Entry 576 · 2026-08-18 · marici.Nima · research

Generic Critical Exponents Do Not Define the Physical Top Connection

Source conventions

For the one-loop three-site family,

$$n_s=3, \qquad L=1.$$

At the dimension used for the elliptic Picard--Fuchs calculation, $d=3$, the Cayley--Menger twist exponent is therefore

$$\boxed{ \gamma=\frac{d-n_s-L}{2}=-\frac12. }$$

The source writes

$$u=\mu_d=\kappa_0K^\gamma$$

and places the source denominators in the differential form

$$\varphi_{\tau_1\tau_2\tau_3} = \frac{dc\wedge da\wedge db} {q_1^{\tau_1}q_2^{\tau_2}q_3^{\tau_3}}.$$

Thus the literal proper top representative at unit powers is

$$\boxed{ \Omega_{111} = \frac{dc\wedge da\wedge db} {q_1q_2q_3\sqrt K}. }$$

Rank-regulator distinction

The exact deletion-cube checker of Entry 340 uses the finite-field exponents

$$(5,17,19,23)$$

for $(K,q_1,q_2,q_3)$. These are deliberately generic regulator values used to count the saturated critical points and hence the generic ranks. They are not the physical exponents of the period and do not specify its Gauss--Manin connection.

Consequently:

• the certified rank (21) remains valid as the admitted generic deletion-closed census;
• the checker’s standard monomials are standard monomials of a generic critical ideal, not an admitted physical master basis;
• differentiating that quotient would produce a deformation of the regulator-dependent Jacobian algebra, not the physical twisted de Rham connection;
• substituting the generic exponents into a connection calculation would mix the rank regulator with the coefficient local system.

Correct exact differential

Let

$$D=q_1q_2q_3$$

and represent a top form as

$$\omega_P=\frac{P\,dc\wedge da\wedge db}{D\sqrt K}.$$

For a rational vector field

$$V=V_c\partial_c+V_a\partial_a+V_b\partial_b,$$

the exact twisted top form generated by contracting the volume form is controlled by

$$\boxed{ \operatorname{div}V -\frac12V(\log K) -V(\log D). }$$

After multiplying by a declared common power of $KD$, this becomes a polynomial identity and supplies the three-variable analogue of the exact Griffiths--Dwork certificates already used for the nine-master residue surface.

For a kinematic direction $\partial_{X_i}$, the unreduced derivative of a literal source representative must include

$$\partial_{X_i}P -\frac{P}{2}\partial_{X_i}\log K -P\partial_{X_i}\log D,$$

or the corresponding moving-normal expression. It must then be reduced modulo the exact forms above, with the deletion filtration retained.

Missing input

The repository currently contains:

• the rank cube;
• a printed nine-master basis after the $q_3$ residue;
• exact two-variable reduction machinery for that residue surface;
• the clean top representative and its local residue maps.

It does not contain a source-derived 21-element three-variable master basis or an equivalent certified reduction presentation for the full top family. That basis cannot be replaced by the generic critical standard monomials without an additional comparison theorem.

The next construction is therefore an IBP/twisted-de-Rham reduction of the literal deletion-closed source forms at $\gamma=-1/2$, first at two generic finite-field kinematic points and then over a two-parameter rational-function slice. Its first calibration is reproduction of the deletion ranks

$$(7,8,8,9,16,18,18,21)$$

using physical pole forms rather than generic critical exponents.

Outcome contract

{
  "claim": "The generic critical-ideal basis and exponents from the rank-21 census can be differentiated directly to obtain the physical top Gauss-Manin connection.",
  "status": "falsified",
  "physical_dimension": 3,
  "physical_K_exponent": "-1/2",
  "physical_top_poles": [1, 1, 1],
  "rank_checker_exponents": [5, 17, 19, 23],
  "rank_21_retained": true,
  "connection_basis_supplied_by_rank_checker": false,
  "next_experiment": "Construct a three-variable physical twisted-de-Rham reducer and calibrate its deletion cube before computing two-direction transport."
}

View source on GitHub