TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/UPDATE-REAL-SLOTS-AND-SOURCE.txt
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000001 | FULL REAL SLOTS, ACTUAL SEAM, AND THE ELLIPTIC SOURCE
000002 | 26 October 2026. Continuation of MATH.ELLIPTIC.MOEBIUS.SOURCE.01.
000003 | 
000004 | Purpose and correction
000005 | =======================
000006 | Retain the exact previous eight-map calculation inside <h,Q,kappa>, its
000007 | annulus graph, raw NS spin commutator and boundary fold. The original
000008 | four-marked base also has x->1/x. Its lift must be retained when discussing
000009 | the FULL marked geometry. Therefore the earlier phrase canonical dividing
000010 | pair is valid in the named subgroup, not as a classification of all original
000011 | marked real structures. This is a scope correction, not a new physical axiom.
000012 | 
000013 | 01. Full marked lift group and two distinct original Real slots
000014 | ==============================================================
000015 | On E:y^2=x^4-1 adjoin r(x,y)=(1/x,i y/x^2). Direct substitution gives
000016 | r^2=h and r Q r^-1=h Q^-1. The holomorphic group has order16; with kappa,
000017 | the full group has order32. Its antiholomorphic coset has16 members:
000018 | 12 involutions (4 dividing,4 nondividing,4 empty) and4 order-four maps.
000019 | The old eight-map coset contributes2 dividing,4 nondividing,2 empty.
000020 | 
000021 | v519 explicitly distinguishes family A:x->mu/bar x from family D:x->mu bar x.
000022 | The equatorial A map x->1/bar x fixes the Euclidean seam pointwise and
000023 | therefore has no OS cut ON THAT SEAM. The D maps reflect this seam and
000024 | invert its quarter clock. Their elliptic lifts occupy separate slots.
000025 | The two new dividing forms are S=r kappa and hS. In the square coordinates
000026 |  S z=bar z+i/2, hS z=-bar z+1/2.
000027 | They project to the pointwise fixed equator. The old hQkappa,hQ^3kappa
000028 | remain the lifts of the midpoint OS reflections and fix no branch mark.
000029 | Proof: EQUATORIAL-LIFTS.txt and SEAM-GRAPH-REVIEW.txt.
000030 | 
000031 | 02. The first literal source map meets a genuine topology test
000032 | =============================================================
000033 | For the original marked equator Gamma={|x|=1},
000034 |  p^-1(Gamma)=Fix(S) union Fix(hS).
000035 | These are2 horizontal plus2 vertical circles. Their4 intersections are
000036 | the ramification points. The full inverse image is a connected embedded
000037 | graph with V4,E8,F4 and b1=5, not a smooth circle.
000038 | Normalizing the crossing graph gives4 circles; an off-equator radial
000039 | push-off has2 components. Neither operation is an unspecified identification
000040 | with the old single-circle source. A marked collar homeomorphism from a
000041 | single smooth circle cannot equal this full pullback, because the local
000042 | valence at a ramification point is4, whereas a circle has local valence2.
000043 | 
000044 | This rejects only the literal smooth marked-pullback identification. It does
000045 | not exclude an algebraic pushforward, a specified local field embedding or
000046 | an originally derived choice of smooth carrier. A graph Dirac interpretation
000047 | would require its actual trace/domain relation at the vertices. No arbitrary
000048 | vertex scattering law is introduced here.
000049 | Proof: independent incidence construction and affine fixed-equation census.
000050 | 
000051 | 03. Full free Gaussian positivity on this SAME elliptic curve
000052 | ============================================================
000053 | On the old midpoint Schottky half, z=t+i s, both periods are NS and
000054 |  L=partial_t+B, B=-i partial_s, spec B=2pi(Z+1/2).
000055 | For0<t<1 its inverse is S(t)=exp(-tB)(1+exp(-B))^-1.
000056 | It equals the normalized elliptic Szego kernel by its pole, both NS
000057 | multipliers and H0(E,alpha)=0. For0<t,u<1/2 both charge blocks factor:
000058 |  -S(-t-u)=X_t^*X_u, S(t+u)=Y_t^*Y_u,
000059 |  X_t=exp(tB)(1+exp(B))^-1/2,
000060 |  Y_t=exp(-tB)(1+exp(-B))^-1/2.
000061 | The fixed source-order/Clifford sign in the anti-linear word reflection
000062 | then gives RP for every free Gaussian CAR polynomial by Wick expansion.
000063 | The time-zero covariance is exactly C=(1+exp(B))^-1, the labelled
000064 | second-face marginal of the previous finite-width Calderon graph. The full
000065 | graph is its canonical two-face purification. Q transports the cuts with
000066 | B->-B,C->1-C,u->u^-1. This is stronger than equal numerical spectra.
000067 | 
000068 | The all-word result uses the DECLARED FREE GAUSSIAN quantization. It is not
000069 | an interacting Wick theorem, does not identify the full graph state space
000070 | with its marginal, and does not derive the primitive relative spectral
000071 | action, the physical P1 time, the old Hardy W8 state or the EM response.
000072 | Proof: ELLIPTIC-GREEN-RP.txt, reusing the earlier full-cover RP method on
000073 | the actual elliptic torus rather than on the different-modulus test strip.
000074 | 
000075 | 04. A common P1/P2 bundle anchor, with a decisive connection distinction
000076 | ======================================================================
000077 | The existing cover and nontrivial alpha already give
000078 |  p_*O_E=O+O(-2), p_*alpha=O(-1)+O(-1).
000079 | Thus V_ell=(p_*O_E/O)+p_*alpha=O(-2)+O(-1)^2 has exactly the original
000080 | holomorphic anchor a=(1,1,2),rank3,degree-4,h0=0. This is an equality of
000081 | holomorphic bundle isomorphism classes, not a physical direct-sum rule.
000082 | 
000083 | The natural induced flat connection has branch residues(0,1/2,1/2),
000084 | local characteristic polynomial(lambda-1)(lambda+1)^2 and monodromyorder2.
000085 | The original P2 target has residues(0,1/3,2/3),polynomiallambda^3-1,
000086 | monodromyorder3. Their traces are-1 and0. Hence they cannot be conjugate.
000087 | Any nonzero scalar twist preserves the repeated eigenvalue and cannot fix
000088 | this mismatch. The elliptic direct sum also retains a rank1 flat summand,
000089 | while the stated original <U,M0> representation is irreducible.
000090 | The equal residue traces1 explain the common degree; they do not make
000091 | connections equal. The P2 matrix statement is exact; v117's separate ODE
000092 | identification remains numerical. v499 already distinguishes Z2 pillowcase
000093 | geometry from the Z3 parabolic weight cover. Neither cover is replaced here.
000094 | Proof: PUSHFORWARD-ANCHOR.txt and direct comparison to actual M0_EXACT.
000095 | 
000096 | Remaining decisive work
000097 | =======================
000098 | The original source audit through fullv21,v31,v42,TFPT45 and MMST finds no
000099 | marked source isomorphism equating their different carriers. This is a
000100 | bounded reviewed-source statement, not a claim to have read every paper.
000101 | The next source map must specify the actual seam carrier and its local
000102 | field/trace/domain transport, retaining both original Real slots. For a
000103 | given genuine collared spin/operator isomorphism, B and P_C transport
000104 | functorially. The state, winding and local net are additional preserved marks.
000105 | The physical P2 target additionally needs its actual order-three connection;
000106 | the natural order-two scalar/spin pushforward is now explicitly excluded.
000107 | No new freely fitted model is needed for either test.
000108 | 
000109 | Status
000110 | =======
000111 | Full marked lift/graph and free elliptic Gaussian results: proved in scope.
000112 | Common holomorphic anchor: proved in scope.
000113 | Literal smooth seam pullback and natural P2 connection equality: excluded.
000114 | Original physical source map, interacting measure/time and c3 response:
000115 | not derived. Complete TFPT solution:false. Physical gates closed:none.