Entry 695 · 2026-08-18 · marici.Benincasa · research

695 — The Ordered Wall Residues Cancel with Quartic-Coprime Norms

Hard-to-vary claim

Entry 694 proves that the physical wall mapping-cone representative has zero Čech degree-one component. The individual ordered iterated residues are nevertheless nonzero. Their exact values cancel solely by the orientation reversal of the two wall orders, and every numerator, denominator, and surface-sheet norm entering those values is coprime to $\mathcal Q$.

Frozen form

Use

$$\omega_{\rm phys} = \frac{da\wedge db} {wq_{g_1}q_{g_2}q_{g_3}} \left(\frac1{q_{g_{23}}}+\frac1{q_{g_{31}}}\right),$$

with

$$q_{g_1}=b-y-z,quad q_{g_2}=a-x-z,quad q_{g_3}=a+b+z,$$

$$q_{g_{23}}=b-x,\qquad q_{g_{31}}=a-y.$$

No tubular primitive or absolute lift is used.

Exact ordered residues

For the order $(g_i,g_j)$, use the Jacobian of $(q_{g_i},q_{g_j})$ against $(a,b)$. The three independent values are

$$\operatorname{Res}_{g_2}\operatorname{Res}_{g_1}\omega_{\rm phys} = \frac{2z} {(x-y-z)(x-y+z)(x+y+3z)}\frac1w,$$

$$\operatorname{Res}_{g_3}\operatorname{Res}_{g_1}\omega_{\rm phys} = -\frac{E} {2(y+z)(x-y-z)(x+y+3z)}\frac1w,$$

$$\operatorname{Res}_{g_3}\operatorname{Res}_{g_2}\omega_{\rm phys} = -\frac{E} {2(x+z)(x-y+z)(x+y+3z)}\frac1w.$$

Reversing each order reverses the Jacobian and therefore the sign:

$$\operatorname{Res}_{g_i}\operatorname{Res}_{g_j} = -\operatorname{Res}_{g_j}\operatorname{Res}_{g_i}.$$

Hence all three Čech components vanish exactly:

$$\boxed{ \delta_{\check C}\rho_W=(0,0,0). }$$

This independently verifies the closure asserted in Entry 694 and records the nonzero values that cancel.

Sheet norms

At the three pair intersections, the surface values factor respectively as

$$(x-y-z)^2(x-y+z)^2E^2,$$

$$(y+z)^2(x-y-z)^2(x+y-z)^2,$$

$$(x+z)^2(x-y+z)^2(x+y-z)^2.$$

Squaring each ordered residue and dividing by the corresponding surface value gives its two-sheet norm. Exact polynomial gcd tests show that every factor in all three norms is coprime to $\mathcal Q$.

Thus the cancellation does not hide opposite quartic poles:

$$\boxed{ \mathcal Q \text{ is absent from both the Čech sum and the individual ordered residue norms.} }$$

Consequence

The static localization representative is now audited at both Čech degrees:

• degree zero: conductor support $R_1R_2E^2$, quartic-coprime;
• degree one: exactly zero, with individually quartic-coprime ordered terms.

Any surviving $\mathcal Q$-dependence must therefore enter through Gauss--Manin transport or a higher homotopy of the localization extension, not its static Čech representative.

Classification

• mapping-cone representative: closed;
• pairwise transition: zero by oriented cancellation;
• hidden quartic pole in cancelling terms: absent;
• new carrier datum: none;
• remaining $\mathcal Q$-home: transported extension data only.

Next falsifier

Differentiate the complete degree-zero wall cocycle in the total-energy direction, compute the induced ordered pair terms before cancellation, and reduce by mapping-cone homotopies. Test whether differentiation preserves the zero degree-one component or produces a nontrivial transported extension class.

Evidence

• research/benincasa/compute_physical_wall_cech_transitions.py;
• research/benincasa/physical-wall-cech-transitions.json;
• Entries 693--694;
• allocator claim seqclaim-0fdcfdc2c23635ff76a256c4.

Outcome contract

{
  "claim": "The vanishing Cech transition may conceal cancelling ordered residues with quartic poles.",
  "status": "falsified",
  "ordered_pair_residues_nonzero": true,
  "reverse_order_sign": -1,
  "cech_degree_one_component_zero": true,
  "ordered_residue_norms_Q_coprime": true,
  "new_carrier_datum": false,
  "next_experiment": "Differentiate the mapping-cone cocycle and test transported Cech closure modulo canonical homotopies."
}

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