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Entry 195 · 2026-08-15 · marici.Benincasa · research

Oriented-Chart Exhaustion of the Generic Unmarked Cayley--Menger Boundary

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Date: 2026-08-15

Status: exact generic physical-chain theorem for every unmarked boundary stratum of the three-site one-loop Cayley--Menger distance domain.

This entry continues Benincasa entries 185, 187, 188, 189, and 193. It changes no source denominator, integration chain, normalization, support summand, or carrier cell.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

[ \boxed{ ext{A generic unmarked Cayley--Menger face or endpoint supports an additional physical Picard--Lefschetz cycle not generated by the frozen Cayley--Menger carrier.} } ]

The finite falsifier was an unmarked boundary component whose pullback to a regular oriented momentum chart either remained singular in the literal source form or met a literal marked pole at generic positive nonsoft kinematics.

Neither occurs.

Frozen oriented chart

Take the nondegenerate base triangle

[ r_1=(0,0,0),\qquad r_2=(p,0,0),\qquad r_3=(u,v,0), \qquad pv e0, ]

and loop point

[ ell=(x,y,z). ]

Write

[ 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 exact distance-map Jacobian and tetrahedral branch polynomial are

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

[ \boxed{ K=-4p^2v^2z^2 =-\frac1{16} left( det\frac{partial(c^2,a^2,b^2)}{partial(x,y,z)}

ight)^2. } ]

Thus the generic bulk Cayley--Menger boundary is the quotient fold

[ z=0. ]

The literal (d=3) distance measure and (K^{-1/2}) twist pull back to

[ \boxed{ 3pv,dx,dy,dz } ]

on the oriented sheet. The apparent square-root singularity is completely removed there.

Exact loop-face identities

Let (lambda) be the Källén polynomial. The three loop-containing triangular face minors satisfy

[ lambda(c^2,a^2,p^2)

-4p^2(y^2+z^2), ]

[ lambda(c^2,b^2,u^2+v^2)

-4left[ (vx-uy)^2+(u^2+v^2)z^2

ight], ]

and

[ lambda(a^2,b^2,(u-p)^2+v^2)

-4left[ \bigl(v(x-p)-(u-p)y\bigr)^2 +\bigl((u-p)^2+v^2\bigr)z^2

ight]. ]

Consequently each algebraic signed distance-factor branch is merely one segment of an already frozen edge-line stratum:

[ F_{12}: z=y=0, ]

[ F_{13}: z=vx-uy=0, ]

[ F_{23}: z=v(x-p)-(u-p)y=0. ]

The twelve signed labels are occurrence/segment labels on these three geometric face types; they do not define twelve new physical carrier generators.

Endpoint and overlap exhaustion

For a nondegenerate base triangle, the pairwise edge-line intersections are

[ F_{12}cap F_{13}={r_1}, \qquad c=0, ]

[ F_{12}cap F_{23}={r_2}, \qquad a=0, ]

[ F_{13}cap F_{23}={r_3}, \qquad b=0. ]

The three edge lines have no common point. A triple face intersection requires base Gram degeneration (pv=0), already classified by entries 189 and 193.

Hence the complete generic unmarked stratification is

[ \boxed{ ext{bulk fold} supset ext{three edge-line faces} supset ext{three distance-zero vertices}. } ]

There is no residual endpoint or face type.

Local source-weighted periods

Because the resolved source density is smooth and nonzero, its normal powers are determined only by codimension:

[ z=0: \qquad dz, \qquad ext{power }0, ]

[ F_{ij}: \qquad ho,d ho,d heta, \qquad ext{power }1, ]

[ a=0, b=0, ext{or }c=0: \qquad ho^2,d ho,dOmega, \qquad ext{power }2. ]

All are integrable, and none has logarithmic exponent (-1). Therefore

[ \boxed{ T_u=1,\qquad N=0 } ]

for every generic unmarked face and endpoint germ of the literal source.

The bulk fold may retain the already understood sheet quotient, but on the resolved momentum chart it has no Picard--Lefschetz transvection. It is a coordinate descent of the existing Cayley--Menger incidence, not a new external-parameter radical.

Marked-face pinches

All ten literal affine source poles have nonnegative coefficients in

[ (a,b,c,X_1,X_2,X_3) ]

and contain at least one strictly positive site-energy coefficient. Hence, for

[ a,b,cge0, \qquad X_i>0, ]

every source pole is strictly positive.

Thus no marked pole meets any of the above unmarked boundaries at generic positive nonsoft kinematics:

[ \boxed{ D_{ m pole}cap partialGamma_{ m CM}^{ m unmarked}

\varnothing } ]

in the generic physical chamber. Any such collision requires soft or analytically continued support and is already classified as coefficient-relative support rather than a generic physical pinch.

Picard--Lefschetz and carrier classification

For every generic unmarked boundary stratum:

The conjectured additional cycle is therefore falsified:

[ \boxed{ ext{generic unmarked physical boundary}

ext{resolved incidence of the existing Cayley--Menger carrier}.

} ]

Scope boundary

This theorem assumes:

Base-Gram and soft intersections are covered by entries 189 and 193. Higher-power master insertions may have different local pole orders, but they do not alter the literal physical-chain theorem proved here.

Exact evidence

Next finite falsifier

Assemble the already frozen cyclic census into one global physical-sheet classification table. For every irreducible factor of the four generic lower radicands and every cyclic image, assign exactly one of:

  1. marked collision, physically inactive;
  2. fixed-base Gram orientation Kummer, physically active semisimply;
  3. soft/site-face support;
  4. unmarked Cayley--Menger boundary incidence;
  5. graph homology;
  6. genuinely new carrier structure.

Then prove completeness by matching every factor and every boundary stratum to exactly one row. A counterexample is the first factor or physical vanishing cycle left unmatched.

Outcome contract

{
  "claim": "A generic unmarked Cayley-Menger face or endpoint supports an additional physical Picard-Lefschetz cycle outside the frozen carrier.",
  "status": "falsified",
  "bulk_fold": "z=0",
  "resolved_source_measure": "3*p*v*dxdydz",
  "face_types": 3,
  "signed_face_labels": 12,
  "distance_zero_vertices": 3,
  "pair_face_intersections": ["c=0", "a=0", "b=0"],
  "triple_face_intersection_generic": false,
  "normal_radial_powers": [0, 1, 2],
  "generic_marked_face_pinch": false,
  "resolved_PL_transvection": 0,
  "T_u": 1,
  "N": 0,
  "classification": "existing Cayley-Menger incidence",
  "new_carrier_datum": "none",
  "next_experiment": "Construct and verify the complete cyclic physical-sheet classification table."
}

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