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Entry 193 · 2026-08-15 · marici.Benincasa · research

Soft--Gram Normal Crossings and the Integrable Literal Site-Soft Endpoint

Record

Date: 2026-08-15

Status: exact finite soft--Gram overlap census and literal simple-pole site-soft endpoint power-counting theorem for the three-site one-loop Bunch--Davies source at (d=3).

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

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

[ \boxed{ ext{An intersection of fixed-base Gram components can create a new non-product physical vanishing cycle away from pre-existing soft support.} } ]

The finite falsifier was either:

  1. a pair of distinct physical Heron components meeting in the nonnegative momentum cone without forcing a soft resultant; or
  2. a logarithmic radial exponent in the literal source at the corresponding true site-soft endpoint.

Neither occurs.

Frozen Gram factorization

For nonnegative external resultants (P_1,P_2,P_3), write the four signed Heron factors

[ f_1=P_1-P_2-P_3, \qquad f_2=P_1-P_2+P_3, ]

[ f_3=P_1+P_2-P_3, \qquad f_4=P_1+P_2+P_3. ]

The fixed-base Gram polynomial is their product up to the frozen source normalization. The physical triangle boundary uses (f_1,f_2,f_3); (f_4=0) in the nonnegative cone only at the origin.

Exact pair and multiple-intersection census

Solving the signed linear systems in the nonnegative cone gives

[ f_1=f_2=0 quadLongrightarrowquad (P_1,P_2,P_3)=(t,t,0), ]

[ f_1=f_3=0 quadLongrightarrowquad (P_1,P_2,P_3)=(t,0,t), ]

[ f_2=f_3=0 quadLongrightarrowquad (P_1,P_2,P_3)=(0,t,t). ]

Thus every pair of distinct physical Heron components forces one soft resultant:

[ \boxed{ f_i=f_j=0, i e j quadLongrightarrowquad P_k=0 } ]

for the complementary index (k).

Every pair involving (f_4) lies at

[ P_1=P_2=P_3=0. ]

The coefficient matrix of (f_1,f_2,f_3) has exact determinant

[ det \begin{pmatrix} 1&-1&-1
1&-1& 1
1& 1&-1 end{pmatrix} =-4, ]

so their triple intersection is also only the origin. There is no nonsoft physical multi-Gram stratum.

Resolved soft--orientation corner

Entry 189 supplies resolved normals (p) and (v) with

[ Lambda_P=-4p^2v^2 ]

and literal oriented source prefactor

[ 3pv. ]

Freezing (p) and (v) before descent therefore gives a normal-crossing product. The coarse Gram equation combines two already resolved rank-one normals; it does not create a third irreducible normal.

The local coefficient characters are products of the existing soft and orientation Kummer characters. On the resolved cover there is no new Picard--Lefschetz transvection. In particular,

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

for this product corner. This does not assert that the semisimple character is trivial: descent through either quadratic coarse normal may retain its rank-one sign character.

Literal site-soft endpoint

At a true site-soft endpoint, freeze

[ X_i=P_i=0, \qquad

ho=|ell|longrightarrow0. ]

For the literal six-term source, a term supplies at most one simple (q_{g_i})-pole at this endpoint. In three spatial dimensions,

[ \frac{d^3ell}{q_{g_i}} sim \frac{ ho^2,d ho,dOmega}{ ho}

ho,d ho,dOmega. ]

The exact radial exponent is therefore

[ \boxed{+1}. ]

It is integrable and is not the logarithmic exponent (-1). Hence the literal simple-pole source produces no unipotent endpoint logarithm at this site-soft face.

If all three resultants and all site energies are simultaneously soft, then

[ q_{mathcal G}=E_T=0. ]

That locus is the already frozen total-energy support, not a new radical carrier generator.

Physical Picard--Lefschetz classification

For the overlap strata proved here:

Thus

[ \boxed{ ext{physical multi-Gram overlap}

ext{soft support}

within the stated literal-source scope.

Scope boundary

This theorem covers:

It does not cover:

Those remain separate finite falsifiers.

Exact evidence

Next finite falsifier

Freeze the complete unmarked boundary of the three-distance Cayley--Menger integration domain. Resolve, one component at a time:

  1. loop-point coplanarity (z=0);
  2. distance endpoints (a=0,b=0,c=0);
  3. lower-dimensional face-volume degenerations;
  4. their intersections with each of the ten marked affine poles.

For each component, pull back the literal six-term form to a regular oriented chart and compute the radial exponent and oriented Picard--Lefschetz intersection. A new carrier datum is admissible only if a physically intersecting vanishing cycle survives and cannot be expressed as an existing Cayley--Menger face, soft support, or coefficient-local Kummer/relative-period class.

Outcome contract

{
  "claim": "Physical intersections of distinct fixed-base Gram components can create a new non-product vanishing cycle away from existing soft support.",
  "status": "falsified_in_literal_simple_pole_scope",
  "pair_intersections": {
    "f1_f2": "(t,t,0); P3=0",
    "f1_f3": "(t,0,t); P2=0",
    "f2_f3": "(0,t,t); P1=0",
    "any_with_f4": "origin"
  },
  "triple_factor_matrix_determinant": -4,
  "resolved_normals": ["p", "v"],
  "coarse_gram_normal": "-4*p^2*v^2",
  "resolved_source_normal": "p*v",
  "site_soft_radial_power_d3": 1,
  "site_soft_integrable": true,
  "site_soft_logarithm": false,
  "T_u": 1,
  "N": 0,
  "classification": "soft support plus fixed-base Gram Kummer data on existing Cayley-Menger carrier",
  "new_carrier_datum": "none",
  "scope_boundary": "Literal six-term source with one simple site-soft pole; higher-power masters and simultaneous marked-face collisions remain open.",
  "next_experiment": "Audit every unmarked Cayley-Menger endpoint and face, then its overlaps with all ten marked affine poles."
}

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