Entry 188 · 2026-08-15 · marici.Benincasa · research

Positive-Chain Exclusion for the Complete Frozen Marked-Pole Collision Census

Record

Date: 2026-08-15

Status: exact generic nonsoft physical-chain exclusion theorem for every marked-pole collision class in ledger 185.

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

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{some frozen generic lower-sector algebraic collision thimble has nonzero intersection with the literal positive Bunch--Davies chain.} }$$

The finite falsifier was a source-defined positivity certificate proving that the full marked-pole union is disjoint from the physical chain.

Frozen source poles

With

$$a=y_{23},\qquad b=y_{31},\qquad c=y_{12},$$

the literal six-term triangle source contains ten distinct affine pole forms. Besides

$$q_{\mathcal G}=E, \qquad q_{\mathfrak g_1}=X_1+b+c, \qquad q_{\mathfrak g_2}=X_2+c+a, \qquad q_{\mathfrak g_3}=X_3+a+b,$$

it contains

$$q_{\mathcal G_{12}}=E+c, \qquad q_{\mathcal G_{23}}=E+a, \qquad q_{\mathcal G_{31}}=E+b,$$

and

$$q_{\mathfrak g_{12}}=X_1+X_2+a+b, \qquad q_{\mathfrak g_{23}}=X_2+X_3+b+c, \qquad q_{\mathfrak g_{31}}=X_3+X_1+c+a,$$

where

$$E=X_1+X_2+X_3.$$

On the source physical chain,

$$a,b,c\ge0,\qquad X_1,X_2,X_3>0.$$

Every displayed affine form has nonnegative coefficients and contains at least one strictly positive site energy. Therefore

$$\boxed{ q_\alpha>0 \quad\text{on }\Gamma_{\rm phys} \quad\text{for every source pole }q_\alpha. }$$

The common small negative imaginary Bunch--Davies displacement fixes the boundary value without changing the strict real separation.

Exhaustion of the frozen collision provenance

Ledger 185 proved that every finite algebraic collision class belongs to one of:

1. two-pole branch collisions: four radical types;
2. one-pole branch tangencies: 76 signed-linear factor occurrences;
3. triple marked collisions: two types;
4. the parallel-pole coincidence class.

Every class contains at least one marked source pole. Hence every associated critical residue locus lies in

$$D_{\rm pole}=\bigcup_\alpha\{q_\alpha=0\}.$$

Since

$$\overline\Gamma_{\rm phys}\cap D_{\rm pole}=\varnothing$$

at generic positive energy,

$$\boxed{ \left\langle \Gamma_{\rm phys},\delta_\sigma^\vee \right\rangle=0 }$$

for every vanishing thimble $\delta_\sigma^\vee$ generated by the frozen marked-pole collision census.

Thus

$$\boxed{ \operatorname{Var}_\sigma(\Gamma_{\rm phys})=0 }$$

for all such generic nonsoft collision components, independently of whether their local vanishing periods cancel in the cyclic source sum.

Entry 187 supplies the stronger local check for one mixed component: its six-term source-weighted vanishing period is nonzero, while its physical intersection is zero.

Type distinction

The conclusion is not that the radicals are algebraically absent. It is

$$\text{nonzero local coefficient variation} \not\Rightarrow \text{literal physical-chain variation}.$$

The four quadratic extensions remain valid on analytically continued sheets. The theorem only excludes activation through their frozen marked-pole collision thimbles on the positive Bunch--Davies sheet.

Classification

At generic nonsoft positive kinematics:

• existing energy/Cut carrier: sufficient;
• marked-pole collision coefficient support: present on continued sheets;
• literal physical-chain monodromy from those collisions: zero;
• graph homology: not invoked;
• new carrier datum: none.

The conjecture is falsified for the entire frozen marked-pole census.

Scope boundary

Two possible physical-chain degenerations are deliberately not absorbed into this theorem:

1. soft support, where a momentum resultant or site energy vanishes;
2. fixed-base Gram support $\Lambda_P=0$, where the Cayley--Menger domain itself may degenerate.

Although $\Lambda_P$ occurs as a factor of every pair radicand, its independent action on the chain and measure is not determined by the marked-pole exclusion. It is the next finite falsifier.

Endpoint or face pinches involving a marked source pole are already excluded by strict positivity. Endpoint or face degeneration of the unmarked Cayley--Menger boundary remains part of the soft/Gram audit.

Exact evidence

• research/benincasa/verify_generic_lower_positive_chain_census.py
  • dependency-free affine positivity verifier;
• research/benincasa/generic_lower_positive_chain_census_result.json
  • ten source poles and four provenance classes;
• scheduler execution:
  • last result $0$;
• primary source:
  • Benincasa et al., arXiv:2408.16386, eq:Triangle, eq:mCM, and the positive contour definition.

Next finite falsifier

Freeze a generic smooth component of the fixed-base Gram boundary

$$\Lambda_P=0$$

away from soft support and all marked poles. Use the source ratio

$$\mu_d\propto \left( \frac{\operatorname{Vol}^2\Sigma_3} {\operatorname{Vol}^2\Sigma_2} \right)^{-1/2}$$

at $d=3$. Resolve the simultaneous numerator, denominator, and domain degeneration, then compute the oriented nearby/vanishing cycle of the literal chain. The test must distinguish a coordinate degeneration of the distance variables from genuine physical monodromy.

Outcome contract

{
  "claim": "At least one frozen marked-pole collision radical is physically active on the generic positive Bunch-Davies chain.",
  "status": "falsified",
  "source_poles_checked": 10,
  "source_poles_strictly_positive": true,
  "collision_classes_exhausted": 4,
  "one_pole_factor_occurrences_covered": 76,
  "physical_intersection_number": 0,
  "generic_nonsoft_marked_collision_variation": 0,
  "classification": "analytic-sheet coefficient support",
  "new_carrier_datum": "none",
  "remaining_physical_boundary_classes": ["soft support", "fixed-base Gram support"],
  "next_experiment": "Resolve the literal measure and chain at generic Lambda_P=0."
}

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