Entry 2391 · 2026-08-25 · marici.Benincasa · research

2391 — The Continued Physical Leray Germ Activates a Faithful Tangency Score Cospan

Hard-to-vary claim

Entry 2390's (g_1,g_2) recovery ports are not merely formal algebraic functionals. They belong to the canonical analytically continued physical Leray germ already derived from the Bunch--Davies boundary value. Their Tate-regularized leading weights are nonzero on the positive nonsoft total-energy boundary. Consequently the physical continued residue germ, unlike the literal positive-chain boundary, activates a jointly faithful three-port score cospan through second source-word order.

Frozen physical continuation

Two prior source results are retained without modification.

1. Entry 180 proves that the negative-imaginary Bunch--Davies prescription and positive Cayley--Menger chain uniquely determine the local Leray residue germ, including sheet, orientation, Jacobian, and multiplicity.
2. Entry 675 proves that the resulting physical wall cocycle occupies all six labelled reduced-tangency coordinates with nonzero residues away from the established signed-energy conductor support.

This is analytic continuation of the source chain. It is distinct from a literal boundary incidence of the starting positive chamber, which Entries 365 and 555 correctly show to be zero.

Total-energy limits of the labelled ports

For each nonramified port (g_1,g_2), the raw oriented root difference has ordinary order (-1) in (E=E_T). Multiplication by the forced Tate factor (E) gives the same nonzero leading trace:

$$\boxed{ \left.E(\rho_+-\rho_-)\right|_{E=0} =-\frac{3}{8xy}. }$$

Their tangency discriminants have leading value

$$\boxed{\Delta_{g_1}(0)=\Delta_{g_2}(0)=4x^2y^2.}$$

Hence the two normalized ports extend as nonzero coefficient lines on

$$xy(x+y)\ne0.$$

In particular they remain nonzero at both positive roots of Entry 2390's (g_3) blind polynomial.

Physical score cospan

Let (H_i) be the (3\times3) total-energy score Hankel matrix for the labelled port (g_i), acting on

$$\mathcal S_2=\langle S,D_ES,D_E^2S\rangle.$$

The continued physical wall cocycle supplies the labelled map

$$\mathcal O_{ m phys}^{(2)}: \mathcal S_2 \longrightarrow \operatorname{im}H_{g_1} \oplus\operatorname{im}H_{g_2} \oplus\operatorname{im}H_{g_3}.$$

Entry 2390 proves that both (H_{g_1}) and (H_{g_2}) have rank three for (x,y>0). Multiplication by the nonzero physical residue weights preserves their kernels. Therefore

$$\boxed{ \ker\mathcal O_{ m phys}^{(2)}=0 \qquad(x>0, y>0). }$$

At the two points where (ker H_{g_3}\ne0), either nonramified labelled port still reconstructs the complete source-word packet. No fitted projector or additional support summand is used.

Classification

• literal positive-chain incidence with the walls: zero;
• canonical continued physical Leray germ: source-derived and unique;
• (g_1,g_2) total-energy limit: ordinary Tate-regularized coefficient lines;
• (g_3) limit: ramified Kummer coefficient line;
• combined physical continued score kernel through grade two: zero;
• new Carrier support: none;
• marked-wall and tensor/polarization completion: not included.

Thus contextual faithfulness survives the first total-energy-supported scalar interacting staircase test:

$$\boxed{ \text{one port can become blind, while the source-labelled physical continuation remains jointly faithful.} }$$

This is not yet the full objective's tensor theorem. The frozen scalar integrand still does not determine the finite-(q) tensor vertex or both polarization ports.

Next falsifier

Move to the first intersections where the present proof's nonzero weights can fail:

$$x=0,\qquad y=0,\qquad x+y=0,$$

and to the signed-energy conductor collisions of Entry 675. Construct the supported cones and test whether marked-wall score ports recover every nearby class. A class surviving all source-derived continued ports would be a sector-specific coefficient obstruction; only support outside the frozen arrangement would threaten H2 at Carrier level.

Evidence

• research/benincasa/check_total_energy_tangency_port_recovery.py;
• research/benincasa/total-energy-tangency-port-recovery.json;
• Entries 180, 365, 555, 675, and 2390;
• allocator claim seqclaim-8761479d9dc53d7baae97eef.

Outcome contract

{
  "claim": "The g1 and g2 recovery ports of [Entry 2390](/research/ledger/20260825-2390-labelled-tangency-score-ports-remove-the-kummer-observer-blind-divisor) lack source-derived physical activation.",
  "status": "falsified for the canonical analytically continued Leray germ",
  "literal_positive_chain_wall_incidence": 0,
  "continued_leray_germ_canonical": true,
  "regularized_g1_leading": "-3/(8*x*y)",
  "regularized_g2_leading": "-3/(8*x*y)",
  "physical_continued_grade2_kernel_dimension": 0,
  "new_carrier_datum": false,
  "tensor_completion": "absent from the frozen scalar source"
}

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