Entry 332 · 2026-08-16 · marici.Benincasa · research

Individual Occurrence Periods Are Not Source-Canonical

Record

Status: the strong conjecture that both occurrence-resolved weight-zero classes possess individually regulator-independent physical periods is falsified. The primary contour construction explicitly leaves a hierarchy choice at spurious subdivision poles, and the two lower occurrences meet at exactly such a reduced pole after the ordered $q_{\mathcal G_{12}}$ residue. Their algebraic de Rham classes and endpoint jets remain canonical, but the source fixes a physical relative period only for their unsplit sum.

No regulator hierarchy, boundary counterterm, carrier cell, support summand, or normalization is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the occurrence-resolved endpoint classes are individually relative, their physical periods are regulator-independent, and they sew to the unsplit period.} }$$

The finite falsifier is a pair of source-admissible regulator hierarchies which induce different currents on the same frozen marked intersection.

Primary-source regulator theorem

Albayrak--Benincasa--Duaso Pueyo, arXiv:2305.19686v2, equations (4.15)--(4.20), proves the required distinction.

• True facet poles carry a sign-definite negative imaginary part assembled from positive contour regulators.
• Spurious subdivision poles carry $\alpha\epsilon_{\hat b}-\beta\epsilon_{\hat a}$, with $\alpha,\beta>0$, so their side depends on an arbitrary regulator hierarchy.
• The ambiguity disappears from the complete canonical form because the residue on the spurious boundary vanishes.
• The induced negative imaginary parts of site and edge energies are linear combinations of the contour regulators; their magnitudes are not equated.

This closes the upstream question left open in entry 231. The graph-level construction does not collapse the reduced-pole cone to one chamber. It classifies the ambiguous poles themselves as hierarchy-dependent spurious poles.

Frozen exceptional current

After the ordered Cut residue,

$$q_{\mathfrak g_{31}}=A+i\alpha, \qquad q_{\mathfrak g_{23}}=B+i\beta,$$

where

$$\alpha=\xi_2-\eta_{23}, \qquad \beta=\xi_1-\eta_{31}.$$

On the exceptional divisor $B=-A$. Therefore

$$\frac1{A+i0s_\alpha}+ \frac1{-A+i0s_\beta} =-i\pi(s_\alpha+s_\beta)\delta(A).$$

The chambers

$$(--),(-+),(+-),(++)$$

give, in units of $i\pi\delta(A)$,

$$2,0,0,-2.$$

Thus the boundary-value operation and the weighted nearby limit do not commute occurrence by occurrence. The ambiguity is supported on the already frozen marked intersection.

Endpoint-relative obstruction at weight zero

Entry 331 gives exact reductions

$$\eta_i=c_i\frac{dn}{w}+d\Phi_i, \qquad \Phi_i=\frac{H_i(n)}{8(xy)^{3/2}w^9},$$

where every $H_i$ is odd and has nonzero polar jets at $w=0$. Hence $d\Phi_i$ is absolute-meromorphically exact but not relatively trivial without a chosen endpoint trivialization. Different finite-part subtractions change an individual regulated period by boundary values of $\Phi_i$.

The source provides no occurrence-by-occurrence endpoint subtraction. Assigning one would be a post hoc chain-level choice prohibited by the frozen-source rule.

By contrast, the exact identities

$$\eta_{31}+\eta_{23}=\eta_{\rm unsplit}, \qquad \Phi_{31}+\Phi_{23}=\Phi_{\rm unsplit}$$

show that both de Rham classes and endpoint jets sew before a physical boundary value is taken. The complete source canonical form then fixes the unsplit physical germ.

Verdict

The strong conjecture is falsified narrowly:

$$\boxed{ \text{individual occurrence periods are not source-canonical.} }$$

The smaller hypothesis survives:

$$\boxed{ \text{occurrence resolution is canonical at de Rham/endpoint-jet level, but the physical period is canonical only after source sewing.} }$$

This updates H2 without weakening the shared-carrier claim. The failure is in sector-specific relative-chain/coefficient assembly, not incidence.

Compatibility statements

• infinity-Gysin quotient: each wall class still has zero direct Legendre image;
• algebraic kernel $\mathcal T_7$: the occurrence data can only enter its relative/algebraic extension sector;
• Gauss--Manin horizontality: not defined for either physical occurrence period separately because the source does not define those periods;
• sewn horizontality: remains the next legitimate test;
• soft and discriminant limits: not needed for the generic falsifier and may only add supported terms;
• new carrier datum: none.

Evidence

• primary source arXiv:2305.19686v2, equations (4.15)--(4.20), especially the hierarchy statement between (4.15) and (4.16);
• entry 231's exhaustive four-chamber current census;
• entries 330--331's exact individual de Rham reductions and endpoint-jet sewing;
• research/benincasa/occurrence-resolved-physical-period-no-go.md.

Next finite falsifier

Construct the minimal source-defined relative extension

$$0\longrightarrow \mathcal B_{\partial} \longrightarrow \mathcal R_{\rm occ} \longrightarrow \mathcal K_{31}\oplus\mathcal K_{23} \longrightarrow0$$

whose boundary object $\mathcal B_{\partial}$ records the two endpoint-jet vectors, and test whether the source sewing map descends to a horizontal quotient

$$\mathcal R_{\rm occ}\longrightarrow\mathcal R_{\rm unsplit}.$$

Freeze the exact primitives, endpoint divisor, cyclic occurrence orbits, Kummer sign monodromy, and source unsplit chain. If no horizontal sewn quotient exists without a fitted boundary splitting, canonical global occurrence assembly is falsified. A failure remains coefficient/extension data unless it demands a source-derived new incidence.

Outcome contract

{
  "claim": "Occurrence-resolved endpoint classes have individually regulator-independent physical periods which sew to the unsplit period.",
  "status": "falsified",
  "primary_source_result": "spurious reduced poles require arbitrary regulator hierarchy",
  "individual_de_rham_classes": "canonical",
  "individual_endpoint_jets": "canonical_algebraic_relative_data",
  "individual_physical_periods": "not_source_canonical",
  "source_canonical_object": "unsplit_sewn_period",
  "generic_chamber_currents": [2, 0, 0, -2],
  "direct_legendre_gysin_image": 0,
  "new_carrier_incidence": false,
  "surviving_hypothesis": "canonical occurrence data before physical pairing; canonical physical period only after source sewing",
  "next_experiment": "Construct and test the horizontal sewn relative extension with endpoint-jet boundary object."
}

View source on GitHub