Axiomatic Bridging
Our method for carrying invariants from the Recognition ledger (balance, symmetry, fairness) into classical mathematics. It is the "translator" that turns ledger constraints into theorems usable in analysis, optimization, and dynamics.
A tautology-to-theorem pipeline. Ledger constraints (double-entry, unitless ratios, cost symmetry) become mathematical laws (conservation, dimensionless invariants, unique even convex cost).
Why it matters: It explains why classical structures (least action, Laplacians, Herglotz/Schur positivity, Jensen convexity) are not empirical accidents, but the only options compatible with recognition's balance rules.
Core outcomes:
- Cost uniqueness on ℝ>0:
J(x) = ½(x + 1/x) − 1as the unique symmetric, unit-normalized benchmark - Golden-ratio fixed point via self-similar scaling and ledger recursion
- Eight-tick minimal traversal on a 3D parity complex (discrete flux quantization)
- Gauge-rigidity: laws depend only on dimensionless ratios (quotient by units)
RDM (Recognition Dynamics & Measurement)
The mathematical skeleton of how recognition "moves" (dynamics) and how it "counts" (measurement). It encodes updates on pattern spaces under ledger constraints and makes the cost J and scaling laws concrete.
A dynamics-and-measure framework on pattern spaces with: ledger balance (double-entry) → conservation, symmetric cost (J) → uniqueness and convexity on the log-axis, and self-similarity → golden-ratio scaling.
Why it matters: It justifies the exact functional forms used across our physics (e.g., J on ℝ>0 and its log-cosh model), ties discrete ticks to continuous limits, and explains why "unit-free" statements are the stable ones.
Key results:
- J-agreement on the exp-axis via Jensen/convexity → J is the unique benchmark
- Even/strict-convex log models become cost functionals compatible with symmetry and balance
- From parity coverage to eight-tick minimality → discrete traversal lower bounds
Core Formulas
The minimal set of ledger‑derived relations that recur across the theory and connect logic to measurement.
A compact toolkit: dual‑balance cost, φ‑ladder mass law, and curvature regulator; each is fixed by invariance and balance, not by fitting.
- Dual‑Balance Cost:
J(x)=\tfrac12(x+1/x)— unique, symmetric, unit‑normalized benchmark - Mass Spectrum:
m = B \cdot E_{coh} \cdot \varphi^{r+f}— runged φ‑ladder with small ledger correction f - Ledger Curvature:
\kappa = \partial^2 S / \partial R^2— universal regulator for discrete→continuum transitions
Riemann (Unconditional Route)
CompleteWe develop a bounded-real (Herglotz/Schur) route to RH on the half-plane Ω = {Re s > 1/2}. The core object is Θ(s), a Cayley transform of a det₂/ξ ratio built from the prime-diagonal operator A(s). Proving |Θ| ≤ 1 (Schur) on Ω is our target.
A stability/contractivity translation of RH:
• det₂(I − A)/ξ → J(s), Θ(s) = (2J − 1)/(2J + 1)
• Contractivity (|Θ| ≤ 1) ⇔ Pick kernel PSD ⇔ Herglotz real part ≥ 0
Why it matters: It replaces "count the zeros" with "certify stability," enabling modular, auditable proofs via positivity and passivity.
The bridge:
- Head–tail split: finite k=1 + archimedean block (exact) vs HS prime tail (continuous)
- Prime-grid lossless realizations with explicit KYP certificates (passivity with diagonal witnesses)
- Alignment + Cayley bounds → finite-to-infinite limit control
Unconditional boundary route:
- Smoothed estimates + de-smoothing for ∂σ Re log det₂(I − A) and ξ
- Poisson–Carleson (P+) certificate with explicit constants → interior Herglotz → Schur/PSD
P vs NP (Recognition Perspective)
We approach P vs NP via the Recognition lens: certificates are "recognized structure," and recognition costs follow the unique J-based calculus and ledger constraints. The aim is to turn "search vs proof" into "enumeration vs recognition with conserved cost."
A program to frame NP certificates as recognition artifacts with bounded recognition cost, and to ask when recognition can be made algorithmic at P-cost.
Why it matters: It reframes NP's asymmetry not as a quirk of Turing models, but as a deep asymmetry between constructing structure and recognizing balanced structure under ledger constraints.
Working hypotheses:
- Proof-as-compression: short certificates reflect intrinsic redundancy; recognition cost is controlled by J and log-convexity
- Ledger-bounded circuits: dimensionless invariants and fairness bounds yield forbidden "free gains," obstructing certain collapses
- Recognition vs enumeration gap: translate into convex/positivity obstructions for uniform families
Near-term steps:
- Formalize recognition cost bounds for canonical NP languages with J-compatible encodings
- Prove no-go inequalities for "free-gain" transformations under gauge-rigidity
- Prototype "certificate-carrying evaluation" where verification is a ledger-preserving contraction