TFPT scientific source excerpt Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/PROVENANCE-AND-ORIGIN.txt Original file SHA-256: 53fe5a1345544d0d864c339115f8be7e12fc49e741300fa4559fdf1d05a300a2 Inclusive original lines: 110-265 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. 000110 | P1: c3 = 1/(8 pi), P2: g_car = 5 000111 | 000112 | are the two operational inputs. Once the degree-four divisor D=mu4 is 000113 | present, two standard functors give 000114 | 000115 | rank H1(P1 minus D) = 3, 000116 | h0(P1,O(D)) = 5. (2.1) 000117 | 000118 | The arithmetic in (2.1) is exact. The same source retains the physical 000119 | carrier identification as conditional and states that AX.P2.01 remains a 000120 | declared axiom. The later seam-forcing result derives y^3=x^4-1 only inside 000121 | the conformal four-equivalent-mark model and explicitly imports the cyclic-3 000122 | choice from the E8 glue. It leaves both that cyclic-3 choice and the 000123 | physical-seam/conformal-seam identification as the named residue. 000124 | 000125 | Hence the loop 000126 | 000127 | E8/J -> Pauli16 and quartic5 -> Pascal/glue -> E8 000128 | 000129 | checks a highly rigid common fixed point after the rank-four symplectic/code 000130 | and degree-four divisor data are present. The displayed feedback loop 000131 | presupposes the rank-four input and does not establish its P1-only derivation. 000132 | In particular, it does not make five follow from the cited P1 statement 000133 | without reusing P2/E8/G31/rank-four information. 000134 | 000135 | The finite-symplectic alternative makes the first extra choice visible. For 000136 | a nondegenerate F2 space of dimension 2g the numbers of odd and even quadratic 000137 | refinements are 000138 | 000139 | n_odd(g) = 2^(g-1) (2^g - 1), 000140 | n_even(g) = 2^(g-1) (2^g + 1). (2.2) 000141 | 000142 | Thus the minimal nondegenerate space, g=1, gives 1 odd and 3 even forms. It 000143 | does not give a six-object family or a standard five. The next possibilities 000144 | are 000145 | 000146 | g=2: 6 odd, 10 even, standard dimension 6-1 = 5; 000147 | g=3: 28 odd, 36 even, corresponding standard dimensions 27 or 35. 000148 | 000149 | Selecting g=2, equivalently A=F2^4 or the two-qubit Pauli rank, is exactly the 000150 | additional rank choice needed for the six-to-five construction. The cited 000151 | original P1 statement does not specify g=2. The numerical five is unique only 000152 | after this rank-four finite symplectic premise or after the degree-four 000153 | divisor/P2 typing has been supplied. This is an assumption audit of the cited 000154 | derivation chain, not a metamathematical non-derivability theorem for every 000155 | possible strengthening of P1. 000156 | 000157 | Original verdict-preserving references: 000158 | 000159 | * documents/origin_theory/sections/ 000160 | 001-section-the-whole-integer-skeleton-from-one-pair-gcar-nf.tex:36-56 000161 | declares P1/P2 and gives the conditional divisor reading; lines 72-80 say 000162 | AX.P2.01 remains declared. 000163 | * The same file, lines 163-174, states that the Riemann-Roch P2 reading is 000164 | conditional on QGEO.SYM.01 and that the later E8 geometric model still 000165 | presupposes E8. 000166 | * documents/origin_theory/sections/ 000167 | 002-section-the-8-is-triply-forced-geometry-lattice-gravity-.tex:12-20 000168 | retains the symmetry-lift bit and AX.P2.01 and leaves the raw-seam 000169 | realization open. 000170 | * documents/tfpt_3_e8_audit_bootstrap/sections/003-source.tex:327-354 and 000171 | verification/v617_seam_cover_forcing.py:2-66 retain the verdict 000172 | SEAM-FORCES-COVER only within the stated seam axioms and leave cyclic 3 plus 000173 | physical/conformal seam identification unresolved. 000174 | * documents/tfpt_3_e8_audit_bootstrap/sections/ 000175 | 013-section-honest-status-bootstrap-not-creation-from-nothin.tex:330-345 000176 | explicitly says the self-consistency loop is not a derivation from zero and 000177 | AX.P2.01 stays declared. 000178 | * verification/v2_carrier_pascal.py:19-35 proves the arithmetic uniqueness of 000179 | g=5 under the Pascal rule. It does not derive that rule from P1. 000180 | * verification/status_ledger.csv:3 preserves AX.P2.01 as a declared, 000181 | over-determined bootstrap input and records the later finite closures without 000182 | changing its axiom typing. 000183 | 000184 | 000185 | 3. The historical Prym produces four clock-quotient labels, not sixteen 000186 | ------------------------------------------------------------------------ 000187 | 000188 | Take the curve supplied by the conditional seam-cover result, 000189 | 000190 | C: y^3 = x^4 - 1, 000191 | rho(x,y) = (i x,y), 000192 | iota = rho^2: (x,y) -> (-x,y). 000193 | 000194 | The quotient by iota is the elliptic curve y^3=u^2-1 with u=x^2. The four 000195 | fixed points of iota are the three points over x=0 and the single point at 000196 | infinity. Riemann-Hurwitz therefore gives genus one for the quotient and a 000197 | two-dimensional Prym P. Its integral homology lattice Lambda has rank four, 000198 | and J=rho|Lambda satisfies J^2=-I. 000199 | 000200 | In the actual CM basis (1,z,z^2,z^3) of verification v784, 000201 | 000202 | J = [[ 0, 1, 0, 1], 000203 | [ 0, 0, 1, 0], 000204 | [ 0,-1, 0, 0], 000205 | [-1, 0,-1, 0]]. 000206 | 000207 | Exact integer reduction gives 000208 | 000209 | SNF(I-J) = diag(1,1,2,2), 000210 | Lambda/(1-J)Lambda = (Z/2)^2. (3.1) 000211 | 000212 | Thus the intrinsic clock quotient has four labels. This is also forced by 000213 | the eigenvalues i,i,-i,-i, since det(I-J)=4. 000214 | 000215 | The full two-torsion P[2]=Lambda/2Lambda is F2^4 and has sixteen points, but 000216 | it is a different quotient functor. The natural Prym polarization has type 000217 | (1,2); its mod-two form has rank two and is degenerate. Replacing it by the 000218 | stored odd type-(1,3) polarization makes the mod-two form nondegenerate, but 000219 | that form is an additional frozen convention. Under it the original v784 000220 | comparison still has zero marked inner transporters: deck tau has fixed-space 000221 | dimension zero and cycle type [3,3] on the six odd theta characteristics, 000222 | whereas code sigma has fixed-space dimension two and cycle type [3,1,1,1]; 000223 | the deck-fixed theta refinement is Arf zero while the code selector is Arf 000224 | one. 000225 | 000226 | A rank-eight doubling such as Lambda direct-sum Lambda^dual would make the 000227 | order of the (1-J) quotient sixteen. No such Prym doubling is an original 000228 | TFPT constructor in the cited chain. Dualization alone retains type (1,2). 000229 | The later outer S6 duad/syntheme bridge transports already existing finite 000230 | objects and does not change (3.1). 000231 | 000232 | Original paths and preserved verdicts: 000233 | 000234 | * verification/v617_seam_cover_forcing.py:145-188 gives the uniform cover, 000235 | the ramified point at infinity and genus three. Its verdict remains 000236 | SEAM-FORCES-COVER with the stated conditional residue. 000237 | * verification/v784_curve_code_2torsion.py:34-67 defines the certified CM 000238 | lattice, J, A[2], and the polarization conventions. 000239 | * The same file, lines 669-818, reconstructs J and tau, finds natural type 000240 | (1,2), chosen odd type (1,3), and proves mod-two ranks 2 and 4 respectively. 000241 | * The same file, lines 840-888 and 940-981, performs the full finite transport 000242 | census. Lines 1108-1154 preserve the verdict CURVE-CODE-PARTIAL and name 000243 | the two failed layers exactly. 000244 | * claim QGEO.SEAMFORCE.01 is currently source-stale at its old monolithic TeX 000245 | wrappers. The statements above therefore cite current v617 and the current 000246 | split modular-paper path rather than treating the stale wrapper annotation 000247 | as a fresh proof. 000248 | 000249 | 000250 | Final classification 000251 | -------------------- 000252 | 000253 | PROVED within the stated finite constructors: 000254 | 000255 | * original recovered-erasure effects span C I5; 000256 | * doily effects span a six-dimensional space and have scalar commutant; 000257 | * the two instruments are different; 000258 | * the displayed E8/J feedback loop presupposes the rank-four finite symplectic 000259 | input and does not establish its P1-only derivation; 000260 | * the Prym clock quotient is F2^2, while its full two-torsion is F2^4 only 000261 | with a degenerate natural mod-two pairing. 000262 | 000263 | The six-effect POVM is therefore new observable structure on the original H5, 000264 | but it is an additional measurement channel rather than the original erasure 000265 | observable.