PPM Framework: All Predictions, Computed Live¶

Every number below is computed from first principles by the ppm Python package. Click show implementation under any prediction to see the function source.

What PPM is. Projective Process Monism is a geometric framework whose entire ontology is one space and one symmetry: complex projective 3-space $\mathbb{CP}^3$ with the Fubini–Study metric, equipped with a $\mathbb{Z}_2$ involution $\tau$ (complex conjugation). The fixed-point set $\mathrm{Fix}(\tau) = \mathbb{RP}^3$ is actuality; the rest of $\mathbb{CP}^3$ is possibility. Every $\tau$-projection event maps a possibility-state onto $\mathbb{RP}^3$, stripping one $\tau$-odd degree of freedom and recording one bit of boundary geometry. Particles are eigenmodes of the resulting boundary spectrum; coupling constants are ratios of geometric quantities on $(\mathbb{CP}^3, \tau)$; gravity arises from the cumulative information loss across many $\tau$-projections; consciousness is the regime where the same projection dynamics support a sustained variational descent on a free-energy functional. This single arena is supposed to derive the Standard Model and consciousness-scale physics.

The empirical anchor. $m_\pi \approx 134.977$ MeV (neutral pion) sets the energy scale. There is one electroweak matching condition that fixes $k_{\rm EWSB} = 44.5$ (any of the Higgs VEV, top mass, or $W$ mass works as the matcher — they all give the same value). Everything else is derived from the geometry of $(\mathbb{CP}^3,\, \tau,\, g_{\rm FS})$.

Where the framework is strongest. Sub-1% matches that come out of the geometry alone, with no per-particle fitting: $\alpha$ (fine-structure constant, 0.16%), $\Lambda$ (cosmological constant, 1.5%), $\sin^2\theta_W$ (weak mixing angle, 0.13% after running), $m_t$ (top quark, 0.15%), $\delta_{CP}$ (CKM phase, within experimental error). Three cosmological observables — $H_0$, $\Lambda$, $G$ — all route through one shared denominator $N_\infty = \varphi^{392}$, which means they move together if any one is wrong. These are the strongest claims and the ones to scrutinize first.

Where the gaps are. The Higgs mass comes out 4.4% high (the discussion in that panel argues it's the tree-level geometric value with higher-order corrections still to be added; whether that holds up is open). The $\tau/\mu$ mass ratio is 6.3% off; atmospheric neutrino mixing $\sin^2\theta_{23}$ is 8.4% off; the leptonic mixing angle $\theta_{13}$ comes out as zero in the framework's zeroth-order PMNS prediction but has been measured nonzero at high significance. Light quark masses sit on the cascade curve by inverse derivation rather than first-principles prediction. None of this is hidden; the panels flag the gaps where they appear.

How to break the framework. The How to Break It section near the end exposes the framework's structural inputs — $k_{\rm EWSB}$, the $N_\infty$ exponent, the pion anchor — to perturbation, showing which predictions break first when you move them. If the framework is right, those knobs produce instability when nudged off canonical values; if any knob has more flex than the framework claims, that's a finding.

How this notebook is organized. Scorecard at the top — every prediction with its error against observation. Then the foundational observation that motivates the whole framework. Then deep-dives grouped by what they buy: Standard Model parameters PPM eliminates, cosmology from one shared boundary capacity, where the golden ratio enters structurally, and an active-inference live demo that puts the consciousness side of the framework into a runnable simulation. A How to Break It section closes with sensitivity tests, and a brief comparison to alternative theories sits at the end.

Contents¶

• Scorecard — every prediction with its error against observation
• The Foundation — the two structural observations the framework rests on
• Three Inputs, Five Predictions — interactive sensitivity panel
• Standard Model Inputs — what PPM eliminates from the SM parameter list
• Cosmology from One Boundary Capacity — $H_0$, $\Lambda$, $G$ from one shared denominator
• Where the Golden Ratio Enters — instanton coincidence and pyramidal-number identity
• Active Inference — runnable demo of the framework's variational principle
• Falsification Tests — six structural counterfactuals
• k-Scale Converter — quality-of-life utility for the cascade
• Compared to Alternatives — string theory, asymptotic safety, LQG, EFT

At a Glance — The Scorecard¶

All numbered predictions plus derived quantities, computed live. The error column shows the percent difference between PPM's value and the measured value. Rows marked — are predictions either below current experimental sensitivity ($\alpha_{GW}$, sterile neutrino window) or stated mechanism-only without a tight numerical target (proton lifetime $> 10^{40}$ years).

The summary at the bottom groups predictions by error magnitude rather than by status taxonomy. Internal labels like "VERIFIED" or "FLAGGED" invite overclaiming; an honest summary just says how far off each prediction is from measurement and lets the reader decide.

Deep-dives for individual predictions follow this scorecard, organized by theme.

The Foundation¶

PPM rests on two structural claims. The first is a numerical observation about the geometry of $\mathbb{CP}^3$ and the golden ratio. The second is that three independent cosmological observables share a single boundary-capacity input. Everything else in this notebook is downstream of these two.

One Coincidence: $e^{-30\pi} \approx \varphi^{-196}$¶

Quantum field theories contain instantons — non-perturbative tunneling configurations between topologically distinct vacuum sectors. Each instanton contributes corrections of the form $e^{-S}$ to physical amplitudes, where $S$ is the instanton's action.

In PPM, the relevant instanton lives on a $\mathbb{CP}^3$ that has been embedded into a larger projective space via a degree-3 Veronese map (a standard algebraic-geometry construction that sends $\mathbb{CP}^3$ into $\mathbb{CP}^{19}$ using cubic monomials). This embedding choice forces the instanton action to $S = 30\pi$ exactly. The number of zero modes — 30 — is the real dimension of $PGL(4, \mathbb{C})$, the automorphism group of $\mathbb{CP}^3$. Both numbers are determined by the geometry, not fitted.

The headline result is the numerical identity $e^{-30\pi} \approx \varphi^{-196}$, matching to $0.07\%$ in the exponent. This isn't a derivation; it's the foundational coincidence the framework rests on. The action $30\pi$ is forced; the appearance of $\varphi^{-196}$ on the right is observed. If the identity is real (rather than a numerical accident), it's evidence that the framework's instanton sector and golden-ratio sector are actually one structure seen two ways. If it's accident, the framework is in trouble. Listed openly so the reader can judge.

Show implementation. Function sources for the $e^{-30\pi}\approx\varphi^{-196}$ check, the action $S=30\pi$, and the 30 zero-mode count.

One Capacity: Three Cosmological Numbers from One Boundary¶

PPM's three cosmological observables — the Hubble constant $H_0$, the cosmological constant $\Lambda$, and Newton's gravitational constant $G$ — all share one denominator:

$$ N_\infty = \varphi^{392} \approx 8.4 \times 10^{81}. $$

This number — the static boundary capacity, fixed by the topology and the golden-ratio invariant — appears in all three:

$$ \Lambda = \frac{2(m_\pi c^2)^2}{(\hbar c)^2\,N_\infty}, \qquad H_0 = \frac{c}{\sqrt{N_\infty}\,\lambda_C}, \qquad G = \frac{(2\pi)^4\,\hbar c\,\alpha}{m_\pi^2\,\sqrt{N_\infty}}. $$

Three independent measurements all routing through one structural input is the kind of relationship that's hard to engineer. If PPM's value of $N_\infty$ is wrong, all three predictions move together; matching all three to within a few percent says either the framework has captured something real or three separate coincidences happen to land in the same place. Each observable's deep-dive sits in Cosmology from One Boundary Capacity below.

Move the Capacity → All Three Move Together¶

If $N_\infty$ were anything other than $\varphi^{392}$, all three cosmological observables would shift in lockstep. The slider below moves $\log_{10}(N)$ across an 8-decade range; the orange line tracks where you are; the green line marks the framework's value; the red line is the observed $\Lambda$. Try moving the slider one or two units off canonical and watch how far $\Lambda$ falls from observation. The sensitivity is the structural argument.

Three Inputs, Five Predictions¶

The framework's structural inputs are three: the cascade location of EWSB ($k_{\rm EWSB} = 44.5$), the boundary-capacity exponent ($N_\infty = \varphi^{392}$), and the pion anchor ($m_\pi \approx 134.977$ MeV). The interactive panel below moves all three and shows how the five flagship predictions — Higgs VEV, top mass, $\Lambda$, $H_0$, $G$ — respond.

The structural claim is that these inputs are not independently tunable. Moving $N_\infty$ shifts all three cosmology observables together because they share the denominator. Moving $k_{\rm EWSB}$ moves the EW masses together because they share the cascade position. Moving $m_\pi$ shifts everything because it's the only dimensionful input. If the framework is right, no slider combination other than canonical hits all five within experimental tolerance — which is what makes the framework falsifiable rather than merely curve-fitting.

Bars turn green within 2%, blue within 10%, amber within 50%, red beyond. The scales with column tells you what each prediction depends on, so you can predict which combinations should move together.

What PPM Says About Standard Model Inputs¶

The Standard Model takes about 19 dimensionless parameters as empirical inputs — the gauge couplings, fermion masses, mixing angles, and CP phases — measured directly and inserted by hand into the Lagrangian. PPM's central claim is that most of these aren't free; they're forced by the geometry of $\mathbb{CP}^3$. This section walks through them prediction by prediction, ordered roughly by how strongly the match holds. Strongest first; known weak points listed honestly at the end.

The Fine-Structure Constant¶

The fine-structure constant $\alpha \approx 1/137$ sets the strength of every electromagnetic interaction. PPM produces it as a ratio of two spectral traces on $\mathbb{CP}^3$ — sums over the eigenvalues of the Laplacian, one twisted by the $\tau$ involution to restrict to its $\mathbb{RP}^3$ fixed-point set, one untwisted. The traces depend on a heat-kernel time $t$, and the ratio is taken at $t^* = 1/32$. That choice is forced by the geometry: $t^*$ is the unique time at which the two traces' variances coincide (the "half-variance" point), so it isn't a free knob. Two computable routes through the framework agree with observation to within 0.5% (the heat-kernel computation here, and a $G$-inversion route that uses observed $G$ and $\Lambda$ to back out $\alpha$); a third instanton route is consistent in form but its prefactor is open. This panel runs the heat-kernel route.

Formula. $\alpha^{-1} = \Theta_{\mathbb{CP}^3}(t^*) / \Theta^\tau(t^*)$ at $t^* = 1/32$.

The two heat traces are sums over the Laplacian eigenvalues $\lambda_k = k(k+3)$ on $\mathbb{CP}^3$ (with multiplicities), the τ-twisted version restricting to the $\tau$-invariant sector: $$\Theta_{\mathbb{CP}^3}(t) = \sum_k m_k\, e^{-\lambda_k t}, \qquad \Theta^\tau(t) = \sum_{k\in\text{Fix}(\tau)} m_k\, e^{-\lambda_k t}.$$

Show implementation. Function source for the heat-trace α computation.

Slider: Why $t^* = 1/32$?¶

The choice $t^* = 1/32$ is the unique time at which the variances of the two heat traces coincide. The slider below moves $\log_{10}(t)$ across multiple decades and shows the $1/\alpha(t)$ curve. The green dashed line marks the half-variance point ($t^* = 1/32$); the red dashed line is the observed $1/\alpha = 137.036$. Try other times — $1/\alpha(t)$ runs from a few to several hundred. Only at $t^*$ does it land near 137.

Why $\mathbb{CP}^3$ and not $\mathbb{CP}^2$ or $\mathbb{CP}^4$?¶

The same heat-kernel computation on any other $\mathbb{CP}^n$ gives a wildly different coupling. Only $n = 3$ places $1/\alpha$ near $137$. The full $n=1\ldots 7$ table — with the spectral intermediate steps — lives in derivations §6 ($\mathbb{CP}^n$ Selectivity).

Show implementation: the function source for the CP³-vs-other-CPⁿ heat-trace family scan.

The Weak Mixing Angle¶

The weak mixing angle $\theta_W$ sets how the photon, $W$, and $Z$ bosons mix when the Higgs field condenses — it's the geometric reason the photon ended up massless and the $W$/$Z$ heavy. The Standard Model treats $\sin^2\theta_W$ as an empirical input.

The formula. PPM gives the GUT-scale value as a ratio of two integer-valued topological invariants of $\mathbb{CP}^3$: $$\sin^2\theta_W \bigm|_{\rm GUT} = \frac{\dim(\mathbb{RP}^3)}{2\,\chi(\mathbb{CP}^3)} = \frac{3}{8}.$$

Why this formula. The $3/8$ value isn't novel to PPM — it's the well-known Pati–Salam / SU(5) GUT prediction from gauge-group embedding. What's specific to PPM is the topological reading of what the trace ratios are counting:

• The numerator $\dim(\mathbb{RP}^3) = 3$ counts the directions on the actuality boundary. SU(2)$_L$ acts on left-handed fermion doublets along these three directions, so the SU(2)$_L$ Casimir trace summed over a generation equals $3$.
• The denominator $2\,\chi(\mathbb{CP}^3) = 2 \times 4 = 8$ counts the total $\tau$-reduction exponent of the $(C, D)$ doublet — the topological "weight" of the U(1)$_Y$ coupling integrated over the multiplet, with the factor of $2$ accounting for both $\tau$-eigenvalues (chirality and intensity sectors).
• The general formula $n / (2(n+1))$ for $\mathbb{CP}^n \to \mathbb{RP}^n$ gives $1/4, 1/3, 3/8, 2/5$ for $n = 1, 2, 3, 4$. The framework's choice of $\mathbb{CP}^3$ — three complex dimensions matching observed three-dimensional space — picks out the $3/8$ value.

Interpreted information-theoretically, $\sin^2\theta_W = 3/8$ is the allocation of the $(C, D)$ channel capacity: $3/8$ assigned to chirality, $5/8$ to intensity. The RG running from $3/8$ at the GUT scale to $0.231$ at $M_Z$ is the $k$-dependent flow of this allocation parameter.

The $3/8$ holds at the Pati–Salam unification scale (the energy where the framework predicts the strong, weak, and electromagnetic forces unify, $\sim 10^{13}$ GeV from $E(k = 16.25)$). Standard SM one-loop running carries the value down from $3/8$ at unification to $0.231$ at the $Z$ pole, matching observation to 0.13%.

Formula. $\sin^2\theta_W = \dfrac{\dim(\mathbb{RP}^3)}{2\,\chi(\mathbb{CP}^3)} = \dfrac{3}{2\cdot 4} = \dfrac{3}{8}$ at the Pati-Salam scale. Standard SM one-loop running carries this down to $M_Z$.

Show implementation. Function source for $\sin^2\theta_W$ with SM running.

CP Violation Phase¶

The CKM matrix — the basis change between quark mass eigenstates and weak-interaction eigenstates — contains one complex phase $\delta_{CP}$ that breaks the symmetry between matter and antimatter (one of the Sakharov conditions for the universe to contain more matter than antimatter, even if not by enough to explain the full asymmetry). The Standard Model treats $\delta_{CP}$ as an empirical input. In PPM it's a Berry phase — the holonomy accumulated when the $\tau$ involution is transported around a closed loop on the $\mathbb{RP}^3$ fixed-point set. The loop's geometry is set by the icosahedral group $A_5$ acting on the framework's moduli space, which is the structural reason $\varphi$ appears in the formula $\delta_{CP} = \pi(1 - 1/\varphi) = \pi/\varphi^2$.

This panel predicts the CKM phase. The leptonic analog (PMNS phase) is a separate observable that DUNE will measure directly over the next decade; the framework predicts both come out the same.

Show implementation. Function source for the neutrino CP phase $\delta_{CP}$.

The Higgs Quartic and Top Yukawa¶

The Standard Model has two free dimensionless parameters in the Higgs sector — the quartic self-coupling $\lambda$ (which sets the Higgs mass via $m_H = \sqrt{2\lambda}\,v$) and the top Yukawa $y_t$ (which sets the top mass via $m_t = y_t v / \sqrt{2}$) — both treated as empirical inputs. In PPM both are fixed by the geometry of how the $\tau$ involution acts on $\mathbb{CP}^3$:

• Quartic: $\lambda_{\rm PPM} = 1/(4\sqrt{\pi})$, from the curvature of the normal bundle to $\mathbb{RP}^3$
• Top Yukawa: $y_t = \pi / (2(2\pi)^{1/4})$, from the $\tau$-involution's geometric structure

Result: the top mass comes out at 173 GeV, matching observation to 0.15%. The Higgs mass comes out at 130.8 GeV — 4.4% above the measured 125.25 GeV. The interpretation panel below addresses why.

Show implementation. Function source for the Higgs quartic $\lambda_{\text{PPM}}$.

Slider: Move $k_{\rm EWSB}$¶

The framework predicts the Higgs VEV $v$, the top mass $m_t$, and the Higgs mass $m_H$ from the cascade location of electroweak symmetry breaking, $k_{\rm EWSB} = 44.5$. Move the slider — all three errors blow up together. The framework hasn't fitted three numbers; it's predicted three numbers from one cascade location.

Interpreting the Higgs Mass Prediction¶

The predicted value m_H = 130.8 GeV is 4.4% above the observed 125.25 GeV. This gap reflects a fundamental distinction: the framework computes the tree-level geometric value from the RP³ normal bundle, not the physical mass including quantum corrections.

The observed Higgs mass includes radiative corrections from the top-quark loop and gauge-loop diagrams, which lower it from the tree-level value. This is expected behavior: tree-level predictions typically differ from the measured value by a few percent in the Standard Model. The 4.4% gap is consistent with quantum effects and RG running from the geometric scale (where λ_PPM applies) down to the electroweak scale. No new physics is required to explain this small discrepancy.

The Mass Hierarchy — All Particles on One Curve¶

$k$ is PPM's cascade index: each unit of $k$ steps the energy down by one factor of $\sqrt{2\pi}$. The pion sits at $k = 51$ (the empirical anchor); the Planck scale is at $k = 1$; the electron is at $k = 57$. Light fermions live at large $k$, heavy bosons at small $k$.

The cascade formula $E(k) = m_\pi\,(2\pi)^{(51-k)/2}$ runs as a single curve from the Planck scale to the electron mass — a span of 23 orders of magnitude — with $m_\pi$ as the only dimensionful input. The strong claim isn't that PPM has fitted 14 particle masses; it's that 14 particles all sit on one curve whose shape is fixed by the geometry of $\mathbb{CP}^3$ and whose absolute scale is set by the pion. The plot below makes the claim visible.

Formula. $E(k) = m_\pi\, (2\pi)^{(k_{\text{ref}} - k)/2}$ with $k_{\text{ref}} = 51$ (pion anchor). One curve, all particle masses, 23 orders of magnitude.

Show the math: the formula $E(k)$ is short enough to write out inline. Every value in the table above comes from one line of code.

Show implementation. Function source for the cascade formula $E(k)$.

Visualize. All particles on the single $E(k)$ curve, observed masses overlaid. The framework's claim is that this curve is geometric (no fitting per particle); each k-level comes from CP³ topology, only the absolute scale $m_\pi$ is empirical.

Slider: Why $g = 2\pi$?¶

The cascade scaling factor $g = 2\pi$ comes from the topology — it isn't fitted to the particle masses. The slider below moves $g$ across $\pm 25\%$. The left panel shows the cascade curve overlaid on 14 observed particle masses; the right panel shows RMS error as a function of $g$. The minimum sits exactly at $2\pi$. Move the slider off and 14 particles slide off the curve simultaneously.

Interpreting the Hierarchy¶

The plot makes the structural claim visible: every Standard Model particle sits on (or near) a single curve $E(k)$ that spans 23 orders of magnitude from the Planck scale to the electron, with one dimensionful input ($m_\pi$).

Tier 1 (filled green circles). Five particles where $E(k)$ alone gives the mass: Planck (k=1), tau (k=48), pion (k=51, anchor), muon (k=51.5), electron (k=57). These are the cleanest first-principles predictions — k from CP³ topology, mass directly from the cascade.

Tier 2 (orange triangles). EW sector at the EWSB anchor k=44.5 — top, Higgs, Z, W. Their cascade rung is framework-predicted, but the masses themselves come through electroweak relations ($v = 2\sqrt{2}(2\pi)^{1/4} E$, $m_t = \pi E$, $m_W = g_2 v / 2$, etc.) — see the Higgs cell above for the explicit derivations. Heavy quarks (bottom k=46, charm k=47.5) sit at integer/half-integer cascade rungs but the full mass requires Kähler-radial $|z|$ corrections that the framework computes separately.

Tier 3 (open grey circles). Light quarks (strange, down, up). The framework predicts they sit on the cascade, but first-principles derivation of their $|z|$ position is on the research agenda. The displayed $k$ is inverted from the observed mass — these don't qualify as parameter-free predictions yet, and the open marker makes that visible.

The honest summary: 5 particles fall directly on the curve via $E(k)$ alone (Tier 1), 6 more anchor at framework-predicted rungs but need additional framework structure for the full mass (Tier 2), and 3 light quarks await first-principles closure (Tier 3). What the plot shows is that all 14 particles cluster on a single ladder set by one geometric input — the framework's structural achievement.

Strong CP: θ = 0 Exactly¶

The strong CP problem: quantum chromodynamics admits a parity-violating term proportional to a number $\theta$, but experimental bounds (from the neutron electric dipole moment) force $|\theta| < 10^{-10}$. The Standard Model offers no explanation for why such fine-tuned cancellation should hold. The leading workaround — Peccei–Quinn symmetry — introduces a new particle, the axion, which has been searched for and not yet found.

PPM's answer is structural. The $\tau$ involution at the heart of the framework forces time-reversal symmetry to hold exactly, and the $\theta$-term is odd under time reversal, so it must vanish identically. $\theta = 0$ by construction. No axion needed.

Neutrino Mixing¶

Neutrinos oscillate between three flavors as they travel — an electron neutrino emitted from the sun arrives at Earth as a mixture of all three types — and the oscillation rates are encoded in the PMNS matrix: three angles plus a CP-violating phase. PPM's zeroth-order prediction is tribimaximal mixing — a specific symmetric pattern with $\sin^2\theta_{12} = 1/3$, $\sin^2\theta_{23} = 1/2$, $\sin^2\theta_{13} = 0$ — falling out of the $Z_2 \times A_4$ symmetry of the lepton sector.

The first two angles match observation to about 10%. The third angle is the problem: tribimaximal predicts $\theta_{13} = 0$ exactly, but reactor experiments (Daya Bay, RENO) have established $\theta_{13} > 0$ at high significance. The framework expects corrections from the $\tau$ involution and renormalization-group running to shift $\theta_{13}$ to the observed value, but that calculation is open. Listed honestly: this is a known weak point.

Formulas (zeroth order, $A_4$ tetrahedral symmetry). $$\sin^2\theta_{12} = \tfrac{1}{3}, \quad \sin^2\theta_{23} = \tfrac{1}{2}, \quad \sin^2\theta_{13} = 0.$$ These define the tribimaximal pattern. RG running and charged-lepton diagonalization shift them toward the observed values.

Show implementation. Function source for the tribimaximal PMNS angles.

Why m_τ/m_μ and sin²θ₂₃ Miss¶

m_τ/m_μ = 15.75 (predicted) vs 16.82 (observed): 6.3% discrepancy

This ratio emerges from bulk spacing in the orbifold eigenvalue spectrum: successive lepton tiers are separated by (2π)^{3/2} ≈ 15.75. The framework captures the structural ratio correctly. The observed value is 6.3% higher, indicating threshold corrections at the Pati-Salam scale (k_break ≈ 16.25). These arise from:

• Matching between the high-scale GUT Yukawa and low-scale physical masses
• Running of the lepton mass hierarchy from k_break to the electroweak scale
• Higher-order corrections to the bulk spacing formula

The discrepancy is small and expected; the framework predicts the zeroth-order coupling structure, not precision values. Detailed threshold-matching calculations (beyond the scope of this notebook) would close the gap.

sin²θ₂₃ = 0.500 (tribimaximal) vs 0.546 (observed): 8.4% discrepancy

Tribimaximal mixing is the zeroth-order prediction from the Z₂ × A₄ family symmetry acting on the neutrino sector. The observed value is slightly higher, indicating corrections from:

• Charged-lepton flavor mixing (diagonalization of the charged-lepton mass matrix modifies the effective PMNS angles)
• RG running from the GUT scale (k_break) down to the low-energy scale where PMNS is measured
• Higher-order terms in the neutrino mass matrix

These corrections are well-known to shift tribimaximal predictions toward the observed values in other GUT models. No contradiction with the framework; these are second-order effects.

Cosmology from One Boundary Capacity¶

Three observables, one denominator $N_\infty = \varphi^{392}$. Predictions are tightly linked: change $N_\infty$ and all three move together. The first deep-dive (Hubble) builds the structural argument; the next two (Dark Energy, Gravity) follow.

The Hubble Constant¶

PPM's three cosmological observables — Hubble's $H_0$, the cosmological constant $\Lambda$, and Newton's $G$ — all share one denominator: $N_\infty = \varphi^{392}$, the framework's static boundary capacity (the maximum number of independent cosmic-horizon tile positions the topology supports). Three independent measurements all routing through one structural input is the kind of relationship that's hard to engineer; if PPM's value of $N_\infty$ is wrong, all three predictions move together.

This panel computes $H_0$ two ways: from the cosmic-age route ($H_0 = 1/T_{\rm universe}$) and from Sidharth's empirical large-number relations ($R = \sqrt{N_\infty}\,\lambda_C$, $T = \sqrt{N_\infty}\,\tau_C$, where $\lambda_C$ is the Compton wavelength of the pion). Both should agree by construction — and they do.

Formula. $H_0 = c\sqrt{\Lambda/3}$ in the de Sitter limit, with $\Lambda$ fixed by $N_\infty = \varphi^{392}$ (see Dark Energy below). Equivalently $H_0 = 1/T_{\text{universe}}$ via the Sidharth scaling.

Show implementation. Function source for $H_0$ from cosmic age / Sidharth scaling.

Dark Energy¶

The cosmological constant $\Lambda$ — the energy density of empty space — is the deepest open puzzle in fundamental physics. Naive quantum field theory overestimates it by roughly 120 orders of magnitude (sometimes phrased: why isn't the universe Planckian?). PPM computes $\Lambda = 2(m_\pi c^2)^2 / ((\hbar c)^2\, N_\infty)$ from the pion mass and $N_\infty = \varphi^{392}$ — the same boundary capacity that fixes $H_0$ in the panel above. No additional inputs.

Show implementation. Function source for Λ from $N_\infty = \varphi^{392}$.

Gravity from Holographic Screening¶

Gravity is the weakest of the four forces by a factor of $\sim 10^{36}$ — a hierarchy the Standard Model treats as input. PPM's claim: $G$ isn't a fundamental constant, it's a derived quantity set by $\alpha$ (electromagnetism), $m_\pi$ (the framework anchor), and $N_\infty = \varphi^{392}$ (the same boundary capacity that fixes $H_0$ and $\Lambda$ above).

Because the cumulative count of actualization events $M(t)$ — distinct from the static $N_\infty$ — sets a screening factor that varies with redshift, different cosmological epochs see slightly different effective values of $G$. The plot below shows this. The framework predicts a small but in-principle measurable shift in the spherical-collapse threshold $\delta_c$ as a function of redshift, distinguishing it from standard Newtonian gravity in cosmological structure formation.

Formula. $G = \dfrac{(2\pi)^4\,\hbar c\,\alpha}{m_\pi^{\,2}\,\sqrt{N_\infty}}$, with $N_\infty = \varphi^{392}$ the static boundary capacity. The cosmological companion is $\Lambda = 2(m_\pi c^2)^2 / ((\hbar c)^2 N_\infty)$ (printed below).

Show implementation. Function sources for $G$/$\Lambda$ from $N_\infty=\varphi^{392}$ and the time-evolving $G_{\rm eff}(z)$, $\delta_c^{\rm PPM}(z)$.

Visualize. $G_{\rm eff}(z)$ running with cumulative actualization count $M(t)$ plotted alongside the collapse-threshold shift $\delta_c^{\rm PPM}(z)$. Today ($z=0$): $G_{\rm eff} = G_0$; toward the high-redshift past, the effective coupling rises as the actualization record was thinner.

Where the Golden Ratio Enters¶

The golden ratio $\varphi \approx 1.618$ has now appeared in three places: the instanton coincidence (Foundation), the cosmological constant ($N_\infty = \varphi^{392}$), and the CP-violating phase ($\delta_{CP} = \pi/\varphi^2$). The reason isn't aesthetic and isn't coincidence — it's forced by the symmetry group acting on the framework's moduli space.

The relevant group is $A_5 \cong PSL(2, 5)$, the rotational symmetry group of the icosahedron. Its representation theory lives over the quadratic number field $\mathbb{Q}(\sqrt{5})$, and $\varphi$ is the generator of the unique unit subgroup of that field. Wherever the framework's geometry routes through an $A_5$-invariant, $\varphi$ shows up as a structural constant.

Show implementation. Function sources for the pyramidal-number identity and the $A_5\to\mathbb{Q}(\sqrt{5})\to\varphi$ chain.

Active Inference: A Live Demo¶

The framework's variational principle — minimize the actualization free energy $\mathcal{F}[\rho, \theta]$ — has a direct interpretation as active inference: an agent updates its measurement frame $\theta$ to descend $\mathcal{F}$, which in PPM is the Helmholtz free energy of the actualization channel and in active-inference terms is variational free energy. The demo below runs FrameFindingLoop on a small $\mathbb{CP}^3$-doublet truncation. An agent starts at a guess for the measurement-frame angles and does noisy gradient descent on the free energy, trying to find a hidden target frame $\theta^*$.

The interesting behavior lives in the interaction between the four knobs, not in any one alone. The defaults below put you in a stressed regime where the agent does NOT trivially converge — the run shows it wandering near the target without settling. Move one knob at a time to find the recovery edge:

• Step size ($\eta$, 0.005–0.50). Too small + noise = slow drift, never arrives. Too big + noise = overshoots and oscillates around the basin. Try $\eta \sim 0.01$ vs $\eta \sim 0.4$ at the default noise.
• Samples per step ($N$, 1–80). How many noisy observations the agent averages before each update. This is the framework's $\Phi$-aggregation scaling: $N$ acts as an effective signal-to-noise multiplier of $\sqrt{N}$. At default noise, the agent fails at $N=1$ and recovers around $N \sim 10–20$.
• Sensor noise ($\sigma_s$, 0–1.5). How noisy the agent's observation of the free-energy landscape is. The world isn't transparent. Default $0.5$ is high enough to force the agent to integrate.
• Motor noise ($\sigma_x$, 0–0.05). How imprecisely the agent moves. Even a perfect plan executes with jitter.

This is a worked example of the framework's dynamics in motion rather than a closed-form prediction; included here so the variational principle has runnable form, and so the user can verify that aggregation actually buys something at low SNR (the prediction the framework makes about consciousness). Lazy-loaded — click the button to build the simulator.

Falsification Tests¶

Six fixed-comparison tests covering structural commitments the framework rests on. Each changes one structural choice — the geometric arena, the involution, the spectral measure choice, the boundary-capacity exponent, the pyramidal-number choice, the running pattern — and computes what the framework would then predict. None of the alternatives survive at order-of-magnitude precision.

F1: $\mathbb{CP}^2$ instead of $\mathbb{CP}^3$. What if the geometric arena were two-dimensional complex projective space rather than three-dimensional?

F2: $\tau$ = identity. What if the involution were trivial — no symmetry breaking?

F4: Pion mass $\pm 10\%$. The pion mass is the framework's one dimensionful anchor. Vary it $\pm 10\%$ and watch which predictions move which way.

F5: $N_\infty \neq \varphi^{392}$. What if the boundary capacity exponent were anything other than 392?

F6: Why $P_4 = 30$? The instanton uses the specific square pyramidal number $P_4 = 1+4+9+16 = 30$ because that's the dimension of the moduli space of a degree-3 Veronese embedding of $\mathbb{CP}^3$ into $\mathbb{CP}^{19}$ — there's a geometric reason for $n=4$. The table below compares $e^{-P_n \cdot \pi}$ against the closest integer power of $\varphi$ for several $n$, just to check that the choice of $n=4$ wasn't a numerical accident.

The k-Scale Converter¶

Quality-of-life utility. Most quantities in PPM flow through the cascade rung $k$ via $E(k) = m_\pi (2\pi)^{(51-k)/2}$ (charged pion anchor, 140 MeV). Drag the $k$ slider to translate to energy, Compton length and time, matching temperature, per-event duration, gravitational decoherence rate, and the resolvability / information variables that depend on an ambient temperature. The second slider sets the ambient $T$ that enters $R$ and $I$. Landmarks from the notecard are marked on the plot.

Compared to Alternatives¶

How does PPM compare to other unified or "theory of everything" approaches? String theory predicts parameters that depend on the choice of vacuum among an estimated $10^{500}$ candidates; concrete numerical predictions for $m_t$, $m_H$, or $\Lambda$ require knowing which vacuum we're in, and no selection principle has been established. Asymptotic safety is in-principle predictive but has not yet produced closed-form derivations of Standard Model parameters. Loop quantum gravity, causal sets, and twistor approaches focus on quantum spacetime structure and largely don't attempt the SM parameter spectrum. The Standard Model + EFT treats those ~19 parameters as empirical and doesn't try to predict them.

PPM's claim is narrower and more falsifiable: closed-form formulas for roughly ten Standard Model and cosmological parameters, all matching observation to within a few percent (most under 1%), from one dimensionful input ($m_\pi$) and one matching condition (electroweak symmetry breaking location). The framework either has captured real geometric structure or it has produced ten independent numerical coincidences. Distinguishing the two requires either showing the structural mechanism is wrong (open research) or finding a parameter PPM gets badly wrong that another framework predicts cleanly (none currently identified).

Companion notebook:

• Technical Derivations — full math: spectral data, heat-kernel coefficients, instanton sector, golden-ratio investigation

All code: ppm/ package. Run ppm.verify.run_all() to confirm 42/42 numerical checks.