This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ The Sorkin quantum measure on causal sets must be covariant -- independent of the label (birth) order in which the causal set is grown (Rideout-Sorkin) -- a requirement that is one part of the thirty-year-old open problem of the canonical causal-set measure. We resolve that part with the central object of the series, and are equally explicit about what remains. PROVED HERE. The Barbu phase factor is the per-element exponential Phi_B(x) = e^{-i m c^2 tau_C(x)/hbar}, the global history factor Phi_B(C) = e^{-i phi_B(C)}, with scalar argument the Barbu action phase phi_B(C) = sum_x omega_B tau_C(x), tau_C(x) = ell_tau depth(x) (the longest chain ending at x, in discreteness units). This action phase is exactly covariant: a maximally added element has its complete causal past, so its depth at birth equals its final depth, and phi_B is birth-order-invariant (verified by enumerating all birth orders). The construction gives a level-2 Sorkin quantum measure (grade-2 sum rule I_3 = 0) whose decoherence functional is strongly positive (an outer product alpha alpha^dagger, Hermitian and positive-semidefinite), and the amplitude is spectator-independent in the Rideout-Sorkin sense (amplitude-level Bell causality). Covariance holds end to end -- every derived observable (proper time, curvature, counts, the cosmological invariant Lambda_C tau_C^2) is isomorphism-invariant -- and the measure extends to matter, staying covariant for any Bell-causal mass field. Uniqueness is physical, not mathematical: an infinite family of past invariants is covariant, but only depth is the order-intrinsic proper time tau_C, and the mass phase must ride on proper time (Feynman S = -m c^2 integral dtau), so phi_B is the unique legitimate mass action phase. Finally the readability support target (r*, v*) -- what makes a base select manifolds -- is a derived renormalization-group fixed point (a theorem by Poisson thinning; in 4D (r_4*, v_4*) ~ (0.062, 0.088)), not a fit. MAPPED, AND OPEN. This is the kinematic half. The measure's scale is cosmic time (a dynamical input -- overturning our own conjecture that the flow or Lambda would fix it), and the sub-distribution converges with a composition-dependent freedom. The dynamical half -- whether the manifold sector actually dominates the Kleitman-Rothschild entropy -- is mapped, not proved: a Bell-causal O(N^2) suppressor equal to the layer entropy gives a manifold-dominance transition at lambda_c = 1, the suppression is symmetry-universal, and its gauge sector points toward Maxwell through the gauge holonomy (holonomy <-> F x area). What is not derived is the origin of that O(N^2) magnitude from the fundamental amplitude |det D| -- the same object the Einstein coefficient zeta_BD requires -- nor the Maxwell 1/4. The kinematic half is a theorem; the dynamical half reduces to one open magnitude. Each result has a reproducible Python script (included).
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ The Sorkin quantum measure on causal sets must be covariant -- independent of the label (birth) order in which the causal set is grown (Rideout-Sorkin) -- a requirement that is one part of the thirty-year-old open problem of the canonical causal-set measure. We resolve that part with the central object of the series, and are equally explicit about what remains. PROVED HERE. The Barbu phase factor is the per-element exponential Phi_B(x) = e^{-i m c^2 tau_C(x)/hbar}, the global history factor Phi_B(C) = e^{-i phi_B(C)}, with scalar argument the Barbu action phase phi_B(C) = sum_x omega_B tau_C(x), tau_C(x) = ell_tau depth(x) (the longest chain ending at x, in discreteness units). This action phase is exactly covariant: a maximally added element has its complete causal past, so its depth at birth equals its final depth, and phi_B is birth-order-invariant (verified by enumerating all birth orders). The construction gives a level-2 Sorkin quantum measure (grade-2 sum rule I_3 = 0) whose decoherence functional is strongly positive (an outer product alpha alpha^dagger, Hermitian and positive-semidefinite), and the amplitude is spectator-independent in the Rideout-Sorkin sense (amplitude-level Bell causality). Covariance holds end to end -- every derived observable (proper time, curvature, counts, the cosmological invariant Lambda_C tau_C^2) is isomorphism-invariant -- and the measure extends to matter, staying covariant for any Bell-causal mass field. Uniqueness is physical, not mathematical: an infinite family of past invariants is covariant, but only depth is the order-intrinsic proper time tau_C, and the mass phase must ride on proper time (Feynman S = -m c^2 integral dtau), so phi_B is the unique legitimate mass action phase. Finally the readability support target (r*, v*) -- what makes a base select manifolds -- is a derived renormalization-group fixed point (a theorem by Poisson thinning; in 4D (r_4*, v_4*) ~ (0.062, 0.088)), not a fit. MAPPED, AND OPEN. This is the kinematic half. The measure's scale is cosmic time (a dynamical input -- overturning our own conjecture that the flow or Lambda would fix it), and the sub-distribution converges with a composition-dependent freedom. The dynamical half -- whether the manifold sector actually dominates the Kleitman-Rothschild entropy -- is mapped, not proved: a Bell-causal O(N^2) suppressor equal to the layer entropy gives a manifold-dominance transition at lambda_c = 1, the suppression is symmetry-universal, and its gauge sector points toward Maxwell through the gauge holonomy (holonomy <-> F x area). What is not derived is the origin of that O(N^2) magnitude from the fundamental amplitude |det D| -- the same object the Einstein coefficient zeta_BD requires -- nor the Maxwell 1/4. The kinematic half is a theorem; the dynamical half reduces to one open magnitude. Each result has a reproducible Python script (included).
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ In causal-set sequential growth, general covariance requires physical quantities to be independent of the birth order in which the causal set is grown. This note gives an elementary construction inside the standard labeled framework: an explicit matter-carrying action phase, built from an order invariant, that is exactly covariant at every finite stage and spectator-independent at the amplitude level. The point is a four-line theorem: an element's depth (its longest ending chain) depends only on its past, and in growth an element is born with its past complete, so depth-at-birth equals final depth. Hence the phase exp[-(i/hbar) m c^2 tau_0 sum_x depth(x)] is birth-order invariant, exactly and at every n. Because spectators do not enter the past of the competing births, the phase ratio is spectator-independent in the Bell sense at the amplitude-phase level. A sharp dichotomy fixes admissible matter: element-intrinsic, past-measurable fields keep covariance, while birth-index fields break it. Given the standard requirement that a mass phase accumulate along proper time, depth is singled out among the standard past-invariants. All statements are verified by exhaustive enumeration; three reproducible Python scripts are included. Scope: this is kinematic and selects no dynamics. Whether the resulting quantum measure makes manifold-like orders dominate is pursued separately in Parts LXIX and LXX.
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ This part proves no new theorem about the measure; it establishes one identification and draws the reading it forces. The Barbu phase of Parts LXIX-LXXI is a single object read in three registers: the internal clock Phi_B = e^{-i omega_B tau_C} (omega_B = mc^2/hbar, with the physical c), the phase of the history amplitude A(C) = W(C) e^{-i phi_B(C)}, and the mass kernel on worldlines. The old role is kept and lifted: the clock of mass becomes the phase of the causal history. The identification is checked numerically -- the history amplitude's phase responds to omega_B with exactly the clock's proper time (d(arg A)/d omega_B = sum_x depth(x), while normalized ratios are nearly clock-free); the dephasing rate between histories is linear in omega_B (Gaussian visibility law, rms 0.002); and branch separation grows with omega_B inside a readable window that closes exactly at omega_B = 2pi, where the integer clock aliases and the overlap revives. The reading it forces: not geometries in superposition, but phased causal histories in superposition, with geometry as the crystallized branch -- C is the combinatorial history, (C, phi_B) the interfering quantum branch, mu selects the readable branches, g_{mu nu} the resulting classical crystal. Mass is the rate at which causal order becomes phase, and numerically the rate at which branches of the blur become distinguishable. The extension breaks nothing: the new objects inherit the relabeling covariance and the quantum-measure structure of the old. The dynamical half is separate and already derived: the crystallized branch obeys the Einstein equation obtained through the readable Dirac operator (coefficient 8 pi G_ind / c_obs^4). The value of m is a declared input throughout; what is new is named as what it is -- an identification, seven gates of measurement, and a reading. Each result has a reproducible Python script (included).
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ Part LXV closed the operator-to-equation arc, ending with the causal Einstein equation G_munu^caus + Lambda_eff g_munu^caus = (8 pi G_ind/c_obs^4) T_munu^read and naming one remaining conceptual gap: the history measure mu(C). Part LXVI carries the programme across that gap and out to phenomenology in one continuous arc. It first defines the physical readability functional whose minimizer selects the physical history and reads gravity as the Green response of an induced causal screen L_Sigma^eff = a I - Pi_SJ (Sec. 1); fixes the normalization principle, the suppression threshold and the critical coarse-graining rule that turn readability into a measure principle (Secs. 2-4); establishes the spectral-gap/leakage bridge, the dynamical saturation principle and the posterior reading of the constants, with their universality and stability (Secs. 5-8); calibrates the Newton/Diosi-Penrose decoherence interface and the matter, gauge and vacuum sectors (Secs. 9-12); and finally assembles the constrained finite quantum measure mu(A) = D(A,A) and reads its experimental windows (Secs. 13-14). Throughout, the Barbu phase Phi_B = e^{-i m c^2 tau_C/hbar} rides on the order-intrinsic proper time and no fundamental graviton is introduced; the structural constant c_C stays dimensionless and only the observed c_obs = lim c_C carries the Einstein coefficient. The result closes the measure gap left open by Part LXV and gives the first phenomenological windows of the causal-gravity programme. This record unifies the former Parts LXXV-LXXXVIII into one article; the attached scripts are self-contained and reproduce the finite results.
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ Part LXVII inherited from causal-set theory the everpresent-Lambda dark-energy scale |Lambda| ~ 1/sqrt(V) ~ 1/tau_C^2. This Part LXVIII asks, as an outlook and a toy mechanism, whether that same everpresent-Lambda fluctuation can also act as the decoherence source that selects classical histories from the Sorkin quantum measure. The picture has two faces. (i) The everpresent Lambda is a fluctuating cosmological phase on the off-diagonal of the decoherence functional; averaging over its sign washes out interference while leaving the diagonal readable weights intact -- decoherence without disturbing which histories dominate. (ii) The accumulated cosmological action S_Lambda ~ Lambda_C V ~ sqrt(V) ~ tau_C^2 grows with cosmic proper time, so the coherence ~ exp(-S_Lambda^2/2) collapses as the universe ages -- an arrow of classicality; for the observable universe the coherence is exp(-O(10^244)) = 0. A decisive test shows this decoherence and the readability selection of the earlier parts are complementary, not redundant: readability acts on the amplitude magnitude (the manifold weight W), the decoherence on its phase (the coherence C); a 2x2 test (neither / readability-only / decoherence-only / both) confirms that only both together give a classical manifold world, each mechanism moving a different axis. A sharp toy classifier further shows that the cosmological phase, built from both the order-intrinsic proper time and the volume with V ~ tau_C^D, is coherent only on causal sets for which tau_C^D ~ V -- the Myrheim-Meyer / Brightwell-Gregory signature of manifold-likeness -- so manifold-like histories are the ones that remain coherent (KR-typical posets are not tested here). Honest scope: this is NOT a derivation but a toy model / outlook with stated assumptions; the Gaussian-dephasing form exp(-S_Lambda^2/2) is an ansatz, the everpresent-Lambda mechanism is Sorkin's (a candidate, not an established causal-set result of this programme), and only the tau_C link and the complementarity structure are the programme's own additions. We do not claim to solve the global history measure. Each result has a reproducible Python script (included).
Parts VIII–XI of this series reduced the program to a single missing object: a quantum measure μ(C) over causal histories. This Part XII specifies that object, realizes it in prototype, and tests whether it selects the observed effective spacetime dimension. The fundamental object is a Hermitian, positive, bi-additive Sorkin decoherence functional D(A,B) with physical measure μ(A)=D(A,A); classical measures obey the two-event sum rule I2=0, while quantum measures have I2≠0 but obey the Sorkin triple sum rule I3=0. An explicit such measure is constructed on the growth histories of small causal sets and verified numerically to satisfy I3=0 to machine precision while I2≠0. The fourth gate is sharpened beyond the standard Lorentzian-continuum requirement: the measure must concentrate on causal orders carrying coherent local proper time (a stable interval volume–proper-time law), the distinctive addition of this program. A systematic test on controlled ensembles then asks whether any non-circular intrinsic ingredient selects the 3+1 sector, with a uniformly negative result. The free measure is dominated by the entropic Kleitman–Rothschild phase (d_MM ≈ 2.37); proper-time coherence with propagation escapes that phase but settles on the effective 2+1 sector; the Benincasa–Dowker quantum phase produces only seed-unstable windows that vanish at the physically natural phase scale; and the Pauli–Jordan operator and the Sorkin–Johnston state, though intrinsic and phase-based, are monotone in dimension. A circularity audit shows that every selector whose optimum lies at four dimensions either inserts the dimension (local gravitational degrees of freedom, bound-matter stability, a (d−4)² term) or was tuned to it. The honest conclusion is that proper-time coherence explains the exit from the entropic phase but not the final 3+1 selection: the bare measure with coherence predicts an effective 2+1 phase, and the open problem is localized as the emergence of the third spatial direction (2+1 → 3+1). This yields a falsifiable prediction and identifies the missing object as a covariant, Bell-causal growth law whose continuum limit would fix the gravitational phase normalization. Supplementary Python scripts reproducing all numerical tests are included.
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ Part LIX serves two roles. First, a status map of the programme: eight results (proper time, the neutralization of Kleitman-Rothschild domination, the Einstein/Lovelock form, the Planck scale, the 3+1 selection, rest mass, quantum interference, gravitational decoherence), each stated with its headline, its logic, an honest status label (exact / analytic / prototype / mechanism), and a link to the Zenodo part where it is built, ordered by position in the series. Second, a focused study of the two frontier items. The history measure mu(C) is taken to be conditioned by the order and the action, mu = mu[C,S], and is shown to factorize, mu(C,S) ~ exp(-E_layer[C]) exp(-E_mat[C,S]): the order capacity E_layer (a log-count in nats, coupling fixed by normalization lambda=1) exactly neutralizes the KR multiplicity (E_layer^KR/N^2 -> (ln2)/4 = ln(#KR)/N^2, so e^{-E_layer} #KR ~ O(1), not a brute suppression); the action phase alone cannot (KR is a phase-coherent plateau); and the 3+1 choice lives in the separate exact term e^{-E_mat} (#E=#B <=> d_sp=3). The matter-sector dynamics is illustrated by a testable screen identity, C_Sigma P_IR ~ gamma_IR L_Sigma^+ P_IR => |D| ~ exp(-gamma_IR E_grav) (finite-N prototype, O(10%)). The single analytic gate that remains open is that this factorized suppressor and this covariance follow from the full measure mu(C) e^{iS/hbar} itself, with the manifold-like sector dominating Z. Results are stated at their honest level (exact, prototype, or mechanism); reproducible Python scripts are included. The part concludes with the interpretation of time as the causal DNA of a 4D world.