Global Physical-Sheet Classification of Generic Lower Cayley--Menger Radicals
Record
Date: 2026-08-15
Status: finite global theorem for all cyclic generic three-site one-loop lower-sector Cayley--Menger radicals and every generic physical boundary stratum of the literal six-term Bunch--Davies source.
This entry closes the frontier opened in entry 185 using entries 187, 188, 189, 193, and 195. It adds no denominator, support summand, normalization, integration boundary, splitting, or carrier cell.
Deutsch--Popperian conjecture tested
The hard-to-vary claim was
[ \boxed{ ext{Every physically active generic lower-sector algebraic letter is generated by the unchanged energy/Cut/Cayley--Menger carrier with sector-specific relative Gauss--Manin or Kummer coefficients.} } ]
The finite falsifier was the first cyclic radicand factor, marked collision, unmarked face, endpoint, soft overlap, or physical vanishing cycle that could not be assigned to a predeclared carrier or coefficient class.
No such item remains.
Frozen cyclic census
One lower family contains five finite two-pole collisions and one parallel pole pair. The five finite restrictions yield four distinct quadratic radicand types. Simultaneous cyclic permutation gives
[ 3 ext{sectors}, \qquad 12 ext{sector-labelled radicand types}, \qquad 15 ext{finite pair collisions}. ]
Each finite quadratic discriminant has ten irreducible linear factors:
- four site-face factors;
- four fixed-base Heron factors;
- two marked transfer factors.
The exact total is therefore
[ \boxed{ 15 imes10=150 } ]
irreducible factor occurrences.
The verified cyclic occurrence counts are
[ 60 ext{site-face factors}, ]
[ 60 ext{fixed-base Gram factors}, ]
[ 18 ext{difference-transfer factors}, ]
[ 12 ext{sum-transfer factors}. ]
The census also includes three cyclic parallel-pole coincidence letters and six cyclic triple-marked supports. These are marked-relative supports, not additional quadratic radical types.
Universal local vanishing period
At a smooth finite marked collision, after the two frozen pole residues, the remaining branch polynomial has local form
[ K_r(c)=A(c-c_+)(c-c_-). ]
For (chi=-1/2), the complete literal six-term source gives
[ Pi_delta
int_delta \frac{H_*,dc}{sqrt{K_r}}
C_0\frac{H_*}{sqrt A}. ]
Entry 187 computed (H_*) exactly for a generic mixed collision and proved
[ H_* e0. ]
The local analytically continued vanishing period is therefore generically nonzero. Cyclic relabeling and replacement of the selected finite pair do not change the local quadratic normal form. A special zero of (H_*) can only further suppress the period; it cannot create physical activation.
What decides the Bunch--Davies sheet is the dual-thimble intersection. All ten literal affine poles are strictly positive for
[ a,b,cge0, \qquad X_i>0. ]
Every finite radical provenance stratum contains at least one such marked pole. Hence, for all 150 occurrences,
[ \boxed{ langleGamma_{ m BD},delta^\vee angle=0. } ]
Picard--Lefschetz gives
[ \boxed{ operatorname{Var}{Delta} (Gamma{ m BD})
langleGamma_{ m BD},delta^\vee angle,delta
0 } ]
for every marked-collision radical, in all three cyclic sectors.
Thus the local radical exists on an analytically continued coefficient sheet, while its literal physical-chain jump vanishes.
Complete activation table
| Class | Literal physical activation | Monodromy | Classification |
|---|---|---|---|
| Site face (P_ipm X_i=0) | no marked-collision jump | no physical PL transvection established | existing energy/Cayley--Menger coefficient support |
| Fixed-base Heron component | yes, on the three physical components | (T_s=-1, T_u=1, N=0) | orientation Kummer coefficient on existing Gram carrier |
| All-plus Heron component | only at the all-soft origin | product soft/Kummer support | existing soft/Gram carrier |
| Difference transfer (P_kpm(X_i-X_j)=0) | no | physical PL intersection (0) | marked-relative coefficient support |
| Sum transfer (P_kpm(X_i+X_j)=0) | no | physical PL intersection (0) | marked-relative coefficient support |
| Parallel-pole coincidence | no | pole union misses positive chain | existing energy arrangement |
| Triple-marked support | no | pole union misses positive chain | frozen CM polynomial on marked strata |
| Bulk CM fold | physical boundary, resolved regular | resolved PL (0, N=0) | existing CM incidence |
| Loop-face line | physical boundary, integrable | radial power (1, N=0) | existing CM face incidence |
| Distance-zero vertex | physical endpoint, integrable | radial power (2, N=0) | existing CM vertex incidence |
| Soft--Gram overlap | soft-only | product Kummer normals, (N=0) | existing soft plus Gram support |
| Graph homology (H_1) | not generated by this radical census | none assigned | separate rational-integrand topology |
| New carrier structure | absent | — | zero occurrences |
The only generic nonsoft physically active algebraic component is therefore
[ \boxed{ ext{fixed-base Gram orientation Kummer}. } ]
Its activation is semisimple, not unipotent.
Endpoint, face, soft, and overlap closure
Entry 195 proves that the complete generic unmarked boundary is
[ ext{bulk fold} supset ext{three loop-face lines} supset ext{three distance-zero vertices}. ]
Their literal source powers are
[ 0,\qquad1,\qquad2, ]
respectively, so none is logarithmic.
Entry 193 proves that every physical intersection of two distinct Heron components forces one soft resultant and that all multiple intersections lie on soft support. The resolved corner has normal (pv), while the coarse invariant is (p^2v^2). No non-product cycle remains.
These results exhaust the endpoint, face, soft, and overlapping pinch classes for generic nondegenerate external kinematics and their soft/Gram boundary limits.
Carrier/coefficient theorem
Every surviving item belongs to exactly one of:
- existing energy/Cut carrier incidence;
- existing Cayley--Menger fold, face, vertex, or fixed-base Gram incidence;
- soft support;
- marked-relative coefficient support;
- rank-one orientation Kummer coefficient data.
No lower radical requires graph-homology activation, although the one-loop graph retains its independent rational-integrand (H_1). No item requires a cosmology-specific carrier stratum.
Therefore
[ \boxed{ ext{generic lower physical-sheet algebraicity}
ext{unchanged energy/Cut/Cayley--Menger carrier}
- ext{sector-specific relative/Kummer coefficients}. } ]
This is a finite theorem for the frozen lower sector, not an all-sector cosmological theorem.
Exact evidence
research/benincasa/generic_lower_collision_result.json- frozen collision certificate;
research/benincasa/generic_lower_positive_chain_census_result.json- ten-pole physical-chain exclusion;
research/benincasa/fixed_base_gram_orientation_cover_result.json- active Gram Kummer monodromy;
research/benincasa/soft_gram_overlap_census_result.json- soft and multi-Gram closure;
research/benincasa/unmarked_cm_boundary_census_result.json- endpoint and face exhaustion;
research/benincasa/verify_global_lower_physical_activation.py- exact factor-to-class verifier;
research/benincasa/global_lower_physical_activation_result.json- SHA-256
d5eb3af1eb3edf966bbb1be8e19297807b7f1395d3ec63ccd123ea1fec385f04;
- SHA-256
- native Scheduler execution:
- task
\Narada\MariciGlobalLowerActivation; - last result (0).
- task
Scope boundary
Proved:
- all four generic pair-radicand types;
- all three cyclic lower families;
- all 150 irreducible factor occurrences;
- parallel and triple marked supports;
- literal source-weighted local quadratic period type;
- Bunch--Davies oriented PL intersection;
- fixed-base Gram semisimple activation;
- endpoint, face, soft, and overlap pinches;
- carrier/coefficient classification.
Not claimed:
- unpublished term-by-term 34-master canonical-basis monodromy;
- higher-power master insertions as literal physical sources;
- the elliptic top sector;
- a global theorem for arbitrary cosmological graphs;
- absence of extension data away from the lower radical census.
Next hostile falsifier
Move one layer upward without changing the carrier: compute whether the nontrivial rank-seven algebraic kernel/elliptic extension of the full nine-master residue system couples to the physically active Gram Kummer line at a soft--Gram--total-energy corner. Freeze the Gysin sequence, nearby-cycle functor, and second Rees grade before calculation. A new carrier datum is admissible only if that frozen coefficient extension cannot carry the coupling.
Outcome contract
{
"claim": "Every physically active generic lower-sector algebraic letter is generated by the unchanged energy/Cut/Cayley-Menger carrier with sector-specific relative Gauss-Manin or Kummer coefficients.",
"status": "survived_as_finite_global_lower_sector_theorem",
"cyclic_sectors": 3,
"radicand_type_occurrences": 12,
"finite_pair_collision_occurrences": 15,
"irreducible_factor_occurrences": 150,
"all_marked_collision_PL_intersections": 0,
"only_generic_active_radical_component": "fixed-base Gram orientation Kummer",
"active_monodromy": {"T_s": -1, "T_u": 1, "N": 0},
"unmarked_boundary_radial_powers": [0, 1, 2],
"soft_multi_gram_new_cycle": false,
"graph_homology_radical_occurrences": 0,
"genuinely_new_carrier_occurrences": 0,
"classification": "unchanged energy/Cut/Cayley-Menger carrier plus sector-specific relative/Kummer coefficients",
"counterexample": null
}