TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/THEOREMS.txt
Original file SHA-256: 648dc0af6af2ad7f31ca6249af644512b829cdd9241deac0250512928a410290
Inclusive original lines: 1-100
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 | ELLIPTIC MOEBIUS SOURCE BRIDGE
000002 | 26 October 2026
000003 | Research contract MATH.ELLIPTIC.MOEBIUS.SOURCE.01
000004 | 
000005 | Purpose
000006 | =======
000007 | Resolve the concrete relation between the existing marked C2 parity curve,
000008 | the Moebius orientable double, the old NS/quarter boundary tuple and its
000009 | finite-width free Cauchy state. This is a scoped mathematical bridge.
000010 | 
000011 | Inputs and first extra premise
000012 | ==============================
000013 | Keep the existing all-J parity curve E:y^2=x^4-1 with h(x,y)=(x,-y),
000014 | Q(x,y)=(i x,y) and kappa(x,y)=(bar x,bar y). It is the supplied square
000015 | carrier. This contract does not rederive tau=i or P2.
000016 | The raw flat spin pullbacks use chi(A)=chi(B)=-1. Uniqueness of this spin
000017 | from the old order-four A4 statement requires a genuine Spin/Pin bundle
000018 | transport, not an operator equality only up to arbitrary U(1) phase.
000019 | The field interpretation uses the explicitly declared free chiral CAR
000020 | quantization of the product Dirac Cauchy graph. Original P1 membership
000021 | in this interpretation is not assumed to be proved.
000022 | 
000023 | 01. Dividing real forms and the same geometric tower
000024 | ===================================================
000025 | Among h^a Q^b kappa,a in Z2,b in Z4, exactly hQkappa and hQ^3kappa have
000026 | two real ovals. Their halves are annuli of width1/2,circumference1.
000027 | Q exchanges these charts. The pair is canonical in the supplied marked
000028 | group. Delta=hQ^2 is the unique free involution among h,Q^2,hQ^2; this
000029 | does not classify all translations on an unmarked elliptic curve.
000030 | Its annulus descent is (t,s)->(1/2-t,s+1/2), with Moebius quotient.
000031 | Proof: GEOMETRY.txt sections1-4; ORIGINAL-RELEVANCE.txt.
000032 | 
000033 | 02. Actual spin central extension
000034 | =================================
000035 | With rho(z)=-bar z,T=(1+i)/2,delta_E(z)=z+T, the raw complex-linear
000036 | coefficient pullbacks obey
000037 |  U_delta^2=+I, U_rho U_delta=-U_delta U_rho.
000038 | The representative g(z)=-bar z+T satisfies g^2(z)=z+i, so U_g^2=-I.
000039 | For the displayed Pauli convention A=sigma_s U_g is complex-linear,
000040 | Dirac covariant and A^2=-I. Its c=+i,-i sectors have involutive quotient
000041 | descent after multiplication by c^-1. Anti-linear Real lifts are separate:
000042 | (sigma_s j_g)^2=+I, and have no ordinary complex +/-i eigenspace split.
000043 | Q-invariance plus genuine transported order-four Spin/Pin gluing selects
000044 | the unique spin character(-,-). No scalar phase is fitted.
000045 | Proof: GEOMETRY.txt section5 and independent review sections3-4.
000046 | 
000047 | 03. Original boundary tuple and fixed geometric KMS normalization
000048 | ================================================================
000049 | On each NS boundary B_Sigma=-i partial_s,Dom H1_NS. The unitary change
000050 | theta=2pi s gives B_Sigma/(2pi)=B_cov. The half shift and winding give
000051 |  V=T_pi, B_cov=2(B_1/4 direct_sum B_3/4),
000052 |  u_cov=[[0,u_base],[I,0]].
000053 | These identities include domains and the winding carry, not just spectra.
000054 | The chiral Cauchy graph at width W=1/2 has selected face covariance
000055 |  C_A=(I+exp(2pi B_cov))^-1.
000056 | Thus beta_angle=2pi in this geometric CAR realization. Overall scale
000057 | cancels. The mixed C_A is a traceclass perturbation of the Hardy sea:
000058 |  ||C_A-P_-||_1 < 2e^-pi/(1-e^-2pi) < 0.086590.
000059 | The relative covariance determinant for u_cov remains 1-z. This is not
000060 | the original electromagnetic/Quillen response or equality of Hardy and
000061 | finite-annulus states. Q, V and A stay different operator slots.
000062 | Proof: BOUNDARY-SOURCE.txt and independent review boundary supplement.
000063 | 
000064 | 04. Decisive restriction on normal crosscap preparation
000065 | ======================================================
000066 | In the separately declared flat Pin model, a finite crosscap Cauchy
000067 | state is not stationary for its signed boundary dynamics. At lambda>0,
000068 | r=e^-2Llambda,t=-c r,P=[[1,bar t],[t,r^2]]/(1+r^2), and
000069 |  ||[diag(lambda,-lambda),P]||=2lambda r/(1+r^2)>0.
000070 | The pair excitation energy variance is4lambda^2 r^2/(1+r^2)^2.
000071 | Normal collar filtering tends to the tree NS vacuum; it loses the c,b
000072 | mark and does not derive the loop quarter Hardy state. This is not a
000073 | no-go for the original P1 theory. Proof: CROSSCAP-TIME-W8-TEST.txt.
000074 | Its older full flat orientation annulus has modulus1, whereas the present
000075 | elliptic annulus has modulus1/2. A change of units cannot identify them.
000076 | 
000077 | Reused results and actual remaining source map
000078 | ==============================================
000079 | v507 already contains the elliptic half-period/CM origin. v510 already
000080 | contains the free Moebius deck and the fermionic nonsplitting mechanism.
000081 | v519 verifies free finite-carrier RP for both relevant axis placements;
000082 | it does not select the elliptic half or identify the finite-width state.
000083 | The generic Cauchy graph reconstruction was known on15Oct; C4 local E8
000084 | extension uniqueness was known on24Oct. Neither is claimed as new here.
000085 | 
000086 | The new content is their concrete realization on the same existing C2
000087 | curve with the full affine word, spin central commutator and fixed-modulus
000088 | boundary source. The primitive P1 source still needs an equivariant map
000089 | preserving the seam, domains, actual Spin/Pin lift, Calderon projector,
000090 | local field products, state and time. The original relative spectral
000091 | action is not silently replaced by a Gaussian action. v813's index-four
000092 | normalization of the physical determinant-line response is still open.
000093 | The established compiler E8 closure and W8 construction are reused.
000094 | 
000095 | Status
000096 | ======
000097 | Mathematical subresults01-03: proved under the stated geometric/CAR inputs.
000098 | Subresult04: scoped source-time exclusion in the declared flat model.
000099 | No physical gate closed. Complete TFPT solution: not proved.
000100 | No verification ledger, paper or website status is promoted.