Entry 224 · 2026-08-15 · marici.Benincasa · research

Marked-Corner Triple-Normal Blowup and the Rank-Two Relative Gate

Record

Status: canonical carrier-level third-Rees interval derived; the induced nine-master Cut--nearby commutator remains uncomputed.

This entry continues entry 318. It changes no denominator, normalization, marked section, support summand, projector, or carrier incidence.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the third-normal marked-corner class is not selected canonically by the frozen source chain and existing flagged carrier.} }$$

The finite falsifier was a source-fixed oriented class on the minimal log blowup. Such a class would show that no support-switch object is required at carrier level.

Frozen local family

Write

$$x=X_1,\quad y=X_2,\quad E=E_T, \quad a=y+A,\quad b=x+B$$

at the positive real marked corner. The exact Cayley--Menger residue family verified in entry 318 has central conductor

$$K_0=R^2, \qquad R=xa^2+yb^2-xy(x+y),$$

and cubic tangent cone

$$-8xy(x+y)E \left( AB+\frac E2(A+B)+E^2 \right).$$

The tangent equation of the conductor is $A+B=0$. Restriction to it gives

$$\boxed{ 8xy(x+y)E(A-E)(A+E). }$$

Thus the frozen third-normal geometry is the triple line arrangement

$$E=0,\qquad A=E,\qquad A=-E.$$

Minimal log blowup

Blow up only the already marked corner $(A,E)=(0,0)$. In the chart

$$r=\frac AE$$

the exceptional divisor is $\mathbf P^1$, and the three strict transforms meet it at

$$p_0:r=\infty,\qquad p_-:r=-1,\qquad p_+:r=1.$$

No fitted divisor has been inserted: these points are the projectivized normal directions of the frozen conductor and its two moving branches.

The exceptional relative group is

$$H_1\bigl(\mathbf P^1,\{p_0,p_-,p_+\};\mathbf Z\bigr) \simeq \widetilde H_0\{p_0,p_-,p_+\} \simeq \mathbf Z^2.$$

Take the basis

$$e_-=[p_-]-[p_0], \qquad e_+=[p_+]-[p_0].$$

Canonical source interval

Entry 180 freezes the continuation through the lower-half energy tube, the positive Cayley--Menger square-root sheet, orientation $da\wedge db$, unit Jacobian, and unit multiplicity. Under $r=A/E$, the two moving boundary branches approach $p_-$ and $p_+$. Their oriented interval therefore has

$$\partial[p_-,p_+] =[p_+]-[p_-] =e_+-e_-.$$

Consequently the frozen source data select

$$\boxed{ \partial\Gamma_{\rm exc}=(-1,1) }$$

in the ordered basis $(e_-,e_+)$. Reversing the already frozen orientation would reverse the sign; it is not an independent admissible choice.

Only $(a,b)=(y,x)$ lies on the positive real loop-edge chamber for $x,y>0$. The other three sign corners are occurrence/deck companions, not four independent physical-chain choices.

Verdict

The carrier-level noncanonicity conjecture is falsified:

$$\boxed{ \text{the existing marked flags and the source orientation canonically generate a nonzero third-Rees relative interval.} }$$

This strengthens the unbounded-grade form of H2. The depth-two form remains falsified.

The result does not yet prove that Cut and nearby cycles commute on the full coefficient system. The rank-two exceptional Tate group must still be mapped into the nine-master algebraic--elliptic extension, and the resulting can--variation class compared with the source Cut residue. No such coefficient-level map is fitted here.

Classification

• existing carrier: total-energy normal, Cut divisor, conductor, two axial marked flags, and their minimal blowup;
• soft support: excluded by $xy(x+y)\ne0$;
• graph homology: none;
• Tate/Kummer coefficient data: the exceptional relative group is rank-two Tate;
• elliptic Gauss--Manin data: unchanged;
• extension data: target of the next comparison;
• genuinely new carrier incidence: none.

Exact evidence

• research/benincasa/check_et_cut_nearby_normal_form.rs;
• research/benincasa/et-cut-nearby-normal-form.json;
• 4,225 exact integer tests of the complete degree-six family;
• warnings-denied optimized Rust compilation and zero-result execution via the governed Scheduler MCP.

Next finite falsifier

Construct the functorial map from the exceptional basis $(e_-,e_+)$ into the specialized nine-master Gysin sequence and evaluate it on $(-1,1)$. Compare the result with the two orders

$$\psi_{E=0}\circ\operatorname{Res}_{q_{\mathcal G_{12}}=0}, \qquad \operatorname{Res}_{q_{\mathcal G_{12}}=0}\circ\psi_{E=0}.$$

If their difference is the image of the frozen third-Rees interval under the predeclared Gysin/excess calculus, H2 survives this coefficient-level gate. If no canonical map exists without a support-switch object or new incidence, the shared-calculus form of H2 fails.

Outcome contract

{
  "claim": "The frozen physical Leray germ canonically selects the third-normal marked-corner relative class on the minimal log blowup.",
  "status": "survived",
  "triple_normal_arrangement": ["E=0", "A=E", "A=-E"],
  "exceptional_points": ["p0=infinity", "pminus=-1", "pplus=1"],
  "relative_rank": 2,
  "canonical_boundary_vector": [-1, 1],
  "physical_real_corner": "(a,b)=(y,x)",
  "new_carrier_incidence": false,
  "full_nine_master_cut_nearby_commutator_computed": false,
  "next_experiment": "Map (-1,1) functorially into the specialized nine-master Gysin extension and compute the Cut-nearby commutator."
}

View source on GitHub