{ "schema_version": 1, "artifact_id": "qft.artifact.holography-quantum-gravity.quantum-cosmology-and-singularity-resolution-programs.relational-clock-packet-control", "title": "Relational Packet and Clock-Rematching Control", "source_revision": 2, "created_on": "2026-08-30", "creator": "OpenAI Codex, for QFT.org", "original_work": true, "reader_question": "What makes a scalar-clock conditional probability reproducible, and what fails if a monotonic clock is relabeled without transforming its generator?", "takeaway": "An exactly normalized positive-frequency packet translates rigidly with the scalar clock. The monotonic clock tau = 2 varphi gives the same relational prediction only when its generator is divided by two; the deliberately unrescaled generator remains normalized but moves the packet twice as far.", "scientific_status": "quantitative analytic fixture and deterministic adversarial control", "claim_ceiling": "The artifact proves unitary same-event equivalence for one exactly deparametrized, exactly chiral minisuperspace fixture and one linear clock relabeling. It does not prove equivalence of arbitrary internal clocks, factor orderings, nonmonotonic clocks, full Wheeler-DeWitt quantizations, inhomogeneous cosmology, or four-dimensional diffeomorphism-invariant observables.", "conventions": { "metric_signature": "(+---)", "logarithmic_scale_variable": "alpha = ln(a/a_star) is the logarithmic scale variable and is not additionally rescaled.", "scalar_clock_rescaling": "The homogeneous scalar field phi, not alpha, is rescaled to varphi = phi/(sqrt(6) M_Pl), giving the unit-speed reduced Klein-Gordon form.", "reduced_constraint": "(partial_varphi^2 - partial_alpha^2) Psi = 0 with Theta = -partial_alpha^2.", "operator_domain": "Theta is the non-negative self-adjoint operator -partial_alpha^2 on H^2(R) in L^2(R,d alpha).", "frequency_sign": "The page convention is -i partial_varphi Psi = sqrt(Theta) Psi. Accordingly positive-frequency modes carry exp(+i |k| varphi).", "chirality": "Only k = -p_alpha > 0 is populated with Fourier basis exp(-i k alpha); this invariant branch translates toward increasing alpha as varphi increases.", "physical_inner_product": "_phys = integral_R d alpha Psi1* Psi2 = integral_0^infinity dk A1* A2.", "probability_measure": "P_alpha is a density with respect to d alpha. For a = a_star exp(alpha), p_a(a|varphi) = P_alpha(ln(a/a_star)|varphi)/a.", "same_event_coordinate": "s = varphi = tau/2 compares the two clock labels at the same event." }, "equations": { "positive_frequency_equation": "-i partial_varphi Psi = sqrt(Theta) Psi, Theta = -partial_alpha^2", "momentum_amplitude": "A(k) = k^2 exp[-k/(2 kappa)] exp(i k alpha_0)/sqrt(24 kappa^5) for k = -p_alpha > 0 and A(k) = 0 for k <= 0", "momentum_normalization": "integral_0^infinity |A(k)|^2 dk = [24 kappa^5]^-1 integral_0^infinity k^4 exp(-k/kappa) dk = 1", "packet_transform": "Psi(alpha,varphi) = (2 pi)^(-1/2) integral_0^infinity A(k) exp(-i k alpha + i k varphi) dk", "closed_wavefunction": "Psi = [12 pi kappa^5]^-1/2 [1/(2 kappa) + i(alpha-alpha_0-varphi)]^-3", "conditional_density": "P_alpha(alpha|varphi) = [12 pi kappa^5]^-1 {[1/(2 kappa)]^2 + (alpha-alpha_0-varphi)^2}^-3", "scale_factor_probability": "Pr(a in [a_1,a_2]|varphi) = integral_ln(a_1/a_star)^ln(a_2/a_star) P_alpha(alpha|varphi) d alpha", "clock_rematching": "tau = 2 varphi implies -i partial_tau Psi = [sqrt(Theta)/2] Psi", "failed_control": "Keeping sqrt(Theta) instead of sqrt(Theta)/2 under tau = 2 varphi gives _naive = alpha_0 + tau = alpha_0 + 2s." }, "fixture": { "kappa": 0.5, "alpha_0": 0, "populated_momentum_domain": "k > 0", "varphi_slices": [ -2, 0, 2 ], "alpha_plot_range": [ -5, 5 ], "event_s_plot_range": [ -2, 2 ], "clock_scale_tau_over_varphi": 2, "window_half_width_in_alpha": 1, "alpha_sample_intervals": 400, "event_sample_intervals": 400, "csv_rows_excluding_header": 401 }, "anchors": { "exact": { "momentum_normalization": 1, "configuration_normalization": 1, "mean_offset": 0, "variance": 0.3333333333333333, "peak": 0.8488263631567752, "window_probability": 0.9244131815783876, "rematched_total_variation_at_s_2": 0, "naive_total_variation_at_s_2": 0.9244131815783876, "baseline_mean_at_s_2": 2, "rematched_mean_at_s_2": 2, "naive_mean_at_s_2": 4 }, "varphi_slices": [ { "varphi": -2, "peak_alpha": -2, "peak_density": 0.8488263631567752, "normalization": 1, "mean_alpha": -2, "variance_alpha": 0.3333333333333333, "window_alpha": [ -3, -1 ], "window_a_over_a_star": [ 0.049787068367863944, 0.36787944117144233 ], "window_probability": 0.9244131815783876 }, { "varphi": 0, "peak_alpha": 0, "peak_density": 0.8488263631567752, "normalization": 1, "mean_alpha": 0, "variance_alpha": 0.3333333333333333, "window_alpha": [ -1, 1 ], "window_a_over_a_star": [ 0.36787944117144233, 2.718281828459045 ], "window_probability": 0.9244131815783876 }, { "varphi": 2, "peak_alpha": 2, "peak_density": 0.8488263631567752, "normalization": 1, "mean_alpha": 2, "variance_alpha": 0.3333333333333333, "window_alpha": [ 1, 3 ], "window_a_over_a_star": [ 2.718281828459045, 20.085536923187668 ], "window_probability": 0.9244131815783876 } ], "same_event_s_2": { "varphi": 2, "tau": 4, "baseline_mean_alpha": 2, "rematched_mean_alpha": 2, "naive_mean_alpha": 4, "baseline_norm": 1, "rematched_norm": 1, "naive_norm": 1, "rematched_total_variation": 0, "naive_total_variation": 0.9244131815783876 } }, "data": { "file": "relational-clock-packet-control.csv", "row_model": "Each row carries one uniformly sampled panel-A alpha coordinate and one uniformly sampled panel-B same-event coordinate s. The three density columns use varphi = -2, 0, 2; the three mean columns use the baseline, correctly rematched tau clock, and deliberately unrescaled tau generator.", "columns": { "alpha": "dimensionless logarithmic scale variable", "p_varphi_minus_2": "P_alpha(alpha|varphi=-2)", "p_varphi_0": "P_alpha(alpha|varphi=0)", "p_varphi_plus_2": "P_alpha(alpha|varphi=2)", "event_s": "same-event coordinate s = varphi = tau/2", "mean_baseline": " under the varphi clock", "mean_tau_rematched": " under tau=2varphi with H_tau=H_varphi/2", "mean_tau_naive": " under the deliberately wrong H_tau=H_varphi" } }, "verification": { "analytic_checks": [ "The k-space gamma integral fixes the state normalization exactly.", "The Fourier-Laplace transform gives the closed wavefunction and conditional density.", "Translation gives mean alpha_0+varphi and invariant variance 1/(12 kappa^2).", "At kappa=1/2, the unit-width conditional probability and the TV distance of two copies separated by two are both 1/2+4/(3 pi).", "The chain rule under tau=2varphi fixes H_tau=H_varphi/2; at tau=2s the rematched state equals the varphi-clock state pointwise." ], "independent_quadrature": { "method": "Composite Simpson quadrature with 200000 intervals. Configuration moments use alpha-alpha_peak = tan(theta), momentum normalization uses k in [0,30], and the TV integral uses the same tangent map.", "momentum_normalization": 0.9999999999999999, "configuration": { "normalization": 1.0000000000000002, "mean_offset": 5.4005663609424147e-20, "variance": 0.3333333333333333, "window_probability": 0.9244131815783877 }, "naive_total_variation_at_s_2": 0.9244131815784543 }, "same_event_pointwise_grid": { "alpha_samples": 401, "event_samples": 401, "comparisons": 160801, "alpha_range": [ -5, 5 ], "event_s_range": [ -2, 2 ], "maximum_density_absolute_residual": 0 }, "acceptance_thresholds": { "momentum_normalization_absolute_residual": 5e-13, "configuration_normalization_absolute_residual": 5e-13, "mean_offset_absolute_residual": 5e-13, "variance_absolute_residual": 5e-13, "window_probability_absolute_residual": 5e-13, "naive_total_variation_absolute_residual": 5e-12, "same_event_mean_rematching_absolute_residual": 0, "same_event_pointwise_density_absolute_residual": 0, "naive_slope_two_absolute_residual": 0, "sampled_density_negative_excursion": 0 }, "maxima": { "momentum_normalization_absolute_residual": 1.1102230246251565e-16, "configuration_normalization_absolute_residual": 2.220446049250313e-16, "mean_offset_absolute_residual": 5.4005663609424147e-20, "variance_absolute_residual": 0, "window_probability_absolute_residual": 1.1102230246251565e-16, "naive_total_variation_absolute_residual": 6.672440377997191e-14, "same_event_mean_rematching_absolute_residual": 0, "same_event_pointwise_density_absolute_residual": 0, "naive_slope_two_absolute_residual": 0, "sampled_density_negative_excursion": 0 } }, "uncertainty_and_failure_cases": [ "The exactly chiral packet is a benchmark state, not a state-selection principle or a prediction for the early universe.", "The physical measure is d alpha. Probabilities in a require the Jacobian 1/a; |Psi(a,varphi)|^2 without the declared measure is not the plotted probability.", "Because P_alpha has polynomial alpha tails, every positive scale-factor moment with n > 0 diverges for this exact fixture; finite alpha-window probabilities remain well-defined.", "The tau = 2 varphi test is a monotonic relabeling. Its success does not establish equivalence for clocks that mix geometry and matter or have quantum fluctuations.", "The unrescaled tau generator is deliberately wrong. Its unit norm shows why normalization alone is an insufficient equivalence test.", "A clock such as chi = varphi^2 is outside this global clock patch: d chi/d varphi vanishes at varphi=0 and chi>0 has two varphi branches unless extra branch data are supplied.", "Changing the factor ordering changes Theta and can change the physical Hilbert space; that separate adversarial test is not performed by this fixture." ], "scientific_sources": [ { "citation": "Page, D. N., and W. K. Wootters. Evolution without Evolution: Dynamics Described by Stationary Observables. Physical Review D 27 (1983): 2885-2892.", "identifier": "DOI:10.1103/PhysRevD.27.2885", "use": "conditional-time framework and the requirement to specify the conditioning clock" }, { "citation": "Craig, D. A., and P. Singh. Consistent Probabilities in Wheeler-DeWitt Quantum Cosmology. Physical Review D 82 (2010): 123526.", "identifier": "DOI:10.1103/PhysRevD.82.123526; arXiv:1006.3837", "use": "direct Wheeler-DeWitt flat-FLRW massless-scalar frequency reduction, physical inner product, and single-clock-time probabilities" }, { "citation": "Ashtekar, A., A. Corichi, and P. Singh. Robustness of Key Features of Loop Quantum Cosmology. Physical Review D 77 (2008): 024046.", "identifier": "DOI:10.1103/PhysRevD.77.024046; arXiv:0710.3565", "use": "massless-scalar relational-time benchmark and positive physical-Hilbert-space evolution in a solvable cosmological model" } ], "accessibility": { "palette": "Explicit white canvas with black and gray strokes; line styles, marker shapes, hatching, and direct labels redundantly encode every distinction.", "reading_order": "Title and fixture, panel A conditional densities, panel B same-event clock comparison, deterministic controls, then claim ceiling.", "page_integration": "Embed in a keyboard-focusable figure-pan-region with visible focus, a full-size SVG link, and CSV/JSON downloads.", "structured_equivalent": "The semantic JSON preserves the model, domain, inner product, equations, parameters, anchors, checks, caveats, caption, alt text, and claim ceiling; the CSV preserves every plotted sample." }, "caption": "Relational packet and clock-rematching control for one exactly chiral massless-scalar minisuperspace fixture. Panel A shows a unit-normalized conditional density translating from alpha = -2 to 0 to 2 with unchanged variance 1/3; the hatched unit-width window has probability 1/2 + 4/(3 pi). Panel B compares the same events using s = varphi = tau/2. The correctly transformed generator H_tau = H_varphi/2 reproduces the varphi-clock mean, whereas the deliberately unrescaled generator has slope two and total-variation distance 0.924413 at s = 2 despite preserving norm one. This is a controlled clock-relabeling benchmark, not general clock-choice independence.", "alt_text": "Two stacked quantitative plots show three equal-area conditional packets centered at logarithmic scale alpha minus two, zero, and two, followed by a same-event clock comparison in which the baseline and correctly rematched tau equals two varphi predictions coincide with slope one while an unrescaled generator follows a dashed slope-two line and differs strongly at s equals two even though all norms are one." }