From a Logical Tautology to Eight Forced Theorems: The Recognition Science Derivation of Reality's Core Structure
Machine-Verified • Zero Parameters • Falsifiable
Summary
We show that a single logical tautology—the Meta-Principle (MP), "nothing cannot recognize itself"—forces eight core theorems (T1–T8) that completely determine Recognition Science's mathematical foundation. These theorems establish the unique cost function, golden ratio scaling, eight-tick cadence, and integer quantization with zero tunable parameters.
Abstract
We show that a single logical tautology—the Meta-Principle (MP), "nothing cannot recognize itself"—forces eight core theorems (T1–T8) that pin down the recognition ledger, the unique convex symmetric cost $J(x)=\frac{1}{2}(x+x^{-1})-1$ (with fixed local scale), the golden ratio fixed point $\varphi$ via $\varphi^2=\varphi+1$, an eight-tick minimal update cycle ($2^3$), coverage lower bounds, and integer $\delta$-units ($\mathbb{Z}$). Formalized in Lean 4, these results constitute a machine-verifiable spine which, combined with bridge factorization through units and the exclusivity/inevitability certificates, yields a parameter-free derivation chain MP $\to$ $\varphi$ $\to$ $(\alpha, C_{\mathrm{lag}}) \to$ gravity $w(r)$ with zero tunable constants.
The Eight Forced Theorems
- T1 (Meta-Principle): Logical tautology $\neg\exists r\in \mathrm{Recognition}(\emptyset,\emptyset)$ — the sole axiom
- T2 (Atomic Tick): Exactly one posting per tick, no concurrency — forces sequential updates
- T3 (Discrete Continuity): Zero flux on cycles — double-entry accounting on recognition graph
- T4 (Potential Uniqueness): Potentials unique up to additive constants — gauge freedom per component
- T5 (Cost Uniqueness): $J(x)=\frac{1}{2}(x+x^{-1})-1$ uniquely — only function satisfying symmetry, convexity, normalization
- T6 (Eight-Tick Minimality): $T_{\min}=2^D=8$ for $D=3$ — exact cover without aliasing
- T7 (Coverage Bound): $T<2^3$ cannot cover all classes — information-theoretic Nyquist limit
- T8 (δ-Units): Ledger increments $\cong\mathbb{Z}$ via $n\mapsto n\delta$ — integer quantization
Mathematical Framework
The eight forced theorems operate on fundamental logical principles:
- Logical Tautology: "Nothing cannot recognize itself" — $\neg\exists r\in \mathrm{Recognition}(\emptyset,\emptyset)$
- Sequential Ledger: Atomic ticks with exact temporal ordering
- Conservation Laws: Zero flux on cycles via double-entry accounting
- Golden Ratio Scaling: $\varphi = \frac{1+\sqrt{5}}{2}$ as unique fixed point
These yield dimensionless parameters:
- Fine Structure Constant: $\alpha = \frac{1-\varphi^{-1}}{2}$
- Lag Constant: $C_{\mathrm{lag}} = \varphi^{-5}$
- Cost Function: $J(x) = \frac{1}{2}(x+x^{-1}) - 1$ (unique)
- Eight-Tick Period: $T_{\min} = 2^3 = 8$ ticks minimum
Machine Verification
Every theorem is formally verified in Lean 4:
Constructive Proofs
- Meta-Principle uses only empty-type elimination (no classical axioms)
- All eight theorems follow constructively from MP
- Complete machine-checked proof spine with #eval reports
Lean 4 Repository
- Full source code: github.com/jonwashburn/reality
- Runnable verification via
lake exe ok - Individual theorem checks with #eval endpoints
Parameter-Free Derivation
The complete chain contains zero adjustable parameters:
Implications
This work establishes Recognition Science as the unique complete framework under logical necessity. Starting from the single tautology "nothing cannot recognize itself," eight theorems follow necessarily, determining the unique structure that any self-consistent reality must have.
The mathematical chain is unbreakable: from pure logic to physical predictions with zero free parameters. Unlike theories with adjustable constants, Recognition Science is maximally falsifiable—every prediction is either precisely correct or the entire framework fails. This vulnerability is a strength: survival of rigorous testing validates logical necessity, not empirical flexibility.
Empirical Tests & Falsification
With zero parameters, the framework provides crisp falsification criteria:
- α⁻¹ Audit: Derive $\alpha=(1-\varphi^{-1})/2$ to full precision vs CODATA
- Rotation Curves: Test $w(r)$ form against galaxy data (ILG vs $\Lambda$CDM)
- Eight-Tick Signatures: Search for neutral window invariants in time-series
- K-Gate Equality: Verify $K_A = K_B$ for independent computational routes
- Bridge Identities: Test $c=\ell_0/\tau_0$, $\hbar=E_{\mathrm{coh}}\tau_0$, etc.