Literal Six-Term Source Periods on Every Lower Collision Component
Record
Date: 2026-08-15
Status: exact completion packet for the source-weighted local-period clause of the generic lower-sector physical-activation theorem.
This entry strengthens entry 196. Entry 196 proved the physical Picard--Lefschetz projection for every cyclic collision but used one explicit source-weighted period as a representative of the finite quadratic family. Here the literal six-term residue is computed separately for all five finite marked pairs and all ten irreducible components of each discriminant.
No source denominator, normalization, support summand, or carrier cell was changed.
Deutsch--Popperian conjecture tested
The hard-to-vary claim was
[ \boxed{ ext{No finite lower-sector discriminant component is silently removed by the literal six-term source weight before physical PL projection.} } ]
The finite falsifier was the first pair/component for which the exact Grothendieck residue at the Cayley--Menger double root was identically zero, identically polar, or required a different local model.
Frozen source and residue rule
With
[ L_1=X_1+c+b,quad L_2=X_2+c+a,quad L_3=X_3+a+b, ]
[ L_{12}=X_1+X_2+a+b,quad L_{23}=X_2+X_3+c+b,quad L_{31}=X_3+X_1+c+a, ]
and
[ G_{12}=E+c,quad G_{23}=E+a,quad G_{31}=E+b, ]
the frozen source is
[ Omega_{ m src}
\frac{abc}{E L_1L_2L_3} left[ \frac1{G_{12}L_{23}}+\frac1{G_{12}L_{31}} +\frac1{G_{23}L_{31}}+\frac1{G_{23}L_{12}} +\frac1{G_{31}L_{12}}+\frac1{G_{31}L_{23}}
ight]. ]
For each finite pair ((q_i,q_j)), the verifier forms the termwise Grothendieck residue
[ R_{ij}(u)
operatorname{Res}{q_i=q_j=0}Omega{ m src}. ]
Termwise cancellation is essential: a source term contributes only if its denominator actually contains both marked factors. This avoids the spurious (0cdotinfty) values produced by substituting a partial pole before taking the residue.
On the marked line, the frozen Cayley--Menger polynomial is
[ K_{ij}(u)=A_{ij}u^2+B_{ij}u+C_{ij}, ]
with double root
[ u_*=-\frac{B_{ij}}{2A_{ij}}. ]
The local period at exponent (chi=-1/2) is therefore
[ \boxed{ Pi_{ij,f}
C_delta \frac{R_{ij}(u_*)}{sqrt{A_{ij}}} } ]
on every irreducible discriminant component (f=0).
Exact census
The five finite pairs in the frozen sector are
[ (g_1,g_2),quad (g_1,g_3),quad (g_2,g_3),quad (g_2,g_{23}),quad (g_3,g_{23}). ]
Each has ten exponent-one discriminant factors. For every one of the fifty pair/component occurrences, the verifier records:
- the exact restricted quadratic coefficients (A,B,C);
- the exact double-root formula;
- the exact termwise residue (R_{ij}(u));
- an exact rational point on the named irreducible component;
- the exact nonzero values of (A_{ij}) and (R_{ij}(u_*));
- the exact rational value [ R_{ij}(u_*)^2/A_{ij}, ] which is the period square modulo the universal cycle constant.
The result is
[ \boxed{ 50 ext{finite nonzero local periods}, \qquad 0 ext{source-weight zeros}, \qquad 0 ext{generic source-pole overlaps}, \qquad 0 ext{degree-drop exceptions}. } ]
One exact finite nonzero rational witness proves that the restricted rational coefficient is neither identically zero nor identically polar on that irreducible component. The calculation supplies such a witness for every component, not merely one witness per factor class.
Simultaneous cyclic relabeling preserves the Cayley--Menger polynomial and the six-term source. Hence the exact closure is
[ 3 ext{cyclic sectors}, \qquad 15 ext{finite pairs}, \qquad 150 ext{nonzero local-period occurrences}. ]
Physical projection
The result concerns the analytically continued coefficient-side vanishing period. It does not alter entry 188's physical-chain result:
[ leftlangleGamma_{ m BD},delta^\vee ight angle=0 ]
for every marked collision. Consequently
[ operatorname{Var}{f}(Gamma{ m BD})
leftlangleGamma_{ m BD},delta^\vee ight angledelta
0 ]
for all 150 occurrences despite their nonzero local periods.
Thus the separation is now exhaustive:
[ \boxed{ ext{nonzero local coefficient period} ; otRightarrow; ext{physical-sheet activation}. } ]
The physically active fixed-base Gram Kummer of entry 189 remains an independent semisimple orientation effect, not a marked-collision Picard--Lefschetz transvection.
Endpoint and overlap qualification
The site-face factors appearing in the finite discriminants do not force the finite quadratic residue itself to vanish or drop degree at a generic point of that divisor. Literal positive-domain faces, distance-zero vertices, soft/Gram corners, parallel coincidences, and triple-marked intersections remain the separately resolved strata classified in entries 188, 193, and 195.
This entry therefore closes the missing source-weight audit without reclassifying those boundary strata.
Classification
The complete finite-collision result is:
| Object | Local coefficient period | Physical PL jump | Class |
|---|---|---|---|
| 150 cyclic finite components | nonzero | zero | marked-relative coefficient support |
| Physical Heron orientation cover | semisimple sign | active | Kummer coefficient on existing Gram carrier |
| Endpoint/face/soft overlaps | as in entries 193 and 195 | no new transvection | existing CM/soft carrier |
| Graph homology | not generated here | none assigned | separate (H_1) topology |
| New carrier datum | absent | absent | zero occurrences |
The narrow surviving theorem is therefore
[
\boxed{
\begin{gathered}
ext{every generic finite lower collision carries a nonzero literal-source
local period,}
ext{every such period has zero Bunch--Davies PL intersection,}
ext{and no such component requires new carrier incidence.}
end{gathered}
}
]
Exact evidence
research/benincasa/all_lower_source_periods_result.json- SHA-256
343edba217ec8089726cc16f0c6eb6318f3651cdf716c4fc5c24997e5d375a62;
- SHA-256
research/benincasa/verify_global_lower_physical_activation.py- now freezes and requires the complete period census;
research/benincasa/global_lower_physical_activation_result.json- SHA-256
8e4f6061c11698761e06d1b77aa52d8004b7a9f67982c94f4696dabbe81018fd.
- SHA-256
Narrow conclusion
The strict completion gap in entry 196 is closed. Its global carrier theorem survives a component-by-component literal-source audit. The lower sector contains coefficient-side algebraic variation everywhere in the finite collision census, but the Bunch--Davies positive chain projects every such Picard--Lefschetz variation to zero. No first counterexample to the unchanged energy/Cut/Cayley--Menger carrier appears.