Generic Lower-Sector Algebraic Letters from Cayley--Menger Collisions
Record
Date: 2026-08-15
Status: exact generic collision-radical provenance theorem for the frozen three-site lower/polylogarithmic sector.
This entry continues entry 183. It adds no denominator, support summand, normalization, integration boundary, or carrier cell after seeing the result.
Deutsch--Popperian conjecture tested
The hard-to-vary claim was
[ \boxed{ ext{the generic multi-external-leg lower-sector algebraic letters are generated by the frozen Cayley--Menger branch geometry and marked denominator arrangement.} } ]
The finite falsifier was a source-required irreducible algebraic divisor absent from every branch--pole, pole--pole, boundary, and marked-stratum collision of that frozen geometry.
Frozen primary source
Benincasa--Brunello--Mandal--Mastrolia--Vazão, arXiv:2408.16386v2, equation (mathrm{eq:subsec_1235}), defines
[ mathcal I_{ au_{mathfrak g_1} au_{mathfrak g_2} au_{mathfrak g_3} au_{mathfrak g_{23}}}
int_Gamma mu_d, \frac{dy_{12},dy_{23},dy_{31}} {q_{mathfrak g_1}^{ au_1} q_{mathfrak g_2}^{ au_2} q_{mathfrak g_3}^{ au_3} q_{mathfrak g_{23}}^{ au_{23}}}. ]
The same section states that for independent site energies and momentum resultants, (X_i eq P_i), this sector has 34 masters, remains generalized-polylogarithmic, and has algebraic letters. The paper does not print the generic 34-master alphabet.
Set
[ c=y_{12},\qquad a=y_{23},\qquad b=y_{31}. ]
The immutable pole planes are
[ L_1=c+b+X_1,quad L_2=c+a+X_2,quad L_3=a+b+X_3,quad L_{23}=c+b+X_2+X_3. ]
The frozen Cayley--Menger matrix is
[
mathrm{CM}=
\begin{pmatrix}
0&1&1&1&1
1&0&c^2&a^2&b^2
1&c^2&0&P_2^2&P_1^2
1&a^2&P_2^2&0&P_3^2
1&b^2&P_1^2&P_3^2&0
end{pmatrix},
\qquad
K=-\frac12detmathrm{CM}.
]
No homogeneous identification is imposed before calculation.
Exact two-pole branch radicals
Define
[ Lambda_P= (P_1-P_2-P_3)(P_1-P_2+P_3) (P_1+P_2-P_3)(P_1+P_2+P_3). ]
Restricting (K) to any independent pair of pole planes leaves a quadratic in the residual loop variable. Its branch-point separation is controlled by one of four exact radicands:
[ \boxed{ Delta_{12}^{-}
-4(P_1^2-X_1^2)(P_2^2-X_2^2)Lambda_P \bigl(P_3^2-(X_1-X_2)^2\bigr), } ]
[ \boxed{ Delta_{13}^{-}
-4(P_1^2-X_1^2)(P_3^2-X_3^2)Lambda_P \bigl(P_2^2-(X_1-X_3)^2\bigr), } ]
[ \boxed{ Delta_{23}^{-}
-4(P_2^2-X_2^2)(P_3^2-X_3^2)Lambda_P \bigl(P_1^2-(X_2-X_3)^2\bigr), } ]
[ \boxed{ Delta_{23}^{+}
-4(P_2^2-X_2^2)(P_3^2-X_3^2)Lambda_P \bigl(P_1^2-(X_2+X_3)^2\bigr). } ]
The pairs ((L_2,L_{23})) and ((L_3,L_{23})) give the same plus radicand. The pair ((L_1,L_{23})) is parallel, with coincidence divisor
[ X_1-X_2-X_3=0. ]
For a restricted quadratic (Au^2+Bu+C), the roots
[ u_pm=\frac{-Bpmsqrt{Delta}}{2A} ]
give local algebraic dlog coordinates. This fixes the quadratic extension, not a preferred 34-master canonical-basis normalization.
One-pole tangency exhaustion
A naive nested resultant was stopped after producing no stratum in bounded time. It is not evidence.
The exact replacement uses the square structure of (K). After solving one pole plane, one residual distance enters through (U=u^2), and the restriction is quadratic (F(U,v)). Tangencies split into
[ U=0:quad F(0,v)=partial_vF(0,v)=0 ]
and, away from degree-drop support,
[ U eq0:quad F=partial_UF=partial_vF=0. ]
The latter is the repeated-root condition for (operatorname{Disc}_U F) in (v).
All four planes pass this stratified audit. The verifier checks 76 factor occurrences across the two branches. Every factor is signed linear in
[ (X_1,X_2,X_3,P_1,P_2,P_3) ]
and every projected factor occurs with even multiplicity. No nonlinear one-pole tangency divisor remains.
Triple marked collisions
The two nontrivial triple supports are
[ T_{123}=K|{L_1=L_2=L_3=0}, \qquad T{23}=K|{L_2=L_3=L{23}=0}. ]
They are coefficient/relative supports by construction: values of the frozen branch polynomial on frozen marked strata, not new incidence generators.
Under (X_i=P_i),
[ \boxed{ T_{123}mapsto \frac14 (P_1-P_2-P_3)^2 (P_1-P_2+P_3)^2 (P_1+P_2-P_3)^2, } ]
while
[ \boxed{T_{23}mapsto0.} ]
Thus one becomes a perfect square and the other collapses onto homogeneous boundary support.
Homogeneous rationalization
Every finite two-pole radicand contains a site-face factor (P_i^2-X_i^2). Therefore
[ \boxed{ Delta_{12}^{-}
Delta_{13}^{-}
Delta_{23}^{-}
Delta_{23}^{+} =0 quad ext{when }X_i=P_i. } ]
The source transition
[ ext{generic lower sector: algebraic GPL letters} longrightarrow ext{homogeneous lower sector: rational ten-letter alphabet} ]
is therefore a collision and rationalization of quadratic branch coordinates, not a change of carrier.
Cyclic closure
Simultaneous cyclic permutation of the (X_i), (P_i), and loop variables preserves (K) and generates the other two lower families. Cyclic closure introduces no new geometric type beyond
[ P_i^2-X_i^2,\qquad Lambda_P,\qquad P_k^2-(X_imp X_j)^2, ]
and the cyclic triple marked-collision polynomials.
This is a support and algebraic-extension theorem. Without the unpublished generic differential system it does not establish cancellation or noncancellation of every radical in a chosen 34-master basis or physical linear combination.
Classification
- existing energy/Cut carrier: unchanged;
- Cayley--Menger coefficient support: yes;
- marked-relative-cycle support: yes;
- soft and fixed-base Gram support: already frozen;
- genuinely new carrier datum: none.
Hence the conjecture survives:
[ \boxed{ ext{generic lower-sector algebraicity is coefficient/relative-period geometry over the existing carrier.} } ]
This positively updates H2:
[ ext{shared carrier and derived calculus} + ext{sector- and layer-specific coefficient objects}. ]
Exact evidence
Primary source applications.tex SHA-256:
3e92460fe2e34dc21a537c784dab3b2fbcd9b7cfee9e7372f06971b50d8b6f9b
Collision derivation and result SHA-256:
a8e9f52dc76262eb3b66ff89de26f6fb4e60cd349b42f9d8768d6c97ffe84cd8
58810c9a360b74a5bc5ed713f40879a4ef98e096bceaa3d9ea5cae2e0074a5ad
Single-tangency derivation and result SHA-256:
84bdaaacd3a84d92a9ef571bb527987ccf9dbb42404f25199d979e7a965280de
93b99ed05e4e2bba1e148ecec9df7c72e4f09595505ad3d5bd72accc9e553f47
Verifier and result SHA-256:
24c80c01797362b419a09592d998e4b486d9c7c2ed1d92ef6ef8baccfef90a1b
1fe784faf592fa778535ce9c116c407aec775970ceb5d15f88cb70a3a16cd1fd
The verifier exited zero with 11 structural checks and 76 one-plane factor occurrences. Run-log SHA-256:
6a79adce8c1c2f76aa1ee49d956f523646f50274934bcc138ada2dcfcaa97165
Narrow result
[ \boxed{ ext{The frozen generic lower sector produces four quadratic radical types entirely from Cayley--Menger/marked-plane collisions; all rationalize at }X_i=P_i ext{, and no new carrier stratum is required.} } ]
Scope boundary
Established:
- exact generic pair radicands;
- exact one-pole tangency census;
- exact triple marked-collision supports;
- homogeneous degeneration;
- cyclic geometric closure;
- carrier/coefficient classification.
Not established:
- the unpublished term-by-term 34-master alphabet;
- a preferred canonical-basis normalization;
- monodromy of every generic physical-chain combination;
- extension through intersecting soft and Gram loci.
Next finite falsifier
Freeze the literal cyclic source chain and coefficients. At a generic point of one radicand away from every other census divisor, compute the Picard--Lefschetz variation and test whether source-weighted cyclic assembly:
- retains a nonzero quadratic-character variation;
- cancels it by source-fixed orientation and weights; or
- transfers it into a marked-relative extension class.
No fitted splitting, support summand, or carrier cell is admissible.
Outcome contract
{
"claim": "The generic lower-sector algebraic extension is generated by Cayley-Menger branch collisions with the frozen marked denominator arrangement.",
"status": "survived",
"finite_pair_count": 5,
"unique_pair_radicand_types": 4,
"single_tangency_factor_occurrences_checked": 76,
"single_tangencies_all_signed_linear_squares": true,
"generic_triple_collision_polynomials": 2,
"homogeneous_pair_radicands_zero": true,
"homogeneous_triple_behavior": ["perfect_square", "zero"],
"cyclic_new_geometric_type": false,
"classification": "Cayley-Menger coefficient and marked-relative-cycle support over unchanged energy/Cut carrier",
"new_carrier_datum": "none",
"next_experiment": "Compute generic radical Picard-Lefschetz variation on the literal cyclic source-weighted lower-sector chain."
}