SPSP–SSC Cheat Sheet — Deriving GR & the SM in Screened Regimes

introduction

what this section sets up: the physical picture (single-sphere microstructure + projection), and the role of Φ as a Lagrange-multiplier–type constraint rather than a dynamical field.

what you do here:

• internalize: Φ enforces balance; it’s elliptic, not hyperbolic—so it does not add a wave.
• note the scope vs. other approaches: this isn’t a UV theory; it’s a derivation/recovery in a regime.

what to verify (quick sanity):

• read how “screened regimes” are referenced and remember you’ll later see explicit inequalities (ε_scr ≪ 1).

outputs you carry forward:

• conceptual model for why no new graviton polarizations appear.
• expectation that exterior solutions look like GR vacuum.

axioms (a1–a6′)

what this section sets up: the minimal ingredients the rest of the paper is allowed to use.

what you do here:

• a1/a2: note the micro space S²×S¹ and a projection Π with weights W(x,σ). observables are averages over σ with normalized W.
• a3: all fields couple universally to g and the same W. you will use this to (i) fix a single Newton constant G and (ii) force a compact Yang–Mills structure with one U(1).
• a4: the cycle phase gives discrete weights (e.g., zeta-like factors) without adding spacetime DOF.
• a5: Φ is a constraint (no kinetic term).
• a6′: boundary value problem: Φ solves Poisson inside sources and homogeneous Laplace outside, with homogeneous boundary data on a closed surface S and at infinity. this will be the hammer that kills Φ in exteriors.

how to check:

• confirm normalization ∫_S W dμ = 1 is stated.
• confirm there’s no (∂Φ)² term anywhere in the action (later).

outputs you carry forward:

• a single G must appear consistently.
• Φ vanishes outside S under the stated boundary data.

pitfalls:

• forgetting that a6′ is an exterior statement. inside S you do not generally have Φ=0.

projection, fields, observables

what this section sets up: precise definitions and the “screening knob”.

what you do here:

• record the explicit projection formula and normalization.
• meet the EFT remainder δℙ and define ε_scr = sup |δℙ|/Λ⁴ over a domain D.
• read the boxed “screening conditions”: in screened domains, Φ obeys elliptic constraints and higher-dim operators are suppressed by ε_scr.

how to use:

• when you later claim GR equivalence, make sure ε_scr ≪ 1 is assumed.
• when you compute anything observable, you’re allowed to drop O(ε_scr) terms.

outputs you carry forward:

• a clean statement that Φ lives in H¹ with the given boundary data (regularity for maximum principle later).
• the dictionary: ρ = rest-mass density; ε = projected “core-energy.”

action with elliptic sorting

what this section sets up: the actual action and why its form is fixed.

what you do here:

• note the “variational selection”: in 4D, second-order, diffeo-invariant ⇒ Einstein–Hilbert (EH); well-posed Dirichlet problem ⇒ add Gibbons–Hawking–York (GHY).
• read the full action S = ∫√−g [ (1/16πG)R + ℒ_m − Φ(ρ−ε) ] + GHY.
• check “no (∇Φ)² term” is explicit (A5).
• normalization lemma: vary w.r.t. Φ to get ∇²Φ = 4πG(ρ−ε). by A3, the same G must appear in the metric equation. this locks G consistently across Newtonian and relativistic sectors.
• do the Newtonian sign check: for ε ≪ ρ, it reduces to ∇²Φ = 4πGρ.

how to verify quickly:

• perform a 1-line variation δS/δΦ to see the Poisson equation.
• match the Newtonian potential normalization to fix the sign and G.

outputs you carry forward:

• the equations of motion you’ll use and the fact G is now fixed once and for all.

pitfalls:

• adding any kinetic or potential term for Φ would break the dof count later.

variation and einstein equations

what this section sets up: the field equations and the “eliminate Φ” move.

what you do here:

• read the off-shell variation: the metric equation is the Einstein equation with the usual stress tensor; the Φ equation is the elliptic constraint.
• inspect the “detailed metric variation”: how ρ, ε shift under metric variation (useful for consistency).
• key step: eliminate Φ by solving its constraint and substituting back. this removes any Φ-stress; remaining corrections are O(ε_scr).
• theorem t6 (exterior GR equivalence): with compact support and homogeneous boundary data, Φ=0 outside S ⇒ exterior dynamics reduce to GR vacuum/matter.

how to verify:

• recall the maximum principle/uniqueness for Laplace’s equation with homogeneous data to justify Φ≡0 in the exterior.
• check boundary conditions match a6′.

outputs you carry forward:

• outside sources, you can work exactly like GR.
• inside sources, Φ is solved instantaneously by Poisson.

pitfalls:

• using exterior equivalence without homogeneous boundary data on S.
• forgetting O(ε_scr) terms if you step outside strictly screened domains.

3+1 (adm) decomposition and dof

what this section sets up: constraint structure and dof counting.

what you do here:

• list primary constraints (π_N≈0, π_i≈0, π_Φ≈0) and secondary ones (𝓗≈0, 𝓗_i≈0, 𝓒_Φ≈0 with 𝓒_Φ: ∇²Φ − 4πG(ρ−ε)).
• see that the (π_Φ, 𝓒_Φ) block is second class/invertible via (∇²)⁻¹ so Φ doesn’t propagate.
• pass to Dirac brackets and confirm the Dirac algebra closes with two tensor modes left.

how to check:

• count: metric 6 config dof → after constraints/gauge, 2 propagating DOF remain.
• ensure no leftover scalar constraint becomes dynamical.

outputs you carry forward:

• linear theory should be pure spin-2 massless (next section).

pitfalls:

• mixing up constraint classes before/after imposing the Green operator.

linearized theory and propagator

what this section sets up: the graviton content.

what you do here:

• linearize in harmonic gauge to get □ \bar h_{μν} = −16πG T_{μν}.
• confirm only two transverse-traceless polarizations propagate.
• note Φ contributes nothing to wave propagation (it’s elliptic).

outputs you carry forward:

• PPN and GW sectors should look like GR at leading order.

post-newtonian (ppn) analysis

what this section sets up: weak-field, slow-motion phenomenology.

what you do here:

• read off the metric potentials to 1PN order: γ=β=1.
• record “no −1PN dipole radiation” since there is no long-range scalar.
• confirm the quadrupole flux is the GR result; waves travel at c in screened regimes.

how to use:

• for timing, lensing, Shapiro delay, perihelion precession—use GR formulas.
• for GWs: use standard PN phasing without dipole terms.

outputs you carry forward:

• the paper’s observational examples (Shapiro, GW phasing) follow directly.

standard model (sm) sector — overview

what this section sets up: why the low-energy gauge group and charges come out as in the SM under minimal assumptions.

what you do here:

• log the assumptions (A3-style universality/locality; compact simple factors × one U(1); observed chiral content; one Higgs doublet; anomaly cancellation; no Witten anomaly; asymptotic freedom of color).
• see the exclusion lemmas removing extra groups/factors at low energy (either extra massless vectors or need for extra fields).
• main claim: only SU(3)_c × SU(2)_L × U(1)_Y works with the observed chiral content.

how to verify:

• sanity-check each exclusion: would an added factor force extra light bosons or new anomaly-cancelling matter? yes → excluded by “minimality.”

outputs you carry forward:

• the SM group is fixed; next you’ll fix the hypercharges.

pitfalls:

• confusing “UV unifications are allowed” with “they must appear here.” the text is agnostic about the UV and only derives the low-energy group.

one-generation multiplets & hypercharge (theorem s1)

what this section sets up: the unique hypercharge assignment.

what you do here:

• write the linear anomaly constraints (SU(3)²U(1), SU(2)²U(1), grav–U(1)) and two ID equations from Q = T₃ + Y (set Y_{q_L}=1/6, Y_{e_R} = −1).
• solve the 5×5 system to get Y(q_L, u_R, d_R, ℓ_L, e_R) = (1/6, 2/3, −1/3, −1/2, −1).
• check that U(1)_Y³ anomaly then vanishes automatically.

how to verify:

• actually row-reduce the given A·Y=b (it’s short).
• plug the solution back into all four anomaly conditions.

outputs you carry forward:

• the charge assignments needed to write the covariant derivatives and Yukawas.

anomaly cancellation (table)

what this section sets up: a compact ledger.

what you do here:

• verify each anomaly sum is zero using the found hypercharges.

outputs you carry forward:

• green light to write the full renormalizable SM lagrangian.

full sm lagrangian (derived content)

what this section sets up: the kinetic terms, covariant derivatives, Higgs potential, and Yukawas.

what you do here:

• write the gauge kinetic terms for SU(3), SU(2), U(1).
• write fermion kinetic terms with D_μ that includes g₃, g₂, g_Y and the right generators/hypercharges.
• write the Higgs sector with one doublet and the mexican-hat potential.
• write the renormalizable Yukawas: −\bar q_L Y_u \tilde φ u_R − \bar q_L Y_d φ d_R − \bar ℓ_L Y_e φ e_R + h.c.

how to verify:

• check operator dimensions ≤ 4 and gauge invariance with the established Ys.
• confirm no bare Dirac mass terms appear because of chirality.

outputs you carry forward:

• the complete low-energy field theory content—ready for EWSB, mixing, and RG.

assumption audit (sm)

what this section sets up: provenance of ingredients.

what you do here:

• tie back to axioms: A3 drives the gauge redundancy and bundle structure; A4 only touches spectral weights; Φ doesn’t enter particle physics.

why it matters:

• to show you didn’t smuggle in the SM—you derived it under stated minimal assumptions.

higgs & yukawa sector (uniqueness)

what this section sets up: why those are the only renormalizable couplings.

what you do here:

• scan allowed gauge-invariant operators with d ≤ 4.
• verify nothing else (beyond kinetic, gauge, φ potential, and the three Yukawas) is allowed.

output:

• uniqueness of the renormalizable flavor-generating interactions.

electroweak symmetry breaking (ewsb)

what this section sets up: symmetry breaking and mass generation.

what you do here:

• take ⟨φ⟩ = (0, v/√2)ᵗ; break SU(2)×U(1) → U(1)_EM.
• get W/Z masses and gauge mixing; confirm photon is massless.

outputs you carry forward:

• fermion mass matrices from Yukawas, and the need to diagonalize them.

masses & mixing (ckm/pmns)

what this section sets up: flavor structure.

what you do here:

• diagonalize Yukawas; note misalignment → CKM in quarks and (if neutrino masses are added) PMNS in leptons.
• the text stays agnostic on neutrino mass mechanism (Dirac/Majorana) since it’s beyond the minimal set shown.

three-generation checks

what this section sets up: anomalies still cancel and global anomalies remain absent.

what you do here:

• recall anomaly cancellation is generation-wise, so 3×(per-gen zero) = zero.
• Witten SU(2) global anomaly remains absent: each generation has an even number of SU(2) doublets.

notes on renormalization

what this section sets up: running and scheme matching.

what you do here:

• note that, in screened regimes, loop integrals can be organized to preserve Ward identities and match an MS-like scheme.
• record that one-loop βs match the SM (spelled out in the appendix).

how to use:

• for rough running, use the standard one-loop coefficients listed later.

cosmological linear perturbations (svt)

what this section sets up: cosmological phenomenology in the screened limit.

what you do here:

• adopt gauge-invariant SVT variables.
• in screened limit: scalar slip Φ−Ψ = 0; vector modes decay; tensor modes propagate luminally with the standard friction term 3H \dot h.
• growth, ISW, and lensing reduce to ΛCDM.

how to use:

• standard ΛCDM linear codes/intuition apply provided screening inequalities hold.

observational fits

what this section sets up: concrete checks.

what you do here:

• Shapiro delay formula with γ=1; the provided solar-system estimate is ∼120 μs (order-of-magnitude match to Cassini).
• GW phasing: no dipole (−1PN) term; leading quadrupole phasing as in GR.

how to use:

• when building constraints, you can port GR pipelines directly for these observables.

predictions & falsifiability

what this section sets up: ways to break the model.

what you do here:

• note hard predictions: (i) dipole radiation fraction ≲10⁻³ of GR; (ii) c_GW = c within experimental bounds; (iii) PPN γ=β=1 to current precision.
• any confirmed deviation in screened environments would falsify the framework as stated.

quantum outlook (non-axiomatic)

what this section sets up: a sketch, not a proof.

what you do here:

• recognize that a constrained path/hamiltonian quantization where Φ remains non-propagating is plausible.
• claim is modest: low-energy EFT matching to GR/SM seems straightforward; no UV completion is asserted.

appendix: constraint algebra

what this section sets up: the brackets and closure.

what you do here:

• read the matrix of Poisson brackets among constraints.
• see the (π_Φ, 𝓒_Φ) block is invertible via the Laplacian Green’s operator.
• pass to Dirac brackets; confirm closure and 2 propagating tensor DOF.

how to use:

• this underpins the “no extra scalar wave” statement in the main text.

appendix: one-loop β-functions (ms-like)

what this section sets up: running of couplings.

what you do here:

• write SU(N) general form, then the SM numbers:
• SU(3): 16π² β_{g₃} = −7 g₃³
• SU(2): 16π² β_{g₂} = −19/6 g₂³
• U(1)Y: 16π² β{g_Y} = +41/6 g_Y³ (and GUT-norm version)
• plus β_{y_t} and β_λ.

quick check: per-generation ∑ Y² = 10/3, Higgs = 1/2 → reproduces 41/6.

how to use:

• for quick RG running or cross-checks, plug these into standard one-loop equations.