Derivation Atlas

An absolute background tick T alongside laboratory proper time tau, related by a smooth positive lapse.

An inverse-square endurance flux that becomes Newtonian gravity in the IR; micro-length identified with the Planck length.

A uniform creation/source rate (White-Void / BLOP events) mimicking a cosmological-constant term.

An internal S1 dial at every address; the quarter-turn generator J (J^2=-1) is what the textbook imaginary unit i was packaging.

A world-cell address is one completed modular-capacity event, not a primitive coordinate-lattice site. Discreteness is earned by completion.

Per-address ceilings on energy, power, and action; no infinite local alphabet at one completed event.

Every physical record closes one full 2-pi modular budget through a visible plus same-universe hidden completion.

E_P = m_P c^2 = h-bar omega_P = rho4 (4 pi l_P^4): mass, frequency, and four-density as four faces of one Planck capacity unit.

A finite-throughput restatement of the uncertainty principle: every laboratory number is an access image of a source object.

Norm preservation makes the history weight orthogonal; ledger additivity under history composition makes it a one-parameter group; A7 dial closure makes it 2-pi periodic. The surviving laboratory weight is the real J-rotor R_J(theta) = cos(theta) I + sin(theta) J. Positive scalar weights cannot interfere, so interference is a closure theorem, not an added quantum axiom.

A short-time completed-history kernel carries a capacity-fixed magnitude and a J-rotor phase R_J(Delta S/hbar – d pi/4). Tick composition plus ledger additivity produces K = integral D_A6 x · R_J(S[x]/hbar). The textbook integral over exp(iS/hbar) is the complex-coordinate shorthand of this real-dial finite path sum; the A6 support is part of the object, not decoration.

In the macroscopic regime |S|/hbar >> 1, neighboring non-stationary histories rotate through rapidly changing J-angles and cancel. Coherent laboratory survival therefore selects delta S_tot = 0. QTT's content is upstream of the standard stationary-phase theorem: it derives the rotor weight and the S/hbar angle, while the final cancellation step is established mathematics. A7U keeps variations closure-preserving across the visible-plus-hidden bundle.

Integrating d theta = dS/hbar around one completed J-dial circle gives Delta S = 2 pi hbar = h. In QTT, hbar is action per radian of the real dial, and h is the cost of closing the dial once. The action quantum is not a separate smallest-action postulate; it is the closure of the dial.

One closure law gives four textbook projections: Planck-Einstein ET = h as the time circle, de Broglie p lambda = h as the space circle, Bohr-Sommerfeld integral p dq = n h as the phase-space loop, and flux quantization q Phi = n h as the gauge-holonomy loop. These are exact identity recoveries, so the node carries no sigma row.

The steady isotropic solution of the A2 endurance flux gives an inverse-square force in the infrared limit.

Newton's second law as the actuation of completed-address ticks; the relativistic gamma emerges from fixed-norm tick-speed sharing rather than being postulated.

Hamilton's principle and the path integral as the stationary real-dial phase over completed-address paths; the action quantum is h-bar per completed event.

Mass-energy equivalence read off the endurance ledger with c as a primitive carrier speed, not from boost algebra.

gamma = (1-v^2/c^2)^{-1/2} emerges from fixed-norm sharing of tick speed between hidden and visible cells, regularizing the v->c divergence.

Source action as a finite ledger over completed A5-X events; the laboratory Lagrangian and least-action integral are Access-Law images of that source ledger, not the constructor of it.

The imaginary unit is the laboratory shadow of the real-dial generator J. Phases, commutators, and path weights are real finite quarter-turns.

The first-order time law as address projection of the real-dial evolution; i d/dt psi = H psi is the lab packaging of J-native transport.

P=|psi|^2 from saturated visible/hidden bundle access and fixed-address counting; the square is forced, not inserted.

Spin quantization and the spinor half-angle from the adjoint action of the real J-dial; the 720-degree return is a rotor theorem.

E = h-bar omega and p = h-bar k as completed-address tick relations on the absolute clock.

Geometric phase as accumulated real-dial holonomy around a closed address loop.

The CHSH bound as balanced orthogonal-rail access geometry; the same equal-capacity rule gives Koide's sqrt 2. P_win = cos^2(pi/8).

The A1 clock-projection constant equals the T-gate magic-state overlap and the symmetric CHSH optimum; pi/8 is derived from the real-dial commutators, not assumed.

Integer flux quanta and the Josephson frequency from 2-pi modular closure of the charge dial.

The single-address pairwise resolution load R_n <= 2-pi forces n in {1,2,3}. The triad exactly saturates the budget (R_3 = 2-pi; R_4 = 4-pi).

The monadic / dyadic / triadic equal-share closures map to U(1), SU(2), SU(3); the e/3 charge quantum follows from the dyad-triad lattice.

After the photon house 8 and neutral projected half-loop rho/2 are quotient out, the residual five-rail edge gives alpha_QTT^-1 = 4*pi*(8 + rho/2 + lambda_gamma) = 137.035999165998, landing -0.523927 sigma against CODATA 2022 without using the observed alpha as a constructor.

NICK ladder: chi_YM = 2/3 + 1/(16 pi^2 cos^2(pi/8)) gives alpha_s = 1/(4 pi chi_YM) = 0.118052, +0.06 sigma. No observed alpha_s used.

Standard 4-loop running of alpha_s(m_Z) gives Lambda_3 = 334.65 MeV (+0.19 sigma); the A6-blocked transfer gives sqrt(sigma) = 444.25 MeV (-0.11 sigma).

Center-neutral adjoint-pair Haar transfer: m(0++)=1.731 GeV (+0.013 sigma), with tensor and pseudoscalar rows from spin-shear and odd-J exposure.

lambda_h = 1/2 + 37/(64 rho^2) gives m_h = 125.204 GeV (+0.038 sigma) as the radial vibration of the scalar-lock ruler.

The monadic U(1) address-transport theorem; Maxwell is the laboratory shadow of single-share modular transport, 1/sqrt(mu0 eps0) = c.

Derives the frozen charged-fermion integer ladder (t,b,tau,c,s,mu,d,u,e) -> (0,4,4,5,8,8,11,12,13) from the 16-sector dial, generation shell, Higgs orientation, and right-handed hypercharge residue. The integer depths are not chosen after looking at masses.

Top is the zero-depth identity channel of the charged MARIAM ladder: ell_t = 0 and T_hat_t = 1. The native anchor is m_t^A = E_H^lab = 174.103584824 GeV; collider-direct, pole, and MS-bar readings require an explicit observation map.

Sets the charged-fermion mass target m_f(mu_obs) = E_H^lab exp(-ell_f) T_hat_f R_f, with the same depth rule, determinant-torsion functor, and QED/QCD boundary transport logic for all nine charged fermions. The depth and transport spine is clean; the all-nine exact-mass theorem waits for determinant torsions and declared scheme windows.

Uses the closed top self-scale anchor, MARIAM depths ell_b = 4 and ell_c = 5, normalized heavy-triad torsions, QTT Yang-Mills transport, and the single-kernel QED window to predict m_b(m_b) = 4.178774016 GeV and m_c(m_c) = 1.271731720 GeV.

Closes the light u,d,s quark masses at 2 GeV with ell_s = 8, ell_d = 11, ell_u = 12, projected-loop torsion actions, common QTT YM transport, and the single-kernel QED window: m_u = 2.156580919 MeV, m_d = 4.704302901 MeV, m_s = 93.790027048 MeV.

Reads bottom/charm as a top-anchored torsion-amplitude triad, not an isolated two-body ratio. The MARIAM phase is phi_Q = -pi/104; with the QCD and photon readout gates the self-scale ratio becomes m_b/m_c = 3.28589273276, a green sigma pass in the book audit.

Uses the MARIAM b/c depth phase as the first quark-address edge: s_23 = sqrt(2) sin(pi/104) = 0.042713531541. This is audited against the global-fit V_cb neighborhood 0.04183(+0.00079/-0.00069) at about +1.1 sigma and deliberately sits on the high/inclusive side of the inclusive/exclusive V_cb tension. The light Cabibbo face gives s_12 = 0.224962553493, or -0.07 sigma against the global CKM lambda comparator 0.22501 +/- 0.00068. The diagonal face prints s_13 and delta_CKM before the precision access readout.

Accepts the flat w-glued Yukawa triangle as the pre-mass birth ledger and rejects disconnected direct-sum MARIAM blocks as an all-nine mass theorem. The next legal object is a coupled w-glued MinCap complex with printed spectra before mass comparison.

From capacity-quantized space plus the UEL: the Einstein equations emerge in the local Einstein gauge from the endurance current and Artian measure.

The Einstein-Hilbert coefficient forced by A2 through the endurance law; numerically the Planck-unit form, but derived prior to it (identical-equation, non-equivalent-theory).

The Lambda branch is consolidated as a Creation Ledger: late-time acceleration is read as an A3 projection effect, the exact vacuum identity is separated from fitted dark-energy fluid language, and epsilon remains a printed ledger-side IOU/falsifier.

Consolidates the dark-matter branch as fixed-coefficient source-kernel gravity: acceleration knee a0_tau = c H_tau/(2 pi), lab floor a0_tau/cos(pi/8), Renewal Dust/lensing readouts, cosmic-dipole reading, and a declared falsifier ledger. The RAR comparison is displayed as a stress row, not promoted into a green sigma claim.

The two-clock projection splits early- and late-time Hubble readouts: sec(pi/8) = 1.0824, matching the SH0ES/Planck ratio at -0.11 sigma under the Atlas convention: QTT minus observed over sigma. S4 remains open: epoch assignment and environment split must stay declared, not absorbed into a dark-energy fit.

The 18-lock 18 Omega_b^ABC (H_tau T0)^2 ~= 1; the lab projection uses Omega_b^lab = (1/18)(63.493001/67.36)^2 = 0.04935999 and is audited against the Planck physical-density row at about +0.18 sigma. The rounded 63.5/67.4 shorthand is not used for the pull.

Baryon-ledger time-drift gives an absolute 15.4 Gyr age, observed as 13.81 Gyr through the same two-clock projection.

The electro-optic Kerr constant as a finite-capacity birefringence readout of the Artian substrate.

Entropy production as anchored modular charge Q_w = 2-pi D(rho||omega); the address-ledger Second Law and a sharpened Page envelope.