Entry 322 · 2026-08-15 · marici.Benincasa · research

Unsplit Occurrence Pair and the Exact Relative Wall Class

Correction: entry 234 supersedes the claimed complete primitive below. The displayed exactness applies to the lower-denominator factor before the nonconstant Cayley--Menger/master factor is restored. The zero logarithmic residue survives, but for the complete-integrand reason proved in entry 234.

Record

Status: the complete common lower-denominator factors have been restored at the exceptional corner. The two individual occurrence terms have hierarchy-dependent simple-pole currents, but their source-defined unsplit sum has zero simple residue and a canonical exact leading two-form. The remaining datum is relative wall support, not an absolute delta class.

No denominator, carrier incidence, support summand, regulator hierarchy, projector, or normalization is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{each lower-denominator occurrence admits an independent canonical physical boundary lift through the exceptional corner.} }$$

The finite falsifier was a hierarchy-dependent individual residue together with a hierarchy-independent zero residue for the frozen source sum.

Primary contour theorem

Albayrak--Benincasa--Duaso Pueyo, arXiv:2305.19686v2, equations (4.14)--(4.18), proves that reduced spurious poles can involve

$$\alpha\epsilon_{hat b}-\beta\epsilon_{hat a}, \qquad \alpha,\beta>0,$$

whose sign requires an arbitrary regulator hierarchy. The corresponding residue of the canonical form is nevertheless zero because the hyperplane is a spurious boundary rather than a facet.

Entry 231 identifies exactly this sign ambiguity for the two lower occurrences after the ordered $q_{\mathcal G_{12}}$ residue. The primary theorem says not to turn those two triangulation terms into independently canonical physical currents.

Complete frozen denominator expansion

At the positive marked corner use

$$E=\tau^2, \qquad X_3=E-x-y,$$

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

On the $q_{\mathcal G_{12}}$-residue surface, the three denominators common to both source occurrences are

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

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

$$\boxed{ q_{\mathfrak g_3} =\tau^2(1+\tau n). }$$

The occurrence-specific denominators are

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

Thus $q_{\mathfrak g_1}$ and $q_{\mathfrak g_2}$ are units at generic nonsoft kinematics, while $q_{\mathfrak g_3}$ supplies one additional existing total-normal factor.

Unsplit cancellation

Each full lower factor

$$\frac1{ q_{\mathfrak g_1}q_{\mathfrak g_2}q_{\mathfrak g_3} q_{\mathfrak g_{31}}} \quad\text{or}\quad \frac1{ q_{\mathfrak g_1}q_{\mathfrak g_2}q_{\mathfrak g_3} q_{\mathfrak g_{23}}}$$

has weight $-4$ before including the already analyzed Cayley--Menger/master factor. Their source-defined plus-plus sum satisfies

$$\frac1{q_{\mathfrak g_{31}}} + \frac1{q_{\mathfrak g_{23}}} = \tau^{-1}\frac{n}{r(-r+\tau n)}.$$

It therefore begins at weight $-3$, not $-4$, after the common $q_{\mathfrak g_3}^{-1}$ factor is restored.

Using

$$q_{\mathfrak g_1}q_{\mathfrak g_2} \longrightarrow 4xy,$$

the leading coefficient two-form is

$$\boxed{ -\frac{n}{4xy,r^2},dr\wedge dn. }$$

It has no simple logarithmic residue at $r=0$. More strongly,

$$-\frac{n}{4xy,r^2},dr\wedge dn = d\left( \frac{n}{4xy,r},dn \right).$$

Hence the leading unsplit pair is zero in absolute de Rham cohomology on the punctured exceptional chart. Its primitive has a pole on the already frozen wall $r=0$, so a relative boundary class may remain.

Verdict

The independent-lift conjecture is falsified:

$$\boxed{ \text{the canonical physical object is the unsplit occurrence pair, not two independently boundary-valued occurrence currents.} }$$

The chamber-dependent delta terms of entry 231 are artifacts of splitting a spurious reduced pole before applying the canonical-form relation. They are not physical coefficient classes by themselves.

The unsplit pair still contains a relative wall datum through the singular primitive $n,dn/(4xy,r)$. Thus this result does not assert that the full physical relative period vanishes. It narrows the next problem from an arbitrary regulator current to a source-defined connecting morphism at the existing wall.

Classification

• existing carrier: $q_{\mathfrak g_3}=0$, the two occurrence divisors, their intersection $r=0$, and the weighted exceptional chart;
• absolute coefficient class: zero at the leading unsplit grade;
• relative coefficient datum: pole of the primitive on $r=0$;
• regulator-dependent individual currents: noncanonical splitting data;
• elliptic Gauss--Manin data: no new image;
• genuinely new carrier datum: none.

Exact evidence

• primary source arXiv:2305.19686v2, equations (4.14)--(4.18), pages 21--23;
• research/benincasa/check_unsplit_occurrence_pair.rs;
• research/benincasa/unsplit-occurrence-pair.json;
• exact weighted denominator tests and exterior-derivative identity;
• warnings-denied optimized Rust compilation and execution.

Next finite falsifier

Compute the relative connecting morphism for

$$d\left( \frac{n}{4xy,r},dn \right)$$

in the frozen marked pair. Evaluate it on the source exceptional chain, with the two occurrence branches retained until after the connecting map.

• A canonical wall class sewing cyclically supports full assembly inside the existing relative coefficient object.
• Zero connecting class means the complete exceptional correction disappears in the unsplit physical source despite its nonzero projected image.
• Dependence on a splitting or regulator hierarchy falsifies canonical occurrence-level assembly but still does not require a new carrier.

Outcome contract

{
  "claim": "Each lower occurrence has an independent canonical physical boundary lift.",
  "status": "falsified",
  "common_total_normal": "q_g3=tau^2(1+tau*n)",
  "individual_weight": -4,
  "unsplit_pair_weight": -3,
  "leading_two_form": "-n/(4*x*y*r^2) dr wedge dn",
  "absolute_class": "exact",
  "relative_wall_datum": "primitive n/(4*x*y*r) dn",
  "individual_delta_currents": "noncanonical splitting artifacts",
  "new_carrier_incidence": false,
  "next_experiment": "Compute the relative connecting morphism on the source exceptional chain before occurrence forgetting."
}

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