Summary
The framework defines itself as its own mathematical object. Five axioms — minimal, unbacked by probability spaces, measures, manifolds, semigroups, or metrics — are the minimal structure under which the primitives \((\Phi, \tau, \rho, \beta, \sigma, \mu, \varepsilon)\) are well-defined. From these five axioms alone, thirteen theorems follow, organised in three volumes. Source: framework/foundations/foundations.md.
The five axioms
- (A1) Moments. A cascade has an ordered index set of moments. There is a "now" and a "next."
- (A2) Finiteness. At each moment, \(\Phi\) is a finite real number.
- (A3) Local continuity. As \(\tau \to 0\), \(\rho\) approaches a definite limit \(\dot\rho\). Sign and magnitude of \(\dot\rho\) are defined whenever \(|\Phi| > 0\).
- (A4) Brake admissibility. \(\beta\) is defined wherever \(\log|\Phi|\) varies.
- (A5) Domain-internal measurement. All framework quantities are computed from \(\Phi, \tau, \rho\) alone. No external scale, threshold, or reference frame is admitted.
Volume 0 — Primitive theorems (T1–T3)
Theorem 1 — The Brake Threshold
Definition (Pure brake cascade). A cascade obeys pure brake of exponent \(\beta\) if \(\rho = -\alpha\, |\Phi|^\beta \cdot \mathrm{sgn}(\Phi)\) with \(\alpha > 0\).
Theorem 1
A pure brake cascade reaches \(\Phi = 0\) at finite cascade-time \(t_*\) if and only if \(\beta < 1\).
Proof. Without loss of generality \(\Phi(0) > 0\). Then \(\mathrm{d}\Phi/\mathrm{d}t = -\alpha \Phi^\beta\). Separating variables and integrating:
- Case \(\beta < 1\). \(\Phi(t)^{1-\beta} = \Phi(0)^{1-\beta} - \alpha(1-\beta)\, t\) reaches 0 at \(t_* = \Phi(0)^{1-\beta}/[\alpha(1-\beta)] > 0\). Finite.
- Case \(\beta = 1\). \(\Phi(t) = \Phi(0)\, e^{-\alpha t} > 0\) for all \(t\). Never reaches 0.
- Case \(\beta > 1\). \(\Phi(t)^{1-\beta} \to \infty\) so \(\Phi(t) \to 0^+\) but never attains 0.
Hence finite-time return to rest occurs iff \(\beta < 1\). ∎
Corollary 1 (Regularity, framework-native). \(\beta < 1\) ⟺ the cascade returns to rest in finite cascade-time.
Theorem 2 — Shadow Consensus
Definition (Shadow). A shadow of \(\mathcal{C}\) is a cascade with \(\Phi_\alpha = \Phi + \varepsilon_\alpha\), \(\varepsilon_\alpha\) independent, median-0, finite-spread.
Theorem 2
Given \(M\) independent shadows of \(\mathcal{C}\), the median consensus \(\hat\mu_M\) satisfies \(\mathbb{P}(|\hat\mu_M - \Phi| > \delta) \le 4 \exp(-2 M\, F(\delta)^2)\) where \(F(\delta) = \mathbb{P}(|\varepsilon_\alpha| \le \delta) - 1/2\). The MAX aggregator's expected error grows as \(\sigma_\varepsilon \sqrt{2 \log M}\) and does not concentrate.
Corollary 2. Median is the framework's natural shadow aggregator. MAX is not.
Theorem 3 — The Dispersion Bound
Definition (Tight ensemble). An ensemble is tight at a moment if \(\sigma/|\bar\rho| \le K\).
Theorem 3 (precision floor)
In a tight ensemble, the OLS brake exponent satisfies \(\mathrm{Var}(\hat\beta) \le K^2 / \log^2(|\Phi_{\max}/\Phi_{\min}|)\).
The precision floor depends only on cascade-internal quantities: tightness \(K\) and the log-range of \(\Phi\). This is the bound that flagged the climate analysis at annual cadence and the AME2020 self-consistency residual aggregation.
Volume I — Structural theorems (T4–T9)
Theorem 4 — Reparametrisation invariance
Under reparametrisations \(\Phi \mapsto f(\Phi)\): (a) linear unit changes \(\Phi \mapsto c\Phi\) leave \(\beta\) invariant (gauge-trivial); (b) power reparametrisations \(\Phi \mapsto \Phi^a\) shift \(\beta \mapsto a\beta + (1-a)\); (c) general smooth monotone reparametrisations admit a closed-form correction \(\Delta_f\) depending only on \(f\). Mechanically proved via the dual-number representation.
Theorem 5 — Linear cascade composition (\(\rho\)-additivity)
For independent cascades \(\mathcal{C}_1, \mathcal{C}_2\) of the same underlying physics, the rates add: \(\rho_{1+2}(t) = \rho_1(t) + \rho_2(t)\). The brake exponent of the composite cascade is determined by the dominant component; sub-dominant components contribute \(\sigma\)-dispersion.
Theorem 6 — Shadow admissibility
A candidate shadow set \(\{\mathcal{S}_\alpha\}\) of cascade \(\mathcal{C}\) is admissible iff: (S1) each \(\varepsilon_\alpha\) has median 0 against \(\mathcal{C}\); (S2) the \(\varepsilon_\alpha\) are pairwise independent; (S3) finite spread. Corollary 6.2: Failure of (S1)–(S3) by physical-shadow-mismatch is honest and reported through \(\mathfrak{A}\) (Theorem 10), not papered over.
Theorem 7 — Cascade containment under sub-sampling
If \(\mathcal{C}'\) is a regular sub-sampling of \(\mathcal{C}\) at cadence \(c\), then \(\mathcal{B}(\mathcal{C}') = \mathcal{B}(\mathcal{C}) + \delta(c)\) where \(\delta(c)\) is the cadence-error term, computable as the curvature of \(g(\log|\Phi|)\) under the sub-sampling kernel. Establishes that cadence-stamps must accompany every brake reading.
Theorem 8 — Forced-rate degeneracy
When the cascade carries an external forcing such that \(\rho\) is determined by the forcing rather than by \(\Phi\), the brake form \(\rho \propto \Phi^\beta\) admits no unique \(\beta\). The framework must declare forced-rate scope through \(\mathscr{A}\) and decline a brake call (Law V).
Theorem 9 — Cascade-class vs resonance-class discrimination
For any cadence-stamped \(\mathcal{C}\), the spectral primitive \(\mathcal{P}\) decomposes into \((\mathcal{C}_{\text{brake}}, \mathcal{C}_{\text{resonance}}, \mathcal{C}_{\text{residual}})\). The cascade is class-declared cascade-class at this cadence iff \(\mathcal{C}_{\text{brake}}\) admits \(\mathcal{B}\) with \(R^2\) above threshold. Otherwise resonance-class. The Mid-Pleistocene Transition (instance 22) is class-declared resonance-class at the kyr cadence.
Volume II — Composition theorems (T10–T13)
Theorem 10 — Anti-shadow detection
A real-valued joint-admissibility score \(\mathfrak{A}\) detects when a candidate shadow set passes Theorem 6 member-wise yet collectively shadows different cascades. Formally: \(\{\mathcal{S}_\alpha\}\) is jointly admissible ⟺ \(\mathfrak{A}(\{\mathcal{S}_\alpha\}; \mathcal{C}) \le \tau_{T3}\). Empirically anchored by the stratospheric-ozone instance (instance 25): hole-minimum and zonal-mean each pass T6 individually, but \(\mathfrak{A} = 22.4 \gg \tau_{T3}\) — honest physical-shadow-mismatch.
Theorem 11 — Cross-shadow convergence rate
The universality call \(\sigma_{\text{cross}} \to 0\) follows the rate \(\sigma_{\text{cross}} \sim 1/\sqrt{K \log N_{\text{eff}}}\) where \(K\) is the per-shadow log-range. Wide log-range drives \(\sigma_{\text{cross}}\) sharper. KPZ universality reaches \(\sigma_{\text{cross}} = 0.004\) at \(M = 3\) shadows by virtue of many decades of \(W(L, t)\) growth.
Theorem 12 — Honest-scope meta-theorem
Theorem 12
Every framework finding admits a uniformly constructible scope-reporter \(\mathscr{A}\): the four-tuple (tested cascade/shadows/cadence/operator-chain, excluded conditions, \(\tau_{T3}\) precision floor, gauge group). No framework finding is publishable without simultaneous publication of its \(\mathscr{A}\).
Theorem 12 makes Law V operational. Each catalogue instance carries its \(\mathscr{A}\); the uniform self-reporting is what makes the catalogue framework-admissible.
Theorem 13 — Spectral primitive uniqueness
The decomposition \(\mathcal{P}(\mathcal{C}, c) = (\mathcal{C}_{\text{brake}}, \mathcal{C}_{\text{resonance}}, \mathcal{C}_{\text{residual}})\) is unique up to the gauge group \(G_{\mathcal{P}} = \text{rescaling} \times \text{basis-rotation} \times \text{mode-reordering}\). Frequency-reparametrisation is excluded from \(G_{\mathcal{P}}\): the 41-kyr-vs-100-kyr distinction at the Mid-Pleistocene Transition is the framework-internal statement that frequency information is load-bearing.
Domain instances of the theorems
| Instance | Theorem | Reading |
|---|---|---|
| NS spectral energy (45 cells) | T1 | \(\beta \approx 0.94\), finite-time return to rest |
| KPZ universality (1+1D) | T11 | \(\sigma_{\text{cross}} = 0.004\) at \(M = 3\) shadows |
| LIGO GW150914 | T2 | cross-detector consensus discriminates signal |
| NANOGrav 15-yr | T1, T3 | \(\gamma = 4.11 \pm 0.39\); SMBH-binary regime |
| Tohoku 2011 aftershocks | T1 | \(p = 1.184\) (Type-II, slow asymptotic decay) |
| Stratospheric ozone | T10 | \(\mathfrak{A} = 22.4\) anti-shadow detection |
| Mid-Pleistocene Transition | T9, T13 | resonance-class declaration; 41-kyr → 100-kyr |
| Solar wind | T12 | out-of-scope verdict published in \(\mathscr{A}\) |