Three-Site Elliptic Coefficient System and Nearby-Cycle Degeneration
Record
Date: 2026-08-14
Status: conditional structural identification of the homogeneous rank-two elliptic block; first-order cosmological first-jet completeness falsified as a sufficient loop test.
Claim
For the homogeneous one-loop three-site cosmological wavefunction, the rank-two elliptic Picard–Fuchs subsystem is not an arbitrary elliptic coefficient system.
It is canonically identified as
[ \boxed{ \mathbb V_{\triangle,\mathrm{ell}} \simeq \mathcal K_{B^{-1/2}} \otimes m^*\mathbb H_{\mathrm{Leg}}, } ]
where
- (\mathbb H_{\mathrm{Leg}}) is the universal Legendre Gauss–Manin variation;
- (m=A/B) is the signed-energy cross-ratio;
- (\mathcal K_{B^{-1/2}}) is a rank-one Kummer twist.
Entry 148 uses the reciprocal Legendre coordinate (m_{148}=B/A=m^{-1}). The two conventions describe the same Legendre variation after the standard permutation of branch points; formulas comparing the entries must apply this reciprocal change of coordinate.
In homogeneous energy variables,
[ A=\ell_1\ell_2, \qquad B=\ell_3\ell_4, ]
where the (\ell_i) are the four signed-energy hyperplanes.
Thus the elliptic subsystem is generated entirely from the existing signed-energy divisor arrangement.
Its singular support satisfies
[ \operatorname{Sing} (\mathbb V_{\triangle,\mathrm{ell}}) \subseteq \mathcal A_{\mathrm{energy}}. ]
No additional carrier divisor is introduced by the pure elliptic block.
The scattering boundary is
[ E_T=\ell_4=0. ]
At this boundary the elliptic curve degenerates to a nodal rational curve and the elliptic variation degenerates through nearby cycles into Tate/Kummer data.
At the physical (B=0) degeneration, the (-1) from Legendre continuation is cancelled by the (-1) Kummer monodromy of (B^{-1/2}). The total twisted system therefore has unipotent monodromy
[ T=\exp N, \qquad \operatorname{rank}N=1, \qquad N^2=0. ]
Hence
[ \boxed{ \psi_{E_T=0} ( \mathbb V_{\triangle,\mathrm{ell}} )
\text{Tate/Kummer variation}. } ]
The algebraic-letter quartic
[ \mathcal Q ]
is constant to first order in the total-energy parameter,
[ \mathcal Q
-16X_1^2X_2^2 + O(E_T^2). ]
Therefore the first genuinely elliptic deformation appears only at second normal order.
A first ordinary normal jet cannot detect it.
Evidence
Direct substitution transforms the published second-order Picard–Fuchs operator into the Legendre hypergeometric equation.
Homogenization identifies the modulus
[ m
\frac{\ell_1\ell_2} {\ell_3\ell_4}. ]
The elliptic discriminant becomes
[ \Delta_E
16AB(A-B)^4, ]
whose support is exactly the signed-energy arrangement together with the ordinary site-energy hyperplanes.
The published scattering degeneration corresponds to
[ E_T=0, ]
where the elliptic curve becomes nodal.
The algebraic quartic expands as
[ \mathcal Q
-16X_1^2X_2^2 +O(E_T^2). ]
Consequently the first nontrivial algebraic deformation is invisible to first-order normal analysis.
Boundary
This entry identifies only the homogeneous rank-two elliptic block.
It does not prove:
- the complete marked relative coefficient system;
- the full Gauss–Manin extension including additional denominator sections;
- the complete nearby-cycle filtration;
- the full Picard–Fuchs system.
The stronger identification
[ \mathbb V_{\triangle}
R^1\pi_* (E_X\setminus D_X) ]
remains conjectural.
Consequence
The first-order cosmological first-jet program is insufficient for integrated loop cosmology.
The correct discriminating object becomes the nearby-cycle and second-Rees filtration of the elliptic coefficient system.
This substantially strengthens the intermediate Marici hypothesis:
[ \boxed{ \text{shared carrier} + \text{shared derived/six-functor calculus} + \text{sector-specific coefficient systems}. } ]
It weakens the hypothesis that a universal first-jet construction alone controls loop cosmology.
Outcome contract
{
"claim": "The homogeneous three-site elliptic subsystem is a Kummer-twisted pullback of the universal Legendre variation over the existing signed-energy arrangement. Its scattering degeneration is a nearby-cycle degeneration to Tate/Kummer data, and the first algebraic elliptic deformation appears only at second normal order.",
"status": "conditional",
"assumptions": [
"Published homogeneous three-site Picard-Fuchs system.",
"Homogeneous energy variables.",
"Only the rank-two elliptic block is identified."
],
"factorization_test": {
"Legendre_identification": "passed",
"existing_divisor_support": "passed",
"scattering_nearby_cycle": "passed",
"first_jet_sufficiency": "falsified",
"full_relative_extension": "open"
},
"next_experiment": "Construct the complete marked Gauss-Manin coefficient object, compute second-Rees nearby cycles, and determine whether the algebraic quartic is entirely coefficient-theoretic or forces a new carrier stratum."
}