DEDUPLICATION AND ORIGINAL ANCHORS 000140 | 000141 | The existing SIX-LINE-THETA-COMPLETION.txt, lines 50-104, already supplies 000142 | c,s,T and the exact q/p Kummer transport. Its result is reused here; R is 000143 | T^-1. SOURCE-SELECTED-THETA-SPIN.txt, sections 1-3, supplies S0 and the 000144 | fixed-spin-square lift convention. MARKED-Q-DIVISOR-CENTRAL-TWIST.txt, 000145 | lines 58-160, distinguishes ordinary point norms, determinant signs, the 000146 | opposite q/p central characters and the paired source transport. 000147 | 000148 | The original manuscript documents/tfpt_5_redteam/sections/002-section-outcome.tex, 000149 | lines 202-242, locates the leading phase in the hexagonal j=0 fiber over 000150 | the square seam and explicitly retains physical CP as conditional. 000151 | verification/v231_cp_mu6_phases.py and v233_cp_triality_phase.py verify 000152 | the mu6 arithmetic and sheet factorization; they do not supply the above 000153 | Hodge-to-flavor contraction. 000154 | 000155 | Targeted Atlas searches for the hexagonal phase fiber and the genus-two 000156 | elliptic quotient did not return an older explicit X=1/w^2,Y=eta/w^3 000157 | operator proof. This is a bounded search observation, not an assertion 000158 | that the entire repository lacks the formula. The algebraic construction 000159 | above is independent of that retrieval boundary. No Atlas, ledger, older 000160 | hashed source, physical state or field assignment was changed.