TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/ELLIPTIC-MODULI-INSERTION.txt
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000001 | ELLIPTIC MODULUS RESPONSE AND RETAINED-COMPILER ADMISSION
000002 | 26 October 2026
000003 | MATH.ORIGINAL-SEAM.ELLIPTIC-MODULUS-INSERTION.01
000004 | 
000005 | 1. Question and verdict
000006 | 
000007 | Can geometric averaging change the existing free elliptic Cauchy source,
000008 | after the correct conformal boundary trace has removed spurious Weyl
000009 | dependence? Derive the first actual response and test original admission.
000010 | 
000011 | Verdict: EXACT SCOPED RESPONSE UNDER A CHANGED COMPLEX-STRUCTURE PREMISE;
000012 | EXCLUDED AS A VARIABLE MODULUS OF THE RETAINED MARKED C2 COMPILER.
000013 | The existing primitive P1-to-C2 realization and measure are not derived.
000014 | No ledger status or physical gate is promoted.
000015 | 
000016 | Read together with:
000017 |   ../elliptic-moebius-source-bridge-20261026/ORIGINAL-RELEVANCE.txt
000018 |   ../elliptic-moebius-source-bridge-20261026/BOUNDARY-SOURCE.txt
000019 |   ../elliptic-moebius-source-bridge-20261026/GLOBAL-SPIN-KERNEL.txt
000020 |   ../native-arc-charge-source-interface-20261026/CONFORMAL-TRACE-AND-WICK.txt
000021 | 
000022 | The last proof establishes support-preserving equality of normalized free
000023 | Cauchy/Wick data for compatible metrics on one fixed complex curve, with
000024 | fixed spin and connection. Earlier bulk Dirac-sea/Schur responses survive;
000025 | they concern a different state and observable. We do not replace that state.
000026 | 
000027 | 2. Actual curve, real structures and first changed premise
000028 | 
000029 | Let E_a: y^2=x^4+a x^2-1; the retained source is E_0.
000030 | Q:(x,y)->(ix,y) maps E_a to E_-a. The exact same-fiber obstruction is
000031 | 
000032 |   F_a(ix)-F_a(x)=-2a x^2.                                  (1)
000033 | 
000034 | The supplied original inversion (x,y)->(1/x,i y/x^2) also maps a to -a.
000035 | For a=i rho, rho real, |rho|<2, both original anti-real maps
000036 | 
000037 |   sigma_+(x,y)=(i bar x,-bar y),
000038 |   sigma_-(x,y)=(-i bar x,-bar y)
000039 | 
000040 | remain involutions of each curve. Their fixed ovals, x=exp(+/-i pi/4)u,
000041 | y=i v, obey v^2=u^4+/-rho u^2+1. These polynomials are strictly positive
000042 | on the real axis for |rho|<2. Each real form has two ovals and remains
000043 | dividing by nonsingular continuation from rho=0. The free glide
000044 | delta:(x,y)->(-x,-y) persists: it has no finite fixed point (x=y=0 is not
000045 | on the curve), and exchanges the two points at infinity and the two ovals.
000046 | 
000047 | The nontrivial even spin partition continues from the original one. If
000048 | r_+,r_- are the roots of z^2+a z-1, its divisor D=R_1+R_2-infty_+-infty_-
000049 | satisfies 2D=div(x^2-r_+). D is not principal: functions with at most simple
000050 | poles at both infinities span {1,x}, and none vanishes at both distinct
000051 | opposite branch points. A nontrivial degree-zero line on genus one has
000052 | H0=0. The inherited boundary NS signs persist under this continuation.
000053 | 
000054 | The changed premise is allowing this complex-structure family and replacing
000055 | the fixed-fiber Q by the explicitly different family action rho->-rho.
000056 | Retaining real cuts, topology, spin partition and glide does not retain Q.
000057 | No original primitive measure support or weight on rho is assumed derived.
000058 | 
000059 | 3. Marked modulus and the stronger local admission obstruction
000060 | 
000061 | For branch labels +/-u,+/-v with uv=i, u^2+v^2=-a, define
000062 | 
000063 |   lambda=((u-v)/(u+v))^2=(a+2i)/(a-2i),
000064 |   j=16(12-a^2)^3/(a^2+4)^2,
000065 |   m=1/(1-lambda)=1/2+i a/4.                                (2)
000066 | 
000067 | Thus lambda(0)=-1, partial_a lambda(0)=i and partial_rho lambda(0)=-1.
000068 | This is a genuine marked modulus despite partial_rho j(0)=0.
000069 | Q sends lambda to 1/lambda. On the one-dimensional deformation space of
000070 | four marked branch points this acts at lambda=-1 by multiplication by -1.
000071 | There is no invariant tangent. Exactly, lambda=1/lambda has only +/-1;
000072 | lambda=1 is degenerate and lambda=-1 is isolated. This extends (1) locally
000073 | beyond this quartic coordinate ansatz: a nearby marked genus-one parity
000074 | cover retaining the same cyclic Q action has no complex modulus.
000075 | 
000076 | The existing C2 compiler already supplies the fixed marked base P1\mu4
000077 | and its peripheral monodromy; ORIGINAL-RELEVANCE.txt explicitly identifies
000078 | this as supplied downstream geometry. Its parity cover is therefore E_0.
000079 | The family rho!=0 is excluded from that retained compiler contract.
000080 | The older registered QGEO.EMERGE.LIGHT.01/v503 already distinguishes this
000081 | clock-given square rigidity from an unproved dynamical selection. We reuse
000082 | that distinction; this is not a new derivation of the square from P1.
000083 | 
000084 | Combining the local modulus obstruction with CONFORMAL-TRACE-AND-WICK:
000085 | with the same marked Q, spin/connection and free Cauchy quantization, all
000086 | nearby admissible compatible metric changes give the same normalized
000087 | transported free boundary functional. Any positive source-independent
000088 | averaging over these metric changes leaves every Wick word unchanged.
000089 | No extra connected fermion four-point contribution follows in this class.
000090 | This is not a no-interaction theorem for other primitive source insertions,
000091 | bundle connections, full marked graph traces or bulk spectral states.
000092 | 
000093 | 4. Explicit response when the fixed-Q premise is relaxed
000094 | 
000095 | For clarity the complete response of the changed family is still recorded;
000096 | it supplies an admission diagnostic, not an approved new TFPT branch.
000097 | With omega=dx/y, a primitive first-oval period, up to its common phase, is
000098 | 
000099 |   I(rho)=2 integral_0^infty du/sqrt(u^4+rho u^2+1).
000100 | 
000101 | The second-oval period has the relative phase i and magnitude I(-rho).
000102 | Choose orientations consistently with tau(0)=i. The primitive cycles
000103 | intersect once at x=0,y=i and meet different points at infinity. Thus they
000104 | give the original A/B basis, with no extra covering factor. Normalize A
000105 | to period one. Then
000106 | 
000107 |   r=I(-rho)/I(rho), tau=i r, W=r/2,
000108 |   I(rho)=2 K((2-rho)/4),
000109 |   I(0)=Gamma(1/4)^2/(2 sqrt(pi)),
000110 |   I'(0)=-Gamma(3/4)^2/(2 sqrt(pi)),
000111 |   r'(0)=2 Gamma(3/4)^2/Gamma(1/4)^2=0.2284732905222318...
000112 | 
000113 | The K identity follows independently by u=tan(theta), reducing to the
000114 | complete elliptic integral with m=(2-rho)/4. Q exchanges A and B, hence
000115 | tau->-1/tau. This remains modular S, not the v117 T generator.
000116 | 
000117 | Use the retained normalized boundary coordinate s in R/Z and transported
000118 | spin half-density. B_Sigma=-i partial_s has NS levels 2pi(n+1/2), and
000119 | the labelled face covariance is C=(1+exp(2W B_Sigma))^-1. Decomposing its
000120 | even/odd NS modes into the retained quarter dictionary gives
000121 | 
000122 |   c_(n,alpha)(rho)=[1+exp(4pi r(rho)(n+alpha))]^-1,
000123 |   alpha=1/4,3/4,
000124 |   partial_rho c|0=-4pi r'(0)(n+alpha)c(0)(1-c(0)).           (3)
000125 | 
000126 | The derivative is trace class by exponential mode decay. For the lowest
000127 | positive mode c(0)=0.04142383216636283, c'(0)=-0.02850113859783212.
000128 | These quarter labels use the retained glide/NS dictionary, not a nonexistent
000129 | fixed-fiber Q away from zero. No EM q or physical time is identified.
000130 | 
000131 | An independent holomorphic check uses the retained global spin kernel:
000132 | g_a(x)^4=(x^2-r_+)/(x^2-r_-),
000133 | S_a(x,z)=sqrt(dx dz)[g_a(x)/g_a(z)+g_a(z)/g_a(x)]/[2(x-z)].
000134 | The same pole, residue, multiplier and H0=0 proof continues near a=0.
000135 | partial_a log(g_a(x))|0=1/[4(x^4-1)]. Its induced kernel derivative is
000136 | nonzero (0.0007573730912245217 at x=2,z=4 in that fixed half-density frame).
000137 | The positive CAR assertion in (3) instead uses the actual real period trace;
000138 | the raw holomorphic kernel is not declared an equal-time covariance.
000139 | 
000140 | 5. Neutral four-field diagnostic and exact positive mixture
000141 | 
000142 | Take the actual lowest opposite NS modes, quarter labels (n,alpha)=(0,1/4)
000143 | and (-1,3/4). At fixed rho their covariance is diag(c,1-c), so their
000144 | occupations N_+,N_- are independent in the given free face state.
000145 | For any positive normalized law on this admitted-as-a-new-premise family,
000146 | the mixture is a positive equal-time CAR state and exactly
000147 | 
000148 |   <N_+N_->-<N_+><N_-> = -Var(c(rho)).                       (4)
000149 | 
000150 | This follows directly from <N_+N_->_rho=c(1-c). It is a neutral fermion
000151 | four-field word; there is no added four-fermion vertex. For nondegenerate
000152 | rho support it is nonzero because I(rho) decreases and r(rho) increases.
000153 | It proves neither interacting spacetime dynamics nor common physical time,
000154 | and no primitive reflection-positive geometric law is inferred.
000155 | 
000156 | Equal weights at +/-epsilon are only diagnostic data. They give exactly
000157 | -(c_+-c_-)^2/4, with a positive normalized two-mode Fock density matrix.
000158 | At epsilon=0.1 the defect is -8.133849896948562e-6; as epsilon->0 its
000159 | quadratic coefficient is -[c'(0)]^2=-0.0008123149013728358.
000160 | In the retained fixed-Q compiler epsilon must be zero and (4) vanishes.
000161 | 
000162 | 6. Original measure boundary and reproducibility
000163 | 
000164 | The archived geometric measure passage in 05_metrology_source/sections/
000165 | 005-section-absolute-dimensionless-metrology-from-the-bounda.tex:136-175
000166 | defines a BRST-reduced geometric measure through a spectral weight and
000167 | reference measure. That passage does not specify rho support/variance or
000168 | identify it with the present changed complex-structure family. The actual
000169 | P1 insertion, marked graph domain and source-to-C2 map still need derivation.
000170 | An inserted positive rho law would assume the missing selection step.
000171 | 
000172 | Checker: elliptic_moduli_response/check_elliptic_moduli_response.py.
000173 | It uses exact symbolic admission/cross-ratio identities, independent direct
000174 | period quadrature and Legendre K evaluation at 70 decimal digits, a direct
000175 | global spin-kernel derivative, and exact symbolic mixed CAR densities.
000176 | Normal and -OO runs must produce byte-identical result JSON. Analytical
000177 | claims are proved above; finite numerical probes only check the formulas.