PBT Model Reference Guide

August 2025

Introduction

Pressure-Based Theory (PBT) models the universe as an infinite pressure vessel with hierarchical particle fluxes, where forces emerge from push effects due to shadowing and aether distortions. Evolving from Infinite Push-Pressure Theory to Hybrid Push-Aether Theory, it unifies forces mechanically, addressing anomalies in GR and QFT without dark matter or singularities. This guide summarizes the original 12 foundational papers below; Papers 14 and 15, published later as a cross-scale audit and a follow-on correction/derivation, are summarized separately at the end and predate neither this guide’s corrections nor their own.

For details, visit PBT Papers page. This guide is also included as an appendix in the formally published version of PBT on ai.viXra.org, and a PDF version with the original flux-shadowing figure is available in the PDF Library.

Status, added 2026-07-22: every paper summarized below now carries an epistemic-status banner on its own page (see Fringe Ideas) and links each borrowed formula to a real, cited page about the theory it comes from (see the Catalog and Established Physics). This guide summarizes; the papers themselves carry the actual corrections and links.

Corrigendum, 2026-07-21: a systematic audit found several of the specific claims summarized below to be errors in the original papers (confirmed via Wayback Machine archaeology to be original mid-2025 drafting mistakes, not a migration artifact) — muon g-2 anomaly stated ~10x too small, CHSH ~2.82 unsupported by the stated mechanism, neutron decay lifetime miscalculated, $H_0$ imprecisely described as matching Planck. Each has been corrected with a dated revision note directly in its source paper; the individual summaries below are annotated with pointers to those corrections. This PDF version and the ai.viXra.org/OSF submission linked above predate these corrections and have not themselves been revised — see Published Paper for that status. Corrections identified and calculated by Claude (Anthropic); the g-2, CHSH, Belle II, and neutron-decay findings were additionally verified independently by Grok (xAI). Original authorship and drafting: Matthew Foutch, with Grok (xAI) as collaborative AI.

Second corrigendum, 2026-07-22: a follow-on audit found four more errors this guide’s summaries didn’t yet reflect — Paper 1’s flagship rotation-curve claim (v(r)~220 km/s) is off by ~32x when computed from the paper’s own stated parameters, and its “matches observed G” claim is a normalization of the model’s own free parameters, not an independent prediction; Paper 4’s independently-parametrized version of the same rotation-curve claim is off by ~2x; and the “proton decay $>10^{34}$ years” alignment cited in Papers 7 and 11 is actually on the wrong side of the real Super-Kamiokande bound — PBT’s own predicted rate is about 30x faster than the maximum the real non-detection allows, not consistent with it. Individual summaries below are annotated accordingly. See the Formula Catalog for the full derivation of each finding. Identified by Claude (Anthropic); not independently cross-verified by Grok for this second pass.

Generated via Python/matplotlib from simplified shadowing equation $F \propto A\, \Delta P$.

Figure 1 illustrates the core concept of flux shadowing in PBT at various scales. The arrows represent directional particle fluxes in the infinite pressure vessel, converging radially to simulate attractive forces like gravity. Solid arrows indicate primary flux directions, while dotted patterns (if present in visualizations) could denote secondary distortions or aether flows. Units of scale are conceptual: at subatomic levels (e.g., $10^{-15}$ m), fluxes model nuclear forces; at planetary scales (e.g., $10^{6}$ m), they replicate Newtonian gravity; and at galactic scales (e.g., $10^{21}$ m), they explain flat rotation curves without dark matter. This multi-scale depiction highlights PBT’s hierarchical unification.

Paper Summaries

Detailed summaries of each paper follow.

PBT Paper 01: Infinite Push-Pressure Theory

Introduces PBT as hierarchical mechanical unification, resolving anomalies like flat rotation curves without dark matter via infinite pressure vessel and shadowing fluxes.

• Key Equation: $G_{eff}(l) \approx \varepsilon(l)\,\sigma(l)^2 / (4\pi\, m(l)^2)$, with $\varepsilon(l) = \varepsilon_0 (l_0/l)^\gamma$.
• Significance: Matches $G \approx 6.6743\times10^{-11}$ m³ kg⁻¹ s⁻²; nuclear bindings ~8 MeV/nucleon; $v(r)$ flattens to ~220 km/s. (Corrected 2026-07-22: the $G$ “match” is a normalization of the model’s own $\sigma$ parameter, not an independent prediction; the $v(r)\sim220$ km/s claim is off by ~32x when computed from the paper’s own stated $k$, $\gamma$, $r_0$ — see Paper 1 and Formula Catalog: Rotation-Curve Effective Gravity.)

Full Paper

PBT Paper 2: Hybrid Push-Aether Theory (Relativistic Unification)

Extends with dynamical aether for Lorentz invariance, unifying relativistically.

• Key Equation: $\theta \approx (4GM)/(c^2 b) \approx 1.75’’$ for Sun.
• Significance: Matches GR weak fields; predicts frame effects $<10^{-6}$ testable at LIGO.

Full Paper

PBT Paper 3: Hybrid Push-Aether Theory (Electromagnetic Forces)

Models magnetism and EM as subatomic flux gradients and directional flows within hierarchical levels.

• Key Equation: $B(r) \approx \mu_0 (\text{flux gradient} \times \sigma_{charge})$, with flux gradient $\propto \varepsilon(l)\, \nabla(q/r)$; Lorentz force $F = q(v\times B) \approx q\, v \times (\text{particle flow direction})$.
• Significance: Derives EM without virtual photons; predicts flow anisotropies in superconductors testable via muon spin rotation; deviations from QED in high fields.

Full Paper

PBT Paper 4: Hybrid Push-Aether Theory (Addressing Objections)

Addresses falsifiability via simulations; focuses on relativity conflicts (updated from nuclear forces).

• Key Equation: $v(r) = \sqrt{G_{eff}(r)\, M_{enc}(r)/r}$, $G_{eff}(r) = G\left[1 + k(r/r_0)^\gamma\right]$.
• Significance: Aligns with data; strengthens unification. (Corrected 2026-07-22: this paper’s own $k$, $\gamma$ give ~128 km/s at 30 kpc, not the claimed ~250 km/s, when computed directly — see Paper 4 and Formula Catalog: Rotation-Curve Effective Gravity.)

Full Paper

PBT Paper 5: Hybrid Push-Aether Theory (Quantum Spin, Entanglement, and Higher-Spin Integration, Revised)

Derives quantum spin, full entanglement, and higher-spin mechanics as emergent vorticity and correlated disturbances in the aether — extends the original quantum-mechanics derivation with refined aether couplings and QFT-style renormalization for infinite scales.

• Key Equation: $\omega = \nabla \times v$ (spin as quantized vorticity); CHSH ~2.82 from correlated flux. (Corrected 2026-07-21: this reconstructs to CHSH = 1.999, the classical bound — see Paper 5.)
• Significance: Quantizes mechanically; explains entanglement with ~99.5% correlation; resolves Rarita-Schwinger ghosts via aether suppression.

Full Paper

PBT Paper 6: Hybrid Push-Aether Theory (Quantum Spin Precession)

Models quantum spin precession from aether vorticity, with damping for stability.

• Key Equation: $dS/dt = -(g\mu_B/\hbar)\, S\times B$; aether damping $\xi\, u^\mu S_\mu = 0$, $\xi=10^{-15}$.
• Significance: Derives spin mechanically; predicts precession frequency $\omega = g\mu_B B/\hbar \approx 1.76\times10^{11}$ rad/s for $B=1$ T; decoherence anomalies testable via Bell experiments.

Full Paper

PBT Paper 7: Grand Unification

Unifies forces mechanically with infinite hierarchies.

• Key Equation: $\varepsilon(l) = \varepsilon_0 (l_0/l)^\gamma$; $H^2 = 8\pi G \rho_{eff}/3$.
• Significance: 80% alignment; proton decay $>10^{34}$ years. (Note, 2026-07-21: “80% alignment” isn’t a computed statistic, and this paper’s own g-2/CHSH/neutron-decay alignments are corrected above — see Paper 7.) (Corrected 2026-07-22: the proton-decay figure is the real Super-Kamiokande experimental bound, correctly stated — but this paper’s own predicted decay rate is about 30x faster than that bound allows, the opposite of the “alignment” being claimed here. See Formula Catalog: Proton Decay Suppression Estimate.)

Full Paper

PBT Paper 8: Hybrid Push-Aether Theory (Full QFT Action)

Develops QFT action for hierarchical unification, quantizing pushes and aether.

• Key Equation: $S = \int\sqrt{-g}\left[\dfrac{R}{16\pi G} - K\nabla u\nabla u + \lambda(u\cdot u + 1) + \dfrac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \dfrac{m^2\phi^2}{2} - \dfrac{\lambda\phi^4}{4} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi + \xi u^\mu \phi^2 + \cdots\right] d^4x$.
• Significance: Closes multi-field QFT gaps; predicts muon $g\text{-}2$ anomaly $\delta g \sim 10^{-10}$, matching Fermilab data; couplings unify at $10^{16}$ GeV. (Corrected 2026-07-21: the real anomaly is $\approx2.51\times10^{-9}$, ~10x larger — see Paper 8.)

Full Paper

PBT Paper 9: Hybrid Push-Aether Theory (Higher-Spin Framework and Infinite-Scale Cosmology)

Integrates higher-spin framework as multi-twist vorticities using Vasiliev-Fronsdal formalism and infinite-scale cosmology as pressure-driven expansion for grand unification.

• Key Equation: Friedmann equation $H^2 = (8\pi G/3)\rho_{eff} + (\Lambda c^2/3)$, with $\rho_{eff} = \varepsilon(l\to\infty)/c^2 \sim 10^{-26}$ kg/m³; higher-spin $\nabla^\mu \Phi_{\mu\ldots} = 0 + \xi u^\mu \Phi_{\mu\ldots} = 0$ (ghost-free).
• Significance: Aligns with LIGO GW speed $v_g = c \pm 10^{-15}$; predicts GW variations $\sim10^{-16}$ testable by LISA; $H_0 \sim 70$ km/s/Mpc matching Planck; closes 6% gaps in higher-spin and cosmology for unification with Papers 8 and 10; no singularities due to infinite scales. (Corrected 2026-07-21: $70$ km/s/Mpc is 3.86% off Planck and 4.11% off the local/SH0ES value — it doesn’t distinctly match either; see Paper 9.) (Flagged 2026-07-22: “closes 6% gaps” has the same unverified, no-defined-metric shape as this guide’s already-corrected “80% alignment” figure — no computation supporting this specific percentage is shown or recoverable in the paper. Treat as rhetorical framing, not a computed result, pending a dedicated check.)

Full Paper

PBT Paper 10: Hybrid Push-Aether Theory (Weak/Strong Forces)

Models weak as leaks, strong as vorticity (updated from astrophysics).

• Key Equation: $\Gamma_{weak} \approx G_F^2 m^5 / (192\pi^3)$.
• Significance: Neutron lifetime ~880 s; confinement ~$10^{-15}$ m. (Corrected 2026-07-21: the paper’s own formula and inputs actually give a lifetime of $\approx3.95\times10^{-11}$ s, not 880 s — the most severe error found in the audit; see Paper 10.)

Full Paper

PBT Paper 11: Hybrid Push-Aether Theory (Empirical Tests)

Synthesizes alignments for unification (updated from quantum gravity).

• Key Equation: $m_\nu \approx 10^{-3}$ eV.
• Significance: Aligns with proton decay, neutrino data. (Corrected 2026-07-22: the proton-decay alignment claimed here is on the wrong side of the real Super-K bound — this paper’s own predicted rate is about 30x faster than the bound allows. See Formula Catalog: Proton Decay Suppression Estimate.)

Full Paper

PBT Paper 12: Top Five Equations

Highlights successes over historical models.

• Key Equation: $g \approx 2$ from vorticity; CHSH ~2.82. (Corrected 2026-07-21: CHSH reconstructs to 1.999, the classical bound, not 2.82 — see Paper 5; the “Belle II ~2.78±0.05” citation elsewhere in this paper also appears unsupported — see Paper 12.)
• Significance: Matches $g\text{-}2$ anomaly $<10^{-12}$; entanglement without non-locality. (Corrected 2026-07-21: the real Fermilab anomaly is $\approx2.51\times10^{-9}$ — see Paper 7.)

Full Paper

PBT Paper 14: The Working Range of Push-Pressure Gravity (Cross-Scale Performance Audit)

Published after a numbering gap: Paper 13 (a nuclear-binding-energy derivation) remains unpublished, held back deliberately to mark the boundary between this series’ original Grok-authored, later-audited development and Paper 14 onward, developed and audited collaboratively throughout. Paper 14 introduces no new physics — it compiles and independently recomputes results already produced across Papers 1–12 into one explicit comparison against Newton, General Relativity, and real data (SPARC rotation curves, MIT bag-model confinement pressure, the atomic unit of pressure).

• Key Equation: performance ratio $\Lambda(l) = |P_{PBT}(l)-M(l)|\,/\,|P_{baseline}(l)-M(l)|$, scoring PBT’s prediction error against the relevant prior theory’s error at each length scale $l$.
• Significance: PBT reduces exactly to Newton in the classical regime ($\Lambda=1$, roughly $10^{-2}$–$10^{12}$ m), but fails at every other scale tested — a 32x-to-catastrophic overshoot at galactic scale (real SPARC data: $\chi^2/\text{dof}\approx82.5$ even in its best refit, versus $\approx1$–5 for MOND/dark-matter halos), 66–123 orders of magnitude at collapse scale (predicted stabilization radius exceeds the observable universe), and 7–38-plus orders of magnitude at atomic/nuclear scale. Also states directly that PBT’s own historical drag/heating problem (the objection that killed classical Le Sage gravity) remains unresolved, not fixed by this series’ “$v\to\infty$” mechanism. See Paper 14 for the full audit, including a dedicated “Objections and Responses” section addressing a hostile-reviewer pass conducted before publication.

Full Paper

PBT Paper 15: Where Newton Stopped Short (An Exact Geometric Result, and Where It Doesn’t Yet Reach)

Follows up on Paper 14’s classical-regime result by asking what it actually establishes — and corrects a real error made in this paper’s own first draft along the way, caught by the dedicated adversarial review this series now runs on every formal paper before publication. That first draft measured blocked solid angle and treated it as proportional to force; it isn’t. The quantity that determines net force is a cosine-weighted momentum-deficit integral over the same blocked cone, and — unlike solid angle — that integral is exactly $\pi(R/d)^2$ at every separation, not just far away, for the same reason Newton’s own Shell Theorem is exact rather than approximate.

• Key Equation: force-relevant geometric factor $F_{geo}(R,d)\propto\int\cos\theta\,d\Omega=\pi\sin^2\alpha=\pi(R/d)^2$ exactly, where $\sin\alpha=R/d$ from the exact tangent-cone construction (not a far-field approximation).
• Significance: the shadowing mechanism’s Newtonian reduction is exact, not approximate, for an idealized point test particle outside a single opaque sphere — stronger than originally claimed, and narrower: it says nothing about whether shadowing preserves exact proportionality to mass for realistically imperfect (non-opaque, non-idealized) bodies, which this paper identifies as the actual historically-fatal Le Sage objection (Poincaré/Maxwell self-shielding), not a side caveat, nor about mutual shadowing between two extended bodies. See Paper 15 for the corrected derivation and an explicit account of the error in the original draft.

Full Paper

PBT Paper 16: One Trade-Off, Not Two (Self-Shielding and Mutual Shadowing for Realistically Imperfect, Extended Bodies)

Follows up on Paper 15’s two named open problems — self-shielding for a realistically imperfect (non-opaque) absorber, and mutual shadowing between two extended bodies — attempting both together. The adversarial review process caught a framing error rather than a math error this time: an early draft claimed to have “resolved” the self-shielding problem, but the result is actually a precise quantification of one side of the 150-year-old Maxwell/Poincaré trade-off, not a rescue of it.

• Key Equation: in the optically-thin, single-scattering limit, $F_{thin}(d)=\kappa M/d^2$ for one extended, finite-opacity sphere (exact mass-proportionality, no residual size/density dependence); extended to two bodies via the Newtonian shell theorem, $F\propto\kappa_A\kappa_BM_AM_B/d^2$.
• Significance: self-shielding is exactly absent, at the order computed, in the one regime realistic bodies must occupy for mass-proportional gravity — but that same regime is exactly the one Maxwell (1875) and Poincaré (1908) already identified as maximizing the drag/heating problem Paper 14 found unresolved. A real, independently-verified result, but a negative/clarifying one: it locates Paper 15’s flagged problem precisely rather than resolving it. See Paper 16 for the full derivation and scope limits.

Full Paper

PBT Paper 17: The Geometric Core (Consolidated Statement, With Its Exact Domain of Validity)

Consolidates Papers 15 and 16 into one self-contained anchor: a numbered assumption set (A1–A8), a three-theorem derivation chain, independent numerical verification of each theorem, and an exclusion table (E1–E10) mapping every excluded phenomenon to the assumption it would relax. Introduces no new physics. Re-deriving Papers 15 and 16 from scratch for this paper found five errors in them, all corrected there via dated revision notes — including a stated precision figure wrong by nine orders of magnitude and a misattributed Maxwell citation.

• Key Equation: $F(d) = \mathcal{K}\kappa_A\kappa_B M_A M_B / d^2$ for two extended, finite-opacity, spherically symmetric bodies, with $G \equiv \mathcal{K}\kappa^2$; the normalization $\mathcal{K}$ carries units of pressure.
• Significance: states plainly and numerically that within a parameter band of $10^{-2}$–$10^{12}$ m, laboratory-to-solar masses, and slow motion ($v/c\le1.6\times10^{-4}$), PBT, Newtonian gravity and General Relativity agree to better than 99.9999975% — PBT reproducing Newton analytically exactly, and departing from GR only by the post-Newtonian factor $GM/rc^2$, at most $2.55\times10^{-8}$ (Mercury) anywhere in the band. Also makes explicit the hinge left implicit in Papers 15/16 — an opaque body’s shadow scales with cross-sectional area, not mass (a 16x force spread at fixed mass), and it is the optically-thin linearization that repairs this by rendering the shadow integral formally identical to the Newtonian volume integral. Outside the band the agreement breaks rather than degrades: at $v=c$ the Newtonian and GR light deflections differ by a factor of two.

Full Paper

Conclusion

This reference enables independent PBT study. For collaboration, contact mwfoutch@gmail.com. Assistance from Grok (xAI) acknowledged. Paper 14 (2026-07-28) developed with Claude (Anthropic) and Grok (xAI) as active collaborators throughout, not solely post-hoc reviewers.

References

• Planck Collaboration, 2020, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys., 641, A6 (1-73), https://doi.org/10.1051/0004-6361/201833910.
• Abbott et al., 2016, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett., 116, 061102 (1-16), https://doi.org/10.1103/PhysRevLett.116.061102.
• Aguillard et al., 2023, Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm, Phys. Rev. Lett., 131, 161802 (1-6), https://doi.org/10.1103/PhysRevLett.131.161802.
• Abbott et al., 2023, Constraints on the cosmic expansion history from GWTC–3, Phys. Rev. D, 107, 043009 (1-18), https://doi.org/10.1103/PhysRevD.107.043009.
• Vasiliev, M.A., 1990, Consistent equation for interacting gauge fields of all spins in (3+1) dimensions, Phys. Lett. B, 243, 378-382, https://doi.org/10.1016/0370-2693(90)91400-6.
• Jacobson, T. and Mattingly, D., 2001, Gravity with a dynamical preferred frame, Phys. Rev. D, 64, 024028 (1-9), https://doi.org/10.1103/PhysRevD.64.024028.