Foundational Constant
The Recognition Length (λ_rec)
The single length that anchors dimensionless logic to physical units
Why this constant matters
A framework is parameter‑free when its laws and dimensionless numbers are fixed by proof, not tuned to fit data. But any contact with the laboratory still requires one bridge from pure, unitless mathematics to meters and seconds. In Recognition Physics that bridge is the Recognition Length, λ_rec — the spatial quantum of a single recognition hop.
Before λ_rec was derived from first principles, the framework’s logic fixed all dimensionless relations, but its dimensional displays were anchored to an external constant (e.g. G). Deriving λ_rec internally was the final step that removed that last dependence. After this, every quantity shown here descends from the eight axioms with no adjustable numbers.
What “parameter‑free” really means
- No fit knobs. Functional forms and dimensionless constants follow from the ledger calculus. There is nothing to twiddle after you see the data.
- One bridge to units. You still need a single conversion between the dimensionless ledger and SI. In our presentation, that anchor is λ_rec.
- Over‑constraints. The same gap and curvature objects reappear across domains (α, spectra, cosmology), leaving no room for independent tweaks.
Two routes to λ_rec
1) Early empirical anchoring (historical)
In the development phase, we inferred a consistent micro‑length by comparing the framework’s global constraints to reality (stellar luminosity balance and vacuum energy density). Those routes yielded the same λ_rec within a few percent and worked as sanity checks — but they left a philosophical loophole: an apparent external dial.
2) Axiomatic derivation (final)
The loophole closes by extremising the ledger cost for a one‑bit recognition in a minimal causal diamond. Two additive pieces compete:
- Bit cost: the unit ledger cost to store a single recognition.
- Curvature cost: the geometric cost of the curvature packet induced by that recognition over a diamond of edge λ.
The balance condition “bit cost = curvature cost” fixes λ without reference to astronomy or cosmology. In SI units this yields a Planck‑scale length.
Result (units restored)
λ_rec ≃ √(ħ G / c³) (Planck length)
≈ 1.616 × 10⁻³⁵ m
Remarks: A closely related normalisation appears if one averages the curvature packet over faces, introducing a factor of π in the denominator,
λ_rec = √(ħ G / (π c³)). We document both conventions; our baseline pages use the standard Planck form and track normalisation choices on the
Proofs page.
The full extremal argument is outlined in the technical note and will be formalised in Lean. See papers and the working draft.
Consequences and cross‑checks
- Speed of light: with one hop per tick,
c = λ_rec / τ_rec. - Planck’s constant: the action quantum follows as
ħ = E_coh · τ_rec / φ. - Gravitational constant:
G = λ_rec² c³ / ħ, expressing G as a composite of recognition units. - Scale closure: dimensional predictions across pages depend on the same λ_rec; change the anchor and all dimensional numbers co‑move, while dimensionless predictions stay fixed.
Status and open technical notes
- Lean mechanisation of the extremal proof (bit vs curvature cost) is in progress.
- Curvature normalisation (face‑averaged vs packet) is tracked explicitly in constants pages; numeric displays note the choice.
- Legacy empirical routes now function only as consistency checks, not inputs.