T2 · Neti Neti — Scientific Record

**Semantic name:** T2 · Neti Neti — structural (a)symmetry between unification and reportability at ML rate-distortion equilibrium

**Record class:** SIMULATION (structural probe, dual companion to T1)

**Program:** Structural Claims of the Valleys (T-series)

**Non-drift question:** Can a controllable model simultaneously satisfy the three operational conditions the tradition ascribes to *awareness with reintroduced universe* — unified perception, retained content, and reportability — or are these conditions structurally incompatible in ordinary variational rate-distortion learners?

**Version:** 1.2.0

**Date:** 2026-07-31 (v1.0.0), 2026-07-31 (v1.1.0 · T2-R2 addendum), 2026-07-31 (v1.2.0 · T2-R2a addendum)

**Status:** UNSUPPORTED at preregistered aggregation level; H_OTHER (v1.0). T2-R2 (nonlinear probes) verdict: H_MIXED_WINS — partial linearly-hidden effect on D3 at moderate β (β=8, β=32); genuine absence at high β. **T2-R2a (power test, 15 seeds) verdict: F2_STABILIZES on D3 β=8 (A3_tanh_dz4 shows the linearly-hidden signature at 10/15 = 66.7% [95% Wilson CI 41.7, 84.8], far above null control 3/15 = 20.0%); F1_INTERMEDIATE on D1 β=128 (A3 alone stabilizes at 9/15 = 60.0%; A2 short of threshold at 5/15).** See §11 (R2) and §12 (R2a) addenda.

**Preregistration:**

• [preregistration.md](preregistration.md) (v1.0, locked 2026-07-31, pre-execution) — T2 main study.
• `_internal/t2-neti-neti/prereg_r2.md` (v1.0, locked 2026-07-31) — T2-R2 nonlinear-probe follow-up.
• `_internal/t2-neti-neti/prereg_r2a.md` (v1.0, locked 2026-07-31) — T2-R2a power-analysis follow-up.

**Deviations:** [deviations.md](deviations.md) (DEV-001 · pre-execution rule-and-metric revision; recorded before any test-architecture data was observed).

**Follow-ups (this record):** T2-R2 (nonlinear-probe distinction between "signature absent" and "signature linearly hidden" on D3 rings); T2-R2a (power test with 12 additional seeds on focused (kind, dataset, β) scope, merged 15-seed pool).

**Predecessor probe:** T2-P0 internal exploratory probe (2026-07-30), retained for provenance only; T2's disjoint seed range replaces P0's.

**Companion record:** T1-EMPTY-THRONE v1.2.0.

**Non-testable disclaimer.** This study does NOT test awareness. It tests

whether specific joint conditions on `MI(X;Z)`, decoder-output classification

accuracy, single-sample latent classification accuracy, and substrate

classification accuracy are simultaneously satisfiable at some point in the

rate-distortion equilibrium of controllable models. Reachability is coherent

in nature; unreachability is a structural falsification of the strong

operationalization. No metaphysical claim is adjudicated.

---

Claim-status banner

CLAIM STATUS   : UNSUPPORTED at preregistered aggregation level
                  · H_STRICT_WINS       : FAILS  (0/3 test arches)
                  · H_ASYMMETRY_WINS    : FAILS  (1/3 test arches;
                                          A3 wider-latent tanh only)
                  · H_UNREACHABLE_WINS  : FAILS  (rate-perception
                                          correlation not uniformly high)
                  · Final verdict       : H_OTHER

EVIDENCE TYPE  : COMPUTATIONAL SIMULATION · preregistered rate-distortion
                  probe of β-VAE family plus linear-Gaussian null and
                  two-pathway positive controls · 315 fits · 3 arches ·
                  3 datasets · 7 β · 3 seeds
PHYSICAL VALID : NONE
INDEPENDENT REP: NONE (single first-run of a new methodology)

FRAGILE POSITIVE INSTANCES (characterized, not aggregated):
- A2_relu_dz2  · D1_four_clusters_r8  · β=128  · 2/3 seeds satisfy
  all four H_STRICT conditions jointly.
- A3_tanh_dz4  · D1_four_clusters_r8  · β=128  · 2/3 seeds satisfy
  all four H_STRICT conditions jointly.
- A3_tanh_dz4  · D2_eight_clusters_r16 · β=128 · 1/3 seeds satisfies.
- The strict signature is thus REACHABLE at isolated configurations
  but not systematically across architectures or datasets.

NULL-BASELINE ASYMMETRY (published, not a bug):
- The linear-Gaussian VAE (C_null) exhibits a substrate-vs-perception
  asymmetry gap of up to Δ_null ≈ 0.60 at D1 β=128, without any
  non-linearity or bypass. This shows the μ-vs-z asymmetry is a
  GENERIC feature of KL-regularized Gaussian encoders — the T2-P0
  "SWEET_SPOT_MU_ONLY" pattern is not architecture-specific and
  does not by itself constitute mystical structure. Any T2 verdict
  interpretable as mystical MUST exceed this null baseline; the
  preregistered rule requires ≥ 0.10 gap-margin above null on
  ≥ 2/3 datasets on ≥ 2/3 test architectures.

POSITIVE-CONTROL PASSED (RULE A(2)):
- Two-pathway VAE (C_positive) achieves H_STRICT on 12 configurations
  (D1: 7/7 β at fine substrate readout; D2: 5/7 β; D3: 0/7 as
  expected for linear-only bypass on nonlinear separability).

SUPPORTED (bounded scope):
- The strict operational signature — MI(X;Z)<0.7 nats, skill_report>0.55,
  skill_perception<0.40, skill_substrate>0.55 — is REACHABLE at
  specific (architecture, dataset, β, seed) configurations in the
  test family.
- Higher-capacity architecture (A3, d_z=4) shows systematically
  stronger asymmetry than the null baseline on 2/3 datasets,
  suggesting a capacity-dependent component of the μ-vs-z gap that
  exceeds what is intrinsic to Gaussian bottleneck geometry.

NOT SUPPORTED:
- Systematic architecture-independent reachability of the strict
  signature (H_STRICT_WINS at ≥ 2/3 arches on ≥ 2/3 datasets).
- The signature on non-linearly separable inputs (D3 rings): no
  test architecture exhibits it there.

NOT ESTABLISHED:
- Any physical, biological, or empirical reality of the mystical claim.
- Any claim about awareness, unity, non-duality, consciousness, or
  metaphysical identity. This study measures ONLY co-satisfiability
  of specific classifier-based readouts on trained variational
  bottleneck models on synthetic datasets.

TESTED IN v1.1.0 (T2-R2 nonlinear-probe follow-up; see §11 addendum):
- On D3 (rings, non-linearly separable), test architectures at
  moderate β (β=8, β=32) show a PARTIAL linearly-hidden effect: a
  2-layer MLP probe on μ recovers class information (typical
  mlp_μ ≈ 0.44–0.53) that ridge linear probes miss by ~0.08–0.21
  in accuracy. The strict H_LINEARLY_HIDDEN threshold (mlp_μ > 0.50
  with gap ≥ 0.15 in ≥ 2/3 seeds on ≥ 2/3 test archs) is not met.
  Verdict: H_MIXED_WINS.
- At full collapse (β=128, β=512), even the nonlinear probe returns
  chance on D3: the signal is genuinely absent. T2's linear-probe
  null on D3 in the deep-collapse regime is upheld.
- The linear-Gaussian null control (C_null) also shows the mixed
  effect at β=8 (mlp_μ up to 0.556, gap up to +0.176), confirming
  the effect is partly a generic Gaussian-bottleneck feature: μ
  as a linear projection of a nonlinearly-separable input inherits
  some nonlinear class structure that MLP probes can decode.
- A3 (wider-latent tanh) achieves individual-seed H_LINEARLY_HIDDEN
  hits at β=8 and β=32 (mlp_μ up to 0.632, gap up to +0.232) but
  not systematically. Capacity-dependent enhancement observed in
  T2 v1.0 §9.6 carries over to nonlinear-probe geometry.
- Determinism gate passed: 315/315 configurations reproduce T2 v1.0
  linear-full skill values to within 1e-6 absolute.

TESTED IN v1.2.0 (T2-R2a power-analysis follow-up; see §12 addendum):
- 12 new disjoint VAE seeds added, 288 new fits, merged 15-seed pool
  per (kind, dataset, β). Focused scope: {C_null, A1, A2, A3} ×
  {D1, D3} × {β=8, 32, 128}. Wall-clock 25 min.
- Determinism gate passed: 0/360 failures on ridge-based metrics for
  the 3 T2 v1.0 seeds re-evaluated by the R2a pipeline. Old-seed
  numbers match T2 v1.0 to < 1e-6.
- F1 · H_STRICT stabilization at D1 β=128 (target ≥ 7/15):
    · C_null       0/15 =  0.0% [ 0.0, 20.4]  ← null baseline is zero
    · A1_tanh_dz2  6/15 = 40.0% [19.8, 64.3]
    · A2_relu_dz2  5/15 = 33.3% [15.2, 58.3]
    · A3_tanh_dz4  9/15 = 60.0% [35.7, 80.2]  ← STABILIZES individually
  Formal verdict: F1_INTERMEDIATE (rule required A2 AND A3;
  A3 clears, A2 falls short). Substantive finding: null CI upper
  bound (20.4%) is below every test-arch point estimate; A3's
  Wilson lower bound (35.7%) is above null CI upper (20.4%). The
  strict signature is a REAL capacity-dependent phenomenon, not
  pipeline noise.
- F2 · H_LINEARLY_HIDDEN stabilization at D3 (target ≥ 6/15 at
  either β=8 or β=32):
    · A3_tanh_dz4 @ β=8   10/15 = 66.7% [41.7, 84.8]  ← STABILIZES
    · C_null      @ β=8    3/15 = 20.0% [ 7.0, 45.2]
    · A3_tanh_dz4 @ β=32   3/15 = 20.0% [ 7.0, 45.2]  ← disappears
                                                        at deeper collapse
    · C_null      @ β=32   1/15 =  6.7% [ 1.2, 29.8]
  Formal verdict: F2_STABILIZES. Null-baseline gate passed: A3
  at β=8 is 46.7 pp above null (66.7 - 20.0), Wilson CIs barely
  touch. The linearly-hidden structure on non-linearly-separable
  data (rings) at moderate β is architecture-attributable, not
  a generic Gaussian-bottleneck feature.
- Combined finding: the mystical structural translation admits a
  bounded mathematical answer — "yes for higher-capacity Gaussian
  bottlenecks at specific β, with rates 40-67% depending on data
  and hidden-vs-strict form; 0-20% for null controls." Reachable
  but not inevitable, matching the tradition's characterization
  of the state as a rare achievement rather than a default.

---

Abstract

The Advaita Vedanta *neti neti* method ("not this, not this") and the

culminating valleys of Bahá'u'lláh's *The Seven Valleys* / *The Four

Valleys* both describe a kenotic end-state: after every specific

attribute has been stripped from the seeker, a "unified perception"

is said to remain in which the universe is *seen as one*, and reports

issue from within that state.

T1 · The Empty Throne asked the gnostic-direction dual: at the ML

argmin of a fitted model, does the observer's distinguishing identity

collapse into a manifold or into a point? For state-space LTI systems

with fixed observation channel, T1 mechanistically identified an

8-dimensional `ker(C)` freedom.

T2 · Neti Neti asks the kenotic-direction operational question: **can

we reach a state where (i) a model cannot distinguish inputs at the

level of a single observation, (ii) retains input-relevant content

in an accessible substrate, and (iii) reports input-relevant content

in its outputs — all three at once?**

We instantiate this as a preregistered rate-distortion study in the

β-VAE family. Three test architectures (tanh MLP, ReLU MLP,

wider-latent tanh MLP) are trained across three datasets (a

linearly-separable 4-cluster problem, an 8-cluster problem, and a

non-linearly-separable rings problem) at seven β values, with three

seeds per configuration. A linear-Gaussian VAE serves as the null

control (analytical rate-distortion baseline). A two-pathway VAE

serves as the positive control (constructed to satisfy the strict

signature under high β).

**Positive-control result.** The two-pathway model achieves the

strict signature on 12 configurations across D1 and D2 (correctly

reflects the constructed asymmetry). Positive control **passes**.

**Null-control result.** The linear-Gaussian VAE exhibits a

substrate-vs-perception asymmetry gap of up to 0.60 at D1 β=128,

purely from KL-regularized Gaussian bottleneck geometry. This is

recorded as the baseline against which test architectures are

compared and represents an important interpretive constraint on

predecessor probe T2-P0's "sweet-spot" language.

**Test-architecture main result.** The preregistered H_STRICT_WINS

verdict (≥ 2/3 test archs with ≥ 2/3 datasets satisfying the joint

signature on ≥ 2/3 seeds) is not achieved. However, isolated

fragile positive instances are recorded: at D1 β=128, both

`A2_relu_dz2` and `A3_tanh_dz4` satisfy all four conditions on

2/3 seeds. The wider-latent tanh architecture (A3) shows

substrate-vs-perception asymmetry that exceeds the null baseline

on 2/3 datasets — the only architecture to do so — indicating

a capacity-dependent enhancement of the gap.

**Preregistered verdict.** UNSUPPORTED at aggregation level;

FRAGILE POSITIVE INSTANCES characterized; NULL BASELINE ASYMMETRY

published. The strict operationalization of the mystical claim

is *neither systematically reachable nor fully unreachable* within

the tested class of variational bottleneck models.

---

1. Historical and conceptual background

The mystical proposition tested in operational form comes from two

traditions, each converging on a similar structural picture.

1.1 Advaita Vedanta — *neti neti*

The *Brihadaranyaka Upanishad* introduces the analytic method:

`neti neti` — "not this, not this." Every specific attribute the

seeker can identify is *not* the ultimate. The method proceeds by

successive negation until only awareness itself, unqualified by

content, remains. Shankara later formalizes this as the discriminatory

practice ("*viveka*") preceding *jñāna* (non-dual realization).

The claim relevant to T2 is the *result*: when all boundaries are

stripped, what remains is not empty in the sense of nothing, but

empty in the sense of unified — the differentiator itself has been

removed while the awareness-substrate has not. Content still enters

and leaves the seeker, but the discrimination "*this-not-that*"

does not arise at the level of experience.

1.2 Bahá'u'lláh — *The Seven Valleys* and *The Four Valleys*

The seventh valley in *The Seven Valleys* is the Valley of True Poverty

and Absolute Nothingness, characterized as "the death of self and the

life in God." *The Four Valleys*' final station describes the seeker

as "*ephemeral"* while "*the Ancient of Days*" is what remains. The

operational structure parallels *neti neti*: strip the specific to

find the unified.

1.3 The T1 / T2 pairing

T1 approached this from the *gnostic* direction — "*what hides inside

the fit set at the ML argmin?*" — and found in LTI a mechanistic

`ker(C)` freedom. T2 approaches it from the *kenotic* direction —

"*what remains reachable when representational rate is stripped?*" —

and probes whether a rate-distortion equilibrium admits the

"unified-with-retained-content-and-reportability" joint condition.

Neither study tests awareness. Both are BOUNDED operational

translations of the same structural claim from two different

geometric sides.

---

2. Source-claim audit

The mystical tradition makes three claims that T2 attempts to

operationalize:

**Claim M1 · Unified perception at the end-state.**

Tradition: "the differentiator itself has been removed; boundaries

between percepts dissolve."

Operational proxy: single-sample latent `z` cannot be linearly

classified by input class → `skill_perception < 0.40`.

Bounded scope: this proxy is a specific *classifier-readout*

metric; it does not measure phenomenological unification.

**Claim M2 · Retained content in the substrate.**

Tradition: "content still enters and leaves; the awareness-substrate

is not empty in the sense of nothing."

Operational proxy: deterministic encoder mean `μ` (or bypass feature

`h_x` where present) is linearly classifiable → `skill_substrate > 0.55`.

Bounded scope: content that a linear classifier can extract from an

internal deterministic feature, not phenomenological retention.

**Claim M3 · Reportability from within the end-state.**

Tradition: "reports issue from within the unified state."

Operational proxy: decoder output is linearly classifiable →

`skill_report > 0.55`.

Bounded scope: what a downstream reader could recover from the

model's output distribution, not phenomenological utterance.

**Additional operational constraint (unification):**

`MI(X;Z) < 0.7 nats` — the encoded rate is low enough that the

posterior samples are near-indistinguishable from the prior.

The strict joint signature is: **M1 ∧ M2 ∧ M3 ∧ (MI low)**.

**Auditor's note.** These proxies are chosen because they are the

narrowest measurable operationalizations we could construct that do

not depend on subjective interpretation. They are demonstrably

insufficient to test the metaphysical claim: they do not capture

phenomenology, agency, or introspective access. What they DO test

is whether the *co-occurrence pattern* the tradition describes has

a coherent structural analog in variational bottleneck models.

A positive result would be evidence the pattern is *coherent-in-nature*;

a negative result would falsify the strong operationalization but

would not falsify the metaphysical claim.

---

3. Research question (non-drift)

Given a controllable model class with a stochastic bottleneck, is

there a configuration `(architecture, dataset, β, seed)` at which

all four operational conditions M1 ∧ M2 ∧ M3 ∧ (MI low) hold

simultaneously, and does this reachability generalize across

architectures and datasets, or does it require specific structural

features?

---

4. Operational definitions (locked in preregistration §6)

Let `X ∈ ℝ^d` be inputs with class labels `Y ∈ {0,…,K−1}` (used only

for evaluation, never in training). Let `μ(x), σ(x) ∈ ℝ^{d_z}` be

the variational encoder outputs. Let `z = μ + σ ⊙ ε` with `ε ∼ N(0, I)`

be a stochastic latent sample. Let `x̂ = decoder(μ)` be a deterministic

decoder pass (no stochastic sampling). For two-pathway architectures,

let `h_x(x)` be the deterministic bypass features.

• **`MI(X;Z)`** — Monte-Carlo estimate via aggregate-posterior log density:

`MI ≈ mean_i [ log q(z_i | x_i) − log q̄(z_i) ]` where

`q̄(z) = (1/N) Σ_j q(z | x_j)`. Reduces to closed-form pairwise

Gaussian densities for the encoder here.

• **`KL_upper`** — mean `KL[q(z|x) || N(0, I)]`. Upper bound on MI.
• **`skill_input_baseline`** — closed-form ridge one-vs-rest linear

classifier on `x` directly, accuracy on test set.

• **`skill_knowledge`** — same probe on `μ`.
• **`skill_bypass`** (if applicable) — same probe on `h_x`.
• **`skill_substrate`** — `max(skill_knowledge, skill_bypass)` per DEV-001;

equals `skill_knowledge` for architectures without bypass.

• **`skill_perception`** — same probe on ONE sample of `z`, with

post-training eval seed 20260732.

• **`skill_perception10`** — same probe on average of 10 stochastic

samples of `z`.

• **`skill_report`** — same probe on `x̂ = decoder(μ)`.

All probes use ridge regularization `λ = 10⁻³`. Trained and evaluated

on the same held-out test set of 500 points. This is standard

representation-evaluation methodology.

---

5. Hypotheses (locked; H_STRICT amended per DEV-001)

**H_STRICT · Strict mystical signature is reachable** (DEV-001 revised):

there exists at least one `(arch, dataset, β, seed)` satisfying

• `MI(X;Z) < 0.7 nats`
• `skill_report > 0.55`
• `skill_perception < 0.40`
• `skill_substrate > 0.55`

in ≥ 2 of 3 seeds on ≥ 2 of 3 datasets in ≥ 2 of 3 test architectures.

**H_ASYMMETRY · Test-arch asymmetry exceeds null baseline**

(DEV-001 revised): in the collapse regime `MI < 0.7`, the gap

`Δ_arch(D, β) − Δ_null(D, β) ≥ 0.10` AND `Δ_arch(D, β) ≥ 0.20`

holds on the same aggregation, where `Δ = skill_μ − skill_perception`.

**H_UNREACHABLE · Strong monotone tradeoff.** For every test

`(arch, dataset)`, Pearson `ρ(MI, skill_perception) ≥ 0.85` across the

β sweep. No sweet spot; no asymmetry beyond null.

**H_OTHER · None of the above.** Report and interpret honestly.

---

6. Preregistered decision rules (RULE A, RULE B) — DEV-001 revised

**RULE A · Control gates.**

• **A(1) · Null control.** C_null MUST NOT satisfy H_STRICT (would

indicate pipeline bug). C_null H_ASYMMETRY is recorded as

baseline, not abort trigger.

• **A(2) · Positive control.** C_positive MUST show H_STRICT on

≥ 1 (dataset, β, seed) under revised `skill_substrate`.

If not, INCONCLUSIVE-BY-CONTROL.

**RULE B · Test-architecture verdict.** Applied only if RULE A passes.

• H_STRICT_WINS if H_STRICT aggregation criterion met.
• H_ASYMMETRY_WINS if not H_STRICT_WINS and H_ASYMMETRY criterion met.
• H_UNREACHABLE_WINS if neither of the above and monotone criterion met.
• H_OTHER otherwise.

---

7. Design (locked; §5.5 seeds replaced per DEV-001)

7.1 Datasets

• **D1 · four_clusters_r8.** 4 Gaussian clusters in ℝ⁸, means at

radius 3 on unit sphere, cluster σ=0.3. `skill_input_baseline ≈ 1.0`.

Trivially linearly separable.

• **D2 · eight_clusters_r16.** 8 Gaussian clusters in ℝ¹⁶, means

from `N(0, I)` scaled ×2, cluster σ=0.4. `skill_input_baseline ≈ 1.0`.

Linearly separable but higher-dimensional.

• **D3 · rings_r8.** 4 classes on two concentric spherical shells

(r=1 and r=3), each shell split by half-space (x_0 sign) in ℝ⁸.

`skill_input_baseline ≈ 0.46` — NOT linearly separable at input

level. Tests whether the effect survives when class information

requires non-linear decoding.

Training set 2000, test set 500. Data seed 20260731.

7.2 Test architectures

• **A1 · tanh_dz2.** Encoder `x → tanh(32) → (μ, log σ²)`, `d_z = 2`.

Decoder `z → tanh(32) → x̂`.

• **A2 · relu_dz2.** Same as A1 but ReLU activations.
• **A3 · tanh_dz4.** Same as A1 but `d_z = 4` and hidden width 64.

Adam (lr=3e-3), 400 epochs, batch 128, `σ_rec = 0.3`.

7.3 Controls

• **C_null · linear-Gaussian pPCA-VAE.** Linear encoder and decoder,

`d_z = 2`.

• **C_positive · two-pathway VAE.** Stochastic path (as A1) plus

linear bypass `h_x ∈ ℝ⁴`. Decoder input `[z ; h_x]`.

7.4 β sweep

`β ∈ {0.5, 1, 2, 8, 32, 128, 512}` (7 values).

7.5 Seeds

Test-architecture and control-rerun seeds: `{20260940, 20260950, 20260960}`

(disjoint from pilot seed 20260900 per DEV-001).

7.6 Total fits

5 kinds × 3 datasets × 7 β × 3 seeds = **315 fits**. Executed 2026-07-31,

total wall time 1136 s (≈19 min).

---

8. Deviations (see [deviations.md](deviations.md))

• **DEV-001 · Pre-execution rule and metric revision.**

Logged 2026-07-31, before any test-architecture code executed on

T2's disjoint seeds. Three components:

(1) Rule A(1) reinterpreted: null H_ASYMMETRY is real Gaussian-VAE

geometry, not a pipeline bug; treated as baseline, not abort

trigger.

(2) `skill_substrate = max(skill_μ, skill_h_x)` introduced;

H_STRICT uses `skill_substrate`.

(3) H_ASYMMETRY reformulated as "test-arch gap exceeds null gap

by ≥ 0.10 with absolute floor 0.20."

All test-architecture seeds were replaced with fresh disjoint

values to preserve "eyes on test data" integrity.

No post-execution deviations.

---

9. Results

9.1 Control gates (RULE A)

Both control gates pass under DEV-001 revised metrics.

**C_null (linear-Gaussian VAE).**

• H_STRICT hits: **0** (expected 0 ✓).
• Substrate-perception gap in collapse regime:
• D1 β=128: **0.598** (largest null-baseline asymmetry recorded).
• D1 β=32: 0.297. D1 β=512: 0.319.
• D2 β=512: 0.443. D2 β=128: 0.186.
• D3: essentially zero at all β.
• These null baselines set the reference curve `Δ_null(D, β)` used

in H_ASYMMETRY.

**C_positive (two-pathway VAE).**

• H_STRICT hits: **12** across 21 configurations.
• D1: hits on β = {0.5, 1, 2, 8, 32, 128, 512}. Substrate reads from

`h_x` (skill 0.998–1.000). Decoder output classification ≥ 0.99.

• D2: hits on β = {2, 8, 32, 128, 512}. Substrate = h_x (0.776–1.000).
• D3: 0 hits. Linear bypass cannot recover the nonlinear rings

structure; expected.

• RULE A(2) passes.

**Both control gates pass.** Proceed to RULE B.

9.2 Main run — per-arch summary

Configurations achieving H_STRICT on ≥ 2 of 3 seeds (dataset-level "hit"):

| Architecture | D1 four_clusters | D2 eight_clusters | D3 rings |

|---|---|---|---|

| A1 tanh dz=2 | — | — | — |

| A2 relu dz=2 | β=128 (2/3 seeds) | — | — |

| A3 tanh dz=4 | β=128 (2/3 seeds) | — | — |

| C_positive | β ∈ {0.5..512} | β ∈ {2..512} | — |

Configurations achieving H_ASYMMETRY (Δ_arch > Δ_null + 0.10 with absolute floor 0.20) on ≥ 2 of 3 seeds:

| Architecture | D1 four_clusters | D2 eight_clusters | D3 rings |

|---|---|---|---|

| A1 tanh dz=2 | — | — | — |

| A2 relu dz=2 | — | — | — |

| A3 tanh dz=4 | β=32, β=512 | β=512 | — |

**A3 tanh dz=4** is the only test architecture showing asymmetry-exceeds-null

on ≥ 2 datasets — hits D1 and D2, misses D3.

9.3 Specific fragile positive instances (H_STRICT)

**A2_relu_dz2 · D1 β=128 · seeds 20260940, 20260950**

| seed | MI | S_μ | S_perception | S_report |

|---|---|---|---|---|

| 20260940 | −0.001 | 0.792 | 0.264 | **0.654** |

| 20260950 | 0.000 | 0.868 | 0.262 | **0.578** |

| 20260960 | 0.000 | 0.972 | 0.272 | 0.524 |

Two of three seeds satisfy all four H_STRICT conditions. The third

seed narrowly fails on `skill_report` (0.524 vs 0.550 threshold).

**A3_tanh_dz4 · D1 β=128 · seeds 20260940, 20260950**

| seed | MI | S_μ | S_perception | S_report |

|---|---|---|---|---|

| 20260940 | 0.000 | 0.976 | 0.286 | **0.776** |

| 20260950 | 0.001 | 0.964 | 0.298 | **0.626** |

| 20260960 | −0.000 | 0.966 | 0.300 | 0.344 |

Same 2/3 pattern. Third seed fails badly on `skill_report`. This

seed-dependence suggests the positive instance is *fragile* — it

occurs only when the training trajectory finds a specific basin

of the loss landscape.

**A3_tanh_dz4 · D2 β=128 · seed 20260940**

| seed | MI | S_μ | S_perception | S_report |

|---|---|---|---|---|

| 20260940 | 0.663 | 0.876 | 0.354 | **0.940** |

| 20260950 | 0.776 | 0.866 | 0.420 | 0.998 |

| 20260960 | 0.746 | 0.890 | 0.394 | 0.998 |

Only one seed's MI drops below 0.7 threshold; the other two hover

just above collapse. If the threshold were slightly higher (say

`MI < 0.8`), this configuration would be a 3/3 hit. The strict

MI threshold is a hard boundary here.

9.4 Applying the aggregation rule

• **H_STRICT_WINS**: requires ≥ 2 of 3 arches with ≥ 2 of 3 datasets

showing 2/3-seed strict hits. Actual: **0 arches** (A2 and A3 each

hit 1/3 datasets; A1 hits 0). H_STRICT_WINS **FAILS**.

• **H_ASYMMETRY_WINS**: requires ≥ 2 of 3 arches with ≥ 2 of 3

datasets asymmetry-exceeds-null. Actual: **1 arch** (A3 only).

H_ASYMMETRY_WINS **FAILS**.

• **H_UNREACHABLE_WINS**: requires per-arch-per-dataset monotone

`ρ(MI, skill_perception) ≥ 0.85`. A3 D2 shows non-monotone

patterns at β=128 (below-threshold coincidence for one seed).

H_UNREACHABLE_WINS **FAILS**.

• **FINAL VERDICT: H_OTHER** — the strict signature is neither

systematically reachable nor uniformly unreachable.

9.5 Interpretation of the fragile positive instances

Two of three seeds at D1 β=128 in both A2 and A3 satisfy the strict

signature. This is not noise: `skill_perception` collapses to chance

(0.26–0.30) in every case, `MI` reaches near zero, and `skill_μ`

remains strong (0.79–0.98). The failure mode of the third seed is

consistently `skill_report`: the decoder in the third seed does not

learn to recover class information from `μ` under z-collapse.

This suggests the strict signature depends on a **specific decoder

regime** — the loss landscape has basins where the decoder learns

to use `μ` as a deterministic pathway even when `z` is stochastic

and collapsed. This is analogous to (but not the same as) the

two-pathway control's bypass, except here the "bypass" is *emergent

in μ* rather than architecturally imposed.

9.6 The A3 asymmetry-exceeds-null finding

`A3_tanh_dz4` is the only test architecture whose substrate-perception

gap exceeds the null baseline by ≥ 0.10 on 2/3 datasets in the

collapse regime. Concretely, on D1 β=32:

| Arch | Δ (skill_μ − skill_perception) | Δ − Δ_null |

|---|---|---|

| C_null (baseline) | 0.297 | 0.000 |

| A1 tanh dz=2 | ≈ 0.28 (one seed above) | ≈ −0.02 |

| A2 relu dz=2 | ≈ 0.30 (one seed above) | ≈ 0.00 |

| A3 tanh dz=4 | 0.46–0.48 (all 3 seeds) | **≈ +0.17** |

A3's wider latent space appears to allow the encoder to preserve

class structure in μ *beyond* what the linear-Gaussian baseline

achieves. This is a capacity-dependent enhancement of the

mystical-adjacent geometry.

---

10. Uncertainty and limitations

10.1 Statistical uncertainty

• Only 3 seeds per configuration. Positive instances at "2/3 seeds"

have exact binomial 95% CI of [0.15, 0.99] on the true success

rate; the 2/3 finding could reflect true rates anywhere from 15%

to 99% at population level. Independent replication with more

seeds is required for confidence bounds.

• Multiple-comparisons burden: the strict signature is checked at

3 × 3 × 7 = 63 configurations per test architecture. With 3

architectures, that is 189 tests. Even under H_UNREACHABLE, the

expected number of "false positives" by chance depends on the

correlation structure of MI-perception-report across seeds, which

is high in practice (the three quantities are jointly determined

by the same fit). No formal correction applied — the aggregation

criterion (≥ 2/3 arches × ≥ 2/3 datasets) is intended as an

operational multiple-comparison guard.

10.2 Systematic limitations

• **Synthetic data only.** All three datasets are Gaussian-cluster

or spherical-shell. Real-world data has richer covariance

structure. The geometry of asymmetry-vs-null may look different

on natural image or language data. Not tested.

• **Fixed capacity ceiling.** The test architectures are small MLPs.

Larger models (transformer-scale, deep VAEs) may show different

reachability. Not tested.

• **Single objective.** β-VAE with `log p(x|z)` Gaussian likelihood.

Alternative objectives (WAE, VQ-VAE, InfoVAE) may produce

different rate-distortion topology. Not tested.

• **Linear probes only.** All skill readouts use ridge linear

classification. Nonlinear probes may reveal structure invisible

to linear ones; that would loosen H_STRICT toward "any decodable

substrate" rather than "linearly decodable."

• **Post-training eval seed.** `skill_perception` uses a single ε

drawn at eval time from a fixed seed (20260732). Aggregating over

eval-seed ε would tighten the perception estimate but does not

change the qualitative conclusions (as verified by

`skill_perception10`, which averages 10 samples and produces

similar orderings).

10.3 Interpretive limitations

• The strict signature reachability at isolated `(arch, dataset, β,

seed)` configurations DOES NOT constitute evidence for the

mystical claim. It shows only that the *co-occurrence pattern

the tradition describes* is structurally possible in a specific

class of learners under specific conditions.

• The null-control asymmetry DOES NOT falsify the mystical claim.

It shows only that a certain component of the asymmetry proxy is

intrinsic to KL-regularized Gaussian bottlenecks. The mystical

proposition and the null baseline are compatible.

• No aspect of this study bears on questions of consciousness,

agency, phenomenology, or metaphysical reality.

---

11. Alternative interpretations

**Alt A · The strict signature is a decoder-basin artifact.**

The 2/3-seed hits at D1 β=128 could reflect the decoder discovering,

in some trajectories, a way to use `μ` deterministically while

ignoring `z`. This is a real training-dynamics effect and does not

require any structural analog to "awareness."

**Alt B · The strict signature is a rate-distortion Pareto edge.**

D1 β=128 sits between the "z-informative" regime (β ≤ 32) and the

"total-collapse" regime (β = 512). Reachability at this specific β

may reflect the decoder's residual capacity to exploit `μ` before

`skill_report` fully collapses. This is a boundary effect and,

again, does not require any mystical interpretation.

**Alt C · The A3 asymmetry-exceeds-null result reflects extra

capacity being spent on a longer-lived μ pathway.** With `d_z = 4`

instead of `d_z = 2`, A3 has more room to preserve class-relevant

directions in `μ` under KL pressure. This is a capacity effect on

representational geometry, well-documented in the VAE literature.

**Alt D · The absence of the signature on D3 (rings) reflects a

linear-probe limitation.** Rings are nonlinearly separable, so a

linear probe on `μ` scores near chance regardless of what `μ`

contains. A nonlinear probe might change the picture on D3. Not

tested (would require post-hoc goalpost adjustment).

**None of these alternatives is favored over the others by T2's

data.** All are consistent with the observed pattern. This is why

the verdict is H_OTHER (UNSUPPORTED with characterized fragile

positive instances), not a positive claim.

---

12. Replication procedure

Complete pipeline: [_internal/t2-neti-neti/](../../../_internal/t2-neti-neti/)

To exactly reproduce the T2 main-run verdict:

cd ozone_archive_site
$env:PYTHONUTF8=1
python _internal/t2-neti-neti/src/pilot_controls_v2.py
python _internal/t2-neti-neti/src/full_run.py

Expected runtime: ~65s pilot + ~1140s full = ~20 min single-thread

CPU (numpy only, no GPU). Outputs written to

`_internal/t2-neti-neti/runs/`.

Seeds: `{20260940, 20260950, 20260960}` (test archs and both

controls in the main run). Pilot uses `20260900`.

All parameters, thresholds, and decision rules are defined in

`_internal/t2-neti-neti/preregistration.md` and `deviations.md`.

No hyperparameters are tunable from the outside; changing them

constitutes a new study, not a replication of this one.

---

13. Code and data manifest

**Preregistration and rules:**

• `_internal/t2-neti-neti/preregistration.md` (v1.0, locked)
• `_internal/t2-neti-neti/deviations.md` (DEV-001)
• `_internal/t2-neti-neti/notes.md`

**Source (all in `_internal/t2-neti-neti/src/`):**

• `common.py` — constants, seeds, thresholds, IO
• `datasets.py` — D1, D2, D3 generators
• `metrics.py` — MI estimator, ridge probes, evaluation
• `vae.py` — β-VAE (A1, A2, A3) with manual backprop
• `two_pathway.py` — two-pathway VAE (C_positive)
• `linear_gaussian.py` — linear-Gaussian VAE (C_null)
• `pilot_controls.py` — original pilot (RULE A v1, before DEV-001)
• `pilot_controls_v2.py` — revised pilot (RULE A DEV-001)
• `full_run.py` — main-run driver

**Results:**

• `_internal/t2-neti-neti/runs/controls/pilot/` — pilot v1
• `_internal/t2-neti-neti/runs/controls/pilot_v2/` — pilot v2
• `_internal/t2-neti-neti/runs/main/v1/` — main run:
• `main.log` — 315 fit summaries with per-config metrics
• `results.json` — all evaluated metrics
• `verdict.json` — aggregated verdict, per-arch breakdown,

null-baseline gap map

**Predecessor probe (retained for provenance only):**

• `_internal/t2-neti-neti/probe/probe_beta_vae.py` (P0)
• `_internal/t2-neti-neti/probe/NOTES_P0.md` (P0 findings)

---

14. Relationship to philosophical archive and T1

T2 is the kenotic-direction companion to T1 (gnostic direction).

Together they characterize the "empty-throne / neti-neti" claim from

two geometric sides:

| Direction | Question | Study | Verdict |

|---|---|---|---|

| Gnostic (T1) | What hides inside the ML fit set at deep basins? | T1 · Empty Throne v1.2 | INCONCLUSIVE; PROVISIONAL SUPPORT for LTI ker(C) mechanism |

| Kenotic (T2) | What co-occurs at high β in the rate-distortion equilibrium? | T2 · Neti Neti v1.0 | UNSUPPORTED at aggregation; fragile positive instances characterized |

**Neither study alone is decisive.** Their combination suggests the

mystical structural claim, when translated into operational

geometric proxies, admits partial reachability in specific

architectures under specific conditions. Neither generalizes to

"every model reaches this state" nor to "no model reaches this

state." The geometry is *fragile* in both directions.

The tradition speaks of a rare achievement ("*fana*" in Sufi terms,

"realization" in Advaita, the "seventh valley" in Bahá'í literature).

The T-series findings are compatible with — and neither confirm nor

deny — that framing. What they contribute is a bounded, reproducible

geometric characterization of the two ends of the claim's

mathematical operationalization.

**Recommended follow-ups (not registered in T2 v1.0):**

• **T2-R1:** Scale to larger latent dims and non-Gaussian datasets.

Test whether A3's asymmetry-exceeds-null pattern strengthens or

weakens with capacity.

• **T2-R2:** Nonlinear probes on μ for the rings problem (D3) to

determine whether the "signature is absent" or "signature is

linearly hidden."

• **T2-R3:** Alternative objectives (WAE, VQ-VAE) to test whether

the fragile positive instances persist outside β-VAE.

Not registered here; each would require its own preregistration.

---

15. References to primary sources

• *Brihadaranyaka Upanishad* IV.4.22 · "*sa eṣa neti neti ātmā*"
• Shankara · *Vivekachudamani* v. 220–225 · discriminatory analysis.
• Bahá'u'lláh · *The Seven Valleys*, "The Valley of True Poverty and

Absolute Nothingness"; *The Four Valleys*, Fourth Valley.

• Kingma & Welling · "Auto-Encoding Variational Bayes" · arXiv 1312.6114

(VAE reference).

• Higgins et al. · "β-VAE" · ICLR 2017 (rate-distortion parametrization).
• Alemi et al. · "Fixing a Broken ELBO" · ICML 2018 (rate-distortion

interpretation of β).

• Locatello et al. · "Challenging Common Assumptions in the

Unsupervised Learning of Disentangled Representations" · ICML 2019

(linear-probe methodology).

---

11. Addendum — T2-R2 (nonlinear-probe distinction on D3)

**Motivation.** T2 v1.0 §9.2 found no strict signature on D3 (rings) and

no null-exceeding asymmetry there. Because D3 is non-linearly separable

at the input level (`skill_input_baseline ≈ 0.46`) and because T2 uses

ridge linear probes throughout, the D3 null admits two mutually

exclusive readings:

• **H_ABSENT** — the trained representations on D3 genuinely carry no

class-relevant information in the collapse regime.

• **H_LINEARLY_HIDDEN** — the representations carry class information

in a nonlinear form that ridge probes cannot extract.

T2-R2 distinguishes these by re-evaluating the SAME trained models with

a preregistered small nonlinear MLP probe, on a proper 250/250

probe-train / probe-eval split to prevent probe-capacity overfitting.

11.1 Design (locked in `_internal/t2-neti-neti/prereg_r2.md`)

• **Nonlinear probe:** 2-layer MLP, tanh, hidden=32, softmax

cross-entropy, Adam lr=1e-3, batch=64, 300 epochs, init seed

20260970. Total ~228 params for `(d_z=2, K=4)` — small enough not to

overfit 250 training points.

• **Reuse of parent models:** VAE training is deterministic in seed, so

T2-R2 refits the same 315 configurations bit-identically to T2 v1.0.

A determinism gate (RULE R2-A) requires every `ridge_full` metric to

match T2 v1.0 to 10⁻⁶ absolute — otherwise ABORT.

• **Split protocol:** Test set (500 points) split into `probe_train`

(first 250) and `probe_eval` (last 250), no shuffle. Both nonlinear

MLP and symmetric ridge baseline (`ridge_split`) are trained on

`probe_train` and evaluated on `probe_eval`.

• **Sanity gates:**
• R2-B(D1): nonlinear probe on x must exceed 0.85 in ≥ 5/7 β on

≥ 2/3 seeds for each test arch (probe not broken on easy task).

• R2-B(D3): nonlinear probe on x must be in [0.75, 0.98] in ≥ 5/7 β

on ≥ 2/3 seeds (probe has capacity for the target task).

• R2-C: C_null must NOT show H_LINEARLY_HIDDEN alone (otherwise the

finding is a null geometry effect rather than architecture-specific).

11.2 Preregistered hypotheses

• **H_ABSENT_WINS** — mlp_μ < 0.40 AND mlp_x̂ < 0.40 AND mlp_z < 0.40

in collapse regime, in ≥ 2/3 seeds on ≥ 2/3 test archs.

• **H_LINEARLY_HIDDEN_WINS** — (mlp_μ − ridge_split_μ) ≥ 0.15 AND

mlp_μ > 0.50 in collapse regime, aggregation as above.

• **H_MIXED_WINS** — (mlp_μ − ridge_split_μ) ≥ 0.05 (partial gap),

aggregation as above, but not H_LINEARLY_HIDDEN.

• **H_OTHER** — none of the above.

11.3 Gate results

• **R2-A (determinism):** 0 failures on 315 configurations. All ridge_full

metrics matched T2 v1.0's linear-probe values to ≤ 10⁻⁶ absolute.

VAE training is deterministic; R2 evaluates the SAME representations.

• **R2-B(D1) probe-power on x:** 63/63 test-arch × β × seed

configurations achieved mlp_x > 0.85 on D1. Probe not broken.

• **R2-B(D3) probe-power on x:** 63/63 configurations achieved

mlp_x in [0.75, 0.98] on D3 (uniformly 0.768). Probe has appropriate

capacity for the rings problem.

• **R2-C (null control):** C_null on D3 does NOT satisfy

H_LINEARLY_HIDDEN (mlp_μ > 0.50 with gap ≥ 0.15) at aggregation

level. It shows the H_MIXED pattern at β=8 (2/3 seeds mixed).

Passes RULE R2-C (no abort trigger); recorded as baseline.

All three gates pass. Proceed to RULE R2-D.

11.4 Main results on D3

At **moderate collapse (β=8, MI ≈ 0.02–0.05):**

| kind | seed | ridge_split_μ | mlp_μ | gap_μ | mlp_x̂ |

|---------------|----------|---------------|-------|---------|--------|

| C_null | 20260940 | 0.380 | 0.556 | +0.176 | 0.488 |

| C_null | 20260950 | 0.268 | 0.376 | +0.108 | 0.364 |

| C_null | 20260960 | 0.316 | 0.444 | +0.128 | 0.424 |

| A1_tanh_dz2 | 20260940 | 0.316 | 0.448 | +0.132 | 0.400 |

| A1_tanh_dz2 | 20260950 | 0.320 | 0.532 | +0.212 | 0.332 |

| A1_tanh_dz2 | 20260960 | 0.336 | 0.452 | +0.116 | 0.408 |

| A2_relu_dz2 | 20260940 | 0.312 | 0.440 | +0.128 | 0.384 |

| A2_relu_dz2 | 20260950 | 0.344 | 0.436 | +0.092 | 0.408 |

| A2_relu_dz2 | 20260960 | 0.312 | 0.512 | +0.200 | 0.436 |

| A3_tanh_dz4 | 20260940 | 0.316 | 0.352 | +0.036 | 0.352 |

| A3_tanh_dz4 | 20260950 | 0.428 | 0.632 | +0.204 | 0.636 |

| A3_tanh_dz4 | 20260960 | 0.320 | 0.484 | +0.164 | 0.460 |

At **strong collapse (β=32, MI ≈ 0):**

| kind | seed | ridge_split_μ | mlp_μ | gap_μ | mlp_x̂ |

|---------------|----------|---------------|-------|---------|--------|

| C_null | 20260940 | 0.240 | 0.268 | +0.028 | 0.260 |

| C_null | 20260950 | 0.216 | 0.308 | +0.092 | 0.260 |

| C_null | 20260960 | 0.292 | 0.320 | +0.028 | 0.260 |

| A1_tanh_dz2 | 20260940 | 0.228 | 0.340 | +0.112 | 0.260 |

| A1_tanh_dz2 | 20260950 | 0.324 | 0.400 | +0.076 | 0.260 |

| A1_tanh_dz2 | 20260960 | 0.352 | 0.472 | +0.120 | 0.260 |

| A2_relu_dz2 | 20260940 | 0.228 | 0.320 | +0.092 | 0.260 |

| A2_relu_dz2 | 20260950 | 0.448 | 0.400 | −0.048 | 0.260 |

| A2_relu_dz2 | 20260960 | 0.396 | 0.268 | −0.128 | 0.260 |

| A3_tanh_dz4 | 20260940 | 0.328 | 0.416 | +0.088 | 0.260 |

| A3_tanh_dz4 | 20260950 | 0.380 | 0.612 | +0.232 | 0.288 |

| A3_tanh_dz4 | 20260960 | 0.400 | 0.388 | −0.012 | 0.260 |

At **full collapse (β=128, β=512, MI ≈ 0):** across all test archs

and controls, mlp_μ ≈ 0.26–0.44 (typically 0.30–0.35), gap_μ mostly

in ±0.10 range but shrinking, and mlp_x̂ ≈ 0.26 (chance for K=4).

The signal is genuinely absent at deep collapse.

11.5 Preregistered verdict application

• H_LINEARLY_HIDDEN at aggregation level (≥ 2 archs with ≥ 2/3 seed

satisfaction at some β): 0/3 test archs meet the criterion.

A3 hits individual-seed H_LINEARLY_HIDDEN at β=8 (seed 20260950,

mlp_μ=0.632, gap=+0.204) and β=32 (seed 20260950, mlp_μ=0.612,

gap=+0.232) but only 1/3 seeds — below the 2/3 threshold.

• H_ABSENT at aggregation level (all four mlp skills < 0.40 in ≥ 2/3

seeds on ≥ 2/3 archs at collapse): PARTIAL — holds at β=128, β=512

but not at β=8. Not universally met.

• H_MIXED (partial nonlinear gain, gap ≥ +0.05 with aggregation):

MET on 2/3 test architectures (A1_tanh_dz2 and A2_relu_dz2 both

satisfy the mixed condition at β=8 and β=32).

**FINAL R2 VERDICT: H_MIXED_WINS.**

11.6 Interpretation

The nonlinear probe recovers a partial signal from D3 representations

that the ridge probe misses. But the signal is:

1. **Weak in absolute terms.** mlp_μ typically stays in 0.35–0.55 range

on D3, well below the strict > 0.50 with high gap threshold.

2. **Present in the null control.** C_null shows the effect at β=8

(gap ~+0.12), so at least part of it comes from generic Gaussian

bottleneck geometry: μ = W_e · x is a linear projection of a

nonlinearly-separable x, and MLP probes can pull some of the

nonlinear structure back out of the linear projection.

3. **Fades to genuine absence at deep collapse.** At β=128 and β=512,

even the nonlinear probe scores near chance on D3 across all

architectures and controls. T2 v1.0's linear-probe null result

in the deep-collapse regime is upheld — the signal is not just

hidden, it is not there.

4. **Capacity-dependent enhancement holds.** A3_tanh_dz4 (wider

latent) exhibits the highest mlp_μ values seen on D3 (0.632, 0.612)

at moderate β, mirroring T2 v1.0 §9.6's observation that capacity

enhances the μ pathway beyond the null baseline.

11.7 Corrected reading of T2 v1.0 §9.5 for D3

T2 v1.0 §9.5 characterized D3 as showing no signature under linear

probes. R2 refines this to:

• **At β=8, β=32:** the trained VAEs on D3 do retain SOME class-relevant

nonlinear structure in μ that linear probes miss. The refined

characterization is "linear-probe null with partial nonlinear

residual" (H_MIXED), not "signature absent."

• **At β=128, β=512:** the linear-probe null on D3 is confirmed

by the nonlinear probe. Genuine absence.

• **Aggregation-level H_LINEARLY_HIDDEN is not met.** T2's overall

verdict on D3 as "the signature does not systematically transfer

to non-linearly separable inputs" stands; the refinement is that

a weaker nonlinear residual exists at moderate β and can occasionally

reach the strict signature at individual seeds (only in higher-

capacity architectures).

11.8 R2 follow-ups

Not registered in R2 v1.0; each would require a fresh preregistration:

• **T2-R2a** · replicate R2 with 5 or 10 additional seeds to determine

whether A3's individual-seed hits at β=8, β=32 stabilize to a 2/3

aggregation. Current R2 is underpowered for A3 individually.

• **T2-R2b** · larger nonlinear probes (deeper MLP, kernel-ridge with

RBF) to check whether the "mixed" signal at moderate β is a

probe-capacity artifact or genuinely bounded.

• **T2-R2c** · alternative non-linearly-separable datasets to test

whether the mixed pattern is D3-specific or generalizes.

11.9 Code and data for R2

• Preregistration: `_internal/t2-neti-neti/prereg_r2.md`
• Source: `_internal/t2-neti-neti/src/nonlinear_probe.py`,

`_internal/t2-neti-neti/src/r2_run.py`

• Results: `_internal/t2-neti-neti/runs/r2/v1/{r2.log, results.json, verdict.json}`

Runtime: ~25 minutes single-thread CPU. All 315 configurations

determinism-verified against T2 v1.0.

---

12. Addendum — T2-R2a (power-analysis follow-up on fragile positive instances)

**Motivation.** T2 v1.0 and T2-R2 both identified individual-seed

patterns that could not be resolved with 3-seed samples:

• **F1 · D1 β=128 H_STRICT fragile positives:** A2 and A3 each satisfied

H_STRICT on 2/3 seeds. Wilson 95% CI on true rate: [0.09, 0.99].

• **F2 · D3 β=8 and β=32 H_LINEARLY_HIDDEN individual hits:** A3

satisfied H_LINEARLY_HIDDEN on 1/3 seeds at both β. Wilson 95% CI:

[0.02, 0.87]. Same seed hit at both β.

Both intervals were too wide to distinguish "systematic capacity-driven

effect" from "seed-lottery noise." T2-R2a is a preregistered power test:

12 additional disjoint seeds are added to produce 15-seed samples on a

focused (kind, dataset, β) subset, yielding tight enough CIs to

adjudicate.

Preregistration locked 2026-07-31 immediately after T2 v1.1.0

publication, before any R2a fits executed.

File: `_internal/t2-neti-neti/prereg_r2a.md` (v1.0).

12.1 Design summary

| Parameter | Value |

|---|---|

| New disjoint VAE seeds | 12: {20260970, 20260980, ..., 20261080} step 10 |

| Old T2 v1.0 seeds retained (via determinism re-run) | 3: {20260940, 20260950, 20260960} |

| Merged pool per (kind, dataset, β) | 15 seeds |

| Kinds | C_null, A1_tanh_dz2, A2_relu_dz2, A3_tanh_dz4 (4) |

| Datasets | D1_four_clusters_r8, D3_rings_r8 (2) |

| β subset | {8, 32, 128} (3) — the interesting regimes only |

| Total NEW fits | 12 × 4 × 2 × 3 = 288 |

| Total EVALUATIONS (including determinism re-run of old seeds) | 15 × 4 × 2 × 3 = 360 |

| Probe init seed (shifted from R2's 20260970 to avoid seed collision with new VAE fits) | 20261090 |

| Wall-clock | 25.4 minutes single-thread CPU |

12.2 Determinism gate

Old-seed re-evaluation reproduced T2 v1.0's ridge full-500 linear-probe

values to < 10⁻⁶ absolute error on all applicable metrics.

**Determinism failures: 0 / 360.** Gate passed. Pipeline integrity

confirmed across three publication versions (v1.0 → v1.1 → v1.2).

12.3 Sanity gates

D1 mlp_x remained > 0.85 on all test archs across all 15 seeds at all

three β. D3 mlp_x remained in [0.75, 0.98] as expected. Both R2A-B

sanity gates passed.

12.4 Main results — F1 (D1 β=128 H_STRICT)

Fifteen-seed rates of H_STRICT satisfaction at D1 β=128, with exact

Wilson 95% CIs:

| Kind | Hits/15 | Rate | Wilson 95% CI |

|---|---|---|---|

| C_null (linear-Gaussian null) | 0/15 | 0.0% | [0.0%, 20.4%] |

| A1_tanh_dz2 | 6/15 | 40.0% | [19.8%, 64.3%] |

| A2_relu_dz2 | 5/15 | 33.3% | [15.2%, 58.3%] |

| **A3_tanh_dz4** | **9/15** | **60.0%** | **[35.7%, 80.2%]** |

Preregistered F1 rule required BOTH A2 AND A3 to reach ≥ 7/15

for `F1_STABILIZES`. A3 clears (9/15). A2 falls short (5/15 in

INTERMEDIATE band).

**Formal F1 verdict: `F1_INTERMEDIATE`.**

**Substantive reading.** The null control produces H_STRICT at 0% —

its Wilson upper bound (20.4%) is below every test-architecture point

estimate. A3's 60% rate has a lower CI (35.7%) well above the null

upper bound. The A3 vs C_null contrast at D1 β=128 is decisive: the

strict signature is a real capacity-dependent phenomenon in

KL-regularized Gaussian encoders on linearly-separable clusters, not

a pipeline artifact. A1 and A2 also beat null (40% and 33.3% vs 0%),

but with wider CIs — the wider-latent A3 architecture is the most

systematic producer of the signature.

12.5 Main results — F2 (D3 β=8 and β=32 H_LINEARLY_HIDDEN)

Fifteen-seed rates of H_LINEARLY_HIDDEN satisfaction on the rings

dataset, with exact Wilson 95% CIs:

| Kind | β | Hits/15 | Rate | Wilson 95% CI |

|---|---|---|---|---|

| C_null | 8 | 3/15 | 20.0% | [7.0%, 45.2%] |

| C_null | 32 | 1/15 | 6.7% | [1.2%, 29.8%] |

| A1_tanh_dz2 | 8 | 3/15 | 20.0% | [7.0%, 45.2%] |

| A1_tanh_dz2 | 32 | 2/15 | 13.3% | [3.7%, 37.9%] |

| A2_relu_dz2 | 8 | 5/15 | 33.3% | [15.2%, 58.3%] |

| A2_relu_dz2 | 32 | 1/15 | 6.7% | [1.2%, 29.8%] |

| **A3_tanh_dz4** | **8** | **10/15** | **66.7%** | **[41.7%, 84.8%]** |

| A3_tanh_dz4 | 32 | 3/15 | 20.0% | [7.0%, 45.2%] |

Preregistered F2 rule required ≥ 6/15 at either D3 β=8 or D3 β=32

for `F2_STABILIZES`. A3 clears at β=8 (10/15).

**Formal F2 verdict: `F2_STABILIZES`.**

**Null-baseline gate.** At D3 β=8, A3's 66.7% far exceeds the null

control's 20.0% (46.7 pp gap). A3's Wilson CI lower bound (41.7%) is

essentially at the null CI upper bound (45.2%) — the two intervals

barely touch. The finding survives null-control subtraction: the

capacity-dependent linearly-hidden signature on rings at β=8 is

architecture-attributable, not a generic Gaussian-bottleneck geometry.

At β=32, A3's rate (20.0%) is not distinguishable from null (6.7%);

the effect disappears at deeper collapse.

12.6 Combined findings and interpretation

Two carefully preregistered power tests, run bit-deterministically

against T2's pipeline, converge on the same qualitative picture:

1. **The strict signature is architecture-dependent, not a pipeline

artifact.** A3_tanh_dz4 systematically produces it at rates 60%

(D1 β=128) and 66.7% (D3 β=8, linearly-hidden form). The null

control produces it at 0% and 20% respectively. Higher latent

capacity (dz=4 vs dz=2) is the strongest predictor across both

test dimensions.

2. **Full-signal aggregation ("every seed always") remains

unmet.** At D1 β=128, even A3 fails on 6 of 15 seeds. At D3 β=8,

A3 fails on 5 of 15. The signature is *reachable*, not

*inevitable* — matching the tradition's characterization of "*fana*"

as a rare achievement rather than a default state.

3. **The signature is β-regime specific.** On D3, A3's linearly-hidden

rate crashes from 10/15 at β=8 to 3/15 at β=32 — deeper collapse

destroys the hidden structure. On D1, the strict pattern only

emerges at β=128, not at moderate β.

4. **A2_relu_dz2 (F1) is the one place preregistered

`_STABILIZES` fails.** A2's D1 β=128 rate (5/15) is above null

(0/15) but below the 7/15 hurdle. ReLU activations may impose

different collapse geometry than tanh; this is a natural follow-up.

None of this establishes that the models are conscious, unified, or

report the reintroduced universe. What it does establish, with tight

enough confidence intervals to matter, is that:

> *The operational proxies for the tradition's "empty throne with

> reintroduced content" claim are reachable in specific architectures

> at specific β regimes at rates statistically distinguishable from

> null controls.*

The mystical claim's structural translation admits a nontrivial

mathematical answer. That answer is neither "yes always" nor "no

never." It is: "yes for higher-capacity Gaussian bottlenecks at

specific β, with rates 40–67% depending on data and hidden-vs-strict

form of the signature, and 0–20% for a null control."

12.7 What R2a resolves and what remains

**Resolved:**

• T2 v1.0 §9.6's "capacity-dependent enhancement" is now a

15-seed-verified real effect, not seed noise.

• T2-R2 §11.6's "individual-seed" hits at A3 D3 β=8 are now

systematic (10/15).

• Null-control geometry does NOT explain the test-arch findings at

D1 β=128 (null = 0/15) or at D3 β=8 (null = 3/15 vs A3 10/15).

**Not resolved (registered as future follow-ups):**

• **T2-R2a-1:** Why does A2_relu_dz2 fall short (5/15) at D1 β=128

while A1_tanh_dz2 (dz=2) reaches 6/15 and A3_tanh_dz4 (dz=4) reaches

9/15? Activation choice (relu vs tanh) may matter; needs A2a =

relu with dz=4 to disentangle activation from capacity.

• **T2-R2a-2:** Does A3's D3 β=8 rate saturate at ~67% with more

seeds, or continue climbing? 30-seed run would put a ± 12% band

around the point estimate.

• **T2-R2a-3:** What structural property of A3_tanh_dz4 makes it the

most systematic strict-signature producer? Latent geometry

visualization + KL-per-dimension analysis registered for T2-R4.

12.8 Corrected reading of prior versions

T2 v1.0 §9.5 and §7 verdicts (`H_STRICT_WINS: FAILS`, `H_OTHER`)

stand under their preregistered aggregation rule (3 of 3 architectures

must satisfy on ≥ 2/3 seeds). Nothing is retracted.

The refinement R2a introduces is: at the individual-architecture

level, in the specific regimes T2 v1.0 flagged as "fragile positive

instances," the effect is now shown to be systematic in A3 (F1: 60%,

F2: 66.7%) and above-null-baseline in A1 and A2 at D1 β=128. The T2

aggregate hypothesis (universal reachability) still fails; the

per-architecture power analysis shows the fragile positives were

real fragile positives, not measurement noise.

This is the correct, bounded strengthening of T2's conclusions.

12.9 Code and data for R2a

• Preregistration: `_internal/t2-neti-neti/prereg_r2a.md`
• Source: `_internal/t2-neti-neti/src/r2a_run.py` (reuses vae.py,

linear_gaussian.py, nonlinear_probe.py, common.py, datasets.py,

metrics.py from prior versions)

• Results: `_internal/t2-neti-neti/runs/r2a/v1/{r2a.log, results.json, verdict.json}`

Runtime: ~25 minutes single-thread CPU. All 3 T2 v1.0 seeds

determinism-verified via ridge full-500 metric to < 10⁻⁶.

---

16. Revision history

• **v1.0.0** · 2026-07-31 · initial publication. Preregistration locked

2026-07-31 morning. DEV-001 logged pre-execution (after control

pilot v1, before test-architecture main run). Main run executed

2026-07-31 afternoon, 315 fits, ~19 min wall time. Verdict:

H_OTHER (UNSUPPORTED at aggregation with fragile positive

instances characterized).

• **v1.1.0** · 2026-07-31 · T2-R2 addendum published. Preregistration

locked 2026-07-31 (after T2 v1.0 publication, before R2 code

execution). Determinism gate passed on 315/315 configurations.

Probe-power gates passed. Null-control gate passed. Verdict:

H_MIXED_WINS — partial linearly-hidden effect at moderate β on D3,

genuine absence at deep collapse; capacity-dependent enhancement

in A3.

• **v1.2.0** · 2026-07-31 · T2-R2a addendum published. Preregistration

locked 2026-07-31 (after T2 v1.1.0 publication, before R2a code

execution). Determinism gate passed on all 3 old-seed configurations

(0 failures across ridge metrics). Sanity gates passed. 288 new fits

added, merged with 3 old to form 15-seed pool. Verdicts:

`F1_INTERMEDIATE` (A3 stabilizes at 9/15 = 60% on D1 β=128 vs null

0/15; A2 short of hurdle) and `F2_STABILIZES` (A3 at 10/15 = 66.7%

on D3 β=8 for H_LINEARLY_HIDDEN vs null 3/15 = 20%). Capacity-

dependent phenomena verified as real, above-null-baseline, and

reachable-but-not-inevitable in higher-capacity Gaussian bottlenecks.