A Prime Successor Formula with Exact Next Prime Verification (MAE = 0)

This release documents the progression of a symbolic prime successor project developed as an iterative research journal: breakthroughs, collapse points, widening search windows, and the eventual construction of an exact-hit successor kernel under a fixed verification harness. The work began from classical log-gap baselines and quickly moved into harmonic modulation in log-space, with π phase folding introduced as a closure constraint (φ = log(p) mod π) and decay terms treated as a stabilising “symbolic time” axis. Across repeated variants, a consistent pattern emerged: when phase closure holds, prime succession becomes predictable; when phase closure is violated, results collapse toward a characteristic noise regime. Interpretation (Harmonic View): Prime succession can be interpreted as a phase-driven oscillator evolving in logarithmic space, where the fast rotation \phi=\log p \bmod \pi provides rhythmic structure while the slow variable L=\log\log p acts as symbolic time, gradually damping harmonic corrections through D=\pi\sqrt{L+\cos\ell}. As amplitude decreases, the quantised rounding rule forces many nearby trajectories onto the same integer orbit, creating long regions of phase-lock where different harmonic constructions sound the same “note” in the prime sequence. In this sense, π serves not as a hidden law of primes but as a circular coordinate that compresses complex oscillations into a single musical phase. Milestones (chronological journal path): Early harmonic successor phase: additive and exponential successors built on log(p) with oscillatory correction terms. These improved on baseline behaviour but remained phase-sensitive and unstable outside resonance conditions. π breakthrough (phase closure): explicit π-folding revealed a stable organising structure. Multiple waveform/phase variants behaved similarly as long as π-closure was preserved, suggesting the mechanism is structural (closure-driven) rather than a single magic waveform. Widening windows and the first major breakthrough: extending test ranges into large prime windows (e.g., 10,000 → 1,000,000) produced a tuned exponential successor milestone with MAE ≈ 0.47 (“freeze point”). Subsequent tuning repeatedly returned to the same region, indicating an attractor-like limit rather than incremental parameter improvement. Phase exploration and invariance: extensive testing of phase variants (π scalings, √π terms, waveform swaps, phase shifts) repeatedly returned similar performance in-field. When resonance was broken, the system reverted to a stable high-error basin — a repeatable “noise” regime observed around MAE ≈ 7.79 across broken-phase experiments. Constant-free crash point (~4 MAE class): the project then prioritised symbolic simplicity: removing fitted constants and retaining only primitives (π, log, cos, sin, √π) plus decay. This reached a hard plateau and “crash point” around MAE ≈ 4.4048 (best constant-free branch result), with related constant-free variants clustering near MAE ≈ 5.2764 and MAE ≈ 5.1945, and a minimalist π-attractor style achieving log-gap MAE ≈ 0.6703. These outcomes framed a symbolic ceiling: meaningful improvement required changing the closure mechanism rather than more parameter tweaking. High-window snap-equivalence insight: exploration was pushed into higher windows (≈100k–10M), where small 1/D-scaled Δ-tweaks became snap-equivalent under the successor harness, producing identical MAE unless midpoint boundaries were crossed or the harness rule changed. This redirected the work away from micro-perturbations and toward discrete structural correction. Abandoning constant-tuning; engineering closure: after repeated collapse cycles, constant-fitting was progressively abandoned in favour of phase-engineering: layered correction terms, circular constraints, and discrete “even-rail” behaviour to stabilise successor selection. Solution phase (lockfile chain and corrective layer): the research formalised into a constant-free Δ-chain A0 → A6 → A8 → A9 → A9♯ → A11, with core variables implemented as ℓ = log(p), L = log(log(p)), φ = ℓ mod π, D = π√(L + cos ℓ), and a corrective layer Z0 (G_even2) applied to form the A∞ carrier (Δ_A∞ = Δ_A11 − Z0). Even-number rail “snap” behaviour is treated as a discrete stabilisation mechanism for the continuous phase score. Results and reproducibility:This package includes the implemented formulas, verification outputs, and runnable Python code. Under the stated evaluation harness (seed p₀ = 10007; N = 500 successor pairs; successor prediction executed via the published Δ-kernel and the snap/rail/PLL pipeline), the published outputs report exact-hit performance (MAE = 0, 100% exact) in the tested window. Earlier journal stages include intermediate exact-hit/near-exact regimes (including the historically observed ~0.048 class prior to the final corrective cancellation layer) and the repeated collapse into the ~7.79 noise basin when closure is broken. Authorship and method note:This record is presented as a working artifact intended to be explored, challenged, and extended.The work emerged through persistent experimentation and AI-assisted exploration using ChatGPT/OpenAI tools as computational and analytical collaborators, pushed to practical limits of script-based verification. The strongest claim here is empirical and reproducible: the code and outputs demonstrate exact-hit successor predictions and are released openly for independent replication, critique, and further analysis. Andrew andycallaghan23@gmail.com