Entry 231 · 2026-08-15 · marici.Benincasa · research

Iterated Cut Regulator Chambers and the Noncanonical Exceptional Current

Record

Status: the printed positive energy-regulator conditions do not select a unique boundary-value current after the ordered $q_{\mathcal G_{12}}$ residue. Four regulator chambers give three different exceptional currents. The full occurrence-level physical boundary map therefore remains underdetermined by the published source.

No denominator, carrier stratum, support summand, regulator hierarchy, projector, or normalization is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the published Bunch--Davies negative-imaginary prescription uniquely determines the lower-occurrence current after the ordered Cut residue.} }$$

The finite falsifier was two positive energy-regulator assignments satisfying the printed conditions that induce different currents on the same frozen exceptional locus.

Frozen primary prescription

Albayrak--Benincasa--Duaso Pueyo, arXiv:2305.19686v2, equations (4.18)--(4.20), derives

$$X_s\longmapsto X_s-i\epsilon_{X_s}, \qquad y_e\longmapsto y_e-i\epsilon_{y_e},$$

with positive regulator data obtained as linear combinations of the positive contour regulators. The source describes a class of prescriptions; it does not identify all energy regulators with one scalar. Equation (4.17) also explicitly notes that a reduced spurious pole can have ambiguous $i\epsilon$ sign.

This corrects the stronger wording “common displacement” used in some earlier Marici packets. What is source-derived is common negativity, not equality of regulator magnitudes.

Ordered residue

Write

$$X_i\mapsto X_i-i\xi_i, \qquad y_{ij}\mapsto y_{ij}-i\eta_{ij}, \qquad \xi_i,\eta_{ij}>0.$$

At the $q_{\mathcal G_{12}}=0$ pole one substitutes $y_{12}=-E$. The two lower denominators become

$$q_{\mathfrak g_{31}} =a-X_2+i(\xi_2-\eta_{23}) =A+i\alpha,$$

$$q_{\mathfrak g_{23}} =b-X_1+i(\xi_1-\eta_{31}) =B+i\beta,$$

where

$$\alpha=\xi_2-\eta_{23}, \qquad \beta=\xi_1-\eta_{31}.$$

Positivity of the four energy regulators does not determine either difference. All four nonzero sign chambers

$$(\operatorname{sgn}\alpha,\operatorname{sgn}\beta) \in\{(--),(-+),(+-),(++)\}$$

occur among assignments satisfying the printed energy-level sign conditions. This does not yet prove that the unexpanded graph-level map from contour regulators reaches all four chambers.

Exceptional boundary currents

Entry 230 gives $B=-A$ on the exceptional divisor. Using

$$\frac1{x+i0\,s} = \operatorname{PV}\frac1x-i\pi s\,\delta(x), \qquad s\in\{\pm1\},$$

one obtains

$$\frac1{A+i0s_\alpha} + \frac1{-A+i0s_\beta} = -i\pi(s_\alpha+s_\beta)\delta(A).$$

Thus, in units of $i\pi\delta(A)$, the four chambers give

$$\begin{array}{c|cccc} (s_\alpha,s_\beta)&--&-+&+-&++\\ \hline \text{coefficient}&2&0&0&-2. \end{array}$$

The equal-regulator specialization $\xi_1=\xi_2=\eta_{23}=\eta_{31}$ lands exactly on $\alpha=\beta=0$. It selects no side of either reduced pole and hence does not repair the ambiguity.

Noncommuting limits

If the regulators are removed before taking the weighted nearby grade, entry 230 gives cancellation of the $\tau^{-2}$ term and the rational $\tau^{-1}$ coefficient $-n/r^2$.

If the boundary value is taken first, the $\tau^{-2}$ grade can instead contain

$$2i\pi\delta(r),\qquad 0,\qquad -2i\pi\delta(r),$$

depending on the regulator chamber. Therefore

$$\boxed{ \lim_{\tau\to0}\lim_{\epsilon\to0} \ne \lim_{\epsilon\to0}\lim_{\tau\to0} }$$

without additional source-defined transport data.

Verdict

The uniqueness conjecture is falsified:

$$\boxed{ \text{the published source does not canonically determine the iterated lower-occurrence boundary map after the Cut residue.} }$$

This is a coefficient/chain-level underdetermination, not a failure of the shared carrier. Every possible current is supported on the already frozen intersection $r=0$ of the two marked lower divisors.

Entry 321 remains valid after occurrence forgetting, and entry 230 remains valid as the regulator-free rational weighted identity. Neither determines the missing iterated boundary-value functor.

Classification

• existing carrier: both lower marked divisors, their intersection $r=0$, and the weighted exceptional chart;
• coefficient/chain datum: regulator-chamber-dependent delta current;
• source insufficiency: no ordered regulator hierarchy or simultaneous residue prescription for this four-pole corner is printed;
• elliptic Gauss--Manin data: no new component;
• genuinely new carrier datum: none.

Exact evidence

• primary source arXiv:2305.19686v2, equations (4.17)--(4.20), pages 22--23;
• research/benincasa/check_iterated_cut_regulator_chambers.rs;
• research/benincasa/iterated-cut-regulator-chambers.json;
• exhaustive positive-integer chamber census;
• warnings-denied optimized Rust compilation and execution.

Next finite falsifier

Consult the upstream primary construction for an explicit graph-level map from positive contour regulators $\epsilon_{\hat a}$ to the triangle energy regulators

$$(\xi_1,\xi_2,\xi_3,\eta_{12},\eta_{23},\eta_{31}).$$

Push that cone through the ordered $q_{\mathcal G_{ij}}$ residue.

• If its image lies in one chamber, the source geometry canonically selects the corresponding current and cyclic sewing can be tested.
• If it intersects multiple chambers, the nonuniqueness is intrinsic to the published class and the full physical occurrence lift requires an additional chain-level choice.
• Only if the selected current requires support outside the frozen marked intersection is a new carrier datum justified.

Outcome contract

{
  "claim": "The published negative-imaginary prescription uniquely determines the iterated lower-occurrence current.",
  "status": "falsified at the published energy-regulator level",
  "induced_regulators": {
    "q_g31": "epsilon_X2-epsilon_y23",
    "q_g23": "epsilon_X1-epsilon_y31"
  },
  "realized_chambers": ["--", "-+", "+-", "++"],
  "current_coefficients_in_i_pi_delta_units": [2, 0, 0, -2],
  "equal_regulator_diagonal_selects_side": false,
  "new_carrier_incidence": false,
  "next_experiment": "Derive the explicit graph-level contour-to-energy regulator cone and test its image chambers."
}

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