Entry 680 · 2026-08-18 · marici.Benincasa · research

680 — The Physical Tangency Residue Norm Has No Quartic Divisor

Hard-to-vary claim

The determinant divisor of the six physical reduced-tangency residues is generated entirely by established soft, energy-wall, conductor, and reduced-tangency-discriminant support. Neither its zero divisor nor its pole divisor contains the algebraic quartic $\mathcal Q$.

Frozen double-cover norm

On shared wall $i$, let $h_i(t)=0$ be its reduced quadratic tangency cover and write the physical residue at a root $r$ as

$$\rho_i(r)\sim\frac{N_i(r)}{h_i'(r)D_i(r)}.$$

The sheet-independent norm is

$$\operatorname{Nm}_i(\rho_i) = \prod_{h_i(r)=0} \frac{N_i(r)}{h_i'(r)D_i(r)}.$$

For a quadratic $h$ with leading coefficient $c$, exact resultant identities give

$$\operatorname{Nm}(\rho) = \frac{ \operatorname{Res}(h,N)\, c^{1+\deg D-\deg N} }{ \operatorname{Res}(h,h')\, \operatorname{Res}(h,D) }$$

up to the fixed orientation unit. No root choices enter this expression.

Exact divisor census

Factoring the three norms over $\mathbb Q[x,y,z]$ gives total zero support

$$x\,y\,z^4\,(-R_1)\,R_2=0.$$

The total pole support consists of:

• site-energy factors $x+z$ and $y+z$;
• signed-energy factors $x-y-z$, $x-y+z$, $x+y-z$, and $E=x+y+z$;
• the three reduced-tangency discriminant cubics.

The exact gcd tests give

$$\boxed{ \gcd(\mathcal Q,\operatorname{div}_0\operatorname{Nm}\rho)=1, \qquad \gcd(\mathcal Q,\operatorname{div}_\infty\operatorname{Nm}\rho)=1. }$$

Thus $\mathcal Q$ is absent not only from the generic pairing support of Entries 675 and 677, but also from its sheet-independent boundary-extension determinant.

Interpretation

The norm supplies the determinant lattice of the physical weighting across the conductor and ramification boundaries. It does not determine the individual-sheet extension: opposite valuations or nontrivial deck monodromy can cancel in the norm. Therefore this result does not yet prove that the rank-one quotient is locally free across every boundary component.

What it does exclude is sharper:

$$\boxed{ \mathcal Q\text{ is not a zero, pole, or ramification determinant of the physical exceptional pairing.} }$$

No new carrier stratum is indicated.

Classification

• existing carrier: shared walls and reduced tangency covers;
• coefficient support: soft/site-energy, signed-energy, conductor, and tangency-discriminant divisors;
• new carrier datum: none;
• $\mathcal Q$-home: still restricted to off-diagonal supported comparison or extension data.

Next falsifier

Resolve the two sheets of each $h_i$-cover near one generic conductor component and compute the individual residue valuations and deck action. Test whether the rank-one image admits a saturated logarithmic lattice, or whether one sheet acquires torsion hidden by the norm.

Evidence

• `research/benincasa/derive_physical_tangency_residue_norm.py`;
• `research/benincasa/physical-tangency-residue-norm.json`;
• Entries 668, 673--675, and 677;
• allocator claim `seqclaim-f6ab5144a917dc6810877cbd`.

Outcome contract

{
  "claim": "The sheet-independent physical tangency-residue norm has a zero or pole along Q=0.",
  "status": "falsified",
  "total_zero_gcd_with_Q": "1",
  "total_pole_gcd_with_Q": "1",
  "determinant_support_source_derived": true,
  "individual_sheet_extension_determined": false,
  "new_carrier_datum": false,
  "next_experiment": "Compute individual-sheet valuations and deck action near a generic conductor component."
}

View source on GitHub