The Numerical Physics Ledger

Status and scope

The Numerical Physics Ledger is a Panta Rhei publication artifact. It records the program’s current internal numerical commitments, calibration cascade, prediction inventory, and falsification targets.

It does not imply external validation, scientific acceptance, or independent confirmation. Its purpose is to make the numerical claims inspectable, comparable, and falsifiable.

The calibration cascade (the architectural spine)

The entire numerical scorecard rests on two inputs:

• One algebraic posit: ι_τ = 2/(π + e) ≈ 0.341304
• One SI measurement: the neutron mass m_n = 1.674927498(95) × 10⁻²⁷ kg

From these two inputs, a four-layer cascade compiles every other quantity:

• L0 — Algebraic layer: ι_τ, κ_D = 1−ι_τ, κ_ω = ι_τ/(1+ι_τ), √3 lemniscate capacity, continued-fraction window sums W_n(k).
• L1 — Dimensionless layer: all mass ratios, all mixing angles, all couplings. Key closed forms: α = (11/15)²·ι_τ⁴ = (121/225)·ι_τ⁴ at 9.8 ppm vs CODATA; R₀ = ι_τ⁻⁷ − √3·ι_τ⁻² is m_n/m_e (neutron/electron ratio) at 7.7 ppm leading order; Koide lepton ratio Q = 2/3 exact at −9×10⁻⁶; sin²θ_W from κ_A/κ_B at ~2.7% (tree); α_s from κ_C(3) at ~2.4%; α_G = α¹⁸√3(1 − 3α/π). These values match Book IV §73’s Predictive Record (Theorem iv-predictive-record) line for line.
• L2 — Single SI anchor: m_n (CODATA 2018). One measurement, theory-forced by the No Knobs Theorem.
• L3 — SI-anchored layer: m_e, G, m_P, ℏ, k_B, ε₀, μ₀ — obtained by the linear rescaling functor M_SI = R_M[M_τ] = M_τ · (m_n^SI / m_n^τ).
• L4 — Verification layer: the 30-item falsification pack N1–N30 + all observational predictions across particle physics, atomic spectroscopy, cosmology.

Three precision bands characterize the predictions: Precision-band (≲ 10 ppm τ-side residual; includes Koide Q exact, R = m_n/m_e at 7.7 ppm, α at 9.8 ppm); Identity-band (∼1–3 per cent τ-side; Weinberg and strong self-coupling at tree level); and Observation-capped (capped by the experimental floor: r, n_s, H₀). A single “0.025 ppm” slogan was used in earlier internal drafts; it is dropped here in favour of per-row residuals, matching Book IV’s Predictive Record exactly. Chapter 58a of the companion contains the complete self-contained cascade architecture.

What the Ledger contains

• Three τ-structural pillars (topologically forced, no fitted parameters):

  gen

  =3 from first homology; θQCD=0 from SA-i mod-3 admissibility; the Cabibbo-complement structure λC=ιτ(1−ιτ).

• A two-tier coherence structure distinguishing global holonomy on H_∂[ω] (which sets cosmic-mean densities like ω_m, ω_b, ℓ₁) from local matter-phase coupling (which sets galaxy-scale dynamics via a trinary D / ω / B sector decomposition).
• A refined BTFR master formula (V.T85) with τ-natural coupling coefficients, tested on the full SPARC-175 rotation-curve sample (Lelli, McGaugh & Schombert 2016).
• A 30-item falsification pack (N1–N30) naming specific τ predictions, experimental instruments, and decisive thresholds for 2025–2035.
• Seven falsifiable seams where τ and orthodox physics make structurally incompatible predictions.
• A tier-1 τ-native holographic structure sketched as a research programme (not mathematically reduced to AdS/CFT; distinct in rank, boundary topology, and metric structure).
• Honest uncertainty bands propagated from integer-choice sensitivity across the 30-item pack, with a per-prediction audit in Appendix 68.A.

Open research targets (stated in the Ledger)

• Pointwise RAR discrimination between τ-natural BTFR coefficient pairs, testable on SPARC-175 HI-dominated dwarfs.
• Elliptical-galaxy BTFR test on ATLAS³D + SLUGGS samples.
• A τ-native CAMB / CLASS Boltzmann pipeline with explicit two-tier source assignment.
• A pre-registered numerical σ₈ prediction from tier-1 holonomy invariants ahead of CMB-S4 and Euclid data.
• Derivation of the central charge, Zamolodchikov c-theorem analogue, and boundary-cut area law for the tier-1 holographic dictionary.
• First-principles derivation of the D / ω / B sector coupling structure from the character theory on H_∂[ω].
• Bullet Cluster lensing-to-baryon offset computation within the capacity-gradient formalism.

Falsification targets on experimental timelines

• d_n = 0 at nEDM/n2EDM (PSI, ~2030)
• r = ι_τ⁴ ≈ 0.0136 at CMB-S4 / LiteBIRD (~2030)
• Σm_ν = 0.089 eV at CMB-S4 + KATRIN / Project 8 (~2032)
• No fourth-generation fermions at LHC Run 3 + FCC (~2040)
• Proton stability at Hyper-K / DUNE (~2040+)

The predictions are bridge-independent: they can be compared directly to experimental measurement without accepting the program’s deeper ontological claims. The number either agrees with experiment or it doesn’t.

Companion to Book V

The τ Numerical Physics Ledger

209 pages · 12 chapters · 4-layer calibration cascade · 3 τ-structural pillars · 30-item falsification pack (tier-tagged A/B/C) · 7 falsifiable seams · two-tier coherence structure · tier-1 τ-native holography (sketched)

Dr. Thorsten Fuchs & Anna-Sophie Fuchs · Second Edition · 2026

Twelve chapters across five physics domains

Precision hierarchy

Tier	Count
Sub-10 ppm	15
10–1000 ppm	11
1–5%	17
Structural	24

Sub-10 ppm (τ-side) includes: Koide Q exact (−9×10⁻⁶), R = m_n/m_e at 7.7 ppm, α at 9.8 ppm, Higgs mass (+8.0 ppm), Silk damping (+9 ppm); spectral index n_s at +13 ppm, Hubble depth-dependent readout (Book V ch45). Per-row residuals matched to Book IV's Predictive Record (Theorem iv-predictive-record).

Falsification Pack (N1–N30)

30 predictions with named experiments and 2025–2035 timelines. The sharpest discriminant: N9 (tensor-to-scalar ratio r = 0.0136), testable at 14σ by CMB-S4.

Seven falsifiable seams where τ and orthodox physics make structurally incompatible predictions — BH topology (T² vs S²), no dark matter (Sector Exhaustion vs WIMP/axion), w₀ vs Λ equation of state, r = ι_τ⁴ vs slow-roll inflation, a ⁷Li framework account (S = 1/3) vs the orthodox lithium anomaly, θ_QCD = 0 (no axion) vs strong-CP tuning, and the holonomy ratio ℓ_D/ℓ_1 as a structural invariant. Twelve of the 30 pack entries carry seam tags in _data/falsifications/falsifications.json (mapping to _data/seams.yml labels); the paired CMB/inflation seam N9 + N10 is the sharpest at ~14σ. Full seam-annotated card grid on Falsifications browse.

Experimental Timeline

• 2025–2027 — DESI, LZ/XLZD, nEDM, ADMX
• 2027–2030 — JUNO, DUNE, nEXO, LEGEND, CMB-S4
• 2029–2033 — ngEHT, BICEP Array, SKA
• 2032–2035 — Einstein Telescope, LISA

Current score (falsification pack, from _data/falsifications/falsifications.json): 4 internally matched to current public data, 26 consistent with current public data, 0 falsified. 30 total. These are program tracking labels, not external acceptance labels.

Inside the document

Interior spread showing the BBN light-element derivations with theorem environments, scope labels, and comparison to observational data.

Interactive versions on this site

The predictions and falsifications are also available as interactive, filterable collections with per-item detail pages:

• Predictions Browse — 67 cards with domain/precision/scope filters, each linking to a detail page with the explicit τ-formula and derivation prose
• Falsifications Browse — 30 cards with domain/status filters, each with named experiments and current observational status

Why this document matters

If even a fraction of the sub-10 ppm predictions hold under independent scrutiny, this would be the first zero-parameter derivation of fundamental constants from a single structural constant — a result with no precedent in theoretical physics.