Compact Proof Of Oph
Author: Bernhard Mueller
First-party HTML and PDF publication page for Compact Proof Of Oph in the OPH paper stack.
Disclosure Day
A short proof that we experience a simulation
Paper release: r1471Released: June 11, 2026
Closure-to-gravity derivation.
This note proves the local OPH gravity readout from imported branch records in the OPH paper stack. The load-bearing local records are \(\mathcal R_P\) and \(\mathcal R_E\). \(\mathcal R_P\) is the local pixel closure record. \(\mathcal R_E\) is the edge-entropy and Einstein-branch record. \(\mathcal R_N\) is the cosmic record-capacity closure record. Its role here is the downstream cosmological display. \(\mathcal R_\gamma\) is the no-\(G\) clock-scale record, used to express the geometric gravity row in SI units.
The theorem proved here is the local gravity readout: \[ \boxed{ \mathcal R_P+\mathcal R_E \Longrightarrow G_{\rm geom}=\ell_\star^2 . } \] The SI display is a separate unit-chart readout: \[ \mathcal R_\gamma \Longrightarrow G_{\rm SI} \mathrel{=} \frac{c^5}{4\pi^2\hbar\nu_{\rm Cs}^2}\varepsilon_{\rm Cs}^2 . \] The global capacity record \(\mathcal R_N\) closes the downstream cosmological term, \[ \Lambda_{\rm CRC} \mathrel{=} \frac{3\pi}{N_\star\ell_\star^2} \mathrel{=} \frac{3\pi c^3}{\hbar G_{\rm SI}N_\star}. \] The structural theorem ends at \(G_{\rm geom}=\ell_\star^2\). The numerical SI \(G\) row is source-predictive only when \(\mathcal R_\gamma\) has a public dependency graph containing no measured \(G\), no Planck area \(\hbar G/c^3\), no measured \(\Lambda\), and no equivalent gravity-calibrated scale. Without that record, the printed SI decimal is a calibration checksum.
OPH starts with zero constants. It starts with closure equations. The local pixel closure computes \(P_\star\). The global screen-capacity closure computes \(N_\star\): \[ P=\Gamma(P)=\varphi+\frac{\sqrt\pi}{A_T(P)},\qquad N=\mathcal C(N). \] Their meanings are concrete: \[ P_\star \quad\hbox{one local pixel},\qquad N_\star \quad\hbox{the total closed-screen capacity}. \] \(P_\star\) says how much area and shared edge entropy one observer cell carries. \(N_\star\) says how many record units fit on the whole closed screen.
These are self-readback equations. A trial value is inserted into the OPH construction. The constructed record system reads the same value back from inside the branch. A valid value is one where input and readout agree. For \(P\), the trial value builds one local cell and its electromagnetic observation scale. For \(N\), the trial value builds a closed screen with that many horizon record units. For the SI scale, the trial value builds the no-\(G\) clock bridge that compares the OPH area unit with the cesium clock.
The OPH method is a targeted fixed-point search from inside the experienced branch. The OPH universe is a closed mathematical readout structure whose observers, records, and branch values coexist in one object. Embedded observers meet record fragments of that branch. Those fragments give educated starting points for the search: the electromagnetic width is near \(1/137\), the screen capacity is near \(10^{122}\), and laboratory clocks supply a scale chart.
Those observations identify the basin of the readback map. The closure equation supplies the exact value as the unique fixed point inside that basin. Observation supplies the address. Banach contraction supplies the number.
The proof obligation is direct: show that the OPH readback map sends the interval \(I\) into itself, contracts distances inside it, and has the quoted candidate as its fixed point. If these statements hold, every starting point in \(I\) lands at the same fixed point. Changing the approximate observation inside \(I\) changes only the starting point. The endpoint is fixed by the equation and uniqueness proof.
A closure equation selects its value as a fixed point. A fixed point is a number that reads back to itself under the rule that defines the branch. For a branch whose readback law is a contraction \(F:I\to I\) on a complete metric interval, \[ F(I)\subseteq I,\qquad d(F(x),F(y))\le q\,d(x,y),\qquad 0\le q<1, \] Banach’s theorem gives exactly one fixed point.
\(I\) is the allowed interval of candidate values. \(F\) is the OPH readback rule. \(d\) is a distance between two candidates. \(q<1\) means the rule pulls candidates closer together each time it is applied. Observation locates the branch interval; the contraction removes numerical freedom inside that interval. The readback loop has one stable value. A different value would fail to read back to itself.
The local pixel \(P_\star\). \(P_\star\) fixes one elementary observer cell. In a simulator it is the cell-area and edge-entropy normalization: \[ a_{\rm cell}=P\ell_\star^2,\qquad \bar\ell_{\rm shared}=P/4 . \] The trial \(P\) is sent through the local source map. That map produces the electromagnetic observation scale of the same cell, written \[ A_T(P)=\alpha_{\rm em}^{-1}(0;P). \] The local cell has two readings. The outer reading is the screen detuning above the self-similar balance point \(\varphi\). The inner reading is the electromagnetic width \(1/A_T(P)\) carried by that same cell. The fixed-point equation for one closed pixel is \[ P=\Gamma(P)=\varphi+\frac{\sqrt\pi}{A_T(P)}. \] The branch contraction has a unique selected value. No other local pixel value solves this OPH readback problem: \[ P_\star=1.6309682094039593\ldots . \] Its meaning is simple: geometry and shared edge entropy name the same local observer cell.
The global capacity \(N_\star\). \(N_\star\) fixes the whole closed screen. It is the dimensionless total observer-facing record capacity. On the declared OPH cosmic record-capacity branch, \(N\) is computed by the readback equation \[ N=\mathcal C(N). \] Here the trial \(N\) builds a closed OPH universe with a horizon record Hilbert space of size \(\log\dim\mathcal H_{\partial,N}=N\). The set \(\Omega_N^{\rm sc}\) contains terminal screen normal forms that are repair-closed, observer-supporting, locally recovered-core closed, and whose own horizon record surface reads back capacity \(N\). The expression \(\log|\Omega_N^{\rm sc}|-N\) counts self-closing normal forms per available screen Hilbert-space size. Equivalently, on the count-density presentation, \[ N_\star=\operatorname{Rep}_{\rm mon}\arg\max_N[\log|\Omega_N^{\rm sc}|-N] \] under the stated strict-concavity proof. Here \(\operatorname{Rep}_{\rm mon}\) means the monotone arithmetic representative: the canonical numeric representative of the maximizing branch. The construction is given in the Observers paper, cited as O1 in Appendix A. The selected display value is \(N_\star\simeq3.31\times10^{122}\). Its meaning is simple: the closed screen has one self-consistent capacity. No other capacity on this branch gives the same closed-screen readback. Once the scale readout is supplied, this same capacity gives the global curvature product \[ \Lambda_\star\ell_\star^2=\frac{3\pi}{N_\star}. \]
The two fixed points determine dimensionless screen geometry: \[ P_\star=\frac{a_{\rm cell}}{\ell_\star^2},\qquad N_\star=\frac{3\pi}{\Lambda_\star\ell_\star^2}, \] \[ \Lambda_\star\ell_\star^2=\frac{3\pi}{N_\star}, \qquad \Lambda_\star a_{\rm cell}=\frac{3\pi P_\star}{N_\star}. \]
On the imported edge-entropy/Einstein branch, the local Newton coupling is the coefficient that converts cell area to edge entropy: \[ G_{\rm geom} \mathrel{=} \frac{a_{\rm cell}}{4\bar\ell_{\rm shared}}. \] On the selected local pixel branch, \[ a_{\rm cell}=P_\star\ell_\star^2, \qquad \bar\ell_{\rm shared}=\frac{P_\star}{4}. \] Therefore \[ G_{\rm geom} \mathrel{=} \frac{P_\star\ell_\star^2}{4(P_\star/4)} \mathrel{=} \ell_\star^2 . \] The numerical value of \(P_\star\) cancels. This cancellation is the local cell identification doing its job: \(P_\star\) ties the cell-area reading to the shared-edge-entropy reading, while \(\ell_\star^2\) carries the area scale.
The global capacity \(N_\star\) closes the cosmological term downstream. The local Newton row has this dependency: \[ (P_\star,\mathcal R_E)\Longrightarrow G_{\rm geom}=\ell_\star^2, \qquad \mathcal R_\gamma\Longrightarrow \ell_\star=\frac{\gamma_\star c}{\nu_{\rm Cs}}. \]
\(P_\star\) and \(N_\star\) are dimensionless. They fix the local pixel ratio and the global screen capacity. SI units require a scale readout. OPH writes that readout as the no-\(G\) cesium clock gap \(\varepsilon_{\rm Cs}\), with \[ \gamma_\star=\frac{\varepsilon_{\rm Cs}}{2\pi}, \qquad \gamma_\star:=\frac{\ell_\star\nu_{\rm Cs}}{c}. \] \(\nu_{\rm Cs}=9\,192\,631\,770~{\rm s^{-1}}\) is the SI cesium frequency. \(\gamma_\star\) compares the OPH length \(\ell_\star\) with the distance light travels during one cesium-clock tick. Its dependency graph contains the unit chart, with no gravity measurement in it. The input list contains no measured \(G\), no \(\hbar G/c^3\), no observed \(\Lambda\), and no \(3\pi c^3/(\hbar G)\). This scale readout follows the same basin logic: observation locates the branch, and the no-\(G\) readback map supplies the selected fixed point.
The trial scale value is a dimensionless clock ratio, not a measured Newton constant. It asks how large the OPH area unit is when expressed through a source-only atomic clock bridge. The bridge may use the SI definitions of \(c\), \(\hbar\), and the cesium transition frequency because those define the human unit chart. It must emit the clock gap without using \(G\), Planck area, or a cosmological formula containing \(G\). Once that no-\(G\) clock gap is emitted, the conversion to \(G_{\rm SI}\) is algebra.
The hierarchy construction enters this note through the public Observers Are All You Need paper, cited as O1 in Appendix A. The imported clock-scale object is the source bridge \[ \mathcal R_\gamma \mathrel{=} \mathcal R_U + \mathcal R_\alpha + \mathcal R_e^{\rm abs} + \mathcal R_{\rm QCD/nuc}^{133{\rm Cs}} + \mathcal R_{\rm atom}^{133{\rm Cs}}. \] \(\mathcal R_U\) is the unified-coupling and weak-scale source record. \(\mathcal R_\alpha\) is the electromagnetic endpoint record. \(\mathcal R_e^{\rm abs}\) is the absolute electron-mass record. \(\mathcal R_{\rm QCD/nuc}^{133{\rm Cs}}\) is the source-side cesium nuclear record. \(\mathcal R_{\rm atom}^{133{\rm Cs}}\) is the cesium hyperfine spectral record. The Observers paper carries the source equations, interval requirements, derivative bounds, and dependency graph. This note uses only the resulting rule: if the full graph contains no measured \(G\), no Planck area, no measured \(\Lambda\), no measured electroweak calibration, and no equivalent gravity-calibrated scale, then the emitted \(\varepsilon_{\rm Cs}\) is a no-\(G\) scale readout. Otherwise the printed SI gravity decimal is a calibration checksum.
The scale record has the form \[ \mathcal R_\gamma \mathrel{=} (I_\gamma,S_\gamma,d_\gamma,q_\gamma,\gamma_0,r_\gamma,\mathcal G_\gamma), \] where \(I_\gamma\) is the branch interval, \(S_\gamma\) is the source readback map, \(d_\gamma\) is the metric, \(q_\gamma<1\) is the contraction constant, \(\gamma_0\) is the displayed candidate, \(r_\gamma=d_\gamma(S_\gamma(\gamma_0),\gamma_0)\) is the residual, and \(\mathcal G_\gamma\) is the dependency graph. The record must satisfy \[ S_\gamma(I_\gamma)\subseteq I_\gamma, \qquad d_\gamma(S_\gamma(x),S_\gamma(y)) \le q_\gamma d_\gamma(x,y). \] The dependency graph must contain no path from \[ G_{\rm exp},\qquad \ell_P^2=\frac{\hbar G}{c^3},\qquad \Lambda_{\rm exp},\qquad B_G=\frac{3\pi c^3}{\hbar G} \] to \(\gamma_0\) or \(\varepsilon_{\rm Cs}\). Approximate observation may locate the branch interval. It may not define \(S_\gamma\), tune its coefficients, choose among multiple roots inside the interval, alter the residual, or update \(\mathcal G_\gamma\) after comparison with measured \(G\).
The selected branch readout is \[ \varepsilon_{\rm Cs} \mathrel{=} 3.113930513416012823901050434132426029496608693041876 \times10^{-33}, \] which gives \[ \gamma_\star \mathrel{=} 4.9559743365484194636657782433696431879484319705825 \times10^{-34}. \] Hence \[ G_{\rm geom} \mathrel{=} \ell_\star^2 \mathrel{=} \left(\frac{\gamma_\star c}{\nu_{\rm Cs}}\right)^2 \mathrel{=} \frac{c^2}{4\pi^2\nu_{\rm Cs}^2}\,\varepsilon_{\rm Cs}^2 . \] The SI display of a geometric length-squared coupling is \[ G_{\rm SI}=\frac{c^3}{\hbar}G_{\rm geom}. \] Therefore \[ G_{\rm SI} \mathrel{=} \frac{c^5}{4\pi^2\hbar\nu_{\rm Cs}^2}\,\varepsilon_{\rm Cs}^2 . \] Substitution gives the selected-branch SI readout \[ G_{\rm OPH} \mathrel{=} 6.674299995910528\ldots\times10^{-11} ~{\rm m^3\,kg^{-1}\,s^{-2}} . \] Comparison with CODATA is listed in Appendix A and excluded from the upstream derivation.
The table collects quantities used or predicted by the same OPH closure architecture. Entries include direct numerical comparisons, exact definitions, structural hits, public upper limits, and values with no public measurement. Each row keeps its measurement type visible without blending unlike evidence types into one number.
| Readout | Value | Meaning |
|---|---|---|
| \(P_\star\) | \(1.6309682094039593\ldots\) | exact local pixel fixed point |
| \(N_\star\) | \(\simeq3.31\times10^{122}\) | global closed-screen capacity fixed point |
| \(\varepsilon_{\rm Cs},\gamma_\star\) | $`\begin{gathered} |
\varepsilon_{\rm Cs}^{\rm cal}=3.11393051342\ldots\times10^{-33}\
\gamma_\star=4.95597433655\ldots\times10^{-34}
\end{gathered}$ | dimensionless clock bridge; source-predictive only after the full $\mathcal
R_\gamma$ chain | | $\ell_\star^2$ | $2.61228030237\ldots\times10^{-70}{\rm
m^2}$ | OPH area quantum | | $a_{\rm
cell}$ | $4.26054612722\ldots\times10^{-70}{\rm
m^2}$ | physical area of one screen pixel | | $\bar\ell_{\rm
shared}$ | $0.407742052350989825\ldots$ | shared edge entropy per local cell | | $G_{\rm
geom}$ | $\ell_\star^2$ | geometric Newton coupling | | $G_{\rm
SI}$ | $6.674299995910528\times10^{-11}{\rm
m^3kg^{-1}s^{-2}}$ | SI branch checksum; source-predictive only after the full clock chain | | $\Lambda_{\rm
CRC}$ | $\dfrac{3\pi}{G_{\rm geom}N_\star}`$ | downstream
de Sitter display |
OPH is a theory of everything. The Newton row is the compact gravity example treated in this note. The wider OPH paper stack uses the same closure architecture for spacetime, gauge structure, particles, and cosmology. Some rows are exact structural readouts. Some rows are numerical values that can be compared with public measurements. Some particle and hadron rows require source-only QCD or nuclear computations that are expensive rather than conceptually free variables.
Appendix A records the surrounding OPH prediction ledger: local pixel, total screen capacity, support-visible spacetime branch, realized gauge branch, and declared quantitative particle branches. The table separates direct measurements, public limits, definitions, and values with no public measurement.
A numerical \(\sigma\) is shown only when the cited public source gives a central value and a one-standard-deviation uncertainty. Exact definitions and rounded matches are marked as exact hits. Limit rows, structural rows, and non-Gaussian global-fit profiles keep their measurement type. The table avoids fake scalar sigma. Rows whose OPH value carries more digits than public measurement permits are sharper OPH readouts if the branch derivation is accepted.
@L0.13L0.24L0.36L0.10L0.10@ Lane & OPH
prediction & Measurement or public
reference & Residual &
Source
Local pixel & \(P_\star=1.6309682094039593\ldots\) & No
direct laboratory measurement. Local cell-area and edge-entropy fixed
point required by a finite OPH screen. & \(n/a\) & O3,O5
Local cell entropy & \(\bar\ell_{\rm
shared}=P_\star/4=0.407742052350989825\ldots\) & No direct
laboratory measurement. It is the entropy weight assigned to one
observer cell by the same fixed point that fixes \(P_\star\). & \(n/a\) & O3,O4
Global capacity & \(N_\star\simeq3.31\times10^{122}\) & No
direct count measurement. Cosmological comparison uses de Sitter
capacity after the scale branch is supplied. & \(n/a\) & O1,O4
Clock bridge proof obligation & \(\begin{gathered}
\varepsilon_{\rm Cs}^{\rm cal}=3.11393051342\ldots\times10^{-33}\\
\gamma_\star=4.95597433655\ldots\times10^{-34}\\
B_\star=3.60787391468\ldots\times10^{70}{\rm m^{-2}}
\end{gathered}\) & The printed decimal is source-predictive
only if the full \(\mathcal R_U+\mathcal
R_\alpha+\mathcal R_e^{\rm abs}
+\mathcal R_{\rm QCD/nuc}^{133{\rm Cs}}+\mathcal R_{\rm atom}^{133{\rm
Cs}}\) chain emits it without a gravity-calibrated parent. &
source-emitted required & O4
Area quantum and pixel area & \(\begin{gathered}
\ell_\star^2=2.61228030237\ldots\times10^{-70}{\rm m^2}\\
a_{\rm cell}=4.26054612722\ldots\times10^{-70}{\rm m^2}
\end{gathered}\) & No direct public measurement of an OPH
screen cell. The inferred area quantum is tested through \(G_{\rm SI}\). & \(n/a\) & O3,O4
Newton coupling & \(\begin{gathered}
G_{\rm geom}=\ell_\star^2\\
G_{\rm SI}=6.674299995910528\times10^{-11}
\end{gathered}\) & Comparison only. \(G=6.67430(15)\times10^{-11}{\rm
m^3\,kg^{-1}\,s^{-2}}\). This CODATA row is excluded upstream by
the no-\(G\) criterion in Section 4.
& source-gated checksum unless \(\mathcal
R_\gamma\) is source-emitted & S1,O3,O4
Cosmological constant & \(\Lambda_{\rm
CRC}\approx1.09\times10^{-52}{\rm m^{-2}}\) & Planck 2018
gives the late-universe \(\Lambda\)CDM
parameter surface; the rounded OPH value agrees at displayed order.
& exact hit at shown precision & S5,O1,O4
Static-patch rate & \(H_{\rm
dS}\approx55.759940256~{\rm km\,s^{-1}\,Mpc^{-1}}\) & De
Sitter static-patch display, separate from the fitted local Hubble
constant. & separate from fitted \(H_0\) & O1,O4
Hubble benchmark & \(H_0=67.4~{\rm
km\,s^{-1}\,Mpc^{-1}}\) in compressed diagnostics & Planck
2018 reports \(H_0=67.4\pm0.5~{\rm
km\,s^{-1}\,Mpc^{-1}}\). This row is a benchmark input in the
compressed comparison. & \(0\sigma\) & S5,O1,O4
Lorentz group & \(\mathrm{Conf}^+(S^2)\cong\mathrm{SO}^+(3,1)\)
& Structural prediction. Direct scalar sigma is the wrong statistic;
the comparison is the Lorentz-kinematics branch. & \(n/a\) & O4
Causal speed & \(c=299792458~{\rm
m\,s^{-1}}\) & Exact SI definition. Display convention for
the Lorentz speed. & exact hit by definition & S3,O4
Einstein branch & Jacobson-type local Einstein equation &
Structural prediction. The recovered local field equation is the
Einstein equation branch; later tests are GR/weak-field/null-energy
comparisons rather than one scalar value. & exact structural hit
& O4
Gauge quotient & \(\mathrm{SU}(3)_c\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y/\mathbb
Z_6\) & Matches the observed Standard Model gauge quotient.
No one-number sigma. & exact structural hit & S6,O4,O5
Hypercharge, color, generations & \(\begin{gathered}
\hbox{one-generation hypercharge lattice}\\
N_c=3,\quad N_g=3
\end{gathered}\) & Matches observed Standard Model charge and
family structure. No one-number sigma. & exact structural hit &
S6,O4,O5
Massless protected carriers & \(\begin{gathered}
m_\gamma=0\\
m_g=0\\
m_{\rm grav}=0
\end{gathered}\) & Photon and gluon masslessness are
gauge-branch structural hits. Graviton masslessness belongs to the
diffeomorphism/Lorentz gravity branch. Direct measurements give public
upper limits; exact mass measurements are absent. & structural hits
& S6,O4,O5
Gauge proton decay & \(\tau_p^{(\rm
gauge)}=\infty\) & Public searches give lower limits on
proton decay channels. The OPH gauge row predicts no simple-GUT gauge
boson decay mode. & \(n/a\) &
S6,O4,O5
Fine-structure source trunk & \(\alpha^{-1}_{\rm
source}=136.994835164621649\ldots\) & No direct public
measurement. It is the source calculation before the declared
hadronic/same-scheme endpoint payload, which is fixed for the displayed
endpoint row. & \(n/a\) &
O5
Fine-structure endpoint & \(\begin{gathered}
\alpha^{-1}(0)=137.035999177(21)\\
\alpha(0)=0.00729735256433\ldots
\end{gathered}\) & CODATA gives \(\alpha^{-1}=137.035999177(21)\). The public
endpoint stays outside the gravity proof. Its hadronic/same-scheme
payload is fixed rather than source-emitted in this compact note. &
\(0\sigma\) & S2,O5
Electroweak source & \(\begin{gathered}
\alpha_2(m_Z)=0.03377843630219015\\
\alpha_Y(m_Z)=0.010131601067241624
\end{gathered}\) & Electroweak-fit comparison row. A scalar
sigma depends on scheme and fit covariances, so this table leaves it
unquoted. & \(n/a\) &
S6,O5
Electroweak scale & \(v=246.76711732749683~{\rm GeV}\) &
Derived from the electroweak branch and displayed in the particle
surface. Direct comparison depends on \(G_F\) convention and loop scheme. &
\(n/a\) & S6,O5
\(W/Z\) validation pair & \(\begin{gathered}
M_W=80.377~{\rm GeV}\\
M_Z=91.18797809193725~{\rm GeV}
\end{gathered}\) & PDG 2024 gives \(M_W=80.3692\pm0.0133~{\rm GeV}\) and \(M_Z=91.1876\pm0.0021~{\rm GeV}\). &
\(\begin{gathered}0.59\sigma\\0.18\sigma\end{gathered}\)
& S6,O5
Higgs/top branch & \(\begin{gathered}
m_H=125.1995304097179~{\rm GeV}\\
m_t=172.35235532883115~{\rm GeV}
\end{gathered}\) & PDG 2024 gives \(m_H=125.20\pm0.11~{\rm GeV}\) and direct
\(m_t=172.57\pm0.29~{\rm GeV}\). &
\(\begin{gathered}-0.004\sigma\\-0.75\sigma\end{gathered}\)
& S6,S7,O5
Charged leptons & \(\begin{gathered}
m_e=0.00051099895\\
m_\mu=0.1056583755\\
m_\tau=1.7769324651340912
\end{gathered}{\rm~GeV}\) & PDG lepton summary gives the
displayed central values. The tau residual is \(0.03\sigma\); \(e,\mu\) match at displayed precision. &
\(\le0.03\sigma\) & S8,O5
Quark sextet & \(\begin{gathered}
u=0.00216,\quad d=0.00470\\
s=0.0935,\quad c=1.273\\
b=4.183\\
t=172.35235532883115\\
{\rm GeV}
\end{gathered}\) & PDG quark summary gives the displayed
light/heavy quark central values. The top comparison is listed in the
Higgs/top row. & \(0\sigma\) at
shown precision & S7,O5
Neutrino masses & \(\begin{gathered}
0.017454720257976796\\
0.019481987935919015\\
0.05307522145074924
\end{gathered}{\rm~eV}\) & No public measurement exists for
the individual absolute neutrino masses. KATRIN gives a direct upper
limit on the effective electron-neutrino mass. & \(n/a\) & S10,O5
Neutrino sum & \(\sum
m_\nu=0.09001192964464505~{\rm eV}\) & Planck 2018 plus BAO
gives \(\sum m_\nu<0.12~{\rm eV}\)
at 95% CL. OPH lies below the public upper limit. & below limit
& S5,S11,O5
Neutrino mixing display & \(\begin{gathered}
\theta_{12}=34.2259^\circ,\ \theta_{23}=49.7228^\circ\\
\theta_{13}=8.68636^\circ,\ \delta=305.581^\circ
\end{gathered}\) & NuFIT 6.0 gives global-fit profiles. The
profiles are non-Gaussian and ordering-dependent, so this table leaves
scalar sigma unquoted. & \(n/a\)
& S9,O5
Majorana phases & \(\begin{gathered}
\alpha_{21}=153.618518^\circ\\
\alpha_{31}=257.003241^\circ
\end{gathered}\) & No public measurement exists for the
Majorana phases. They require neutrinoless double-beta or equivalent
absolute-phase information. & \(n/a\) & S11,O5
Compressed matter budget & \(\begin{gathered}
\Omega_{\Lambda,\rm OPH}=0.684423332\\
\Omega_A=0.264114401\\
\Omega_m=0.315484968
\end{gathered}\) & Planck 2018 gives \(\Omega_m=0.315\pm0.007\). Flat \(\Lambda\)CDM gives \(\Omega_\Lambda\simeq0.685\pm0.007\). &
\(\begin{gathered}\Omega_m:0.07\sigma\\
\Omega_\Lambda:-0.08\sigma\end{gathered}\) & S5,O1,O4
Measurement source keys.
\(\mathrm{S1}\): NIST CODATA 2022 \(G\).
\(\mathrm{S2}\): NIST CODATA 2022 \(\alpha^{-1}\).
\(\mathrm{S3}\): NIST CODATA \(c\), exact.
\(\mathrm{S4}\): NIST CODATA \(\hbar\), exact. Used in the arithmetic check.
\(\mathrm{S5}\): Planck 2018 cosmological parameters.
\(\mathrm{S6}\): PDG 2024 gauge and Higgs summary.
\(\mathrm{S7}\): PDG 2024 quark summary.
\(\mathrm{S8}\): PDG 2024 lepton summary.
\(\mathrm{S9}\): NuFIT 6.0 oscillation fit.
\(\mathrm{S10}\): KATRIN 2025 direct neutrino-mass limit.
\(\mathrm{S11}\): PDG 2024 neutrino-mass review.
OPH paper keys.
\(\mathrm{O1}\): Observers Are All You Need.
\(\mathrm{O2}\): Reality as Consensus Protocol.
\(\mathrm{O3}\): Screen Microphysics and Observer Synchronization.
\(\mathrm{O4}\): Recovering Relativity and Standard Model Structure from Observer Overlap Consistency.
\(\mathrm{O5}\): Deriving the Particle Zoo from Observer Consistency.