TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ACTUAL-FUCHS-PATH-C6-HARDY-CONSUMER.txt
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000001 | ACTUAL FUCHS PATH, C6 GRAM KERNEL AND THE CERTIFIED HARDY CONTRIBUTION
000002 | 30 October2026. Source/operator continuation. No Atlas or original edits.
000003 | 
000004 | 1. Positive result, exact scope
000005 | 
000006 | The original C6 has a REALIZED path representation on two fibers of the
000007 | original v117 Fuchs system, stronger than equality of six eigenvalues.
000008 | The existing executable path is tfpt_explorer/flavor_path_transport.py:
000009 | original_fuchs_connection:28-35, half_arc:38-51,
000010 | integrate_path_transport:70-130, exact_clock_dictionary:133-154,
000011 | finite_kernel_comparison:157-199. Graph-first code discovery was used.
000012 | 
000013 | Its exact unitary finite dictionary and its numerical raw ODE frames were
000014 | previously separate. The ALREADY PROVED actual algebraic G now closes that
000015 | frame comparison with the original flat metric. This is no new fitted
000016 | metric and no identification of U6 with the geometric spin R or native J.
000017 | 
000018 | The original primitive Wedgeconsumer and our nonzero kappa contribution
000019 | remain positively established. Below an actual EVEN C6 word reads that
000020 | contribution, without identifying it with the whole Yukawa matrix.
000021 | The original selection/insertion of this contribution in the joint source
000022 | functional is not inferred from a valid matrix product alone.
000023 | 
000024 | 2. The actual two paths and their source metric
000025 | 
000026 | Take x=1/2, x'=1/x=2. P is the original connection's lower half-circle
000027 | x->x' around+1; Q its reciprocal upper return. Their concatenation winds
000028 | once positively about+1 and zero times about i,-1,-i. Write Hloop=QP and
000029 | 
000030 |  Traw=[[0,Q],[P,0]].                                      (1)
000031 | 
000032 | Let Gx be the actual original algebraic fundamental matrix at x, with
000033 | old rho principal and tau=i omega². Let Gy=P Gx: this is the continuation
000034 | of THAT SAME G along P, not an independently chosen endpoint frame.
000035 | Set Cpair=diag(Gx,Gy), Hx=Gx^(-*)Gx^-1, Hy=Gy^(-*)Gy^-1.
000036 | ODE uniqueness and the algebraic fundamental solution imply exactly
000037 | 
000038 |  P*Hy P=Hx,  Q*Hx Q=Hy,
000039 |  Cpair^-1 Traw Cpair=[[0,Cs],[I,0]].                       (2)
000040 | 
000041 | The second equality uses the actual loop Cs below. Consequently the
000042 | adjoint appropriate to this source is A^sharp=Hpair^-1 A*Hpair,
000043 | Hpair=diag(Hx,Hy), and Traw^sharp=Traw^-1. The raw Euclidean adjoint
000044 | must not replace it. The independent exact finite comparison alone did
000045 | not establish that raw-adjoint identity; (2) supplies the actual frame.
000046 | 
000047 | The existing source G, H and W transport here are reused from
000048 | ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt:39-146 and
000049 | FUCHS-FRAME-INDEPENDENT-TRACE-REVIEW.txt. The metric is unique up to the
000050 | already understood common positive scale; that scale cancels from(2).
000051 | 
000052 | 3. The signs are fixed by the old branch, not removed by a fit
000053 | 
000054 | In this G calibration the loop is NOT literally the printed bare Pc.
000055 | Along the real approach x->1 from inside, w->0 through negative real w,
000056 | 
000057 |  h0->-omega, h1->-omega, h2->+omega.
000058 | 
000059 | Thus the order3 lift stabilizing the reached Kummer point acts
000060 | 
000061 |  w->omega w, h0->-h2, h1->h0, h2->-h1.
000062 | 
000063 | For G columns g(w,h0),g(omega w,h2),g(omega²w,h1), this gives exactly
000064 | 
000065 |  Gloop=G Cs,
000066 |  Cs=[[0,0,1],[-1,0,0],[0,-1,0]]=S Pc S,
000067 |  S=diag(1,-1,1), Pc=[[0,0,1],[1,0,0],[0,1,0]].             (3)
000068 | 
000069 | Cs is real orthogonal, Cs³=I. These two minus signs are the actual V4*c
000070 | lift reached by the chosen path. The canonical abstract c lift in the old
000071 | G report remains Pc; it is not being overwritten by this based loop.
000072 | A direct independent execution of the existing ODE and literal algebraic
000073 | Gx gave ||Gx^-1 QP Gx-Cs|| below1e-12. The determinant of Gx matched the
000074 | original exact Wronskian. Parent's independent full metric calculation is
000075 | saved as workspace ROOT-ACTUAL-PATH-G-METRIC-CHECK.json; it found source
000076 | isometry errors below1.3e-12. No exact monodromy was substituted for P/Q.
000077 | 
000078 | 4. Original C6 words, transported basis and structural restriction
000079 | 
000080 | Put Ts=[[0,Cs],[I,0]] and
000081 |  Js=(1/sqrt2)[[I,I],[Cs,-Cs]]. Then EXACTLY
000082 | 
000083 |  Js*Js=I, Js*Ts^-1 Js=diag(Cs,-Cs)=U6s,
000084 |  Js*Ts³ Js=diag(I,-I),
000085 |  Dpath=yI-delta Traw^-1,
000086 |  (Cpair Js)^-1 Dpath(Cpair Js)=yI-delta U6s.               (4)
000087 | 
000088 | Thus the original D_y, its H=D^sharp D, its Green inverse and its positive
000089 | inverse square root are all the actual path operators in transported
000090 | source frames. This closes the raw-ODE versus exact-unitary gap.
000091 | The last equation for Ts³ identifies the LINEAR PATH sheet parity. It
000092 | does not identify that parity with P1's Calderon polarization.
000093 | 
000094 | For a completely diagonal six-phase frame set
000095 |  fk=(1,omega^(-k),omega^(-2k))/sqrt3, k=0,1,2.
000096 | Then Cs(S fk)=omega^k(S fk). The actual normalized endpoint vectors are
000097 | 
000098 |  e_(+,k)^path=Cpair Js (S fk,0),
000099 |  e_(-,k)^path=Cpair Js (0,S fk),                            (5)
000100 | 
000101 | with eigenvalues omega^k,-omega^k and orthonormality in Hpair. There is
000102 | no unidentified Euclidean normalization in these canonical e vectors.
000103 | Every GIVEN structural projector Pi is transported by the SAME map
000104 | (Cpair Js) Pi (Cpair Js)^-1; the operator inverse square root commutes
000105 | with that unitary change of Hilbert frame. This does not assign a new
000106 | six-site matrix to iC, P_Theta or P_sing. The original Pstr must retain
000107 | its actual actions; no equality iC=Ts³ is inferred from dimension/parity.
000108 | The path fiber pair may be a finite coupling space inside the theory.
000109 | It need not equal the F5 face pair to be a valid original path consumer.
000110 | 
000111 | 5. An original word now actually reads the certified source component
000112 | 
000113 | The certified finite Hardy contribution is
000114 | 
000115 |  B02=kappa E02 P_gamma,+, kappa!=0.
000116 | 
000117 | Its indices are the ACTUAL G-column/induced-Hardy basis. Equation(3),
000118 | with no new matrix reader, gives
000119 | 
000120 |  Cs B02=-kappa E12 P_gamma,+,
000121 |  Cs² B02=+kappa E22 P_gamma,+,
000122 |  Tr_family(Cs² B02)=kappa P_gamma,+.                       (6)
000123 | 
000124 | This is an even C6 word on both transport branches:
000125 | U6s²=diag(Cs²,Cs²). It uses neither a freely chosen sheet nor a claim that
000126 | geometric R is U6. In the ORIGINAL finite resolvent EM4:368-371, this is
000127 | the m=2 word with the already fixed weight
000128 | 
000129 |  w2=y³ delta²/(y^6-delta^6).                              (7)
000130 | 
000131 | For the original closed cusp y=2/3 and 0<delta<2/3, w2>0. Its contribution
000132 | in(6) is w2 kappa P_gamma,+, nonzero. This is the certified component's
000133 | contribution; other components of the full response are NOT assumed zero,
000134 | and no nonzero total trace after other slots/selectors is claimed.
000135 | 
000136 | The actual bare positive K consumer also retains it. On the two signed
000137 | three-dimensional branches, with a=y²+delta²+eps²,b=y delta,
000138 | 
000139 |  K_plus=uplus I+vplus(Cs+Cs²),
000140 |  vplus=[(a-2b)^(-1/2)-(a+b)^(-1/2)]/3>0,
000141 |  K_minus=uminus I+vminus(Cs+Cs²),
000142 |  vminus=[(a+2b)^(-1/2)-(a-b)^(-1/2)]/3<0.                  (8)
000143 | 
000144 | Here uplus=[(a-2b)^(-1/2)+2(a+b)^(-1/2)]/3, and uminus
000145 | has respectively a+2b,a-b. Formula(8) follows from Cs³=I, with eigenvalues
000146 | 1,omega,omega². Therefore Tr_family(K_plus/minus B02)=vplus/minus
000147 | kappa P_gamma,+, before any additional original restriction/holonomy.
000148 | This is a concrete kernel-word response, not a renewed CP classification.
000149 | 
000150 | 6. The whole already given source is retained
000151 | 
000152 | In the already fixed inner analytic G chart, Cs acts on all three family
000153 | coefficients and preserves the weighted L2 and H1_NS domains. On the named
000154 | free Phi=0 source it commutes with the family-blind h0=A_NS gamma_F;
000155 | therefore it also commutes with its entire graph covariance and time.
000156 | Its real generational completion commutes with the actual J_F and grading.
000157 | The selfdual fields and all CAR products follow by the existing whole
000158 | Cliffordchart. The p endpoint action is transported with the EXISTING
000159 | q->p j/Rx map; it is not a second independent triplet or state.
000160 | On the full genuine C3 pullback image one uses W Cs W^-1 with its signed
000161 | coefficient cocycles. A naked constant Cs on every unquotiented sheet is
000162 | not being substituted for this transported operator. For a nontrivial
000163 | coupled Phi this gives the usual family covariance Phi->Cs Phi Cs* and
000164 | its correspondingly transported G/R/time, not an unproved fixed-Phi symmetry.
000165 | The preserved Hardy reader still has NS energy1/2 and ell_NS, as before.
000166 | 
000167 | 7. Actual original source call and the remaining first identification
000168 | 
000169 | The original blocked-T consumer QFT4:111-155 uses local positive C6 kernels,
000170 | diagonal weights and Padm. The original Zadm source reconstruction QFT8:
000171 | 143-168,179-253 uses the whole source family and its insertions. Their
000172 | prescription does not explicitly substitute the above Hardy B02 into D_f,
000173 | V_adm, a specified local transfer tensor, or a differentiated Zadm coupling.
000174 | Our original primitive Wedgeconsumer DOES evaluate this coefficient once
000175 | it is present: +kappa in the exterior convention and -kappa in exact KO.
000176 | The present result adds the genuine original path action and its nonzero
000177 | weighted word/kernel values(6)-(8). It does not turn the composability of
000178 | these maps into an assertion that Zadm has already executed that insertion.
000179 | The precise next source equality is the original local transfer/insertion
000180 | whose actual coefficient is this B02 contribution, or another original
000181 | producer of the same branch tensor. Hardy=entireY is not a requirement.
000182 | No new state, time, normal integration weight or fixedpoint rule was added.
000183 | 
000184 | Durable exact five-identity check:
000185 | workspace outputs/multi-model-source-search-20261029/
000186 | ACTUAL-FUCHS-CYCLE-HARDY-WORD-CHECK.json,
000187 | SHA256 46bd0d58d91ac445821593e7a02e375f4af991ff3daf7d21d868f7753cbf26d2.
000188 | Original EM3/EM4/EM5 paths, source IDs and hashes are recorded in the
000189 | unchanged C6-GRAM-PRODUCER-AND-ORIGINAL-SOURCE-RESPONSE.txt. Reuse the
000190 | existing geometric-six-spin-clock versus marked-native-clock finding.