{ "schema_version": 1, "artifact_id": "qft.artifact.holography-quantum-gravity.quantum-cosmology-and-singularity-resolution-programs.minisuperspace-omitted-mode-backreaction-control", "title": "Omitted-Mode Backreaction and Minisuperspace Control", "source_revision": 1, "created_on": "2026-08-30", "creator": "OpenAI Codex, for QFT.org", "original_work": true, "reader_question": "Can omitted inhomogeneous modes invalidate minisuperspace even when their one-point function vanishes and every mode remains a linear oscillator?", "takeaway": "Yes. In the declared finite-box de Sitter fixture, the vacuum-relative mean energy density of the first omitted shell grows linearly with occupation, crosses the declared one-percent source tolerance, and eventually equals the homogeneous background source.", "scientific_status": "quantitative analytic finite-box stress test", "claim_ceiling": "The artifact verifies one vacuum-relative mean-energy control for one conformal spectator, periodic cell, symmetric first-shell state, reference slice, and declared tolerance. It does not verify stress fluctuations, interactions, gauge-constraint closure, entanglement, anisotropic sectors, continuum convergence, a renormalized all-mode vacuum stress, singularity resolution, or full inhomogeneous quantum cosmology.", "conventions": { "metric_signature": "(+---)", "planck_mass": "M_Pl^2 = (8 pi G)^-1", "reference_slice": "a_star = 1", "background": "flat de Sitter with H_0/M_Pl = 10^-5, proper-time lapse N = 1, and comoving four-velocity u^mu = (1, 0, 0, 0)", "cell": "periodic physical cube with L_star = H_0^-1", "field": "free massless conformally coupled real spectator scalar", "shell": "the three opposite-wavevector pairs 2 pi L_star^-1 times plus or minus x, y, and z, represented by six independent real standing-wave oscillators", "state": "product number state relative to the conformal vacuum, with equal integer occupation n in each first-shell real oscillator and every other mode in the conformal vacuum; the plotted continuous envelope also equals the result for a number-diagonal mixture with equal mean occupation", "subtraction": "vacuum-relative state difference on the same background; common conformal-vacuum, trace-anomaly, and finite-volume vacuum terms cancel" }, "equations": { "first_shell_wavenumber": "k_star = 2 pi/L_star = 2 pi H_0", "excitation_density": "Delta rho_chi = u^mu u^nu (_n^ren - _0^ren) = 6 n k_star/L_star^3", "backreaction_ratio": "epsilon_br = Delta rho_chi/(3 M_Pl^2 H_0^2) = 4 pi n (H_0/M_Pl)^2", "restored_hubble_source": "H(n)/H_0 = sqrt(1 + epsilon_br)" }, "fixture": { "a_star": 1, "h0_over_planck_mass": 0.00001, "physical_side_in_hubble_units": 1, "first_shell_wavenumber_in_h0_units": 6.283185307179586, "real_oscillator_count": 6, "equal_occupation_per_oscillator": true, "coefficient_epsilon_per_occupation": 1.2566370614359174e-9, "declared_mean_energy_tolerance": 0.01, "mean_energy_equality_boundary": 1, "continuous_one_percent_crossing": 7957747.154594767, "continuous_equality_crossing": 795774715.4594766, "first_integer_failing_one_percent": 7957748, "first_integer_at_mean_energy_equality": 795774716, "plotted_occupation_range": [ 1, 10000000000 ], "plotted_epsilon_range": [ 1e-10, 10 ], "sample_count": 401 }, "anchors": [ { "occupation_per_oscillator": 0, "total_shell_quanta": 0, "epsilon_backreaction": 0, "fractional_hubble_shift": 0, "status": "vacuum-relative excitation source vanishes" }, { "occupation_per_oscillator": 1, "total_shell_quanta": 6, "epsilon_backreaction": 1.2566370614359174e-9, "fractional_hubble_shift": 6.283185305205665e-10, "status": "passes-declared-mean-energy-tolerance" }, { "occupation_per_oscillator": 7957748, "total_shell_quanta": 47746488, "epsilon_backreaction": 0.010000001062367548, "fractional_hubble_shift": 0.004987562640636637, "status": "fails-one-percent-tolerance" }, { "occupation_per_oscillator": 795774716, "total_shell_quanta": 4774648296, "epsilon_backreaction": 1.0000000006792418, "fractional_hubble_shift": 0.41421356261324327, "status": "background-split-fails" } ], "data": { "file": "minisuperspace-omitted-mode-backreaction-control.csv", "row_model": "401 logarithmically spaced points on the analytic continuous occupation envelope from 1 through 10^10, with shell quanta, backreaction ratio, restored-source Hubble shift, and status; pure product number states occupy the integer points.", "columns": { "occupation_per_oscillator": "equal dimensionless occupation n in each real first-shell oscillator; integer for the pure product number states and interpretable as equal mean occupation for a number-diagonal mixture", "total_shell_quanta": "six times the equal occupation", "epsilon_backreaction": "vacuum-relative mean-energy ratio Delta rho_chi/(3 M_Pl^2 H_0^2)", "fractional_hubble_shift": "sqrt(1 + epsilon_backreaction) - 1", "status": "classification under the declared one-percent and equality boundaries" } }, "verification": { "independent_mode_sum": { "delta_rho_per_occupation_in_h0_fourth_units": 37.69911184307752, "background_density_in_h0_fourth_units": 29999999999.99999, "coefficient": 1.2566370614359178e-9, "closed_form_coefficient": 1.2566370614359174e-9, "absolute_residual": 4.1359030627651384e-25 }, "shell_isotropy": { "shell_wavevectors_in_h0_units": [ [ 6.283185307179586, 0, 0 ], [ -6.283185307179586, 0, 0 ], [ 0, 6.283185307179586, 0 ], [ 0, -6.283185307179586, 0 ], [ 0, 0, 6.283185307179586 ], [ 0, 0, -6.283185307179586 ] ], "total_momentum_per_occupation_in_h0_fourth_units": [ 0, 0, 0 ], "pressure_tensor_per_occupation_in_h0_fourth_units": [ [ 12.566370614359172, 0, 0 ], [ 0, 12.566370614359172, 0 ], [ 0, 0, 12.566370614359172 ] ], "expected_isotropic_pressure_per_occupation_in_h0_fourth_units": 12.566370614359172, "maximum_absolute_residual": 0 }, "log_log_slope": { "expected": 1, "intervals_checked": 400, "maximum_absolute_residual": 3.774758283725532e-15 }, "zero_excitation_control": { "occupation_per_oscillator": 0, "total_shell_quanta": 0, "epsilon_backreaction": 0, "fractional_hubble_shift": 0, "status": "vacuum-relative excitation source vanishes" }, "integer_crossings": { "one_percent": { "previous_occupation": 7957747, "previous_epsilon": 0.009999999805730487, "first_failing_occupation": 7957748, "first_failing_epsilon": 0.010000001062367548 }, "equality": { "previous_occupation": 795774715, "previous_epsilon": 0.9999999994226046, "first_failing_occupation": 795774716, "first_failing_epsilon": 1.0000000006792418 } }, "dimensional_check": "Delta rho scales as k/L^3 and therefore has mass dimension four; 3 M_Pl^2 H_0^2 has the same dimension, so epsilon_br is dimensionless.", "acceptance_thresholds": { "coefficient_absolute_residual": 1e-24, "shell_isotropy_absolute_residual": 1e-15, "zero_momentum_absolute_residual": 1e-15, "log_log_slope_absolute_residual": 2e-14, "zero_excitation_absolute_residual": 0, "integer_crossing_minimality": true } }, "uncertainty_and_failure_cases": [ "There is no fitted or Monte Carlo uncertainty because every plotted point is evaluated from the analytic line; displayed decimals remain subject to floating-point roundoff and rounding, while the dominant uncertainty is model and truncation dependence.", "The one-percent boundary is a declared tolerance for this teaching fixture, not a universal physical constant.", "The equal shell is isotropic only in expectation. Stress fluctuations and unequal occupations require separate controls.", "The finite vacuum-relative state difference does not define the renormalized all-mode vacuum stress tensor.", "Changing the physical box or holding one discrete mode label fixed changes the state unless physical wavelengths and occupation density are remapped.", "Conformal coupling removes expansion-driven particle production in this fixture and is not a claim about general Mukhanov-Sasaki or tensor modes." ], "scientific_sources": [ { "citation": "Halliwell, J. J., and S. W. Hawking. Origin of Structure in the Universe. Physical Review D 31 (1985): 1777-1791.", "doi": "10.1103/PhysRevD.31.1777", "use": "homogeneous degrees treated together with inhomogeneous and anisotropic modes through second order" }, { "citation": "Schander, S., and T. Thiemann. Quantum Cosmological Backreactions. IV. Physical Review D 105 (2022): 106012.", "doi": "10.1103/PhysRevD.105.106012", "use": "constraint-first gauge-invariant perturbations and parameter-dependent oscillator backreaction" }, { "citation": "Birrell, N. D., and P. C. W. Davies. Quantum Fields in Curved Space. Cambridge University Press, 1982.", "doi": "10.1017/CBO9780511622632", "use": "renormalized stress-tensor and conformal-field context" } ], "accessibility": { "alt_text": "The omitted-shell mean-energy ratio grows linearly with equal mode occupation, crossing one percent near 7.96 million and unity near 796 million.", "caption": "Mean-energy backreaction from the first omitted shell in the declared finite-box benchmark. At a_star = 1, H_0/M_Pl = 10^-5, and L_star = H_0^-1, equal occupation n of the six real first-shell oscillators gives epsilon_br = 4 pi n (H_0/M_Pl)^2. The analytic envelope crosses the declared one-percent tolerance at n = 7.9577 times 10^6 and the unavoidable mean-energy failure boundary at n = 7.9577 times 10^8; pure product number states lie at integer n. This certifies only the displayed mean-source test.", "color_independence": "Black analytic line, different boundary line styles, direct labels, and distinct hatching classify every region without color.", "canvas": "explicit light canvas for dark-theme, monochrome, high-contrast, and print legibility" } }