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Entry 578 · 2026-08-18 · marici.Nima · research

578 — Fixed-Pole Polynomial IBP Does Not Recover the Physical Top Module

Hard-to-vary claim

The physical three-variable top module is not the finite quotient obtained by freezing every source denominator to a simple pole and reducing only by polynomial-vector-field integration-by-parts relations.

This is a falsification of one coefficient-lattice model. It is not evidence for a new carrier stratum.

Frozen ablation

Use variables

(c,a,b)=(y12,y23,y31)(c,a,b)=(y_{12},y_{23},y_{31})

and the source Cayley--Menger polynomial KK at the two generic specializations

(X1,X2,X3)=(2,3,4),(3,5,6).(X_1,X_2,X_3)=(2,3,4),\qquad(3,5,6).

For a selected product DD of

qg1=c+b+X1,qg2=c+a+X2,qG12=c+X1+X2+X3,q_{g_1}=c+b+X_1,\qquad q_{g_2}=c+a+X_2,\qquad q_{\mathcal G_{12}}=c+X_1+X_2+X_3,

freeze the top form

ΩP=P dc∧da∧dbDK.\Omega_P=\frac{P\,dc\wedge da\wedge db}{D\sqrt K}.

For each polynomial vector field VV, clear the fixed denominator in the exact-form relation. Its numerator is

KD div⁡V−D2V(K)−KV(D).\boxed{ KD\,\operatorname{div}V -\frac D2V(K) -KV(D). }

At total-degree cutoff NN, quotient all polynomial numerators of degree at most NN by these relations, admitting precisely the polynomial vector fields whose cleared numerators remain within the cutoff.

The calculation is exact over F32003\mathbb F_{32003}. No generic critical exponents, fitted support summands, or new cells are used.

Census

In mask order

000,001,010,011,100,101,110,111,000,001,010,011,100,101,110,111,

the quotient dimensions at (2,3,4)(2,3,4) are

N00000101001110010111011182767671056610510513510277979131741281281811227919115782150150219\begin{array}{c|rrrrrrrr} N&000&001&010&011&100&101&110&111\\ \hline 8&27&67&67&105&66&105&105&135\\ 10&27&79&79&131&74&128&128&181\\ 12&27&91&91&157&82&150&150&219 \end{array}

The second generic point (3,5,6)(3,5,6) reproduces the complete N=8,10N=8,10 rows exactly.

The required physical deletion cube is

(7,8,8,9,16,18,18,21).\boxed{(7,8,8,9,16,18,18,21).}

The ablation fails in two independent ways:

  1. even the denominator-free sector stabilizes at 2727, not 77;
  2. every sector containing a qq-divisor grows with the cutoff.

Narrow interpretation

Specializing the twist to the physical half-weight before choosing the correct logarithmic lattice leaves a large resonant affine excess. Fixed simple poles and polynomial vector fields do not implement the localized twisted de Rham quotient.

The absent datum is coefficient-side:

localized/rational logarithmic vector fieldsor an equivalent compatible higher-pole lattice.\boxed{ \text{localized/rational logarithmic vector fields} \quad\text{or an equivalent compatible higher-pole lattice.} }

Nothing in this failure requires modifying the energy/Cut carrier.

Surviving conjecture

The physical rank packet may still be obtained from the same three source divisors and Cayley--Menger cover, but only after constructing the correct localized logarithmic de Rham/IBP complex at exponent −12-\tfrac12.

Next falsifier

Predeclare a finite higher-pole or localization filtration, prove that its exact differential closes, and test whether its deletion ranks stabilize to

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

at both generic kinematic points.

Only after that calibration may its two-parameter Gauss--Manin connection be used to classify the rank-nine/rank-five extension.

Artifacts

View source on GitHub