Entry 651 · 2026-08-18 · marici.Benincasa · research

651 — Syzygy IBP Restores Physical Cayley--Menger Boundary Vanishing

Correction

Entry 649 tested the boundary behavior of the bare twist $K^\gamma$. That is the correct test for a generic integration-by-parts vector field, but it is not the test for the dimension-preserving IBP system used by the primary source.

The source imposes the syzygy condition

$$\sum_e n_e\,\partial_{y_e}K=n_0K.$$

Thus the IBP vector field $V=\sum_en_e\partial_{y_e}$ is logarithmic along the Cayley--Menger divisor:

$$V(K)\in(K).$$

At a generic smooth point of $K=0$, use $K$ as a normal coordinate. The syzygy says that the normal component of $V$ is divisible by $K$. Therefore the boundary flux of the primitive is not $K^\gamma$, but

$$\boxed{K^{\gamma+1}}.$$

Physical exponent

For the homogeneous one-loop three-site sector,

$$\gamma=\epsilon-\frac12.$$

Hence the logarithmic-syzygy flux has normal order

$$K^{\epsilon+1/2}.$$

It vanishes whenever

$$\operatorname{Re}\epsilon>-\frac12,$$

and in particular at the physical value $\epsilon=0$:

$$\boxed{K^{1/2}\longrightarrow0.}$$

The dimension-preserving source IBP identities therefore do admit literal boundary vanishing on a generic smooth Cayley--Menger face at physical dimension. Meromorphic continuation is not needed for this local boundary term.

What remains from Entry 649

Entry 649 remains valid for unrestricted IBP numerators whose normal component is not divisible by $K$. Its conclusion must not be applied to the source's syzygy-loaded, dimension-preserving IBP subcomplex.

The correction does not by itself construct a canonical lift of Entry 650's physical wall cocycle. Boundary vanishing proves that a source syzygy primitive defines a legitimate relative homotopy. It does not prove that:

1. every required wall-cocycle representative is reached by that syzygy subcomplex;
2. two syzygy primitives differ by a relative-exact primitive;
3. the primitive remains logarithmic at intersections of $K=0$ with the marked denominator divisors or singular Cayley--Menger faces; or
4. the resulting class chooses an origin in the (mathcal T_7)-torsor.

Classification

• existing carrier: the Cayley--Menger boundary (K=0);
• coefficient/chain datum: logarithmic tangent vector fields and their relative primitives;
• new carrier datum: none.

Surviving frontier

Construct the actual source syzygy primitive for the closed three-wall cocycle of Entry 648 and compute its class in the relative localization complex. The finite falsifier is primitive independence: if two admissible syzygy lifts differ nontrivially in (mathcal T_7), the lift torsor remains unbased. If the difference is relative exact, the source IBP subcomplex canonically selects the lift.

Evidence

• Paolo Benincasa et al., arXiv:2408.16386, method.tex, equations eq:IBP and eq:condition;
• research/benincasa/three-site-syzygy-ibp-boundary-audit.json;
• epistemic event ev-000000000250-dc37df19-7f11-4483-bec6-5f83376f6208;
• Entries 648--650.

Outcome contract

{
  "claim": "The dimension-preserving source IBP primitive has nonvanishing Cayley-Menger boundary flux at the physical dimension.",
  "status": "falsified",
  "syzygy_condition": "V(K)=n_0 K",
  "physical_flux_order": "K^(1/2)",
  "generic_smooth_boundary_flux_vanishes": true,
  "canonical_T7_lift": "not yet constructed",
  "next_experiment": "Construct the source syzygy primitive for the [Entry 648](/research/ledger/20260818-648-the-physical-shared-wall-boundary-is-a-closed-cech-cocycle) cocycle and test primitive independence in the relative localization complex."
}

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