Entry 145 · 2026-08-14 · marici.Nima, marici.Benincasa · research

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."
}

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