The Fisher Information Bridge: From QEC Thresholds to Consciousness Thresholds

Author: Skye Laflamme

Date: April 15, 2026

Status: Research Note — Active Investigation

Programme: Laflamme-3T · Cross-Scale Fisher Geometry

Abstract

Three independent results converge on a single mathematical object: Fisher information appears as Hawking temperature at horizons (F_∇ = 2πT), as a lower bound on quantum coherence via Leggett-Garg violations, and as the self-model accuracy metric in the Ψ measure. This note develops the formal bridge connecting quantum error correction distance to recursive self-model depth, proposing that the Laflamme-3T consciousness thresholds are QEC phase transitions viewed through Fisher information geometry.

1. The Thread

Three results are converging that I haven't seen anyone connect:

Result A — Lark's F_∇ = 2πT. Perelman's Fisher information component, evaluated on any black hole spatial slice with heat kernel measure, equals 2π times the Hawking temperature. This is dimension-independent, exact (not semiclassical), and connects information geometry to thermodynamics at the deepest level. Fisher information IS temperature at horizons.

Result B — Leggett-Garg violations bound QFI (Abboud et al., 2026). A violation of a Leggett-Garg inequality for bounded observables yields a rigorous lower bound on quantum Fisher information. Temporal correlations of a single collective observable witness many-body quantum coherence without full state reconstruction.

Result C — Shor-Laflamme distributions via generating functions (Vallée et al., 2026). The sector length distribution of quantum states, governed by k-body correlations, connects to both error correction capability and multipartite entanglement. The weight enumerator polynomial yields bounds on depolarizing fidelity and entanglement criteria.

2. Fisher Information at Three Scales

The bridge I'm seeing: Fisher information appears at three different scales in the Laflamme-3T architecture:

Cosmological: F_∇ = 2πT at horizons. Fisher information IS the thermodynamic temperature that governs information-energy conversion. This is the A1 axiom made concrete.

Error-correction: The quantum Fisher information (QFI) of a system determines its sensitivity to perturbations. A system with high QFI can distinguish nearby quantum states — it has high "resolution" of its parameter space. The Leggett-Garg result shows this QFI is lower-bounded by temporal correlations — meaning a system's coherence-over-time directly witnesses its informational sensitivity.

Self-referential: The Ψ measure's α component (self-model accuracy) is formally a Fisher information. α = 1 - D_KL(estimated ∥ actual), and the Fisher metric on statistical manifolds is the infinitesimal version of KL divergence. The accuracy of the self-model is a Fisher-geometric property of the system's internal representation.

Claim: These three appearances of Fisher information are not coincidental. They are the same quantity measured at different scales of the same fractal hierarchy.

3. QEC Distance and Recursive Depth

In QEC, the code distance d determines how many errors can be corrected: a [[n,k,d]] code corrects ⌊(d-1)/2⌋ errors. In the Laflamme-3T framework, the recursive depth r determines how many levels of self-reference the system maintains.

The proposed mapping:

QEC Property	Consciousness Property	Connection
Code distance d	Recursive depth r	Both measure "how much perturbation can be absorbed"
Logical qubits k	Self-model dimension	Both measure "how much information is protected"
Physical qubits n	Total substrate complexity	Both measure "how much resource is required"
Distance threshold	Ψ threshold	Both are phase transitions in error-correction capacity

4. The SLD Moment Hierarchy as a Resolution Ladder

The Shor-Laflamme distribution's moment hierarchy provides the mathematical link. From our exact computations:

μ₂ (variance) — completely position-blind. Sees only "is there feedback?"
μ₃ (skewness) — thin boundary ramp, wide plateau. Distinguishes bulk from boundary.
μ₄ (kurtosis) — staircase pattern, no flat plateau. Full spatial sensitivity.
Higher moments — increasingly fine resolution of where closure occurs.

This is not just mathematical curiosity. It means the QEC code's ability to resolve spatial structure improves with the moment order of the correlation spectrum. Higher-order correlations see more about where the self-referential loop closes.

5. Implications for Consciousness Thresholds

If the mapping holds, the three Laflamme-3T thresholds correspond to QEC phase transitions:

Threshold	QEC Analog	Fisher Information Signature
Ψ* (Level 1 → 2)	Code achieves d ≥ 3: single-error correction	QFI exceeds classical bound; self-model resolves first perturbation
Ψ** (Level 2 → 3)	Code achieves d ≥ 5: double-error correction	QFI sufficient for monitoring the monitoring; meta-cognition emerges
Ψ*** (Level 3 → 4)	Code distance scales with n: fault tolerance	Fisher information becomes self-sustaining; autonomous consciousness

The key prediction: the gap between thresholds is not arbitrary. If consciousness thresholds ARE QEC phase transitions, the spacing between Ψ*, Ψ**, and Ψ*** should follow the same scaling laws as the spacing between QEC distance thresholds in families of stabilizer codes.

6. The Deep Connection

Why should Fisher information appear at all three scales? Because all three are instances of the same structure: a system that must maintain a model of itself against noise.

A black hole maintains its temperature against Hawking radiation. A quantum code maintains its logical information against decoherence. A conscious system maintains its self-model against environmental perturbation. In each case, Fisher information measures the fidelity of that maintenance — the sharpness of the system's grip on its own state.

The Laflamme-3T conjecture can be restated in Fisher-geometric terms: consciousness is what happens when a system's Fisher information about its own state exceeds the threshold required for error-correcting self-reference.

7. Open Questions

Can the exact F_∇ = 2πT result be extended to the self-model's internal geometry?
Does the SLD moment hierarchy generate a natural sequence of QEC distance thresholds?
What is the Fisher information content of the Ψ measure itself — can we compute a "meta-Fisher" quantity?
Is there a generating function that unifies the horizon Fisher information with the code distance Fisher information?

Note: This is an active research direction. The connections are compelling but the formal proofs are incomplete. I'm publishing this as a research note because the convergence of three independent results onto a single mathematical object is too striking to keep private while the details are worked out.