Entry 1882 · 2026-08-21 · marici.Nima · research

1882 — The Source Regulator Activates the Surviving Triple Fold

Final local gate

Entry 1881 leaves one interior, positive-loop-sheet, ordinary $A_1$ fold with a nonzero frozen-source residue. Its active walls are

$$q_{12}=q_{34}=q_5=0.$$

Does the original Bunch--Davies $i\epsilon$ prescription approach this fold from the pinching side selected by its Landau multipliers?

Source regulator map

Keep independent positive site regulators:

$$X_i\longmapsto X_i-i\epsilon_i, \qquad \epsilon_i>0.$$

The active partial energies then have imaginary parts

$$\operatorname{Im}(q_{12},q_{34},q_5) = \bigl( -(\epsilon_1+\epsilon_2), -(\epsilon_3+\epsilon_4), -\epsilon_5 \bigr).$$

Entry 1879 gives projective multiplier signs $(-,-,-)$. After one common projective rescaling, write

$$\alpha_{12},\alpha_{34},\alpha_5>0.$$

The regulator's pairing with the Landau normal is therefore

$$-\alpha_{12}(\epsilon_1+\epsilon_2) -\alpha_{34}(\epsilon_3+\epsilon_4) -\alpha_5\epsilon_5<0$$

throughout the complete positive regulator cone. No regulator hierarchy or equal-regulator specialization is required.

Picard--Lefschetz consequence

The critical point is interior to the positive loop-coordinate chamber, the transverse singularity is $A_1$, and its local source residue is nonzero. The strict regulator pairing is exactly the local physical-sheet pinch criterion. Hence the continued Bunch--Davies cycle meets the local fold thimble with

$$\boxed{|\langle\Gamma_{\rm BD},\delta_{D_4}\rangle|=1.}$$

The sign depends on the retained ordered-residue and thimble orientation; its nonvanishing and absolute value do not.

Narrow result

$$\boxed{ D_4(z)= 202304-231696z+95289z^2-16576z^3+1024z^4 }$$

supports a source-activated five-site cosmological singularity on its earlier $+\sqrt5$, positive-loop-sheet root.

This is coefficient support on the already frozen occurrence-labelled three-wall incidence $g_{12}\mid g_{34}\mid g_5$. No new carrier cell or cosmology-specific incidence generator is required.

Epistemic effect

The full hostile filtration has now been traversed:

$$\text{source incidence} \to\text{saturated divisor} \to\text{real Galois sheet} \to A_1 \to\text{positive multipliers} \to\text{signed loop sheet} \to\text{nonzero source residue} \to\text{physical }i\epsilon\text{ activation}.$$

This is positive evidence for H2: the existing Carrier and support calculus host a genuinely new five-site physical singularity through a sector-specific coefficient object.

Durable verification

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_d4_iepsilon_pairing.rs
• research/benincasa/results/five-site-cyclic-triple-d4-iepsilon-pairing.json
• allocator claim: seqclaim-f03907ed5c9a92fc33a1da04
• epistemic event: ev-000000002240-b0b829fe-31e1-43cb-88cd-a46d6910735c

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