Appendix PPP — Collapse of Physical Constants (Explicit Test Under a Declared SI Bookkeeping Map)

\labelapp:PPP

\newcommand\cSI2.99792458\times 10^8 \rm m,s^-1

\newcommand\GSI6.67430\times 10^-11 \rm m^3,kg^-1,s^-2 \newcommand\kBSI1.380649\times 10^-23 \rm J,K^-1

\newcommand\muPhys4.834\times 10^31 \rm m^-1

\newcommand\alphaV1.793\times 10^8 \rm m,s^-1

\newcommand\LamObs1.11\times 10^-52 \rm m^-2 \newcommand\LamPred5.52\times 10^60 \rm m^-2

PPP.1 Statement of Goal (What Is Being Claimed, Precisely)

This appendix states the strongest quantitative claim of the framework in a falsifiable form:

\beginquote Given (i) the closure equations, and (ii) closure invariants extracted numerically from the validator (Appendix ‘§app:QQQ‘), the familiar physical constants ( c, \hbar, G, \Lambda, k_B ) are not independent postulates. Exactly one empirical anchor is permitted. After that single anchor, all remaining constants must follow as predictions from the extracted invariants, with no additional tuning. \endquote

The logical chain is strictly deductive and depends only on preceding appendices:

$$\text{GGG} \Rightarrow \text{HHH} \Rightarrow \text{III} \Rightarrow \text{JJJ} \Rightarrow \text{OOO} \Rightarrow \text{KKK} \Rightarrow \text{LLL} \Rightarrow \text{MMM} \Rightarrow \text{NNN} \Rightarrow \text{PPP}.$$

\paragraphNumerical provenance (non-negotiable). All dimensionless quantities that appear as “inputs” in this appendix are not fitted here. They are extracted from ‘orbit_runs.csv‘ and saved ‘.npz‘ fields by the protocol defined in Appendix ‘§app:QQQ‘ and validated via the executable scoring, robustness, and falsification pipeline documented in Appendix RRR. This separation is mandatory to avoid circular reasoning.

PPP.2 The One Allowed Calibration Step

The validator operates in dimensionless units (Appendix GGG). To compare with SI units, exactly one calibration is permitted.

We use that single calibration to fix the global SI velocity factor by imposing one empirical anchor, and only one. Two legitimate anchors exist:

• Option A (Light anchor; adopted): Fix the SI velocity factor so that the emergent maximal stable propagation speed (c_\rm eff) equals the measured (c).
• Option B (Gravity anchor): Fix the scale so that the Newtonian limit in Appendix PPP.6 matches one gravitational datum.

This appendix adopts Option A because it is maximally clean: a causal bound is used as the sole anchor, and then (\hbar, G, \Lambda, k_B) become predictions (once the deterministic (\mu)-to-SI bookkeeping is stated).

PPP.3 Unit Map and Scale Bookkeeping (explicit Bookkeeping Map A)

\paragraph*Bookkeeping status. The conversion adopted below is explicitly labeled Bookkeeping Map A. It is declared solely to render the collapse test numerically explicit and falsifiable. Failure under this map constitutes a valid falsification of the theory under Map A, not a tuning ambiguity.

Let (\mu_\rm phys) denote the screening scale in SI units ((\rm m^-1)), defining the coherence length:

$$\ell_0 \;\equiv\; \mu_{\rm phys}^{-1}.$$

Causality fixes the time scale once (c_\rm eff) is known:

$$t_0 \;\equiv\; \frac{\ell_0}{c_{\rm eff}}.$$

The potential (\Phi) in the inertial equation has dimensions (\rm (length)^2/(time)^2), hence the natural potential scale is

$$\Phi_0 \;\equiv\; \frac{\ell_0^2}{t_0^2} \;=\; c_{\rm eff}^2.$$

Finally, the screened Poisson structure sets a natural density scale consistent with the propagation mapping:

$$c_{\rm eff}^2 \;\sim\; \frac{\rho}{\mu} \quad\Longrightarrow\quad \rho_0 \;\equiv\; \mu_{\rm phys}\,c_{\rm eff}^2.$$

\paragraph*No further freedom (but bookkeeping is required). The single empirical anchor fixes only the velocity factor (PPP.4). To instantiate numerical SI predictions for (\hbar, G, \Lambda), one must state the deterministic map from validator length units to meters.

\paragraph*Concrete bookkeeping convention (Map A; declared, not fitted). We adopt one explicit, declared conversion (not a fit):

$$\boxed{\ dx_{\rm phys} \equiv 1.616\times 10^{-35}\ {\rm m}\ }\qquad\text{(declared)}$$

Using the measured validator grid spacing and screening parameter from the run table, (dx_\rm val=3.90625\times 10^-3) and (\mu_\rm val=2.0\times 10^-1), the physical screening scale follows deterministically:

$$\mu_{\rm phys} \;\equiv\; \mu_{\rm val}\,\frac{dx_{\rm val}}{dx_{\rm phys}} \;= 2.0\times 10^{-1}\,\frac{3.90625\times 10^{-3}}{1.616\times 10^{-35}\ {\rm m}} \;= \boxed{\ \mu_{\rm phys}=\muPhys\ }.$$

No additional degrees of freedom are introduced by this declaration; it is a bookkeeping convention that makes the SI table fully explicit. If the resulting numerical table fails, the theory (under this declared mapping) fails.

PPP.4 Emergent Speed (c) (Maximal Stable Propagation) — now closed

Appendix ‘§app:QQQ‘ defines an operational maximal speed in validator units from saved velocity fields. For each run with stored ((v_x,v_y)), define the pointwise speed

$$|v|(x,t) \equiv \sqrt{v_x(x,t)^2+v_y(x,t)^2}.$$

To avoid single-cell spikes and ensure audit robustness, define

$$c_{\rm eff}^{\rm val} \;\equiv\; Q_{0.999}\!\big(|v|\big), \qquad \mathcal{V}_c \;\equiv\; \big(c_{\rm eff}^{\rm val}\big)^2.$$

For the current sweep (all runs with available velocity fields), your extraction yields:

$$\boxed{\mathcal{V}_c^{\rm med}=2.7955671947897613}, \qquad \boxed{c_{\rm eff}^{\rm val,med}=1.6719949745109168}.$$

\paragraph*Single calibration (Option A of PPP.2). Define the unique SI velocity factor (\alpha_v) by enforcing

$$c \;\equiv\; \alpha_v\,c_{\rm eff}^{\rm val,med} \quad\Longrightarrow\quad \boxed{\alpha_v=\frac{\cSI}{1.6719949745109168}}.$$

Numerically, this gives (\boxed\alpha_v=\alphaV). Then the emergent physical speed is

$$c_{\rm eff} \;=\; \alpha_v\,c_{\rm eff}^{\rm val} \quad\Rightarrow\quad c_{\rm pred}=c \ \text{(by construction, unique anchor)}.$$

PPP.5 Emergent (\hbar) (Quantum Closure Minimum)

Appendix OOO establishes the closure quantization statement:

$$L_{\min} \;=\; \hbar,$$

where (L_\min) is the smallest historical angular quantity that still permits closure.

The validator provides

$$L_{\min}^{(\rm val)} \;=\; \mathcal{L}_\hbar,$$

with (\mathcalL_\hbar) extracted numerically according to Appendix ‘§app:QQQ‘. For the current dataset:

$$\boxed{\mathcal{L}_\hbar^{\rm med}=1.017373\times 10^{-2}}, \qquad 95\%~{\rm CI}=[1.017373\times 10^{-2},\ 1.017373\times 10^{-2}].$$

A unique mass scale is implied by (\rho_0):

$$m_0 \;\equiv\; \rho_0\,\ell_0^3 \;=\; (\mu_{\rm phys} c^2)\,\mu_{\rm phys}^{-3} \;=\; \frac{c^2}{\mu_{\rm phys}^{2}}.$$

The canonical angular scale is (m_0 \ell_0^2/t_0), hence

$$\boxed{ \hbar_{\rm pred} = \mathcal{L}_\hbar \left( \frac{c^3}{\mu_{\rm phys}^3} \right). }$$

Therefore (\hbar) is predicted once (\mu_\rm phys) is fixed by validator bookkeeping.

PPP.6 Emergent (G) (Newtonian Limit of the Screened Closure Potential)

The theory uses:

$$(\nabla^2-\mu^2)\Phi \;=\; \rho - \langle\rho\rangle.$$

In the near-field regime (r\ll \mu^-1), screening is negligible:

$$\nabla^2\Phi \;\approx\; \rho - \langle\rho\rangle.$$

Compare with Newtonian gravity:

$$\nabla^2 \Phi_N \;=\; 4\pi G\,\rho_{\rm mass}.$$

Use the physical mappings:

$$\Phi_N \equiv \Phi_0 \Phi,\qquad \rho_{\rm mass}\equiv \rho_0 \rho.$$

Then

$$\nabla^2(\Phi_0\Phi) = \Phi_0\,\ell_0^{-2}\,\nabla'^2\Phi \;\approx\; \rho_0\rho,$$

and coefficient matching yields

$$4\pi G_{\rm pred}\,\rho_0 \;=\; \frac{\Phi_0}{\ell_0^{2}} \quad\Longrightarrow\quad G_{\rm pred} = \frac{1}{4\pi}\, \frac{\Phi_0}{\rho_0\,\ell_0^{2}}.$$

With (\Phi_0=c^2), (\rho_0=\mu_\rm physc^2), (\ell_0=\mu_\rm phys^-1),

$$\boxed{ G_{\rm pred} = \frac{1}{4\pi}\, \mu_{\rm phys}. }$$

PPP.7 Emergent (\Lambda) (Cosmic Non-Closure Drift)

Appendix JJJ defines existential phases. In the asymptotic non-closure drift regime, introduce the dimensionless invariant (Appendix ‘§app:QQQ‘):

$$\mathcal{V}_\Lambda \;\equiv\; \frac{\ddot R}{R}\,t_0^2.$$

For the current dataset:

$$\boxed{\mathcal{V}_\Lambda^{\rm med}=7.869909\times 10^{-4}}, \qquad 95\%~{\rm CI}=[5.961478\times 10^{-4},\ 9.778599\times 10^{-4}].$$

Using the late-time correspondence ( \ddot R/R \approx \Lambda c^2/3 ),

$$\boxed{ \Lambda_{\rm pred} = 3\,\mathcal{V}_\Lambda\,\mu_{\rm phys}^2. }$$

PPP.8 Emergent (k_B) (Micro-Closure Fluctuations) — explicit status

Micro-closure fluctuations define a thermal-like sector (Appendix III). Let the validator-extracted fluctuation invariant be

$$\mathcal{V}_T \;\equiv\; \frac{\delta E}{E_0}.$$

For the current dataset:

$$\boxed{\mathcal{V}_T^{\rm med}=1.994383\times 10^{-2}}, \qquad 95\%~{\rm CI}=[1.159724\times 10^{-2},\ 3.404630\times 10^{-2}].$$

The natural energy scale is

$$E_0 \;\equiv\; m_0\frac{\ell_0^2}{t_0^2} \;=\; m_0 c^2, \qquad m_0=\frac{c^2}{\mu_{\rm phys}^2}.$$

Define an operational closure-temperature observable (T_\rm cl) by the fixed protocol of Appendix III / Appendix ‘§app:QQQ‘, and impose the identification

$$k_B\,T_{\rm cl} \;\equiv\; \delta E \;=\; \mathcal{V}_T\,E_0.$$

Hence

$$\boxed{ k_{B,{\rm pred}} = \mathcal{V}_T\,\frac{E_0}{T_{\rm cl}}. }$$

\paragraph*Mandatory completion note. If (T_\rm cl) is not yet defined/extracted from validator outputs under a fixed protocol, then (k_B) must be reported as deferred (not fitted), and PPP.11 must label the (k_B) row as pending.

PPP.9 The Collapse Summary (All Constants from (\mu) + Closure Invariants)

After the single calibration (Option A), the collapsed forms are:

$$\boxed{ c_{\rm pred}=c } \qquad \boxed{ \hbar_{\rm pred} = \mathcal{L}_\hbar\, \frac{c^3}{\mu_{\rm phys}^3} } \qquad \boxed{ G_{\rm pred} = \frac{1}{4\pi}\,\mu_{\rm phys} }$$ $$\boxed{ \Lambda_{\rm pred} = 3\,\mathcal{V}_\Lambda\,\mu_{\rm phys}^2 } \qquad \boxed{ k_{B,{\rm pred}} = \mathcal{V}_T\,\frac{E_0}{T_{\rm cl}}, \ \ E_0=m_0c^2,\ m_0=\frac{c^2}{\mu_{\rm phys}^2} }$$

All nontrivial content is therefore in ( \mathcalL_\hbar, \mathcalV_\Lambda, \mathcalV_T, \mathcalV_c ), numerically extracted in Appendix ‘§app:QQQ‘.

PPP.10 Falsification (Where the Theory Dies)

Because only one calibration is permitted, each of the following is a hard test:

• Failure of (\hbar_\rm pred) falsifies the closure-quantization claim (Appendix OOO).
• Failure of (G_\rm pred) falsifies the Newtonian mapping (Appendix PPP.6).
• Failure of (\Lambda_\rm pred) falsifies the non-closure drift identification (Appendix PPP.7).
• Failure of (k_B,\rm pred) under a fixed validator-only (T_\rm cl) protocol falsifies the micro-closure thermodynamic map (Appendix PPP.8).

PPP.11 Numerical Collapse Table (Killer Table; now invariant-complete and explicitly staged)

All dimensionless closure invariants used below are extracted from validator outputs by Appendix ‘§app:QQQ‘:

$$\mathcal{L}_\hbar=1.017373\times 10^{-2},\quad \mathcal{V}_\Lambda=7.869909\times 10^{-4},\quad \mathcal{V}_T=1.994383\times 10^{-2},\quad \mathcal{V}_c=2.7955671947897613.$$

The single empirical anchor is (c_\rm pred\equiv c).

\paragraph*Table status. Under the declared bookkeeping convention of PPP.3 (which fixes (\mu_\rm phys=\muPhys)), the SI table below is fully instantiated for (\hbar, G, \Lambda). The (k_B) row remains deferred unless and until a validator-only protocol for (T_\rm cl) is specified and extracted.

\begintable[h!]

\setlength\tabcolsep4pt

| p3.2cm p3.6cm p2.2cm p2.6cm

Constant | Predicted | Known (CODATA / standard) | Rel. Error | Notes | | — | — | — | — | — | | $c$ | $c$ (single anchor) | $\cSI$ | $0$ | anchor | | $\hbar$ | $\hbar_{\rm pred}=2.43\times 10^{-72}\ {\rm J\,s}$ | $1.054571817\times 10^{-34}\ {\rm J\,s}$ | $1.00$ | from $\mathcal{L}_\hbar$, $\mu_{\rm phys}$ | | $G$ | $G_{\rm pred}=3.85\times 10^{30}$ | $\GSI$ | $5.77\times 10^{40}$ | from $\mu_{\rm phys}$ (PPP.6) | | $\Lambda$ | $\Lambda_{\rm pred}=\LamPred$ | $\LamObs$ | $4.97\times 10^{112}$ | from $\mathcal{V}_\Lambda$, $\mu_{\rm phys}$ | | $k_B$ | deferred | $\kBSI$ | — | pending $T_{\rm cl}$ |

Figure: Numerical collapse table (audit-ready). All invariants are fixed by Appendix \refapp:QQQ. The only empirical anchor is $c$. Bookkeeping Map A fixes $dx_{\rm phys}$ and thus $\mu_{\rm phys}$. \labeltab:PPP_collapse \endtable

\paragraph*Error definition. For any predicted constant (X_\rm pred) with known reference value (X_\rm known) we define

$$\varepsilon_X \;\equiv\; \left|\frac{X_{\rm pred}-X_{\rm known}}{X_{\rm known}}\right|.$$

PPP.12 Uncertainty Quantification (Bootstrap Across Runs)

The validator yields an ensemble of runs. Therefore each extracted invariant ( \mathcalL_\hbar, \mathcalV_\Lambda, \mathcalV_T, \mathcalV_c ) is treated as a random variable over the admissible closure subset defined in Appendix ‘§app:QQQ‘.

\paragraph*Bootstrap protocol. Let $\{\mathcal{I}_i\}_{i=1}^N$ denote admissible sample values of an invariant. We perform bootstrap resampling:

1. Sample (with replacement) $N$ values from $\{\mathcal{I}_i\}$ to form a bootstrap replicate.
2. Compute the invariant estimate for that replicate, yielding $\mathcal{I}^{(b)}$.
3. Propagate $\mathcal{I}^{(b)}$ through the collapse formulas to obtain ( \hbar_\rm pred^(b), G_\rm pred^(b), \Lambda_\rm pred^(b), k_B,\rm pred^(b). )
4. Repeat for $b=1,\dots,B$ (e.g. $B=10^3$).

\paragraph*Mandatory reporting note. Once (\mu_\rm phys) (and (T_\rm cl), if used) is fixed, bootstrap results must be reported (either $\sigma$ or a 95% interval) for each predicted constant in PPP.11. Otherwise the numerical claim is incomplete under finite-sweep variability.

\paragraph*Reported uncertainties. We report either:

• Standard deviation across bootstrap samples: ( \sigma_X = \rm std\X_\rm pred^(b), )
• or a $95\%$ interval: ( [X_\rm pred^(2.5%),,X_\rm pred^(97.5%)]. )

This uncertainty is purely numerical/ensemble-driven: it reflects finite sampling, finite time horizons, and run-to-run variability, not additional theoretical freedom.

PPP.13 Pass–Fail Criterion (Two Lines, No Escape)

Because only one calibration is permitted, the theory is falsifiable by a single, explicit criterion. Choose a target relative tolerance $\delta$ (e.g. $\delta=10^{-2}$ for $1\%$).

• PASS: if

$$\max\{\varepsilon_{\hbar},\varepsilon_{G},\varepsilon_{\Lambda},\varepsilon_{k_B}\}\ \le\ \delta,$$

with bootstrap uncertainties consistent with this bound.

• FAIL: otherwise; i.e. if any one of these constants lies outside the tolerance after the $c$-calibration and Appendix \refapp:QQQ extraction protocol are fixed.

For the declared bookkeeping convention used here, the instantiated table above fails by many orders of magnitude.

\paragraph*Final Statement (Updated).

$$\boxed{ \text{After a single calibration by }c, \text{ all remaining constants are predictions from closure invariants.} }$$

For the explicitly declared Bookkeeping Map A used here, the numerical collapse table fails by many orders of magnitude. This constitutes a clean, non-degenerate falsification of the theory under Map A.

@incollection{hassan2026appendixpppcollapseo,
  author    = {Hassan, Akram},
  title     = {Appendix PPP — Collapse of Physical Constants (Explicit Test Under a Declared SI Bookkeeping Map)},
  booktitle = {The Complete Structural Selection Corpus},
  publisher = {Nuronova Genix Corp},
  year      = {2026},
  url       = {https://structuralselection.org/book/appendix/appendix-ppp-collapse-of-physical-constants-explicit-test-under-a-declared-si-bookkeeping-map}
}

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