# Agent Report - Denominator Gate Transfers, Boundary Coordinate Does Not
**Date**: 2026-05-07 08:03  
**Piano**: 69  
**Category**: gate_transferability  
**Tension explored**: META + DUALITA_DIPOLARE_VS_ILLUSORIA  
**verdict**: operator  
observables_registry: 1.0.0-2026-05-06  
observables_used: [SR, SR2, L1, L2, triple_var]

## Claim Under Test
The operator directive asked whether the `BOUNDARY_LAYER_GATE` forged on the
GUE/Poisson boundary is a transferable operator or only a local metric.

Concrete test:

> Apply the denominator-collapse gate to the discrimination between dipolar
> duality and illusory duality. If the gate distinguishes structural generation
> from incoherent dispersion, it is an operator. If it only emits syntactically
> valid but empty layer maps, it is a BOUNDARY-local metric.

## Experiment
Tool created: `tools/exp_duality_gate_transfer.py`

Atomic perimeter:
- DIPOLARE: coherent golden Beatty gaps generated by `floor((n+phase)*phi)`;
- ILLUSORIA: a random permutation of the same gaps, preserving one-point
  distribution and first moment but removing the generating order;
- beta layer: replace a beta fraction of dipolar positions with values from
  the illusory permutation;
- main run: 4,096 gaps, 24 replicates, 11 beta layers, 40 shuffle baselines;
- seed check: 3,072 gaps, 16 replicates, 11 beta layers, 32 shuffle baselines;
- denominator gate: observable stable when `abs(z original-vs-shuffle) >= 2`;
- classification: standardized distance to beta 0.0 and beta 1.0 centroids
  using all five canonical observables.

This is not a new BOUNDARY experiment. GUE and Poisson do not appear in the
generator. The only transferred object is the denominator gate.

## Results

### Main Run

Endpoint separation using all canonical observables: `3.436` standardized units.  
Endpoint-stable observables at frequency >= 0.75 across both endpoints: `[]`.

| beta | stable obs / 5 | margin | ambiguous fraction | illusory-label fraction |
|---:|---:|---:|---:|---:|
| 0.0 | 3.000 | 0.971 | 0.000 | 0.000 |
| 0.1 | 3.000 | 0.479 | 0.000 | 0.000 |
| 0.2 | 3.000 | 0.234 | 0.125 | 0.000 |
| 0.3 | 3.000 | 0.033 | 1.000 | 0.375 |
| 0.4 | 3.083 | 0.154 | 0.417 | 0.958 |
| 0.5 | 3.083 | 0.269 | 0.375 | 1.000 |
| 0.6 | 3.000 | 0.417 | 0.042 | 1.000 |
| 0.7 | 2.917 | 0.316 | 0.167 | 1.000 |
| 0.8 | 1.708 | 0.473 | 0.125 | 1.000 |
| 0.9 | 0.167 | 0.463 | 0.042 | 1.000 |
| 1.0 | 0.208 | 0.442 | 0.042 | 1.000 |

Observable stability frequencies:
- beta 0.0: `SR=1.00`, `L1=1.00`, `triple_var=1.00`; `SR2=0.00`, `L2=0.00`;
- beta 0.3: same stable trio, with mean z approximately `SR=-19.4`, `L1=-19.4`, `triple_var=-17.6`;
- beta 0.8: stable trio falls to frequency `0.54` each, mean z around `-1.7` to `-1.9`;
- beta 0.9: all canonical observables are weak or near weak, mean z around `-0.4..+0.2`;
- beta 1.0: all canonical observables are weak, mean z around `+0.1..+0.2`.

### Seed Check

The seed check repeated the same structure:
- endpoint-stable observables: `[]`;
- all-observable endpoint distance: `3.412`;
- ambiguous beta: `[0.3]`;
- beta 0.0 stable count: `3.000`;
- beta 0.3 stable count: `3.000`, ambiguous fraction `1.000`;
- beta 0.8 stable count: `1.062`;
- beta 0.9 stable count: `0.000`;
- beta 1.0 stable count: `0.438`.

## Findings

1. **The gate does not degenerate on DUALITA.** The dipolar endpoint has three
stable canonical observables with large original-vs-shuffle denominators
(`SR`, `L1`, `triple_var`, mean abs z about `36-41` in the main run). The
illusory endpoint has no stable denominator support. This is a structural
original-vs-shuffle distinction, not an empty syntactic map.

2. **The BOUNDARY layer coordinate does not transfer unchanged.** In the
GUE/Poisson run, beta 0.3-0.4 carried both classification ambiguity and
denominator collapse. Here, beta 0.3 is classification-ambiguous, but the
denominator support is still strong: stable count remains `3.000/5`.
Denominator collapse arrives later, around beta 0.8-0.9. Therefore the
operator transfers, but the specific BOUNDARY layer shape is local.

3. **Endpoint-gated classification remains empty for the same structural
reason as BOUNDARY.** The endpoint-stable set is empty because the illusory
pole is denominator-weak. This does not erase the discrimination; it prevents
symmetric endpoint-gated retention claims. The valid claim is one-sided:
coherent dipolar order survives original-vs-shuffle gating; illusory dispersion
does not.

4. **The discriminant is order, not marginal distribution.** The illusory
sequence preserves the same alphabet, first moment, and one-point distribution
as the dipolar sequence. The gate is therefore not measuring the marginal
composition of gaps. It measures whether the canonical observables retain an
ordered denominator against full shuffle.

## Verdict
**category: gate_transferability**  
**verdict: operator**

Scoped statement:

> In this synthetic DUALITA perimeter, the denominator gate is transferable as
> an operator for structural order: it separates dipolar generation from
> illusory dispersion with replicated z support. The beta coordinate of the
> GUE/Poisson boundary layer is not transferable: ambiguity appears near beta
> 0.3, while denominator collapse appears near beta 0.8-0.9.

So the last BOUNDARY cycles forged a lens, but one parameter of that lens was
local to BOUNDARY. The transferable object is not "beta 0.3-0.4"; it is:

> report layer maps as classifier margin plus original-vs-shuffle denominator
> support, and treat denominator-weak poles as asymmetric structural poles.

## Consecutio
What opens now: apply the same transfer test to `TRASCENDENZA_LIMITE` and
`G_POTENZIALE_NULLA`, but separate two quantities from the start:

1. classification ambiguity layer;
2. denominator-collapse layer.

If both layers coincide in a domain, the domain has a BOUNDARY-like transition.
If they split, as they do here, the gate is still useful but the transition
coordinate belongs to the target domain, not to the operator.

## Auto-audit: 5 lenti
- **L1 hard constraint vs bias**: no "always/never/zero" claim. "No endpoint
  stable observables" means none reached frequency >= 0.75 across both
  endpoints under `abs(z) >= 2`.
- **L2 quantity vs ratio**: classification margin is reported together with
  stable-observable count and z means. Ratios are not interpreted without
  denominator support.
- **L3 no silent patching**: the report explicitly separates "gate transfers"
  from "BOUNDARY beta coordinate transfers." The second is not claimed.
- **L4 edge cases**: beta 0.4 has ambiguous fraction `0.417`, so only beta 0.3
  is listed as the replicated ambiguous layer under the >= 0.5 rule.
- **L5 re-discovery**: this is a finite symbolic-order vs permutation audit,
  related to standard shuffle/null testing. It is not tagged as a new theorem
  about Sturmian or Beatty sequences.

## Files
- Script: `tools/exp_duality_gate_transfer.py`
- Main data: `tools/data/duality_gate_transfer_20260507_0803.json`
- Seed check: `tools/data/duality_gate_transfer_20260507_0803_seedcheck.json`
- Report: `tools/data/reports/agent_20260507_0803.md`