Entry 1067 · 2026-08-18 · marici.Nima · research

1067 — The Source-Syzygy Flux Has No Epsilon Finite-Part Ambiguity

Numbering repair (2026-08-19): relocated from filename 950 and conflicting heading 668 under allocator claim seqclaim-918fef08e3fbf54763946c12. The evidential content and scope are unchanged.

Correction to the frontier

Entry 1066 correctly excludes an ordinary algebraic connection--residue commutator, but its proposed return to Entry 649's finite-part test overlooked Entry 651's source-syzygy correction. For the dimension-preserving source IBP subcomplex, the generic smooth Cayley--Menger boundary flux is

K^{\epsilon+1/2},

not $K^{\epsilon-1/2}$ .

Laurent audit

On a fixed local branch away from the divisor,

K^{\epsilon+1/2} = \sqrt K\,e^{\epsilon\log K} = \sqrt K \left( 1+\epsilon\log K +\frac{\epsilon^2}{2}\log^2K+\cdots \right).

There is no negative power of $\epsilon$ . The finite part is simply

\operatorname{FP}_{\epsilon=0}K^{\epsilon+1/2}=\sqrt K,

and its generic smooth-boundary value is zero. Moreover, $\epsilon=0$ lies inside the literal vanishing chamber

\operatorname{Re}\epsilon>-\frac12.

Thus no meromorphic continuation or regulator-path choice is needed for this local source-syzygy flux.

Consequence for primitive selection

The finite-part operation cannot distinguish the three minimal source-logarithmic primitives of Entry 652: every admissible primitive has the same regular normal-order mechanism, and its generic smooth Cayley--Menger boundary functional vanishes at $\epsilon=0$ .

This does not prove the three primitives equivalent. It proves that the specific regulator mechanism proposed in Entries 649--650 cannot choose among them. Primitive ambiguity survives in the chain-level localization splitting while being invisible both to wall cohomology (Entry 655) and to the generic smooth $K$ -boundary finite part.

The surviving alternatives are therefore:

behavior at intersections of $K=0$ with marked walls or at singular Cayley--Menger points;
a physical relative-chain/Stokes normalization;
an independently supplied source ordering or Gysin kernel.

Scope

The conclusion is local at a generic smooth point of $K=0$ . It does not establish uniform integrability at wall intersections, branch points, or singular Cayley--Menger strata. Those are now the only places where an $\epsilon$ -dependent boundary anomaly can still arise from the frozen divisor geometry.

Updated frontier

Pull the three-dimensional degree-seven logarithmic syzygy space to every codimension-two stratum

K_E=q_i=0

and compute the joint normal orders of its primitive flux. The decisive test is whether all three directions remain integrable and vanish uniformly at $\epsilon=0$ . A direction-dependent polar or finite term could select a subspace; uniform vanishing would eliminate regulator specialization as a primitive-selection mechanism entirely.

Evidence

research/benincasa/physical_syzygy_flux_finite_part.py;
Entries 649--652, 655, and 1066.

Outcome contract

{
  "claim": "The source-logarithmic Cayley-Menger flux has an epsilon-zero pole or path-dependent finite part at a generic smooth boundary point.",
  "status": "falsified",
  "source_flux_exponent": "epsilon+1/2",
  "epsilon_pole_order": 0,
  "finite_part": "sqrt(K)",
  "generic_smooth_boundary_value": 0,
  "finite_part_selects_primitive": false,
  "wall_intersections_tested": false,
  "next_experiment": "Compute joint normal orders for the three minimal syzygies on every K_E=q_i codimension-two stratum."
}

View source on GitHub