Entry 189 · 2026-08-15 · marici.Benincasa · research

Fixed-Base Gram Kummer Activation on the Orientation Double Cover

Record

Date: 2026-08-15

Status: exact generic smooth fixed-base Gram monodromy theorem for the three-site one-loop physical chain at (d=3).

This entry continues entries 185, 187, and 188. It changes no source denominator, normalization, chain, support summand, or carrier cell.

It also repairs the source-pole census in entry 188 from six displayed varying denominators to all ten distinct affine pole forms. The positivity theorem survives unchanged.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \Lambda_P=0 \text{ is merely a singular distance-coordinate chart and produces no physical coefficient monodromy.} }$$

The finite falsifier was a nontrivial character of the literal source form after pulling it back to a regular oriented momentum-coordinate chart.

The claim is falsified: the resolved chart is regular, but descent to the coarse Gram coordinate carries the sign character.

Frozen oriented chart

Choose base vertices

$$r_1=(0,0,0),\qquad r_2=(p,0,0),\qquad r_3=(u,v,0),$$

where (p=P_2>0) and (v) is the signed altitude of the external momentum triangle. Let the loop point be

$$\ell=(x,y,z).$$

The three loop distances are

$$c^2=x^2+y^2+z^2,$$

$$a^2=(x-p)^2+y^2+z^2,$$

$$b^2=(x-u)^2+(y-v)^2+z^2.$$

The coarse fixed-base Gram coordinate is

$$\boxed{ \Lambda_P=-4p^2v^2. }$$

Thus (v) is the resolved linear normal and (Lambda_P) is its quadratic coarse invariant.

Exact Jacobian and volume cancellation

Direct determinant evaluation gives

$$\boxed{ \det \frac{\partial(c^2,a^2,b^2)} {\partial(x,y,z)} = 8pvz. }$$

Consequently,

$$abc\,da\,db\,dc = pvz\,dx\,dy\,dz$$

on the oriented (z>0) chart.

The base area and tetrahedron volume obey

$$A_{\rm base}^2=\frac{p^2v^2}{4}, \qquad V_{\rm tet}^2=\frac{p^2v^2z^2}{36}.$$

Therefore

$$\frac{V_{\rm tet}^2}{A_{\rm base}^2} = \frac{z^2}{9}.$$

For (d=3,n_s=3,L=1), the source exponent is

$$\chi=-\frac12.$$

The literal distance measure and normalized Cayley--Menger twist combine as

$$abc\,da\,db\,dc \left( \frac{V_{\rm tet}^2}{A_{\rm base}^2} \right)^{-1/2} = \boxed{ 3pv\,dx\,dy\,dz }$$

on the oriented cover.

Using (Lambda_P=-4p^2v^2), the coarse-normal factor is

$$3pv = \frac32\sqrt{-\Lambda_P}$$

on the physical orientation sheet.

Literal six-term source parity

Under

$$(v,y)\longmapsto(-v,-y),$$

all three squared distances (a^2,b^2,c^2) are unchanged. Hence every one of the ten affine source denominators, the complete six-term cyclic source weight, and the half-space (z>0) are preserved.

Write the pulled-back period as

$$I(v)=3pv\,J(v).$$

The residual integral satisfies

$$J(-v)=J(v).$$

At generic positive nonsoft kinematics its literal integrand has one sign, so (J(0) eq0) after the same source regularization used away from the Gram boundary.

Therefore

$$\boxed{ I(-v)=-I(v). }$$

Monodromy and physical activation

A loop around the coarse divisor (Lambda_P=0) exchanges the two orientation sheets,

$$v\longmapsto-v.$$

Thus the semisimple and unipotent parts are

$$\boxed{ T_s=-1, \qquad T_u=1, \qquad N=0. }$$

On the resolved orientation cover, the chain and residual coefficient extend regularly and have no Picard--Lefschetz transvection:

$$\operatorname{Var}_v(\widetilde\Gamma_{\rm phys})=0.$$

After descent to the coarse Gram coordinate, the physical coefficient has the nontrivial rank-one Kummer character. This is genuine physical-sheet activation, but it is not an additional unipotent vanishing cycle.

The appropriate distinction is

$$\boxed{ \text{resolved-chain regularity} + \text{coarse orientation Kummer monodromy}. }$$

Classification

At a generic smooth component of (Lambda_P=0), away from soft support and every marked pole:

• existing carrier incidence: fixed-base Cayley--Menger Gram boundary;
• coefficient support: rank-one orientation Kummer system;
• literal source coefficient: nonzero;
• semisimple monodromy: (-1);
• unipotent logarithm: (N=0);
• graph homology: not invoked;
• new carrier datum: none.

This is the first physically active class in the generic lower-radical census. It is generated by the unchanged Cayley--Menger carrier together with its native orientation double cover.

Relation to the marked-pole radicals

Every two-pole radicand contains (Lambda_P), but entry 188 proves that its marked-pole collision thimble has zero intersection with the positive chain. The present result is a different mechanism on the same divisor:

$$\text{marked-pole collision contribution}=0,$$

while

$$\text{independent fixed-base orientation contribution}=\mathcal K_{\sqrt{-\Lambda_P}}.$$

These must not be conflated.

Scope boundary

The theorem is generic along one smooth Gram component. It excludes:

• (p=0) and other soft loci;
• intersections of two signed factors of (Lambda_P);
• simultaneous Gram and marked-boundary pinches;
• endpoint strata where the oriented chart itself changes rank.

Those are the next overlap tests.

Exact evidence

• research/benincasa/verify_fixed_base_gram_orientation_cover.py
  • sparse-polynomial Jacobian, volume, and reflection verifier;
• research/benincasa/fixed_base_gram_orientation_cover_result.json
  • monodromy and classification packet;
• scheduler execution:
  • last result (0);
• primary source:
  • Benincasa et al., arXiv:2408.16386, eq:mCM, eq:ukchi, and eq:Triangle.

Next finite falsifier

Blow up a soft--Gram corner, beginning with

$$p=P_2=0, \qquad \Lambda_P=0.$$

Freeze (v) and (p) as separate resolved normals before passing to the coarse invariant (-4p^2v^2). Determine whether the source form is a normal-crossing product of Tate/Kummer factors or whether a new non-product vanishing cycle appears. Then repeat at the intersections of two signed Gram factors and at unmarked Cayley--Menger endpoint faces.

Outcome contract

{
  "claim": "Generic fixed-base Gram support is only a coordinate singularity and has no physical coefficient monodromy.",
  "status": "falsified",
  "resolved_normal": "v",
  "coarse_normal": "Lambda_P=-4*p^2*v^2",
  "jacobian": "8*p*v*z",
  "physical_form_prefactor": "3*p*v",
  "residual_source_integral_parity": "even",
  "oriented_period_parity": "odd",
  "T_s": -1,
  "T_u": 1,
  "N": 0,
  "resolved_cover_variation": 0,
  "classification": "fixed-base Gram Kummer/orientation support on existing Cayley-Menger carrier",
  "physically_active": true,
  "new_carrier_datum": "none",
  "next_experiment": "Resolve soft--Gram and multi-Gram overlap corners."
}

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