Self-Reinforcing Information Dynamics in Conscious Systems: Why Rτ/I = 1 and How Consciousness Accumulates

This paper closes two gaps in the m ⇄ E ⇄ I ⇄ C framework left open by Papers 8 and 9: (a) Paper 8 observed Rτ/I ≈ 1. 004 from three independent paths but did not derive this convergence, and (b) Paper 9 defined the condensation quality W = 1 − Sbasis/Sₘax but addressed only millisecond attention switching, leaving long-term W accumulation unresolved. By isolating the Bose-Einstein stimulated-emission part of Fröhlich's 1968 rate equation and translating it into the information domain, a single self-reinforcing term emerges: dI/dt = R − Γ + α · I₀ · Φₑff (Eq. 1) Three derivations follow: - Theorem 1 — Rτ/I = 1 is the unique globally asymptotically stable fixed point of the consciousness cycle (4-path proof: arithmetic, Lyapunov, thermodynamic, numerical). - Theorem D. 1 — The dominant mode fraction p₀ follows exact Verhulst logistic dynamics: dp₀/dt = α · Φₑff · p₀ · (1 − p₀), derived from Bose statistics with no approximation beyond Lemma A. 1. - Theorem C. 1 — The composite four-arrow feedback gain |g'| = 3. 0 × 10⁻¹⁰ < 1, making the loop a Banach contraction (βₛub < 1 structural origin). A timescale separation of 2. 88 × 10¹³ emerges between the microsecond coherence cycle and the multi-month accumulation dynamics — the fast loop remains locked at Rτ/I = 1 while informational order accumulates beneath it, like a thermostat regulating fast around a slowly-shifting set point. ZERO NEW FITTING PARAMETERS. The daily self-reinforcing rate αdaily = 0. 0024 is derived from three already-established quantities: Fröhlich pump efficiency αF (Paper 9, Eq. A. 8), feedback loop gain g' (this paper, §3. 6), and coherence cycles per waking day Nwake (Paper 8, τ = 1 μs). The framework is explicitly falsifiable. Six falsification conditions (F1–F6) are stated. Six predictions (P1–P6) are classified as discriminating (P1, P2, P3), accumulation-unique (P4, P5), and scaling (P6). Sixteen internal cross-validation checks pass (16/16): analytical consistency, numerical robustness, and published-data directional agreement (Lutz 2004, Shamatha 2007/2022, Bandura 1977, Choking literature, Identity-Before-Growth). Series context. This paper is the Crown of the Trilogy (Existence × Direction × Growth) and the sixth element of the ten-element Information Physics Series Dectet. Theorem D. 1 functions simultaneously across multiple roles: universality catalog (Paper 11), cross-substrate validation (Paper 15 with Paper 12 parallel instance, Bucher et al. Nature 2026 forward-prediction confirmation), synthesis (Paper 16, κ ≡ g'), and thermodynamic constitution (Paper 17, canonical local expression of the generalized Second Law). The series-wide IVP record stands at 154/154 across heterogeneous tests; this is reported as contextual coherence, not pooled evidence. Supplementary materials (CC BY-NC 4. 0): - paper10ₛimulation. py — 16 internal checks, generates simdata. json- makeₐllfigures. py — all 9 figures from simdata anchors- verifyₐll. py — 3-axis validation (math / Paper 8-9 consistency / public data) Keywords: Fröhlich condensation, Bose-Einstein stimulated emission, Verhulst logistic, Banach contraction, m ⇄ E ⇄ I ⇄ C cycle, Rτ/I fixed point, sigmoid learning curves, gamma spectral entropy, meditation EEG, self-efficacy, consciousness dynamics, Information Physics Series.