This notebook walks each numerical claim PPM makes and shows the math that produces it. Where the geometry does substantial work, the computation is written out inline — a reader sees the heat-trace sum, the topological routes, the closed-form formulas evaluated step-by-step. Where a value is a one-line algebraic consequence of values shown above, it stays a one-line check. Each section closes with a card comparing the result to observation: predicted, observed, percent deviation, paper anchor, function call. Six convergence and robustness probes are inlined for the load-bearing computations.
The math itself lives in core/ppm-technical.tex — references like §T.11.4
point into the chapter and section structure of the Technical Reference.
predictions.ipynb is the engagement companion: sliders, exposition,
falsification framing. This notebook stays close to the data.
Where referenced, nSM_params = 26 is the standardized count under the νSM
Dirac convention.
Imports, ppm package modules, and the rendering helpers.
ppm package loaded.
ppm.predictions.build_table() is the package's authoritative registry. Each
row is tagged by what kind of comparison it encodes:
The chart filters to empirical rows only, so the spread shown is honest PPM-vs-nature deviation. The table below carries all 43 rows with the type column visible — identity and derived rows would otherwise inflate the chart with synthetic 0% matches.
Of 43 total registry entries: • 19 empirical (shown above) — PPM vs measurement • 6 identity — structural LHS vs RHS claims • 18 derived — closed-form values, ranges, or bounds All 43 rows appear in the table below; the chart shows only empirical comparisons.
The arena is fixed once: $\mathbb{CP}^3$ with the antiholomorphic involution $\tau$. Its scalar Laplacian carries spectral data — a zeta value at $s=0$ and the Seeley–DeWitt heat-kernel asymptotics — that feed downstream into the fine-structure derivation in §T.11 and the prediction registry in §T.15.
a₀ = 0.0208333333 a₂ = 0.0833333333 a₄ = 0.1666666667 a₆ = 0.2243386243
The variational structure introduced in §T.2 has two closed-form children that appear in the registry: the Higgs quartic $\lambda$ and the top Yukawa $y_t$. Both are derived in §T.9 (Particle Masses) and feed PRED.4 (Higgs VEV) and PRED.5 (top mass). The closed forms are short — one line of arithmetic each.
Electroweak symmetry-breaking sector. The Higgs VEV and W mass live here; the weak coupling is included for completeness; the Weinberg angle row is grouped here for sector affinity but anchored to §T.8 where the geometric value $3/8$ is derived as a topological ratio, with §T.7 carrying the SM running down to $M_Z$.
The three-generation structure of the Standard Model has no SM-internal explanation. PPM grounds the count in $\mathbb{CP}^3$ topology via the Euler characteristic.
Per-event entropy production sets the thermodynamic accounting for the actualization process and underwrites the framework's thermodynamic predictions.
The hierarchy is geometric: a single coupling $g = 2\pi$ generates an $\exp(-g)$ ladder that places the Planck, Pati-Salam, EWSB, and confinement scales at integer (and half-integer) levels $k$. The first row inlines the two independent topological derivations of $g = 2\pi$. The boundary capacity $N_\infty = \varphi^{392}$ (DER.17) and the $T^2$ partition function value (DER.6) follow with their actual derivation sites annotated. The cascade table closes the panel.
The bridges are six master relations linking the arena's Hodge data, the fundamental coupling, the boundary capacity, and the fine-structure constant. Two ledger rows live here: the master self-consistency identity computed LHS vs RHS, and the bridge sum rule expressed as the explicit V₄-orbit sum. Convergence probe #3 inlines the precision of $(2\pi)^{27}\sqrt{\alpha} \approx \varphi^{98}$.
LHS = (2π)^27 · √α = 3.0307298661e+20 RHS = φ^98 = 3.0254413932e+20 LHS − RHS = 5.2885e+17 Mismatch = 0.174800% Match within 1%: True
Top quark mass from $y_t$ and $v$, charged-lepton mass ratios, Higgs mass. Light-quark masses are an open item: the wall-sector mass-ratio mechanism is identified but the first-principles wavefunction overlaps that fix $|z|$ values are not yet derived.
The CKM and PMNS matrices, the strong-CP angle, the CP phase, and the proton lifetime gate. Convergence probe #5 inlines the sterile-neutrino Berry-K sensitivity demo.
The load-bearing chapter, and the marquee inline computation. Three routes converge on $\alpha$. The first cell below performs Route I from primitives: the CP³ Laplacian eigenvalues, the τ-trace and full-degeneracy multiplicities, the half-variance window $t^* = 1/32$, and the heat-trace ratio that is $\alpha$. The reader can watch $\alpha$ emerge from CP³ representation theory line by line. Route II (cogito self-consistency) and Route III (instanton) follow with their respective receipts. The pyramidal identity (DER.4) and instanton triplet (DER.1, DER.2, DER.3) close the panel as integer arithmetic.
Stable to ≥4 sig figs by nmax = 200; converges essentially instantly.
The spectral-gap window t* = 1/(2(n+1)²) with n=3 picks 1/32. 1/α ranges from ~267 to ~68 across the surrounding t* values, so the half-variance condition is selecting the answer, not happening to land near it.
S = 30π = 94.247780
R_τ_bare = (1/2)·e⁻³⁰π = 5.857056e-42
α_observed = 7.297353e-03
Prefactor gap = α / R_τ_bare = 1.246e+39
(≈ 10^39.10)
Probe #2 above is the receipt: outside this window, 1/α takes very different values.
Newton's $G$, the cosmological constant $\Lambda$, the Hubble parameter $H_0$, the sterile-neutrino dark-matter mass window, the dark-energy equation of state, and the gravitational-wave dispersion coefficient. The $G$ and $\Lambda$ formulas are evaluated step-by-step inline so the reader can watch the boundary capacity $N_\infty = \varphi^{392}$ enter through $\sqrt{N_\infty}$ and $1/N_\infty$ respectively. Convergence probe #6 inlines the $w_\text{eff}(\varepsilon)$ family showing where the framework leaves $\Lambda$CDM behavior.
Formula: w_eff = -1 + (2/3)(Ω_δ/Ω_DE) where Ω_δ ~ ε
Four consciousness-related quantities the framework derives: the hierarchy level $k_\text{conscious}$ where geometric energy matches body-temperature thermal energy, the integration time at that level, the integrated information $\Phi$ for an awake brain, and the predicted scaling exponent of $\Phi$ with system size.
The 42-check self-consistency suite. The same suite runs in CI on every
commit and via python -m ppm.verify locally; this button is a convenience
for inspecting it interactively. Output is grouped by sector (geometric
constants, instanton, gauge, cosmology, quantum dynamics, active inference,
Kähler spectrum, …) with a one-line description of what each sector tests
and the per-check status and percent deviation.
Coverage statement and cross-references back to the canonical sources of truth.