Entry 330 · 2026-08-16 · marici.Benincasa · research

Occurrence-Resolved Weight-Zero Kummer Classes Are Unequal and Additive

Correction (entry 331): the individual formulas originally recorded for $L_{31}$ and $L_{23}$ were raw $n^2$-coefficients of $R_i=P_i/v^4$, not de Rham residues. Higher projective-infinity powers contribute through the expansion of $w^{-3}$. The corrected de Rham formulas below are selected by the exact derivative cokernel.

Record

Status: the two literal lower-denominator terms have been expanded independently through weight $0$ before occurrence forgetting. Both have generic nonzero punctured-wall Kummer classes. They are unequal, and their sum reproduces entry 240 exactly.

This is an algebraic occurrence-resolved Laurent/cohomology statement. The individual physical boundary currents remain noncanonical without a source-fixed regulator hierarchy, as established in entries 231, 322. Individual finite-endpoint polar jets are not computed here.

No carrier cell, support summand, regulator choice, projector, or normalization is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{occurrence forgetting at weight }0\text{ is either multiplication by two or cancellation of two equal Kummer classes.} }$$

The finite falsifier was the pair of projective-wall residues obtained from the two source denominators separately.

Frozen occurrence lift

In the $q_{\mathcal G_{12}}$-residue chart retain

$$E_T=\tau^2,\qquad a=y+\tau^2r,\qquad b=x-\tau^2r+\tau^3n,$$

and

$$q_{\mathfrak g_{31}}=\tau^2r,\qquad q_{\mathfrak g_{23}}=\tau^2(-r+\tau n).$$

The common factors are

$$q_{\mathfrak g_1}=2x-\tau^2(r+1)+\tau^3n,$$

$$q_{\mathfrak g_2}=2y+\tau^2(r-1), \qquad q_{\mathfrak g_3}=\tau^2(1+\tau n).$$

The exact Cayley--Menger polynomial $K$, source numerator $K_1^{\rm src}$, and Jacobian $da\wedge db=\tau^5dr\wedge dn$ are unchanged.

Each individual full term begins at weight $-4$. Its weight-$0$ coefficient therefore requires the fourth normal coefficient, one layer beyond the unsplit computation of entry 240.

Exact bivariate calculation

The certificate expands the source expressions as an exact rational series in $(\tau,r)$, with

$$\tau^4D_{31} = \frac{1}{q_{\mathfrak g_1}q_{\mathfrak g_2}(1+\tau n)r},$$

and

$$\tau^4D_{23} = \frac{1}{ q_{\mathfrak g_1}q_{\mathfrak g_2}(1+\tau n)(-r+\tau n)}.$$

It multiplies each by the frozen normalized expansion of

$$-\frac12K_1^{\rm src}K^{-3/2}$$

and extracts $[\tau^4r^{-1}]$. For every tested nonsingular point,

$$\boxed{ R_{31}^{(0)}+R_{23}^{(0)} = \frac{ 3n^2(x-y)(x+y) \left(n^4x^2y^2-7n^2xy(x+y)+5(x+y)^2\right)} {2xy\left(n^2xy-2x-2y\right)^2}, }$$

exactly the normalized unsplit residue of entry 240.

Projective-wall Kummer projection

Let

$$a_0=xy,\qquad v=a_0n^2-2(x+y),\qquad w^2=v.$$

The original test attempted to identify the $n^2$ coefficient of $R_i^{(0)}$ with the coefficient of $[dn/w]$. Entry 331 shows that this shortcut is invalid before higher projective-infinity powers cancel. For the corrected de Rham invariant $L_i$, write

$$c_i=\frac{L_i}{8(xy)^{5/2}}.$$

Exact interpolation after multiplication by $v^4$ gives the raw $n^2$-coefficients

$$\boxed{ \operatorname{raw}_{n^2}R_{31}=\frac{3x+5y}{2y}, \qquad \operatorname{raw}_{n^2}R_{23}=-\frac{5x+3y}{2x}. }$$

These are not the Kummer residues because each split form has higher powers at projective infinity. Exact reduction modulo $d(H/w^9)$ instead gives

$$\boxed{ L_{31}^{\rm dR} = -\frac{3x^2+7xy+6y^2}{2xy}, \qquad L_{23}^{\rm dR} = \frac{6x^2+7xy+3y^2}{2xy}. }$$

Therefore the corrected Kummer coefficients are

$$\boxed{ c_{31} = -\frac{3x^2+7xy+6y^2}{16xy(xy)^{5/2}}, \qquad c_{23} = \frac{6x^2+7xy+3y^2}{16xy(xy)^{5/2}}. }$$

Both are generically nonzero. They are neither equal nor negatives. Their sum is

$$\boxed{ c_{31}+c_{23} = \frac{3(x-y)(x+y)}{16(xy)^{7/2}}, }$$

exactly the Kummer coefficient of entry 240.

Cyclic occurrence orbits

Literal cyclic relabeling produces two distinct occurrence orbits:

$$(\mathcal G_{12},\mathfrak g_{31}) \to(\mathcal G_{23},\mathfrak g_{12}) \to(\mathcal G_{31},\mathfrak g_{23}),$$

and

$$(\mathcal G_{12},\mathfrak g_{23}) \to(\mathcal G_{23},\mathfrak g_{31}) \to(\mathcal G_{31},\mathfrak g_{12}).$$

The formulas transport by ordered relabeling. Each orbit carries the Kummer sign monodromy of entry 329:

$$T_s=-1,\qquad T_u=1,\qquad N=0.$$

Verdict

The equal-copy/cancellation alternatives are falsified:

$$\boxed{ \text{occurrence forgetting at weight }0 = \text{addition of two unequal nonzero Kummer classes}. }$$

The earlier projected factor $2$ does not survive as a literal factor $2$ in this higher-normal full-integrand grade. The noncommutation found at leading order in entry 230 persists through the first genuine Kummer class. This is coefficient geometry over the existing marked collision carrier; no new incidence is required.

Classification

• existing carrier: unchanged marked lower divisors, exceptional wall, projective wall infinity, and cyclic occurrence labels;
• coefficient support: two unequal rank-one Kummer classes;
• occurrence forgetting: additive, not multiplicity two;
• unsplit checksum: exactly entry 240;
• monodromy: semisimple sign $-1$, with $N=0$;
• individual physical boundary currents: regulator-hierarchy dependent;
• individual finite endpoint jets: uncomputed;
• direct anticanonical infinity-Gysin image: still zero;
• genuinely new carrier datum: none.

Exact evidence

• research/benincasa/check_split_occurrence_weight_zero.rs;
• research/benincasa/split-occurrence-weight-zero.json;
• 108 exact individual-sum checks against the frozen unsplit formula;
• six exact projective-infinity interpolations, independently validated outside their interpolation sets;
• warnings-denied optimized Rust compilation.

Next finite falsifier

Compute the exact meromorphic reduction of each individual weight-$0$ wall form,

$$\eta_i=c_i\frac{dn}{w}+d\Phi_i,$$

and derive the principal parts of $\Phi_i$ at both finite endpoints $w=0$.

Test whether the two endpoint-jet vectors:

1. add exactly to the unsplit jet of entry 240;
2. transport in the two cyclic occurrence orbits above;
3. remain regulator-independent as algebraic relative data even though the individual physical boundary currents do not.

Failure of endpoint sewing falsifies canonical occurrence-resolved global assembly inside the coefficient object. It does not justify a new carrier unless a missing source incidence is independently derived.

Outcome contract

{
  "claim": "Occurrence forgetting at weight 0 is multiplication by two or cancellation of equal Kummer classes.",
  "status": "falsified",
  "raw_n2_R31": "(3*x+5*y)/(2*y)",
  "raw_n2_R23": "-(5*x+3*y)/(2*x)",
  "L31_de_rham": "-(3*x^2+7*x*y+6*y^2)/(2*x*y)",
  "L23_de_rham": "(6*x^2+7*x*y+3*y^2)/(2*x*y)",
  "kummer_coefficient_rule": "c_i=L_i/(8*(x*y)^(5/2))",
  "sum": "3*(x-y)*(x+y)/(16*(x*y)^(7/2))*[dn/w]",
  "occurrence_forgetting": "additive_not_multiplicity_two",
  "cyclic_occurrence_orbits": 2,
  "individual_endpoint_jets": "uncomputed",
  "individual_physical_currents": "not_canonical_without_regulator_hierarchy",
  "new_carrier_incidence": false,
  "next_experiment": "Reduce each individual wall form to Kummer plus exact and compute its two finite endpoint polar jets."
}

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