JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$

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Phase N (01b): LaTeX restoration — promote Unicode display-math to <math>; lint-clean per tools/wiki_latex_lint.py

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= Sanity-Check Limits =

{{Audience_Sidebar
| difficulty = Intermediate
| reading_time = 12 minutes
| prerequisites = [[Psionics_Primer]]; [[Psionics]] equation set; calculus; familiarity with the limits being recovered.
| if_too_advanced_see = [[Psionics_Primer]]
| if_you_want_the_math_see = [[5D_Action_Principle]]
}}

{{Notation
| psi_convention = ψ = scalar field amplitude.
| signature = Mostly-plus (−,+,+,+).
| units = ℏ = c = 1 unless explicitly noted.
}}

A theoretical framework is only as credible as the limits in which it reduces to known, well-tested physics. The '''sanity-check programme''' is the exhaustive list of regimes where the [[Psionics|psionic equation set]] must — and does — recover an established theory.

Every entry below is a falsifier in disguise: if any of these reductions fails, the framework is broken.

== Master table ==

{| class="wikitable" style="width:100%;"
! # !! Limit (parameter regime) !! Recovered theory !! Recovery is via
|-
| 1 || All ψ fields off (<math>\psi = 0</math> everywhere) || Standard Einstein gravity + Maxwell EM + matter || Trivial: all ψ-dependent terms vanish.
|-
| 2 || No 5th dimension (<math>L \to 0</math>, zero-mode only) || 4D scalar-tensor theory with ψ as standalone scalar coupled to EM and gravity || Kaluza–Klein reduction; see [[5D_Action_Principle]] §"Step 3".
|-
| 3 || <math>\lambda \to 0,\ J_\psi \to 0,\ F_{\mu\nu} \to 0</math> || Free Klein–Gordon equation <math>(\Box - m^2)\psi = 0</math> || Drop nonlinear and source terms.
|-
| 4 || <math>\lambda \to 0,\ J_\psi \to 0,\ F_{\mu\nu} \to 0,\ m \to 0</math> || Free massless wave equation <math>\Box\psi = 0</math> || Continue from limit 3.
|-
| 5 || Non-relativistic + static + linear || [[Yukawa_Potential|Yukawa equation]] <math>\nabla^2\psi - m^2\psi = -4\pi G_\psi \rho_\psi</math> || Drop <math>\partial_t^2</math>; drop <math>\lambda\psi^3</math>; drop <math>F^2</math>.
|-
| 6 || Non-relativistic + static + linear + <math>m\to 0</math> || Poisson equation <math>\nabla^2\psi = \text{source}</math> || Continue from limit 5.
|-
| 7 || <math>T^\psi_{\mu\nu} \to 0</math> (ψ extremely weak) || Standard Einstein equations <math>G_{\mu\nu} = 8\pi G\,(T^{\text{matter}}_{\mu\nu} + T^{\text{EM}}_{\mu\nu})</math> || ψ contribution becomes negligible source.
|-
| 8 || <math>\alpha \to 0,\ J_\psi \to 0</math> || Decoupled standard model + free ψ sector || Two non-interacting subsystems.
|-
| 9 || Slow ψ variation, non-relativistic || Schrödinger-like equation <math>i\hbar\,\partial_t\psi = -\tfrac{\hbar^2}{2 m_{\text{eff}}}\nabla^2\psi + V_{\text{nonlin}}(\psi)</math> || Standard non-relativistic reduction of a relativistic scalar.
|-
| 10 || Linearised gravity, slow source || Newtonian gravity for matter, [[Gravitoelectromagnetism|GEM]] for currents || See [[Gravitoelectromagnetism]] §"Sanity Checks".
|-
| 11 || Linearised GR, mass current <math>\rho_m \mathbf{v}</math> rotating rigidly || Lense–Thirring frame-dragging field || See [[Lense-Thirring_Frame_Dragging]] / [[Gravity_Probe_B]].
|-
| 12 || Test particle, <math>p = 0</math> || Standard geodesic equation <math>D^2 x^\mu/d\tau^2 = 0</math> (free fall) || ψ-force term <math>p\,\partial^\mu\psi</math> vanishes.
|-
| 13 || Test particle, <math>p = 0,\ q \ne 0</math> || Lorentz force law <math>q F^\mu{}_\nu\,(dx^\nu/d\tau)</math> || Standard charged-particle dynamics.
|-
| 14 || <math>\psi \to</math> constant (homogeneous everywhere) || ψ acts as a renormalisation of <math>\alpha_{\mathrm{EM}}</math> via <math>e^{k\psi}</math>; otherwise inert || Constant ψ contributes only a constant shift.
|-
| 15 || No coherent neural firing (<math>J_\psi = 0</math>) || ψ excited only by ambient EM (<math>\alpha F^2</math>) — no biological pump || "No psionics" baseline.
|-
| 16 || Vanishing ψ–EM coupling (<math>k = 0,\ \alpha = 0</math>) || ψ propagates independently of EM; brain cannot source ψ || Establishes that the framework's empirical content comes from the coupling.
|}

== Detailed treatment of each recovery ==

=== 1. All ψ off → standard physics ===

Setting <math>\psi \equiv 0</math> in the 4D effective action:

:<math>S = \int d^4x\,\sqrt{-g}\left[\frac{R}{16\pi G} - \tfrac{1}{4} F_{\mu\nu}F^{\mu\nu} + \mathcal{L}_{\text{matter}}\right]</math>

This is the action of standard physics: Einstein–Hilbert + Maxwell + matter. Variation gives the ordinary Einstein equations and Maxwell equations. No new physics. Pass: trivial.

=== 2. No 5th dimension ===

Setting the compactification radius <math>L \to 0</math> with all KK-tower modes ignored leaves the zero-mode action — which after dimensional reduction is exactly the 4D scalar-tensor theory we use. See [[5D_Action_Principle]] §"Derivation of the 4D effective theory".

=== 3. Free Klein–Gordon ===

With <math>\lambda = 0,\ J_\psi = 0,\ F = 0</math> the master ψ equation reduces to

:<math>\Box\psi - m^2\psi = 0</math>

which is the relativistic free-field Klein–Gordon equation — the same equation governing the Higgs field, the inflaton, and the QCD axion. ψ is just another scalar field in this limit. Pass: by construction.

=== 4. Free massless wave equation ===

Further setting <math>m \to 0</math>:

:<math>\Box\psi = 0</math>

The Lorentz-invariant wave equation. ψ disturbances propagate at exactly c. This limit is required for the framework to be compatible with special relativity in the high-frequency / short-distance regime.

=== 5. Yukawa equation ===

In the static (<math>\partial_t\psi = 0</math>), weak-field (drop <math>\lambda\psi^3</math>), no-EM (<math>F = 0</math>) limit, the master equation becomes

:<math>\nabla^2\psi - m^2\psi = -4\pi G_\psi\,\rho_\psi</math>

with point-source solution

:<math>\psi(r) = -\frac{G_\psi M_\psi}{r}\,e^{-m r}.</math>

This is identical in form to [[Hideki_Yukawa|Yukawa's]] 1935 meson potential — the recovery is exact. The shielding range <math>1/m</math> gives finite-range ψ effects and the rigorous basis for personal-shield phenomenology.

=== 6. Poisson equation ===

In the massless limit of Yukawa:

:<math>\nabla^2\psi = -4\pi G_\psi\,\rho_\psi.</math>

This is Poisson's equation — the equation of Newtonian gravity and electrostatics. Solutions go as <math>1/r</math>.

=== 7. Vanishing ψ stress-energy → ordinary Einstein equations ===

When the ψ field is dynamically negligible (low amplitude, small gradients), <math>T^\psi_{\mu\nu} \to 0</math> and the modified Einstein equations reduce to

:<math>G_{\mu\nu} = 8\pi G\,(T^{\text{matter}}_{\mu\nu} + T^{\text{EM}}_{\mu\nu})</math>

— standard GR with EM source. All [[Gravity_Probe_B|Gravity-Probe-B-class]] tests pass without modification. The framework only deviates from GR when ψ is dynamically significant.

=== 8. Decoupled sectors ===

If <math>\alpha = 0</math> and <math>J_\psi = 0</math>, the ψ sector and the standard-model sector are uncoupled. They are independent free theories: standard model on one side, free Klein–Gordon ψ on the other. This is the limit in which the framework would be "ψ exists but doesn't matter" — and would be empirically indistinguishable from no-ψ baseline.

=== 9. Schrödinger-like non-relativistic limit ===

For ψ varying slowly compared to c, writing <math>\psi = e^{-i m t}\,\chi(\mathbf{x},t)/\sqrt{2m}</math> with χ slowly varying gives, to leading order,

:<math>i\,\partial_t \chi = -\frac{1}{2m}\nabla^2 \chi + V_{\text{nonlin}}(\chi)</math>

— the same procedure that derives the non-relativistic Schrödinger equation from the Klein–Gordon equation. The nonlinear potential <math>V_{\text{nonlin}}</math> captures the <math>\lambda\psi^4</math> self-interaction, supporting [[Soliton_Solutions_of_Psi_Field|soliton]] solutions.

=== 10. GEM in the linearised-gravity limit ===

In the linearised-gravity weak-field, slow-motion limit, the modified Einstein equations reduce to a set of Maxwell-analog equations for gravitoelectric and gravitomagnetic fields. The ψ-coupling contributes an additional source term proportional to <math>T^{00}_\psi</math> on the gravitoelectric side. See [[Gravitoelectromagnetism]] §"GEM Maxwell-Analog Equations" and §"Coupling to ψ".

=== 11. Lense–Thirring frame-dragging ===

For a rotating massive body of angular momentum <math>\mathbf{J}</math> the off-diagonal <math>\bar{h}_{0i}</math> components of the metric perturbation generate a gravitomagnetic field

:<math>\mathbf{B}_g(\mathbf{r}) = \frac{G}{c^2 r^3}\bigl[\,3(\hat{\mathbf{r}}\cdot\mathbf{J})\,\hat{\mathbf{r}} - \mathbf{J}\,\bigr].</math>

This is the Lense–Thirring 1918 result, confirmed by [[Gravity_Probe_B]] in 2011 to 19% accuracy. The ψ-framework does not modify this prediction in the regime of [[Gravity_Probe_B]].

=== 12. Geodesic equation for p = 0 ===

For a particle of zero psionic charge (<math>p = 0</math>) and zero electric charge (<math>q = 0</math>), the equation of motion reduces to

:<math>\frac{D^2 x^\mu}{d\tau^2} = 0</math>

— pure free fall along a geodesic of the (possibly ψ-modified) metric. No anomalous forces.

=== 13. Lorentz force for q ≠ 0, p = 0 ===

For a particle of electric charge <math>q</math> but no psionic charge, the equation of motion is the standard Lorentz force law

:<math>\frac{D^2 x^\mu}{d\tau^2} = q\,F^\mu{}_\nu\,\frac{dx^\nu}{d\tau}.</math>

Recovery is automatic.

=== 14. Homogeneous constant ψ ===

If <math>\psi(\mathbf{x}) = \psi_0</math> (constant everywhere), all derivatives vanish; the effective fine-structure constant is shifted to <math>\alpha_{\mathrm{EM}}\cdot e^{-k\psi_0}</math> uniformly; there are no propagating ψ waves; no forces; ψ is observationally inert except via this fine-structure shift. This is the cosmological-background limit.

=== 15. No coherent neural firing (<math>J_\psi = 0</math>) ===

The "baseline" universe with no biological pump. ψ is still excited by ambient EM through <math>\alpha F^2</math> (lightning, magnetospheric currents, etc.) but no concentrated source. Sets the noise floor against which biologically-driven <math>J_\psi</math> must compete.

=== 16. Vanishing ψ–EM coupling (<math>k = 0,\ \alpha = 0</math>) ===

Without ψ–EM coupling, the brain cannot source ψ (no <math>\alpha F^2</math> term), and ψ cannot modify EM (no <math>e^{k\psi}</math> shift). The framework reduces to an inert isolated scalar that ordinary matter cannot see. This is the "null hypothesis" of psionics. Setting both couplings to zero turns the framework off and recovers standard physics + a hidden sector.

== Cross-checks across pages ==

Each numbered limit corresponds to a specific page in the wiki where it is exercised:

{| class="wikitable"
! Limit !! Discussed on
|-
| 1, 7, 14 || [[Psionics]] §"Sanity-Check Limits"; [[Modified_Einstein_Equations_with_Psi]]
|-
| 2 || [[5D_Action_Principle]]; [[Compactification_in_Kaluza-Klein]]; [[Kaluza-Klein_Unification]]
|-
| 3, 4 || [[Klein-Gordon_Equation]]; [[Quantization_of_the_Psi_Field]]
|-
| 5, 6 || [[Yukawa_Potential]]; [[Psi_Field]] §"Three field equations"
|-
| 9 || [[Soliton_Solutions_of_Psi_Field]]
|-
| 10, 11 || [[Gravitoelectromagnetism]]; [[Lense-Thirring_Frame_Dragging]]; [[Gravity_Probe_B]]
|-
| 12, 13 || [[Geodesic_Equation]]; [[Psionics]] §"Geodesic equation with psionic fifth force"
|-
| 15, 16 || [[Wilson-Cowan_Coupled_to_Psi]]; [[Falsification_Criteria_for_Psionics]]
|}

== Falsification implications ==

The framework is falsified by any one of the following experimental observations:

# A measurement of psionic phenomena in any of the "null" limits 1, 2, 3, 4, 7, 8, 14, 15, 16 — i.e. effects when the framework predicts none.
# A failure of one of the recovered limits (5–13) under conditions where the standard theory has been confirmed (Newtonian gravity, GR, Maxwell, Lorentz force, Lense–Thirring, etc.).

In the second case the failure is of the framework, not of the standard theory.

== See Also ==

* [[Psionics]] — equations whose limits are tabulated here.
* [[5D_Action_Principle]] — origin of every equation.
* [[Modified_Einstein_Equations_with_Psi]] — limit 7 in full detail.
* [[Gravitoelectromagnetism]] — limits 10 + 11.
* [[Yukawa_Potential]] — limit 5.
* [[Klein-Gordon_Equation]] — limits 3 + 4.
* [[Falsification_Criteria_for_Psionics]] — operational falsifiers.
* [[Symbol_Glossary]] — all symbols used above.

[[Category:Psionics]]
[[Category:Equations]]
[[Category:Reference]]

Sanity Check Limits - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$