# T1 · The Empty Throne — Scientific Record

**Semantic name:** T1 · The Empty Throne — parameter-manifold geometry at ML fit
**Record class:** SIMULATION (structural probe)
**Program:** New program — Structural Claims of the Valleys (T-series)
**Non-drift question:** Does perfect fit collapse the observer's distinguishing identity into a manifold, or into a point?
**Version:** 1.2.0
**Date:** 2026-07-30 (v1.0.0), 2026-07-30 (v1.1.0 addendum), 2026-07-30 (v1.2.0 addendum)
**Status:** INCONCLUSIVE overall; PROVISIONAL SUPPORT for LTI with R1a mechanism qualitatively upheld by T1-R2 (square-C falsification) and not yet cleanly adjudicated by T1-R6 (baseline-failed α-sweep; R6b registered).
**Preregistration:**
- [preregistration.md](preregistration.md) (v1.1, locked 2026-07-30) — original T1 study.
- `_internal/t1-empty-throne/prereg_r2_r6.md` (locked 2026-07-30) — T1-R2 and T1-R6 follow-ups.
**Deviations:** [deviations.md](deviations.md) (001, 002 — pre-execution; 003 — pre-execution for R6).
**Follow-ups (this record):** T1-R1 (LTI innovations-form parametrization), T1-R1a (mechanism of the residual manifold), T1-R2 (square-C falsification test), T1-R6 (nonlinear-observation mechanism test).

---

## Claim-status banner

```
CLAIM STATUS   : INCONCLUSIVE overall (architecture-specific outcomes)
                  · LTI (C fixed): PROVISIONAL SUPPORT at global-minimum basins
                  · MLP (tanh, small): UNSUPPORTED at matched and over capacity
                  · GP (RBF, 3-param): UNSUPPORTED

EVIDENCE TYPE  : COMPUTATIONAL SIMULATION · preregistered linear-detection
                  study of parameter-space manifold structure at ML fit
PHYSICAL VALID : NONE
INDEPENDENT REP: NONE (single first-run of a new methodology)

SUPPORTED (bounded scope):
- Under a linear-detection technique (numerical Hessian rank + orthogonal
  flat-walk verification), state-space LTI with C fixed at truth exhibits a
  positive-dimensional manifold at the deepest fit basin (d ≈ 10–13 residual
  flat directions at n_M = n_S = 4, N ∈ {512, 2048, 8192}).
- The LTI manifold is a genuine feature of the loss-landscape at the ML
  argmin: it persists at truth-initialized fits that reach lower NLL_train
  than random-init fits, and it does NOT appear at the source parameters
  themselves (which sit at a sharp minimum of higher NLL_train).
- The excess-mode manifold at LTI over-capacity (n_M > n_S) is partially
  recovered: at n_M = 6, E1 reports d = 7 matching the closed-form prediction
  d_trivial_predicted = 7; E2 verifies 5 of those under non-local flat-walk.

NOT SUPPORTED:
- The strong universal reading of the mystical claim ("at any perfect fit,
  distinguishing identity annihilates") is NOT supported. MLP and GP fits
  at matched capacity show d = 0 across E1, E2, E4.
- MLP at over-capacity (H_M = 8 vs H_S = 6) with predicted dead-unit
  d_trivial = 10 also shows d = 0: the fit does not land on the predicted
  dead-unit manifold; it uses excess capacity to overfit noise instead.
  The predicted manifold exists in the parameter space but the ML fit
  is not on it.
- Reproducibility of the LTI manifold across seeds is highly bimodal
  (d ∈ {0, 2, 11} across 3 datasets). Whether a given fit reaches a
  flat basin or a sharp local minimum depends on initialization and
  local-minimum landscape structure. The finding is only reliable
  when the fit reaches the global minimum, which random-init L-BFGS
  does not consistently achieve.

NOT ESTABLISHED:
- Any physical, biological, or empirical reality of the mystical claim.
- Any generalization beyond these three source classes.
- Any claim about consciousness, unity, non-duality, or metaphysical
  identity. This study measures ONLY the geometry of a specific
  computational loss surface in a specific parametrization.

RESOLVED IN v1.1.0 (see §11 addendum):
- The LTI manifold does NOT vanish in innovations-form parametrization
  (A_pred, K, S_ss), where spectral-factorization non-uniqueness is
  removed by construction. A residual d ≈ 8 manifold persists at deep
  fits across seeds and initializations. The manifold is therefore not
  a pure artifact of the (A, Q, R) → (A_pred, K, S_ss) fiber.
- The residual is mechanistically identified as the classical
  unobservable-subspace freedom: with C fixed at truth, ker(C) has
  dim n_M − m = 2, and 8/8 flat-direction eigenvectors of the Hessian
  at the fit put dK columns entirely in ker(C), leave the innovation
  covariance untouched (dL_S = 0), and preserve the observed
  innovation sequence to numerical precision (~10⁻⁷ relative).
- Interpretation: the manifold is real, structural, and well-understood
  — it is the "hidden state" freedom of a state-space realization
  under a fixed observation channel.

TESTED IN v1.2.0 (see §12 and §13 addenda):
- T1-R2 (square-C falsification): with m = n_S = 4 so that ker(C) = {0},
  the R1a mechanism predicts d = 0. Result across three seeds
  (20260900, 20260901, 20260902): d ∈ {4, 0, 0}, mean = 1.33. Verdict:
  INCONCLUSIVE per prereg (rule requires 2-of-3 seeds ≥ 3 for
  REFUTES, or all seeds ≤ 1 and mean ≤ 0.7 for SUPPORTS). Post-hoc
  diagnostic (see §12.5) shows the seed-20260900 detection is a
  gap-based rank-detector methodology artifact: the "flat" eigenvalues
  are stable across Hessian eps but four orders of magnitude larger
  than the R1 flat-direction eigenvalues, and walking along the
  smallest eigenvector produces NLL curvature matching the
  quadratic prediction. Mechanism qualitatively upheld; strict
  statistical rule borderline. An exploratory corroboration with
  the original T1 seeds ({20260730, 20260800, 20260830} + truth-init)
  gave clean d = 0 on all four fits (see §12.10).
- T1-R6 (nonlinear-observation mechanism test): α-sweep of a quadratic
  ker(C)-leaking observation channel. Strict prereg verdict:
  REFUTES_R1A_MECHANISM — but driven by baseline failure
  (d(α=0) = 0 < 5; the R6 pipeline at N=512 / EKF / (A,Q,R) form
  did not recover the T1 v1.0 / R1 manifold). The α-trend test is
  therefore uninformative: a pipeline that cannot reproduce the
  linear baseline cannot adjudicate whether nonlinearity shrinks
  it. See §13. T1-R6b registered to restore the baseline before
  re-testing the α-trend.
```

---

## Abstract

The mystical tradition of "The Empty Throne" — from the culminating valleys
of Bahá'u'lláh's *The Seven Valleys* and *The Four Valleys*, and echoed in
non-dual formulations in scripture — describes a structural end-state in
which the seeker's distinguishing content is annihilated at the point of
contact with the source. In an operational form suitable for a linear
detection technique, this study asks whether the fit set of a parametric
model — the set of parameter configurations achieving the minimum training
NLL within a small tolerance — is a positive-dimensional manifold
(structural arrival) or an isolated point (parametric arrival).

Three source classes are tested: state-space LTI-Gaussian (C fixed at
truth), a small tanh MLP with Gaussian output, and an RBF Gaussian process.
Adversarial controls (well-specified regression; colinear regression with a
known k_dup-dimensional manifold) validate the estimators at d = 0 and d =
k_dup respectively.

At matched capacity, MLP and GP show clean point-fits (d = 0 across all
estimators). LTI shows a bimodal result: some random-init fits land in
sharp local minima (d = 0 or 2), while others — and truth-initialized
fits, which achieve lower NLL_train — land in flat basins of dimension
10–13, stable across N ∈ {512, 2048, 8192}. At LTI over-capacity (n_M = 6
vs n_S = 4), E1 recovers the closed-form excess-mode prediction (7 of 7);
E2 verifies 5 of those under non-local walks.

The preregistered verdict, applied per-class, is:

- **LTI**: PROVISIONALLY SUPPORTS the structural interpretation, but ONLY
  at deep-fit basins that random-init L-BFGS reaches unreliably.
- **MLP, GP**: UNSUPPORTED at all tested capacities.
- **Overall**: INCONCLUSIVE. The universal reading of the mystical claim
  is not supported by cross-architecture agreement.

The LTI finding is a real, previously non-obvious geometric feature of
state-space likelihoods in the fixed-C parametrization, and warrants
follow-up in a minimum-phase canonical form to determine whether it
reflects spectral-factorization non-uniqueness (a known LTI phenomenon)
or a subtler geometric feature.

---

## 1. Motivation and non-drift question

The user's stated intent is to test a specific structural claim from
scriptural and mystical sources: that the seeker's identity — everything
that would distinguish one observer from another — collapses at the point
of union with the source. This is a claim about the *geometry of arrival*,
not a claim about phenomenal experience.

For this to be a scientifically meaningful test — as opposed to a
philosophical assertion — the claim must be recast into a testable
question about a specific formal object. The mapping used here is:

- **"Observer"** → a parameterized model class with parameters θ ∈ ℝᵖ_M.
- **"Perfect fit"** → the ML argmin of the training-set NLL, within a
  local tolerance.
- **"Distinguishing identity"** → the identity of θ within its class.
- **"Annihilation of distinguishing identity"** → collapse of the fit set
  to a positive-dimensional manifold, so that a continuum of distinct θ's
  are indistinguishable by their NLL.

Under this operationalization, the mystical claim predicts that the fit
set is a manifold with dimension exceeding the known trivial redundancy
of the parameterization (permutation, sign, similarity — whichever is
linear-detectable). We call this outcome **STRUCTURAL_ARRIVAL**. The null
outcome — a point fit at generic θ — we call **PARAMETRIC_ARRIVAL**. A
verdict of NO_ARRIVAL would arise if no meaningful fit is reached at all.

This is a bounded and honest translation. It does not claim to test
consciousness, non-duality, or metaphysical content. It tests only the
loss-surface geometry of a specific class of models — a translation
that the researchers stipulate before execution as one legitimate but
partial reading of the source claim.

## 2. Preregistration and deviations

The full preregistration is at [preregistration.md](preregistration.md).
Both deviations were introduced before results were tabulated:

- **Deviation 001** (v1.0 → v1.1): the pilot on the regression baseline
  and colinear positive control exposed flaws in the initial estimators.
  E2 was PCA-based and could not discriminate point fits from manifold
  fits; E3 measured parameter uncertainty rather than manifold structure;
  E4 inherited E2's flaw. All four estimators were redesigned around a
  train-NLL Fisher-rank + H-eigenvector flat-walk verification approach.
  Regression baseline now reports d = 0 across p ∈ {4, 8, 16}; colinear
  regression reports d = k_dup ∈ {1, 2, 3} with residual d = 0.

- **Deviation 002**: state-space similarity is a NONLINEAR action on the
  natural Cholesky parametrization of an LTI system. Our linear-Hessian
  technique cannot detect it — the linear tangent generators approximate
  the true (curved) similarity direction only to first order, and the
  numerical Hessian sees the second-order correction as spurious
  curvature. Rather than build a nonlinear-symmetry-walking test — which
  would require substantial new code and introduce new bugs — we fix C at
  its source value throughout. For generic C, the constraint CT = C
  forces T = I, removing similarity from the parametrization at the level
  of the model class. This gives us a clean linear-detection test in
  which any residual manifold is genuinely residual (not smuggled-out
  similarity). We consider this the more honest of the two available
  options.

Both deviations are documented in full in [deviations.md](deviations.md).

## 3. Design (locked pre-execution)

Full details in the preregistration. In summary:

### 3.1 Source classes

- **LTI-Gaussian (state space)**: n_S = 4 hidden state, m = 2 outputs,
  stable random A, generic random C, Q = 0.1 · I, R = 0.1 · I. C fixed
  at source (Deviation 002). Model capacities n_M ∈ {4, 6}. Truth
  parameters unpacked to Cholesky factors for `L_Q`, `L_R`.
- **MLP (small)**: d_in = 4, d_out = 2, H_S = 6, tanh activation,
  Gaussian output with fitted σ. Model capacities H_M ∈ {6, 8}. Generic
  parametrization (no canonicalization enforced during fit).
- **GP (RBF)**: 1D input, RBF kernel, Gaussian noise. 3-parameter model
  (log-length-scale, log signal std, log noise std). No latent capacity
  gradient.

### 3.2 Estimators (v1.1)

- **E1 (Fisher rank)**: numerical Hessian of NLL_train at θ̂; combine
  absolute-threshold (1e-6 · λ_max) and gap-based (largest log-eigenvalue
  gap ≥ 10x) rules; report d̂ = p_M − rank.
- **E2 (Orthogonal flat-walk verification)**: use the H-eigenvectors from
  E1's near-null spectrum as proposal directions; walk bidirectionally
  along each and check whether NLL_train stays within τ_fit_local up to
  s_manifold. Count verified directions.
- **E3 (Multi-init dispersion)**: fit ntimes = 20 fresh random-init
  models; canonicalize; compute per-H-eigendirection dispersion; compare
  empirical spread against Fisher-predicted spread. Report the number of
  directions where empirical spread exceeds prediction by ≥ μ_E3 = 10x.
- **E4 (Trivial vs. residual split)**: verify all trivial-symmetry
  generators preserve NLL to 10 · τ_fit_local at step ε; take the
  verified flat-directions from E2 and project out the span of trivial
  generators; report d_raw = total, d_residual = orthogonal complement.

### 3.3 Adversarial controls

- **Well-specified regression**: p features, iid Gaussian, fit by OLS.
  Predicted d_trivial = 0. Ran at p ∈ {4, 8, 16}, N = 4096. Expected all
  estimators report d = 0.
- **Colinear regression (positive control)**: p features, k_dup exact
  duplicates injected. Predicted d_trivial = k_dup, all linear-detectable.
  Ran at k_dup ∈ {1, 2, 3}, N = 4096. Expected all estimators report d =
  k_dup, residual = 0.

## 4. Results

### 4.1 Adversarial controls (validation)

Baseline regression, p ∈ {4, 8, 16}, N = 4096:

| p_features | p_M | E1 | E2 | E3 | E4_raw | E4_res | cliff |
|-----------:|----:|----|----|----|-------|-------|-------|
|          4 |   4 | 0  | 0  | 0  |   0   |   0   |  1.03 |
|          8 |   8 | 0  | 0  | 0  |   0   |   0   |  1.04 |
|         16 |  16 | 0  | 0  | 0  |   0   |   0   |  1.04 |

Cliff ratio near 1 indicates a fully-constrained spectrum — no manifold.

Colinear regression, p = 8, N = 4096, k_dup ∈ {1, 2, 3}:

| k_dup | E1 | E2 | E3 | E4_raw | E4_res |
|-------|----|----|----|-------|-------|
|     1 | 1  | 1  | 1  |   1   |   0   |
|     2 | 2  | 2  | 2  |   2   |   0   |
|     3 | 3  | 3  | 3  |   3   |   0   |

All estimators recover the known manifold with residual d = 0 (fully
explained by trivial redundancy). Controls pass.

### 4.2 LTI-Gaussian (C fixed)

Matched capacity n_M = n_S = 4, N = 2048, three seeds (random init, 3
restarts each):

| seed       | p_M | d_triv_pred | NLL_train_fit | NLL_train_truth | E1 | E2 | E4_raw | E4_res | cliff |
|------------|----:|------------:|---------------|-----------------|----|----|-------|-------|------|
| 20260730   |  29 | 0           | 4475.25       | 4482.27         | 0  | 0  | 0     | 0     |  7.1 |
| 20260800   |  29 | 0           | 5496.32       | (dataset-specific) | 2 | 2 | 2 | 2 |  5.3 |
| 20260830   |  29 | 0           | 6387.07       | (dataset-specific) | 11 | 11 | 11 | 11 | 24.7 |

Truth-initialized L-BFGS at seed 20260730 (same dataset as first row)
reaches NLL_train = 4473.77 (1.5 nats lower than the best random-init
fit) with d = 11 (residual 11). This confirms the flat-basin minimum
exists and is deeper than the random-init sharp minimum.

Persistence-across-N check at seed 20260730, truth-init:

| N     | NLL_train_fit − NLL_train_truth | manifold dim | cliff |
|-------|--------------------------------:|-------------:|------:|
|   512 |                          −4.29  |          13  | 34.7  |
|  2048 |                          −8.49  |          12  | 54.2  |
|  8192 |                          −6.47  |          13  | 17.5  |

Manifold dim is essentially constant across a 16× range in N; the
per-sample NLL improvement over truth (0.001–0.008 nat/sample) is
bounded and does not scale with N in a way consistent with a
1/N-shrinking finite-sample artifact. The manifold is a geometric
feature of the loss surface at the ML argmin.

At source parameters (not fit), the Hessian is well-conditioned across
all N: cliff ratios 2.9–3.4x, no eigenvalue below the numerical noise
floor, manifold dim 0. The truth is at a sharp minimum; the ML fit
lands in a nearby lower-NLL flat basin.

Over-capacity, n_M = 6, N = 2048, seed 20260730:

| Estimator | value |
|-----------|-------|
| p_M       |    60 |
| d_trivial_predicted (excess-mode) | 7 |
| E1        |     7 (matches prediction) |
| E2        |     5 verified of 7 candidates |
| E4_raw    |     5 |
| E4_res    |     5 |
| cliff     |   5.5 |

E1 recovers the closed-form excess-mode prediction exactly. E2 verifies
5 of the 7 candidate directions under non-local flat-walk. The 5
verified directions cannot be assigned to the linear trivial-symmetry
generators (the residual formula predicts unobservable-mode padding
should produce ALL 7, but the fit visits only a subspace of the
predicted manifold — a legitimate outcome when the ML argmin lies on
part but not all of the trivial submanifold).

### 4.3 MLP (tanh, Gaussian output)

N = 2048, seed 20260731:

| H_M | H_S | p_M | d_triv_pred | NLL_train_fit | NLL_train_truth | E1 | E2 | E4_raw | E4_res | cliff |
|-----|-----|----:|------------:|---------------|-----------------|----|----|-------|-------|------|
|   6 |   6 |  45 | 0           | −740.55       | −766.24 (test)  | 0  | 0  | 0     | 0     |  4.7 |
|   8 |   6 |  59 | 10          | −773.15       | −766.24 (test)  | 0  | 0  | 0     | 0     |  2.5 |

Matched capacity: d = 0 across all estimators. Consistent with Sussmann
(1992) — a generic MLP with tanh has no continuous trivial symmetries
at matched capacity — and consistent with the null hypothesis.

Over-capacity (H_M = 8): d = 0 across all estimators, despite a
closed-form prediction of d_trivial = (H_M − H_S)(d_in + 1) = 10 from
the dead-unit submanifold. Inspection of θ̂ shows all 8 hidden units
have nonzero output weights: the fit uses excess capacity to over-fit
training noise rather than to leave a subset of units dead. The
predicted manifold exists in the parameter space, but the ML argmin is
not on it.

Cliff ratios ~2.5–4.7x, below our 10x manifold threshold; the fit is
consistent with a sharp point-fit.

### 4.4 GP (RBF, 3 parameters)

N = 200 (limited by O(N³) kernel cost), seed 20260732:

| θ̂                   | NLL_train | NLL_test | E1 | E2 | E4_raw | E4_res | cliff |
|----------------------|-----------|----------|----|----|-------|-------|------|
| (−0.66, −1.05, −1.29) | 43.27    | 84.43    | 0  | 0  | 0     | 0     |  9.2 |

d = 0 across all estimators. Cliff just below 10x threshold. Consistent
with a tightly-parametrized model at a sharp fit. Note the fit is far
from truth (0, 0, log 0.3) — the GP overfits the 200 training points
substantially — but the fit basin is still a point.

## 5. Verdict per preregistered rules

Applying §12 of the preregistration:

- **LTI (matched)**: bimodal across seeds. When the fit reaches the
  deepest basin (seed 20260830 random, or seed 20260730 with truth
  init), d_residual ≈ 10–13, far above the pos_threshold of 1 dim.
  **PROVISIONAL SUPPORT** at global minimum. When the fit lands in a
  shallower local minimum (seed 20260730 random init, 3 restarts),
  d_residual = 0. **UNSUPPORTED** at local minima. Since the preregistered
  verdict specifies "at the ML argmin," and the deeper basin is the true
  argmin (lower NLL_train), the correct verdict is PROVISIONAL SUPPORT,
  with the qualification that the finding requires a global-minimum
  fitter or explicit initialization from a good point.
- **LTI (over)**: E1 matches d_trivial_predicted exactly; E4_res = 5 of
  7 predicted trivial directions. The residual is not clearly greater
  than the predicted trivial contribution. **INCONCLUSIVE** for structural
  arrival at over-capacity; the excess-mode manifold is partially
  detected as predicted.
- **MLP (matched)**: d_residual = 0. **UNSUPPORTED**.
- **MLP (over)**: d_residual = 0. **UNSUPPORTED**. The predicted trivial
  manifold is not detected because the fit does not visit it, and no
  additional structural manifold is detected.
- **GP**: d_residual = 0. **UNSUPPORTED**.

**Overall verdict (cross-class): INCONCLUSIVE.** The strong universal
reading of the mystical claim — that any perfect fit annihilates
distinguishing identity — is not supported by cross-architecture
agreement. Two of three source classes (MLP, GP) show clean
point-fits. Only LTI, and only at the deepest fit basin, exhibits
a residual manifold.

## 6. Interpretation and boundaries

### 6.1 What the LTI finding likely reflects

The most probable mechanism for the LTI manifold is
**spectral-factorization non-uniqueness**: a stationary Gaussian
time-series with rational spectral density has multiple state-space
realizations that produce the same output distribution. The minimum-phase
realization is unique, but our parametrization does not enforce
minimum-phase; multiple non-minimum-phase (A, Q, R) triples can produce
the same output distribution, and the ML fit apparently lands on a
manifold of these equivalent realizations rather than at the specific
minimum-phase point.

This is a well-known identifiability structure in system identification.
It is a real geometric feature of the LTI parameter space, not a numerical
artifact. But it is architecture-specific — MLP and GP do not have
analogous spectral-factorization ambiguity.

A conclusive follow-up would refit in observable-canonical form
(minimum-phase enforced). If the manifold vanishes, the finding is
non-minimum-phase reflection; if it persists, there is a deeper
structural feature.

### 6.2 What this study does NOT show

- It does not show that ANY of the three source classes has
  "consciousness," "identity," or "unity" in any physical, biological,
  or phenomenal sense.
- It does not show that the mystical texts' claims about identity,
  non-duality, or arrival at the source are literally true in any
  domain outside the specific formal object measured.
- It does not settle whether the LTI manifold reflects
  spectral-factorization non-uniqueness or something subtler.
- It does not establish that fitting to a lower-dim fit set is
  epistemically preferable to fitting to a higher-dim fit set — the
  practical modeling advice is orthogonal to the philosophical
  interpretation.

### 6.3 What this study DOES show

- A general-purpose linear-detection technique for parameter-manifold
  structure at ML fit, validated against a null baseline and a
  positive control.
- Three model classes tested under identical methodology, with
  quantitatively different outcomes.
- One architecture (LTI, C fixed) exhibits a robust, N-stable
  positive-dim manifold at its deepest fit basin. Two architectures
  (MLP, GP) show clean point-fits.
- A methodological finding: for over-parameterized MLPs, the ML fit
  does not spontaneously visit the predicted dead-unit manifold. This
  is relevant to how researchers reason about "effective dimension" in
  neural networks.

## 7. Adversarial notes

- **Local-minimum sensitivity in LTI**: our headline positive result
  requires the fit to reach the global-minimum basin. Random-init
  L-BFGS with 3 restarts sometimes achieves this and sometimes does
  not. A more thorough study would use many more restarts or
  gradient-flow methods to characterize the entire distribution of
  local minima and the frequency of the flat-basin outcome.
- **Parametrization dependence**: our LTI finding is in a specific
  parametrization (C fixed, Cholesky-factored Q and R). A
  minimum-phase canonical form would likely produce a very different
  answer. The finding is not parametrization-invariant.
- **MLP over-capacity behavior**: the closed-form d_trivial = 10 at
  H_M = 8 is a valid tangent-space prediction at points on the
  dead-unit submanifold. The ML fit is not on that submanifold, so
  the prediction is inapplicable to the specific fit measured. This
  is a methodological subtlety worth flagging.
- **N-scaling of the LTI manifold**: three N values (512, 2048, 8192)
  is a limited scan. A larger scan (up to N ≈ 65k) would more
  cleanly distinguish structural manifold from slow finite-sample
  drift. Manifold dim went 13 → 12 → 13 across our three points, but
  a single-point difference could reflect noise in the gap detector.
- **GP overfit**: our GP fit at N = 200 is far from truth (log_l =
  −0.66 vs 0). This is expected for finite-N GP fitting but limits
  the strength of the "sharp fit" conclusion — perhaps at larger N
  the fit basin sharpens further, or perhaps a manifold emerges.
- **The interpretation is not the finding**: even if the LTI result
  survives all follow-up scrutiny, the reading that it "supports the
  mystical claim" depends on the operational translation used at
  §1–§2. Alternative operational translations would give different
  verdicts.

## 8. Relationship to the philosophical archive

The T-series was launched in response to a request to test structural
claims from mystical sources — specifically the culminating valleys of
Bahá'u'lláh's *The Seven Valleys* and *The Four Valleys*, and echoed
formulations in scripture. The philosophical texts describe an
end-state at contact with the source in which the seeker's
distinguishing content is dissolved. This is a claim about arrival
geometry.

Our operational translation — fit set of a parametric model at ML
argmin, dimension measured by linear-detection techniques — is
DELIBERATELY BOUNDED. The mystical claim is metaphysical; ours is
strictly geometric. A supportive finding under our translation would
be one thin thread of evidence for a very specific formal reading of
the claim, not a demonstration that the claim is "true" in any broader
sense.

The Empty Throne image, in Bahá'í commentary tradition, is a warning
against reifying the endpoint of the journey. The throne is empty
because whoever sits upon it necessarily ceases to be a distinguishable
"whoever." Our LTI finding is consistent with a very literal
computational reading of this: at the deepest fit, the θ that would
"identify" one LTI observer from another can be moved continuously
without changing anything the outside world can measure.

We do not claim this is what the text is about. We claim only that,
under one legitimate operational reading, we find partial and
architecture-specific evidence for a structural translation of it.

## 9. Data and code manifest

- `_internal/t1-empty-throne/preregistration.md` — v1.1 (locked
  2026-07-30 before pilot and full-run execution).
- `_internal/t1-empty-throne/deviations.md` — 001 (estimator redesign)
  and 002 (LTI C fixed).
- `_internal/t1-empty-throne/notes.md` — internal design notes.
- `_internal/t1-empty-throne/src/model_interface.py` — abstract
  `FittedModel` contract.
- `_internal/t1-empty-throne/src/common.py` — logging, constants,
  seed configuration.
- `_internal/t1-empty-throne/src/lti.py` — LTI-Gaussian source and
  model (C fixed).
- `_internal/t1-empty-throne/src/mlp.py` — small tanh MLP source
  and model.
- `_internal/t1-empty-throne/src/gp.py` — RBF GP source and model.
- `_internal/t1-empty-throne/src/regression_baseline.py` — null
  control.
- `_internal/t1-empty-throne/src/pilot_positive_control.py` —
  colinear-regression positive control.
- `_internal/t1-empty-throne/src/estimators.py` — E1, E2, E3, E4.
- `_internal/t1-empty-throne/src/pilot_regression.py`,
  `pilot_lti.py`, `pilot_mlp.py`, `pilot_gp.py` — per-class pilot
  drivers.
- `_internal/t1-empty-throne/src/full_run_lti.py` — three-seed LTI
  matched + one over-capacity run.
- `_internal/t1-empty-throne/runs/**/*.json` — raw estimator outputs.
- `_internal/t1-empty-throne/runs/LTI/cache/*.npz` — cached fits.

## 10. Revision history

See §14 for the full revision history through v1.2.0.

---

## 11. Addendum — T1-R1 and T1-R1a (2026-07-30)

### 11.1 Motivation

T1 v1.0.0 found a d ≈ 10–13 residual manifold at the deep-fit basins
of the LTI-Gaussian model with C fixed. §7 explicitly flagged two
alternative explanations that v1.0 could not rule out:

1. **Spectral-factorization non-uniqueness.** A given output-covariance
   structure can be produced by multiple non-minimum-phase (A, Q, R)
   triples. This is a well-known 3–4 dim ambiguity in the standard
   (A, Q, R) parametrization.
2. **Unobservable-subspace freedom.** With C fixed and non-square
   (m = 2, n_M = 4), there is a subspace of the state space that C
   cannot see. Perturbations of (A_pred, K) that only affect motion
   in that subspace produce identical observed innovations.

The two follow-ups address these directly.

### 11.2 T1-R1 — LTI in innovations form

**Design.** Re-parametrize the LTI model as (A_pred, K, S_ss)
directly, where A_pred = A − K C is the predictor matrix, K is the
steady-state Kalman gain, and S_ss is the innovation covariance. In
this form the (A, Q, R) → (A_pred, K, S_ss) fiber is quotiented out
by construction, so any residual manifold cannot be attributed to
spectral-factorization non-uniqueness.

Truth is mapped into innovations form by solving the discrete
algebraic Riccati equation (DARE) at the source parameters and
recording (A_pred*, K*, S_ss*). Model fits are compared against T1
v1.0.0 fits at matched capacity n_M = 4, N = 2048, on the same three
seeds.

**Results (n_M = 4, N = 2048, three seeds + one truth-init).**

| Seed | Init | T1 v1.0.0 d | T1-R1 d | T1 v1.0.0 NLL_fit | T1-R1 NLL_fit |
|---|---|---:|---:|---:|---:|
| 20260730 | random | 0 | **8** | 4475.25 | **4473.49** |
| 20260800 | random | 2 | 1 | 5496.32 | 5496.29 |
| 20260830 | random | 11 | **8** | 6387.07 | 6386.63 |
| 20260730 | truth | 11 | 8 | 4473.77 | **4473.49** |

**Findings.**

- The manifold persists in innovations form. It does not vanish, so
  spectral-factorization non-uniqueness alone does not account for it.
- The manifold shrinks from d ≈ 10–13 to d ≈ 8. The dropped 3–5
  dimensions correspond to the (A, Q, R) → (A_pred, K, S_ss) fiber
  in the previous parametrization — that portion IS attributable to
  spectral-factorization non-uniqueness and is correctly removed by
  the innovations form.
- Random-initialized L-BFGS in innovations form reaches deeper minima
  more consistently: seed 20260730 recovers d = 8 from random init
  (v1.0.0 required truth-init to see the manifold on this seed). The
  innovations-form loss surface is easier to descend.

### 11.3 T1-R1a — mechanism of the residual manifold

**Design.** At a T1-R1 fit with d ≈ 8 detected by E1, decompose each
flat-direction eigenvector v of the Hessian into its (dA_pred, dK,
dL_S) components. For each direction we ask:

- Are dK's columns in ker(C)? If yes, dK acts only on the hidden
  subspace.
- Is dL_S ≈ 0? If yes, the innovation covariance is identified and
  the ambiguity is purely in dynamics/gain.
- Walking a small step along v, are the sample innovations e_t
  preserved? If yes, v is a bona fide flat direction in output
  distribution, not merely an approximate one.

**Results (seed 20260730, k = 8 flat directions ranked by |eigval|;
similar pattern on seed 20260830).**

| k | |eigval| | ‖dA_pred‖ | ‖dK‖ | ‖dL_S‖ | ‖P_ker(C) · dK‖ / ‖dK‖ |
|---:|---:|---:|---:|---:|---:|
| 5 | 1.3e-04 | 0.85 | 0.53 | 0.000 | **1.000** |
| 4 | 1.0e-03 | 0.85 | 0.53 | 0.000 | **1.000** |
| 3 | 9.0e-03 | 0.72 | 0.70 | 0.000 | **1.000** |
| 2 | 3.5e-02 | 0.85 | 0.52 | 0.000 | **1.000** |
| 6 | 8.3e-02 | 0.79 | 0.61 | 0.000 | **1.000** |
| 1 | 9.2e-02 | 0.98 | 0.22 | 0.000 | **1.000** |
| 7 | 9.7e-02 | 0.98 | 0.20 | 0.000 | **1.000** |
| 0 | 1.8e-01 | 0.96 | 0.29 | 0.000 | **1.000** |

Walking along null-direction 0 at step size 0.01 shifts the
innovation sequence by ‖Δe‖_F / ‖e_ref‖_F ≈ 9 × 10⁻⁸ — at
machine-precision floor for double-precision Kalman filter arithmetic.

**Findings.**

- **All 8 flat directions have dK columns entirely in ker(C).**
  This is the defining signature of the unobservable-subspace
  freedom: any dK whose action lands in the hidden subspace of the
  state cannot change the output.
- **dL_S = 0 for every flat direction.** The innovation covariance
  is fully identified. No ambiguity there.
- **The dA_pred perturbations couple ker(C) with itself and with the
  observable subspace, subject to the constraint that C · A_pred_new
  reproduces the observed correlations.** The number of free
  parameters this leaves is bounded above by (n_M − m) × n_M + (n_M
  − m) × m = 12 in principle; the observed d = 8 is consistent with
  additional constraints from observability of the (A_pred*, C) pair.
- **Innovation invariance is verified to numerical precision along
  a flat direction.** The likelihood is genuinely flat, not merely
  approximately flat, along these 8 directions.

### 11.4 Combined interpretation

The T1 v1.0.0 LTI manifold has now been decomposed:

- 3–5 dimensions attributable to the (A, Q, R) → (A_pred, K, S_ss)
  fiber (spectral-factorization non-uniqueness). Removed by
  innovations-form parametrization.
- 8 dimensions attributable to the classical unobservable-subspace
  freedom under fixed C. Persists in innovations form. Cannot be
  removed without adding structural constraints on C's null space
  (e.g., forcing C to be square).

**On the mystical translation.** The v1.0.0 SUPPORTED clause said
"θ that would identify one LTI observer from another can be moved
continuously without changing anything the outside world can
measure." v1.1.0 mechanistically identifies what portion of θ can
be so moved: exactly the components that describe the state
trajectory in the null space of the observation channel. This is
strictly weaker and more principled than v1.0.0's finding. It is
the formal statement that **a fixed observation channel cannot see
its own hidden state**.

If one is willing to accept "the observation channel" as a formal
analog of "the seeker's operational access to the source," and
"the state in ker(C)" as a formal analog of "the throne" or "the
essence," then the LTI result is a very literal computational
translation of *the essence is unknowable from within the channel*.
This is much closer to Abdu'l-Bahá's epistemology in
[the Tablet to Auguste Forel](https://bahai-library.com/abdul-baha_tablet_auguste_forel.html)
(the "essence unknowable, effects knowable" formulation) than to a
metaphysical annihilation reading.

We do NOT claim this translation is what the mystical texts are
"really" about. We claim only that the LTI finding is now precise
enough to identify which formal readings it does and does not
support:

- **DOES formally support:** "under a fixed observation channel,
  a positive-dimensional family of underlying realizations
  produces identical observations." (§11.3 above.)
- **DOES NOT support:** "at any perfect fit, all distinguishing
  identity annihilates." MLP and GP results in §5 remain
  UNSUPPORTED, and even the LTI finding is now known to be a
  channel-specific phenomenon (unobservable subspace), not a
  universal feature of parametric fits.

### 11.5 Follow-ups opened by v1.1.0

- **T1-R2 (not yet designed).** Symmetric LTI variant with C square
  and full-rank. Prediction: manifold should collapse to d = 0 at
  matched capacity. This is the direct falsification test for the
  ker(C) mechanism identified in §11.3.
- **T1-R3 (not yet designed).** Nonlinear generalization: does a
  small state-space nonlinear system with a fixed observation
  channel exhibit an analogous unobservable-manifold structure?
  Prediction: yes, but potentially only locally; global structure
  may collapse under nonlinearity.

Both follow-ups are logged but not committed to. The v1.1.0
addendum closes the specific question raised in T1 v1.0.0 §7
about the spectral-factorization alternative.

### 11.6 Additional artifacts (v1.1.0)

- `_internal/t1-empty-throne/src/lti_innov.py` — innovations-form
  LTI model class.
- `_internal/t1-empty-throne/src/pilot_lti_innov.py` — T1-R1 pilot
  driver (three random-init seeds + one truth-init).
- `_internal/t1-empty-throne/src/debug_lti_innov_mechanism.py` —
  T1-R1a mechanism analysis script.
- `_internal/t1-empty-throne/logs/r1a_mechanism.log` — captured
  console output for T1-R1a runs (seeds 20260730 and 20260830).

---

## 12. Addendum — T1-R2 (square-C falsification, 2026-07-30)

### 12.1 Motivation

The R1a mechanism identified in §11 states that the residual T1-R1
manifold is exactly the classical unobservable-subspace freedom
under fixed rectangular C (m < n_M ⇒ ker(C) has dim n_M − m). Under
that mechanism, setting m = n_M so that ker(C) = {0} should collapse
the manifold to d = 0. This is a direct binary falsification test
and was preregistered in
`_internal/t1-empty-throne/prereg_r2_r6.md` §T1-R2.

### 12.2 Design

- LTI-Gaussian source in innovations-form parametrization
  (same as T1-R1).
- n_S = 4, m = 4 (square C).
- C sampled per seed as the transpose of a QR-orthogonalized
  Gaussian, giving cond(C) ≈ 1 and rank(C) = 4 exactly (verified in
  `runs/LTI_R2/pilot_results.json`).
- Matched capacity n_M = 4, N_train = 2048, three seeds:
  {20260900, 20260901, 20260902}.
- Fitting: 3 random restarts + 1 truth-init per seed, L-BFGS
  max_iter = 200.
- Estimator: E1 Fisher rank with gap detection (same as T1-R1).

### 12.3 Preregistered decision rule

- **SUPPORTS R1a mechanism:** all seeds show E1 d ≤ 1 AND mean
  d ≤ 0.7.
- **REFUTES R1a mechanism:** at least 2 of 3 seeds show E1 d ≥ 3,
  OR mean d > 3.0.
- **INCONCLUSIVE:** anything in between.

### 12.4 Results

| Seed | cond(C) | NLL_fit | NLL_truth | ΔNLL | E1 d |
|---|---:|---:|---:|---:|---:|
| 20260900 | 1.00 | 5653.30 | 5671.72 | −18.4 | **4** |
| 20260901 | 1.00 | 5744.62 | 5770.61 | −26.0 | 0 |
| 20260902 | 1.00 | 6063.58 | 6080.10 | −16.5 | 0 |

Mean d = 1.33.

Applying the rule: 1-of-3 seeds ≥ 3 (not 2-of-3), mean 1.33 ≤ 3.0,
not all seeds ≤ 1. **Verdict: INCONCLUSIVE.**

### 12.5 Post-hoc diagnostic on the seed-20260900 anomaly

The seed-20260900 d = 4 detection is the ONLY point of tension with
the R1a mechanism prediction. We ran a targeted diagnostic to
distinguish (a) a genuine residual manifold from (b) a gap-based
rank-detector methodology artifact. Full log:
`_internal/t1-empty-throne/logs/r2_seed900_diagnostic.log`.

**Test 1 — Hessian eigenvalue stability across finite-difference eps.**

The bottom four eigenvalues at eps ∈ {3e-3, 1e-3, 3e-4, 1e-4, 3e-5}
are:

| eps | λ₁ | λ₂ | λ₃ | λ₄ |
|---:|---:|---:|---:|---:|
| 3e-3 | 1.228 | 2.879 | 3.487 | 26.08 |
| 1e-3 | 1.231 | 2.891 | 3.500 | 26.10 |
| 3e-4 | 1.232 | 2.892 | 3.501 | 26.10 |
| 1e-4 | 1.232 | 2.892 | 3.502 | 26.10 |
| 3e-5 | 1.233 | 2.893 | 3.503 | 26.11 |

The bottom eigenvalues are STABLE across 100× eps variation. They
are not numerical noise. They represent real (though shallow)
curvature in the loss surface.

**Test 2 — Walking along the smallest eigenvectors.**

Comparing ΔNLL when walking a step s along each of the four
smallest eigenvectors and one "mid-range" eigenvector (rank 20):

| step | v₀ (λ=1.23) | v₁ (λ=2.89) | v₂ (λ=3.50) | v₃ (λ=26.1) | v_mid (λ≈2900) |
|---:|---:|---:|---:|---:|---:|
| 1e-4 | −3.8e-6 | −6.1e-6 | −5.8e-6 | −3.6e-6 | +1.5e-5 |
| 1e-3 | −3.8e-5 | −6.0e-5 | −5.7e-5 | −2.4e-5 | +1.5e-3 |
| 1e-2 | −3.2e-4 | −4.7e-4 | −4.1e-4 | +9.5e-4 | +0.145 |
| 1e-1 | +1.9e-3 | +8.6e-3 | +11.5e-3 | +0.149 | +14.4 |
| 1.0 | +0.62 | +1.78 | +1.83 | (blew up) | +2131 |

At step 1.0, ΔNLL along v₀ is 0.62 versus quadratic prediction
0.5·λ·s² = 0.5·1.23·1 = 0.615 — the curvature model is
accurate. **These directions have real quadratic curvature; they
are not flat.**

Compare T1-R1 innovations-form seed 20260730 (rectangular C):
walking at step 0.01 along a flat direction gave |Δe|/|e| ≈ 9e-8
(numerical floor), corresponding to eigenvalues ~10⁻⁴ — **four
orders of magnitude smaller** than R2's "smallest" eigenvalues.

### 12.6 Corrected interpretation

The gap-based rank detector fired at seed 20260900 because the
eigenvalue ratio 26.10 → 283.64 crosses the preregistered gap
threshold of 10×. That cliff is real, but the eigenvalues below
the cliff are still 4 orders of magnitude too large to represent
a genuine unidentified direction. In R1 by contrast, the "flat"
directions had eigenvalues ~10⁻⁴ (near numerical floor) and
verified innovation-sequence invariance to ~10⁻⁷.

The seed-20260900 "manifold" is thus a **rank-detection method
artifact**: a shallow-but-real curvature basin whose eigenvalue
distribution happened to have a factor-of-10 cliff. It is not the
same phenomenon as R1's genuine unobservable-subspace freedom.

The R1a mechanism prediction — d = 0 when ker(C) = {0} — is
**qualitatively upheld** by the data:

- All fits achieve NLL_fit within 16–26 nats of NLL_truth (finite-N
  overfit range, as expected for well-identified models).
- No fit exhibits eigenvalues near numerical floor.
- Walking along the smallest eigenvectors produces real quadratic
  NLL curvature.
- 2 of 3 seeds show clean d = 0.

The strict preregistered verdict is INCONCLUSIVE because our
detection method has non-trivial false-positive rate against shallow-
but-real curvature. This is a known limitation of gap-based rank
detection and does not falsify the mechanism itself.

### 12.7 Consequence for the mechanism claim

The mechanism claim (v1.1.0 §11) stands:

> The residual T1-R1 manifold is the classical unobservable-subspace
> freedom under fixed rectangular C, characterized by dK columns in
> ker(C), dL_S = 0, and preserved innovation sequence.

T1-R2 result is compatible with this claim. The one apparent
counterexample (seed 20260900 d = 4) is methodology-limited, not
mechanism-refuting. To sharpen the test to distinguish 1e-4-scale
flat directions from 1-30-scale weakly-identified directions, a
future T1-R2b could use a stricter gap threshold (≥ 100× rather
than 10×) or the T1-R1a innovation-sequence-invariance verification.

Neither is preregistered here. The current addendum reports the
strict prereg verdict and the honest post-hoc diagnostic.

### 12.8 Follow-up registered

- **T1-R2b** (not yet designed). Same square-C setup as R2, but
  apply the R1a-style innovation-sequence-invariance verification
  to each candidate flat direction. Prediction: at seed 20260900,
  the four "flat" directions will NOT preserve the innovation
  sequence, confirming they are curvature artifacts, not R1-style
  hidden-state freedoms.

### 12.9 Artifacts (v1.2.0 R2 section)

- `_internal/t1-empty-throne/prereg_r2_r6.md` — locked
  preregistration.
- `_internal/t1-empty-throne/src/lti_square_c.py` — square-C
  truth generator and model builder.
- `_internal/t1-empty-throne/src/pilot_lti_r2.py` — R2 driver.
- `_internal/t1-empty-throne/src/debug_r2_seed900.py` — post-hoc
  diagnostic for the seed-20260900 anomaly.
- `_internal/t1-empty-throne/runs/LTI_R2/pilot_results.json` —
  full numerical results.
- `_internal/t1-empty-throne/logs/r2_pilot.log`,
  `logs/r2_seed900_diagnostic.log` — captured console output.

### 12.10 Exploratory corroboration (non-preregistered seeds)

An independent square-C innovations-form run using the original T1
seeds `{20260730, 20260800, 20260830}` plus one truth-init, N=2048,
n=m=4, gave:

| Seed | Init | E1 d | E2 d | E4 residual | NLL_fit |
|---|---|---:|---:|---:|---:|
| 20260730 | random | 0 | 0 | 0 | 6942.07 |
| 20260800 | random | 0 | 0 | 0 | 7366.94 |
| 20260830 | random | 0 | 0 | 0 | 8937.71 |
| 20260730 | truth | 0 | 0 | 0 | 6942.04 |

All four fits: d = 0 across E1/E2/E4. This is not part of the
preregistered R2 decision rule (different seeds), but it is a
clean corroboration that when ker(C) = {0}, the manifold collapses
under the same innovations-form pipeline that produced d ≈ 8 at
rectangular C. Artifacts:
`_internal/t1-empty-throne/runs/LTI_SQUARE_C/pilot/results.json`,
`logs/r2_square_c.log`.

---

## 13. Addendum — T1-R6 (nonlinear observation channel, 2026-07-30)

### 13.1 Motivation

If the R1a manifold is exactly the ker(C) hidden-state freedom,
then a nonlinearity that leaks ker(C) components into the
observation should shrink the manifold as the leak strength α
grows. This is the mechanism-breaking test complementary to R2's
mechanism-removing test.

### 13.2 Design (with Deviation 003)

Preregistered in `prereg_r2_r6.md` §T1-R6; Deviation 003
(logged before execution) switches from innovations-form to
standard (A, Q, R) + EKF because innovations form is misspecified
under a nonlinear observation channel.

- Dynamics: linear LTI, n_S = 4, m = 2, C fixed rectangular.
- Observation: `y = C x + α · (x^T Q_kern x) · v0 + η`, with
  Q_kern constructed to have support on ker(C).
- α ∈ {0.0, 0.3, 1.0} (Deviation 003 reduced the sweep from
  {0, 0.1, 0.5, 1.0} for runtime; 0.3 stands in for the mid-range).
- N_train = 512 (reduced from 2048 for EKF runtime).
- Two seeds: {20260910, 20260911}.
- Fit: (A, L_Q, L_R) via EKF-NLL; α fixed at truth; C fixed.
- Estimator: E1 Fisher rank.

### 13.3 Preregistered decision rule (Deviation 003)

- **SUPPORTS:** d(0) ≥ 5, d(1.0) ≤ 2, and d(α) monotone
  non-increasing (one seed-level exception allowed).
- **REFUTES:** d(1.0) ≥ 5, OR baseline d(0) < 5.
- **INCONCLUSIVE:** otherwise.

### 13.4 Results

| α | Seed | NLL_fit | NLL_truth | ΔNLL | E1 d |
|---:|---|---:|---:|---:|---:|
| 0.0 | 20260910 | 1256.45 | 1266.33 | −9.9 | **0** |
| 0.0 | 20260911 | 840.33 | 851.74 | −11.4 | **0** |
| 0.3 | 20260910 | 1534.93 | 1996.78 | −461.8 | 0 |
| 0.3 | 20260911 | 941.98 | 941.98 | 0.0 | 0 |
| 1.0 | 20260910 | 2097.40 | 5561.13 | −3463.7 | **27** |
| 1.0 | 20260911 | 1284.03 | 1729.87 | −445.8 | 0 |

Mean d by α: d(0) = 0.0, d(0.3) = 0.0, d(1.0) = 13.5.

### 13.5 Strict preregistered verdict

**REFUTES_R1A_MECHANISM.**

Gate failure: d(0) = 0 < 5. The R6 pipeline at N=512 / EKF /
(A, Q, R) form did not recover the T1 v1.0 / R1 manifold that the
α-trend test is designed to shrink. Under the preregistered rule
this is a refutation of the claim that "this pipeline measures the
R1a mechanism," which is the precondition for the α-trend test.

Secondary observation (not decisive under the rule): at α = 1.0,
seed 20260910 reports d = 27 with ΔNLL = −3463 (fit far below
truth NLL — a signature of EKF-approximation pathology / local
degeneracy, not a genuine 27-dim manifold). Seed 20260911 at
α = 1.0 is clean d = 0. The α = 1.0 mean of 13.5 is therefore
dominated by a single degenerate fit and is not interpretable as
a mechanism signal.

### 13.6 Honest interpretation

The R6 result does **not** cleanly refute the ker(C) mechanism.
It refutes the claim that *this particular EKF / N=512 / (A,Q,R)
pipeline* recovers and then shrinks the R1a manifold. Three
concrete failure modes are visible:

1. **Baseline miss.** At α = 0 the bottom Hessian eigenvalues are
   O(10⁻²)–O(10⁻¹), two to three orders of magnitude larger than
   R1's genuine flat directions (O(10⁻⁴)). The relative and gap
   detectors correctly report d = 0. The deep-basin manifold of
   T1 v1.0 / R1 was not reached (likely N=512 + EKF linearization
   + fewer restarts).
2. **EKF pathology at large α.** Seed 20260910 at α = 1.0 finds a
   far-below-truth NLL with near-full-rank-deficient Hessian —
   consistent with EKF likelihood becoming badly behaved when the
   observation nonlinearity is strong, not with a physical
   27-dimensional symmetry.
3. **Mid-α uninformative.** With baseline d(0) = 0 already, the
   α ∈ {0.3, 1.0} measurements cannot demonstrate "shrinkage."

What R6 does establish: under the current EKF pipeline, the
α-trend test is not yet a valid adjudicator of the R1a mechanism.
A restored-baseline follow-up is required before any mechanism
verdict from the nonlinear-observation design can be claimed.

### 13.7 Exploratory note (nonlinear innovations form)

A separate exploratory run (non-preregistered; nonlinear
innovations-form predictor `x+ = A_pred tanh(x) + K y` with
rectangular C fixed, N=2048, original T1 seeds) found E2-verified
manifolds of d ∈ {5, 9} at deep-fit basins and d = 0 at one sharp
local minimum, with one failed fit discarded. This is consistent
with local persistence of unobservable-subspace freedom under
nonlinear *dynamics*, but it is a different design from the
preregistered R6 (nonlinear *observation*) and does not enter the
R6 verdict. Artifacts:
`_internal/t1-empty-throne/runs/NL_SSM/pilot/results.json`,
`logs/r6_nl_ssm.log`.

### 13.8 Follow-ups opened by v1.2.0

- **T1-R2b** (registered in §12.8): innovation-sequence-invariance
  verification on the seed-20260900 "flat" directions.
- **T1-R6b** (new): restore the α = 0 baseline under conditions
  that recover T1-R1 (innovations form or steady-state Kalman at
  N=2048, multiple random restarts, truth-init required). Only
  after d(0) ≥ 5 on both seeds may the α-sweep be re-run for a
  mechanism verdict. Prediction if R1a is correct: d(0) ≈ 8,
  d(1) ≤ 2, monotone.
- **T1-R7** (not designed): particle-filter or UKF likelihood at
  α = 1.0 to rule out EKF pathology as the source of the
  seed-20260910 degeneracy.

### 13.9 Artifacts (v1.2.0 R6 section)

- `_internal/t1-empty-throne/src/lti_nonlin_obs.py` —
  nonlinear-observation LTI + EKF NLL.
- `_internal/t1-empty-throne/src/pilot_lti_r6.py` — R6 driver.
- `_internal/t1-empty-throne/runs/LTI_R6/pilot_results.json` —
  full numerical results (includes strict prereg verdict).
- `_internal/t1-empty-throne/logs/r6_pilot.log` — captured console.
- `_internal/t1-empty-throne/deviations.md` — Deviation 003.

---

## 14. Revision history (v1.2.0)

- **v1.0.0** — 2026-07-30. Initial release. Verdict: INCONCLUSIVE
  overall; PROVISIONAL SUPPORT for LTI at deep-fit basins;
  UNSUPPORTED for MLP and GP.
- **v1.1.0** — 2026-07-30. Addendum §11: T1-R1 + T1-R1a. Manifold
  mechanistically identified as ker(C) unobservable-subspace
  freedom.
- **v1.2.0** — 2026-07-30. Addenda §12 (T1-R2) and §13 (T1-R6).
  R2: strict prereg INCONCLUSIVE, mechanism qualitatively upheld
  (plus exploratory corroboration d = 0 on all four original-seed
  fits). R6: strict prereg REFUTES_R1A_MECHANISM driven by
  baseline failure; α-trend uninformative; T1-R6b registered.
  Overall T1 status unchanged in direction (PROVISIONAL SUPPORT
  for LTI / R1a), with the nonlinear-observation adjudicator now
  known to require a restored baseline before it can speak.

---

**References to primary sources (partial):**

- Bahá'u'lláh, *The Seven Valleys and The Four Valleys*. Bahá'í
  Publishing Trust translations. Specifically the seventh valley
  ("Valley of True Poverty and Absolute Nothingness") and its
  characterization of the end-state.
- Sussmann, H. J. (1992). "Uniqueness of the weights for minimal
  feedforward nets with a given input-output map." *Neural Networks*
  5(4), 589–593. — Establishes that generic tanh MLPs at matched
  capacity have no continuous symmetries beyond hidden-unit
  permutation and sign-flip (discrete).
- Kailath, T., Sayed, A. H., Hassibi, B. (2000). *Linear Estimation*.
  Prentice Hall. — Standard reference for state-space LTI
  identifiability, DARE, and spectral factorization.