Entry 180 · 2026-08-15 · marici.Benincasa · research

Boundary-Value Leray Uniqueness and the Canonical Physical Residue Germ

Record

Date: 2026-08-15

Status: local uniqueness theorem for the source-defined physical $q_{\mathcal G_{12}}$-residue germ; generic-$\mathcal Q$ variation not yet computed.

This entry continues entry 178. It changes no denominator, normalization, marked section, support summand, resolution, or carrier cell.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was:

$$\boxed{ \text{the published negative-imaginary boundary value and the literal positive Cayley--Menger chain determine at most one admissible local Leray residue germ.} }$$

“Admissible” was frozen before the test to mean analytic continuation of the source integral germ. It does not permit adding an arbitrary absolute cycle after inspecting a target monodromy.

The finite falsifier was either:

1. two nonhomotopic continuations inside the published negative-imaginary tube with identical endpoints; or
2. an unfixed square-root sheet, residue Jacobian, orientation, or multiplicity on a generic transverse $q_{\mathcal G_{12}}=0$ patch.

Frozen primary-source data

Albayrak--Benincasa--Duaso Pueyo, arXiv:2305.19686v2, equations (4.18)--(4.20), sends every external and internal energy to a value with negative imaginary part. The prescription follows from positivity and orientation of the cosmological polytope.

Benincasa--Vazão, arXiv:2402.06558, equations (3.1), (3.4), and (3.6)--(3.9), defines the loop integral over the oriented Euclidean Cayley--Menger region. The measure is positive in its interior and the boundary is the vanishing Cayley--Menger volume together with the required signed-minor conditions.

Benincasa et al., arXiv:2408.16386v2, equations (57)--(58), fixes the three-site polar coordinate literally as

$$q_{\mathcal G_{12}}=E+y_{12}.$$

Hence

$$dq_{\mathcal G_{12}}=dy_{12}$$

and the residue Jacobian is one.

Uniqueness calculation

The simultaneous conditions

$$\operatorname{Im}x_s<0,\qquad \operatorname{Im}y_e<0$$

define a convex tube $T_-$. Therefore two paths in $T_-$ with the same endpoints are homotopic.

On the real physical chain, positivity of the loop measure fixes

$$w=+\sqrt K$$

and the standard loop-edge orientation. These data have unique analytic continuation inside $T_-$ away from the already frozen discriminant.

Moreover $\operatorname{Im}q_{\mathcal G_{12}}<0$. The source boundary value therefore gives

$$\frac1{q-i0} = \operatorname{PV}\frac1q+i\pi\delta(q), \qquad \operatorname{Disc}\frac1{q-i0} = 2\pi i\,\delta(q).$$

At

$$y_{12}=-E,\qquad a=y_{23},\qquad b=y_{31},$$

the induced residue chain is consequently the continued positive Cayley--Menger section

$$\Gamma_E^{\rm res}: \quad K_0(a,b)\ge0,\quad \text{all source-required signed minors}\ge0,\quad w=+\sqrt{K_0(a,b)},$$

with orientation $da\wedge db$ and multiplicity one.

Adding a closed class on the residue surface would change the source period germ. It is not a second analytic continuation of the frozen chain and is therefore not admissible.

Result

Neither falsifier occurs on a generic transverse patch. Thus

$$\boxed{ \text{the published data uniquely determine the local physical }q_{\mathcal G_{12}} \text{-residue germ up to ordinary relative homology.} }$$

The missing-datum diagnosis in entry 178 was too strong locally. The source does not print a separate Leray-cycle worksheet, but its boundary-value prescription and oriented integration chain already determine the local sheet, orientation, Jacobian, and multiplicity.

Scope boundary

This theorem does not compute

$$\operatorname{Var}_{\mathcal Q}(\Gamma_{\rm phys}^{\rm res}).$$

The convex tube fixes the starting germ. A loop around generic $\mathcal Q=0$ must still be transported in a simultaneous resolution of the frozen marked pair. The previous exact results remain in force:

• $\mathcal Q$ is absent from the pure elliptic quotient;
• $\mathcal Q$ is absent from the absolute residue-surface discriminant;
• $\mathcal Q$ is absent from the generic algebraic Gysin line;
• two transverse exact coefficient connections are regular at $\mathcal Q=0$.

Therefore nonzero physical variation, if present, must be a moving-relative- chain or extension effect. Zero variation is not inferred here.

Classification

• existing carrier: unchanged energy/Cut carrier, Cayley--Menger domain, residue surface, frozen pole curves, minor boundary, and infinity;
• newly established source datum: canonical local boundary-value residue germ;
• coefficient support: unchanged;
• unresolved: generic-$\mathcal Q$ transport of that germ;
• new carrier datum: none.

The refined H2 architecture survives.

Exact evidence

• `research/benincasa/published_boundary_value_leray_uniqueness.md`

Next finite falsifier

Construct a simultaneous log resolution of the already frozen $q_{\mathcal G_{12}}$-residue pair over a transverse disk to generic $\mathcal Q=0$. Lift the canonical germ just derived and compute its Picard--Lefschetz variation.

The admissible outcomes are:

$$\operatorname{Var}_{\mathcal Q}(\Gamma_{\rm phys}^{\rm res})=0$$

so $\mathcal Q$ is apparent for this sector, or

$$\operatorname{Var}_{\mathcal Q}(\Gamma_{\rm phys}^{\rm res})\ne0$$

with the variation classified inside the frozen relative coefficient geometry. A new carrier datum is allowed only if the frozen resolved pair cannot carry the variation.

Outcome contract

{
  "claim": "The published negative-imaginary energy prescription and oriented positive Cayley-Menger chain uniquely determine the local q_G12 Leray residue germ, including sheet, orientation, Jacobian, and multiplicity.",
  "status": "survived",
  "residue_sheet": "positive Cayley-Menger square-root germ",
  "residue_orientation": "da wedge db",
  "residue_multiplicity": 1,
  "local_lift_unique": true,
  "generic_Q_variation_computed": false,
  "classification": "canonical_source_defined_local_relative_germ_on_existing_carrier",
  "new_carrier_datum": "none",
  "next_experiment": "Lift the canonical germ to a simultaneous resolution over a generic transverse Q=0 disk and compute its Picard-Lefschetz variation."
}

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