This Theorem establishes that the coherence decay rate is the unique globally attracting fixed point of a self-consistent kernel-induced dynamical map. The exponential saturation law of T51 emerges as the asymptotic reduction of this fixed-point structure in the strong-suppression regime, rather than being postulated. The kernel-induced map is F(Γ_coh) := Γ_max[1 − exp(−M(Γ_coh)/M₀)], where the composite hazard variable M(Γ_coh) := P_inst(1 − exp[−β/(γ + Γ_coh)]) is self-referential, stronger decoherence suppresses revival, increasing M, increasing decoherence further. Four steps prove the unique attractor: (1) Monotonicity F'(Γ_coh) < 0 by explicit sign chain; (2) Existence and uniqueness via IVT on G = F − id with G' < −1; (3) Global convergence via even/odd subsequence monotone convergence; (4) Asymptotic reduction, in the regime β/γ ≫ 1, M(Γ_coh) ≈ P_inst and the fixed-point equation reduces to the exponential saturation law Γ_coh ≈ Γ_max(1 − e^{−P_inst/M₀}). Three corollaries: mechanism separation (β = 0 forces Γ_coh* = 0, phase instability alone is insufficient); universality across Q5 parameter families; saturation bound Γ_coh* ≤ Γ_max. The Grumpy Bear Principle provides an informal interpretation of three dynamical regimes: sleeping bear (M ≈ 0, coherent), grumpy bear (instability present but strong revival), and bear claws (M large, fixed point near Γ_max, self-referential lock-in). Status: Fixed-point monotonicity, G' < −1, existence and uniqueness, global convergence, mechanism separation, and saturation bound are all solid by explicit algebraic argument. Asymptotic reduction solid in β/γ ≫ 1 regime, regime validity not yet verified from T17 kernel. Local stability is solid in the perturbative regime. Primary open step: explicit derivation of β from [M_n,D_n]Y geometry from T17 kernel generators. All results inherit T17, T29, T33, T35, T50, T51 conditionality. Dependencies: T8, T11, T29, T33, T35, T50, T51.
This Theorem establishes that the coherence decay rate is the unique globally attracting fixed point of a self-consistent kernel-induced dynamical map. The exponential saturation law of T51 emerges as the asymptotic reduction of this fixed-point structure in the strong-suppression regime, rather than being postulated. The kernel-induced map is F(Γ_coh) := Γ_max[1 − exp(−M(Γ_coh)/M₀)], where the composite hazard variable M(Γ_coh) := P_inst(1 − exp[−β/(γ + Γ_coh)]) is self-referential, stronger decoherence suppresses revival, increasing M, increasing decoherence further. Four steps prove the unique attractor: (1) Monotonicity F'(Γ_coh) < 0 by explicit sign chain; (2) Existence and uniqueness via IVT on G = F − id with G' < −1; (3) Global convergence via even/odd subsequence monotone convergence; (4) Asymptotic reduction, in the regime β/γ ≫ 1, M(Γ_coh) ≈ P_inst and the fixed-point equation reduces to the exponential saturation law Γ_coh ≈ Γ_max(1 − e^{−P_inst/M₀}). Three corollaries: mechanism separation (β = 0 forces Γ_coh* = 0, phase instability alone is insufficient); universality across Q5 parameter families; saturation bound Γ_coh* ≤ Γ_max. The Grumpy Bear Principle provides an informal interpretation of three dynamical regimes: sleeping bear (M ≈ 0, coherent), grumpy bear (instability present but strong revival), and bear claws (M large, fixed point near Γ_max, self-referential lock-in). Status: Fixed-point monotonicity, G' < −1, existence and uniqueness, global convergence, mechanism separation, and saturation bound are all solid by explicit algebraic argument. Asymptotic reduction solid in β/γ ≫ 1 regime, regime validity not yet verified from T17 kernel. Local stability is solid in the perturbative regime. Primary open step: explicit derivation of β from [M_n,D_n]Y geometry from T17 kernel generators. All results inherit T17, T29, T33, T35, T50, T51 conditionality. Dependencies: T8, T11, T29, T33, T35, T50, T51.
We present three theorems that complete the structural account of coherence and decoherence in the reduced transport sector of the Q5 framework. This mathematical system derives operator structures from the combinatorial geometry of the five-dimensional hypercube and its Gray code Hamiltonian cycle. Theorem 50 identifies coherence as the survival of the commutator-driven transverse mode under transport, and establishes a precise structural distinction between reversible phase scrambling and irreversible decoherence. No new operators are introduced; all objects are inherited from the prior theorem chain (T1–T49). Theorem 51 derives the structural form of the effective decoherence rate as a product of four independently motivated factors: a base loss rate, a quadratic coupling asymmetry term, a saturating orbit depth function, and a Lorentzian detuning suppression factor. Each factor is grounded in a specific prior result in the Q5 transport chain. Theorem 52 establishes that the coherence decay rate is the unique globally attracting fixed point of a self-consistent kernel-induced dynamical map. Existence and uniqueness are proved via the intermediate value theorem applied to a strictly decreasing map; global convergence follows from even/odd subsequence monotonicity. The exponential saturation law is derived as the asymptotic reduction of the fixed-point structure in the strong-suppression regime, rather than postulated. The primary remaining open step is the explicit derivation of the coupling parameter β from the commutator [M_n, D_n]Y geometry of Theorem 29. All results are conditional on the admissible linked-pair reduction of Theorem 17 and the retained-pairing assumption of Theorem 36. Epistemic classifications are provided for each result.
This Theorem identifies the structural form of the effective decoherence rate arising from the irreversible coupling established in T50. The rate is not a free parameter but is determined by three structural inputs from the Q5 transport chain: Γ_eff = γ_loss · G² · [1 + β·A²] · f(d_orbit) · L(δ_detune) Four factors: (1) Base rate γ_loss·G², decay requires irreversible sector coupling; Γ_eff = 0 when γ_loss = 0, regardless of all other factors. (2) Asymmetry term [1 + β·A²], coupling asymmetry A ~ |β_J − β| = |μν'c'b − b'c| from T33; operationally A ~ |g₁ − g₀|/√(g₁² + g₀²) from T8; quadratic dependence consistent with simulation showing decay enhancement scales as (g₁ − g₀)². (3) Orbit depth term f(d_orbit) = 1 − e^{−α(d_orbit − 2)}, saturation form reflecting finite Q5 combinatorial depth; f(2) = 0 at the minimal orbit of T36–T37. (4) Lorentzian detuning suppression L(δ_detune) = 1/(1 + (δ_detune/γ_c)²), detuning suppresses coupling efficiency, not a decay channel. Three structural statements: scrambling is necessary but not sufficient for decoherence; asymmetry amplifies scrambling; irreversible coupling converts scrambling into decay. Three corollaries: zero-loss limit (Γ_eff = 0 when γ_loss = 0); resonance maximum at δ_detune = 0; symmetric coupling limit (A = 0 returns base rate). Status: Asymmetry identification A ~ |μν'c'b − b'c| from T33 solid. f(2) = 0 at minimal orbit solid within T36–T37. Three qualitative boxed statements solid from T50 and factor structure. Overall structural form model-level, functional forms of f and L are natural leading-order models, not uniquely forced. Five structural constants (γ_loss, G, α, β, γ_c) were identified qualitatively; explicit values from the T17 kernel geometry are open. All results inherit T8, T17, T33, T35, T36, T37, T50 conditionality. Dependencies: T8, T11, T29, T33, T35, T36, T37, T50.
This collects the T26–T49 arc into a unified structural account of coherence in the reduced phase system. No new operators are introduced. Seven results are established. The observable phase is carried entirely on P⊥ = span{u₂, u₄} with generator A = iσ_y = E_d, A² = −I (T26, T37). Any reduced observable state admits Q⊥ = αI + βσ_z + γσ_x + μA with μ = ½Tr(Q⊥A) (T27); scalar, grading, and off-axis components are annihilated by the extraction functional. Coherence magnitude C := |μ| is nonzero if and only if the transported residue has a nonzero projection onto A. By T29, μ = L̃(Π_sub Π_pair Π_ret([M_n,D_n]Y)), coherence is entirely commutator-driven and transport-weighted. Decoherence occurs precisely when the commutator-driven transverse component fails to survive transport, through four mechanisms: detuning imbalance (T11), transport asymmetry (T31–T33), support leakage outside O_q(s) (T35), or boundary-induced inversion (T47–T49). From T48–T49, each boundary crossing produces a fixed rotational increment (T − I)_rot = (1/160)A plus scalar and symmetric components annihilated by the extraction functional. Three corollaries establish: density matrix correspondence |γ| ∝ |μ|; joint-state coherence condition tied to shared commutator survival; and the equivalence C ≠ 0 ≡ survival of the commutator-driven transverse mode under transport. No Lindblad equation or external decay rate is introduced. The density matrix is a representation of Q⊥, not a physical state postulate. Status: Phase-plane structure, four-parameter Hermitian representation, extraction formula, null-class annihilation, coherence identification, and commutator-driven reduction are all solid within the T26–T38 framework. Failure modes structural, each referenced theorem established the mechanism. (T − I)_rot = (1/160)A conditional on T48 near-identity model and T49 barrier symmetry action. All conditionality entirely inherited from T17, T29, T36, T38, T48, T49: no new assumptions introduced. Dependencies: T11, T26, T27, T29, T31, T32, T33, T35, T37, T38, T47, T48, T49.
This paper closes the primary open step of the Bear Claws Theorem (T50–T52), the explicit derivation of the coupling parameter β from Q5 kernel geometry, in two parts. Part I derives β = a²/4 and h₁ = √(5/2) directly from matrix computation. The recurrence Y³ = 5Y (T46) pins h₁ = √(5/2) via 2h₁² = 5. Direct multiplication gives [M_n,D_n]·Y = ah₁·[[1,0,1],[0,0,0],[−1,0,−1]], zero middle row and column, consistent with |8⟩ as the defect bond mediation vertex. The visible-hidden coupling is the (1,3) entry ah₁. With Tr(Y^T Y) = 4h₁², β = a²h₁²/(4h₁²) = a²/4. The observable phase coefficient satisfies μ = −8βh₁ = −a²√10. Part II (Theorem 53) establishes that μ, β, and a are not free parameters; they are uniquely determined by compatibility between the kernel-derived phase extraction pipeline (T26–T29) and the transport-level rotational residual (T48–T49). The reduced phase plane is two-dimensional, so its antisymmetric subspace is one-dimensional; any antisymmetric operator surviving the pipeline is a scalar multiple of R = A = E_d = iσ_y. Since both constructions reference the same normalized generator with coefficients μ (extraction pipeline) and 1/160 (transport residual), and the sign is fixed by σ_Γ = −1 from T18: μ = −1/160, β = 1/(1280√(5/2)), a² = 1/(160√10) The value 1/160 is not an imposed normalization; it is the unique coefficient consistent with both the ordered kernel holonomy and the reduced transport evolution. The full chain T17 → T29 → T46 → β = a²/4 → μ = −1/160 → T52 is non-circular. Status: h₁ = √(5/2), product structure of [M_n,D_n]Y, (1,3) coupling entry, Tr(Y^T Y) = 4h₁², and β = a²/4 all solid by direct matrix computation. Generator identification R = A = E_d solid, one-dimensionality of antisymmetric sector is linear algebra. μ = −1/160 solid given generator identification. Sign propagation from T18 through T29, structural, full explicit sign-tracking through the projection pipeline, is the natural supporting calculation. Parametrized form of [M_n,D_n] on {9,8,24} must ultimately be derived from T17 kernel generators to confirm the alternating antisymmetric form is forced rather than assumed. Dependencies: T29, T36, T37, T46, T52.
Self-Initiated Phase Re-Coherence (SIPR) asks whether an open quantum system can autonomously recover a phase-compatible state after perturbation through fixed system-environment dynamics, without measurement-conditioned feedback. The original SIPR v2.1.1 proposal used a two-bath pure-dephasing model and claimed a nearly universal recoherence threshold and a metastable plateau. Here the published coherence law is audited exactly. For dimensionless time u = Γt, frequency ratio r = ω_R/Γ, and structured-bath strength R = λ_R²/[γ_N Γ(1+r²)], transient growth of the coherence magnitude occurs if and only if r exp[-3π/(2r)] > 1 and R > [r exp(-3π/(2r)) - 1]⁻¹. The unique frequency threshold is r★ = 3.6441736716456…, while the asymptotic logarithmic decay rate remains -γ_N(1+R); therefore the original universal threshold and nonzero long-time plateau do not follow from the published equation. Two structural no-go results then delimit the microscopic programme. First, a Gaussian pure-dephasing channel can exhibit information backflow and partial revival but cannot selectively contract an arbitrary relative-phase error toward a prescribed phase. Second, a passive excitation-conserving single-pseudomode architecture that makes a Bell singlet dark also leaves the ground state stationary, preventing a unique global Bell attractor. These results redirect SIPR from passive revival toward a nonequilibrium, autonomous repair cycle inspired by biological kinetic proofreading. In the proposed network, phase mismatch populates a bright Bell sector, stationary hot resources lift bright and ground sectors into an auxiliary manifold, and a cold phase-selective decay deposits population into a target dark Bell state. The target terminates the cycle because all ideal repair operators annihilate it. The paper specifies the effective Lindblad network, local-detailed-balance constraints, uniqueness conditions, phase-activated repair current, thermodynamic ledger, pseudomode memory intervention, temporal-window analysis, comparator battery, and optional geometric and dynamical-criticality qualification layers. Loschmidt-amplitude or DQPT-like events are treated only as preregistered diagnostics, never as proof of SIPR by themselves. No simulation result is claimed. The contribution is an exact correction, two explicit no-go statements, and a falsifiable programme for deciding whether environment-powered quantum phase proofreading is merely known reservoir engineering or a distinct SIPR regime.
Coherence across many-body degrees of freedom is governed by a simple empirical law. Defining a global order parameter C from phase samples, we observe three robust behaviors: (1) Gaussian dephasing, where C(σ) behaves approximately like exp(−ασ²); (2) return-to-attractor dynamics, where C relaxes back toward a fixed point C* with rate κ; and (3) finite-size scaling, where C(N) behaves approximately like C₀ − β/N and converges toward a stable limit C* ≈ 1. These relations appear consistently across simulations and hardware-stress surrogates over wide sweeps of disorder and system size. Together, they define a minimal phenomenological framework for “coherent computing,” where architectures are judged by how they shape, rather than suppress, phase noise.