IO Framework The Interior Observer Framework

Reference page

Calculator Theorem Dictionary

Standalone reference for the Interior Observer Framework black hole cosmology calculator. This theorem dictionary carries the self-contained theorem text behind the zero-fitted-parameter cosmological calculator and its theorem-grade predictions.

Back to output cards

About this calculator

This theorem dictionary is the public reference surface for the Interior Observer Framework cosmological calculator. It keeps the theorem text, scope boundaries, and claim statuses visible to readers and crawlers without requiring the paper archive.

What can this predict

  • T_CMB (0.3σ from FIRAS)
  • H0 (0.35σ from Planck)
  • Omega_k within 1σ of Planck CMB-only
  • BBN triple (chi^2 = 1.13)
  • Hubble tension resolved (max 0.57σ across 6 methods)
  • 40-year lithium problem resolved
  • CMB first peak ell = 224

The calculator page carries derivation chains inside each output card. This reference page is the standalone theorem dictionary for the live bundle.

premisePremise 1
premise

Statement

We live inside a black hole, and the CMB is the event horizon, with Hawking radiation falling inward and being observed from the interior.

Node id. premise.1

Scope summary. Global working assumption for IO model-building in this lab.

Proof outline

  • This node is declared as a working premise of the calculator rather than proved internally.
  • Any downstream node that lists premise.1 is explicitly conditional on this premise.

Scope boundary

  • Applies only as a lab working assumption for IO model-building.
  • Not presented here as an empirically established theorem.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

PREMISES.md
premisePremise 2
premise

Statement

The physics inside our black hole are the same as the physics outside our black hole.

Node id. premise.2

Scope summary. Global working assumption for IO model-building in this lab.

Proof outline

  • This node is declared as a working premise of the calculator rather than proved internally.
  • Any downstream node that lists premise.2 is explicitly conditional on this premise.

Scope boundary

  • Applies only as a lab working assumption for IO model-building.
  • Not presented here as an empirically established theorem.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

PREMISES.md
theoremPaper 12 Baryon Dictionary Fraction Theorem
derived / scoped

Statement

The Baryon Dictionary Principle fixes the framework baryon inventory fraction to f_b = 2 gamma_BI / x = 0.312708336215025 on the standard minimal-coupling matter class.

Node id. paper12.baryon_dictionary_fraction

Scope summary. Framework baryon inventory fraction before observable-class slot transport.

Depends on. premise.1, premise.2

Premises

  • premise.1 and premise.2 fix the IO horizon setting in which the baryon dictionary is posed.
  • The baryon fraction is an inventory-class selection statement, not a late-time one-number density theorem for every observable.

Proof outline

  • Identify baryons as the dust subset coupled to the boundary gauge sector rather than as an arbitrary fitted matter fraction.
  • Use the surviving line-scale exponent to select the alpha = 1 branch of the geometric inventory map.
  • Read off the exact inventory fraction f_b = 2 gamma_BI / x without introducing a fitted baryon parameter.

Scope boundary

  • Inventory fraction only.
  • Does not by itself determine every observable-class baryon slot or the full perturbation-era hierarchy loading.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

Interior_Observer_Paper12_v3_9.docx
theoremPaper 17 GTTP Thermal Readout Theorem
derived / scoped

Statement

The observer thermal readout obeys T_obs = T_IO x^K_gauge with K_gauge = ln(1 + gamma_BI^2) = 0.054872817742915. On the carried active thermal slot T_IO = 2.6635 K, this gives T_CMB = 2.7253 K.

Node id. paper17.gttp_thermal_readout

Scope summary. Observer-side thermal transfer law on the IO active branch.

Depends on. premise.1, premise.2

Premises

  • premise.1 identifies the observed CMB with the interior horizon readout problem.
  • premise.2 licenses the local thermal transfer class used to promote GTTP to theorem grade.

Proof outline

  • Use KMS rigidity to fix exact Planck-form preservation under uniform frequency rescaling.
  • Combine multiplicative horizon gauge data with additive transfer generators to force a logarithmic homomorphism.
  • Fix the coefficient on the Schwarzschild S^2 horizon and evaluate the resulting thermal map on the carried IO temperature slot.

Scope boundary

  • Thermal readout law only.
  • Does not by itself determine every late-time background or perturbation observable on the active branch.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

Interior_Observer_Paper17_v1_4.docx
theoremPaper 21 TIO Branch Assignment
derived / scoped

Statement

The active calculator branch is the carried Paper 10 legacy projected branch used by the public runtime surface.

Node id. paper21.branch_assignment

Scope summary. Calculator branch selection and carried active package.

Depends on. premise.1, premise.2

Premises

  • premise.1 and premise.2 fix the IO setting in which a carried active branch is meaningful.
  • The calculator publishes one carried active package rather than dynamically averaging over multiple branch candidates.

Proof outline

  • Read the carried runtime package selected by the Paper 21 branch-assignment result.
  • Identify that public runtime package with the legacy Paper 10 projected branch used by the live calculator constants and derived outputs.
  • Once this identification is fixed, all downstream calculator outputs inherit one branch label instead of refitting branch choice per observable.

Scope boundary

  • Applies only to the calculator's carried active package.
  • Does not assert uniqueness of the branch outside the published runtime surface.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper21_tio_branch_assignment_theorem_report.txt
theoremPaper 22 Spatial Mode Ladder
derived / scoped

Statement

The closed S^3 spatial carrier decomposes into scalar, vector, and TT tensor ladders with lambda_n^(S) = n(n+2) / a^2, lambda_n^(V) = (n+1)^2 / a^2, lambda_n^(TT,rough) = (n(n+2)-2) / a^2, multiplicities (n+1)^2, 2n(n+2), 2(n-1)(n+3), and diagonal-spin floors J_min = 0,1,2 respectively.

Node id. paper22.spatial_mode_ladder

Scope summary. Spatial harmonic carrier on closed S^3.

Depends on. premise.1, premise.2

Premises

  • premise.1 and premise.2 place the perturbation problem on the accepted IO closed-space setting.
  • Paper 22 constructs the scalar/vector Hodge carriers and the TT extension on round S^3.

Proof outline

  • Decompose the round-S^3 spatial Hilbert space into scalar, coexact-vector, and TT tensor branches.
  • Read off the exact eigenvalues, multiplicities, and diagonal-spin supports for each branch.
  • Use those branch laws as the theorem-grade spatial carrier for later perturbation work.

Scope boundary

  • Spatial carrier only.
  • Does not by itself determine the dynamical perturbation/source equations.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper22_spatial_hodge_complex_report.txt
theoremPaper 23 Closed Scalar Operator Theorem
derived / scoped

Statement

On the closed scalar hierarchy, the first curvature correction is lambda_n - 3 = (n-1)(n+3), scalar hyperspherical support obeys ell = 0,1,...,n, and the physical inhomogeneous scalar spectrum begins at n >= 2 with n = 1 pure gauge.

Node id. paper23.closed_scalar_operator

Scope summary. Closed scalar shell operator and physical mode support.

Depends on. paper22.spatial_mode_ladder

Premises

  • paper22.spatial_mode_ladder fixes the scalar harmonic carrier on S^3.
  • The closed scalar perturbation problem lives on that discrete shell basis rather than a flat continuous k basis.

Proof outline

  • Start from the scalar S^3 harmonic ladder with lambda_n = n(n+2).
  • Insert the closed-space scalar curvature correction to obtain lambda_n - 3 = (n-1)(n+3).
  • Classify n = 0 as background, n = 1 as gauge, and n >= 2 as physical scalar support.

Scope boundary

  • Closed scalar shell operator and support only.
  • Does not by itself close the full metric-plus-fluid source/acoustic hierarchy.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper23_scalar_perturbations_report.txt
theoremPaper 23 Bridge Uniqueness Theorem
derived / scoped

Statement

The accepted one-slot background-contraction bridge sends a boundary vector shell n only to adjacent scalar shells N = n - 1 or N = n + 1, so the bridge image is multiplicity-one and adjacent-shell by construction.

Node id. paper23.bridge_uniqueness

Scope summary. Adjacent-shell bridge grammar on the accepted Paper 23 one-slot scalar sector.

Depends on. premise.1, premise.2

Premises

  • premise.1 and premise.2 fix the IO horizon setting in which the bridge is posed.
  • Paper 23 works on the one-slot background-contraction bridge rather than a generic multi-shell source operator.

Proof outline

  • Start from the accepted Paper 23 background-contraction bridge operator on the boundary vector carrier.
  • Project that bridge onto scalar shells and read off the exact selection rule.
  • Kill non-adjacent shell targets, leaving only the multiplicity-one adjacent-shell image N = n - 1 or N = n + 1.

Scope boundary

  • Accepted one-slot bridge grammar only.
  • Does not by itself fix the downstream source coefficients or the full perturbation/readout solver.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper23_bridge_uniqueness_report.txt
theoremPaper 28 Closed S^3 Shell-power Definition
derived / scoped

Statement

The exact observer-side closed-S^3 shell covariance obeys Var_n(X) = ((n+1)^2 / (2 pi^2 R^3)) P_X(n), with dimensionless shell conventions Delta_q^2(n) = ((n+1)^3 / (2 pi^2 R^3)) P_X(n), Delta_scalar^2(n) = (n(n+1)(n+2) / (2 pi^2 R^3)) P_X(n), and Delta_MS^2(n) = (((n-1)(n+1)(n+3)) / (2 pi^2 R^3)) P_X(n).

Node id. paper28.closed_s3_shell_power

Scope summary. Observer-side shell covariance and power-spectrum conventions on closed S^3.

Depends on. paper22.spatial_mode_ladder, paper23.closed_scalar_operator

Premises

  • paper22.spatial_mode_ladder fixes the exact scalar-shell degeneracy (n+1)^2 on closed S^3.
  • paper23.closed_scalar_operator fixes the physical scalar shell variable k_MS(n) = sqrt((n-1)(n+3)) / R.

Proof outline

  • Expand the observer-side scalar field in orthonormal closed-S^3 hyperspherical harmonics.
  • Write the equal-point variance shell by shell using the scalar degeneracy (n+1)^2 and the volume 2 pi^2 R^3.
  • From that invariant shell variance, read off the exact q, scalar-Laplacian, and Mukhanov-Sasaki Delta^2 conventions without importing the flat k^3 law as primitive.

Scope boundary

  • Observer-side closed-shell power definitions only.
  • Does not by itself derive the physical IO source-side shell covariance law or the missing bridge typing.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper28_closed_s3_power_definition_report.md
theoremPaper 28 Boundary Fixed-point Scalar Tilt Law
conditional / scoped

Statement

Conditional on the Boundary Fixed-point Principle, the active scalar tilt closes as n_s = 1 - K_gauge / x = 0.9639.

Node id. paper28.boundary_fixed_point_scalar_tilt

Scope summary. Active scalar-tilt closure on the published Paper 28 scalar sector.

Depends on. premise.1, premise.2

Premises

  • premise.1 and premise.2 fix the IO boundary/source setting for the scalar sector.
  • The Boundary Fixed-point Principle is the surviving scalar-sector premise replacing the older boundary-covariance route.

Proof outline

  • Use the Paper 28 boundary audit to kill the ordinary local shell-blind tilt mechanisms.
  • Retain the fixed-point route as the surviving scalar-sector coefficient principle.
  • Evaluate the resulting tilt law n_s = 1 - K_gauge / x on the active IO constants.

Scope boundary

  • Conditional scalar-tilt closure only.
  • The current stack does not license this law as an unconditional theorem independent of the Boundary Fixed-point Principle.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

Interior_Observer_Paper28_v1_3.docx
theoremPaper 30 Legacy Recompute Surface
derived / scoped

Statement

The active branch supports the late-time closed-FRW background surface with H(z) = H_0 sqrt[Omega_r (1+z)^4 + Omega_m (1+z)^3 + Omega_k (1+z)^2 + Omega_Lambda], D_M(z) = R_c sin(chi(z)), chi(z) = ∫_0^z dz' H_0 / H(z'), and t(z) = ∫_z^∞ dz' / ((1+z') H(z')).

Node id. paper30.background_surface

Scope summary. Observer-side background geometry and BAO-side runtime surface.

Depends on. paper21.branch_assignment

Premises

  • paper21.branch_assignment fixes the active branch package.
  • Closed-FRW background evolution on that branch is already part of the accepted runtime surface.

Proof outline

  • Evaluate the active-branch closed-FRW integrals for expansion, transverse distance, radial distance, volume distance, lookback time, and age.
  • Export those quantities directly in the calculator without delegating geometry to a flat-space backend.
  • Use the carried branch value consistently across the public background and BAO surfaces.

Scope boundary

  • Observer-side closed-FRW background geometry on the fixed active branch only.
  • Does not by itself determine a full perturbation transfer solver.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper30_full_recompute_legacy_branch_results.json
theoremPaper 30 Active-branch Parameter Package
derived / scoped

Statement

The carried public runtime branch fixes H0 = 67.575856535826 km/s/Mpc, Omega_m = 0.348683950676, Omega_k = -0.045791125760, Omega_Lambda = 0.697015757616, T_CMB = 2.7253 K, and Y_p = 0.2477.

Node id. paper30.active_branch_parameter_package

Scope summary. Fixed active runtime parameter package carried by the public calculator.

Depends on. paper21.branch_assignment

Premises

  • paper21.branch_assignment fixes the carried active branch rather than refitting a branch per observable.
  • The public calculator exposes one reviewed runtime package as its active numerical surface.

Proof outline

  • Read the active branch constants from the reviewed Paper 30 runtime package.
  • Carry those values unchanged into the calculator constants layer.
  • Expose them as public theorem-grade or scoped-active package values rather than hiding them behind an opaque backend.

Scope boundary

  • Fixed active runtime package only.
  • Does not claim that the active package is the unique surviving branch outside the reviewed public calculator surface.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

Interior_Observer_Paper30_v1_3.docx
theoremPaper 30 Bare Master-clock Theorem
derived / scoped

Statement

The all-epoch local IO master clock is the bare FRW proper-time integral t_bare(z) = integral_z^infinity dz' / [(1+z') H_bare(z')] with E(z)^2 = Omega_r (1+z)^4 + Omega_m (1+z)^3 + Omega_k (1+z)^2 + Omega_Lambda, equivalently t_bare(a) = H0_bare^-1 integral_0^a da' / sqrt(Omega_r + Omega_m a' + Omega_k a'^2 + Omega_Lambda a'^4), yielding t_bare(z=0) = 19.181055510227 Gyr on the carried bare branch with exact radiation.

Node id. paper30.bare_master_clock

Scope summary. Bare local-clock theorem on the Paper 30 master-clock branch.

Depends on. premise.1, premise.2, paper17.gttp_thermal_readout

Premises

  • premise.1 and premise.2 fix the IO local-clock setting and allow the standard local radiation-density input.
  • paper17.gttp_thermal_readout fixes the carried observer CMB temperature entering the exact radiation density.
  • The Paper 30 master-clock correction replaces the old dust cycloid as the all-epoch local clock.

Proof outline

  • Derive the exact bare radiation density from the carried thermal readout and standard-neutrino slot.
  • Insert that radiation term into the bare FRW proper-time integral instead of dropping to the dust cycloid approximation.
  • Evaluate the corrected integral on the carried bare branch and expose the resulting present-day local age.

Scope boundary

  • Bare local-clock branch only.
  • Not the projected observer-side age already exposed by the active closed-FRW background snapshot.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper30_master_clock_correction_report.txtInterior_Observer_Paper30_v1_3.docx
theoremPaper 30 Active Deuterium Scorecard
derived / scoped

Statement

The active repaired BBN scorecard carries the fixed deuterium prediction D/H = 2.509000000000e-05. The Paper 30 absorber compilation verifies that this fixed IO prediction survives the current precision sample cleanly.

Node id. paper30.deuterium_scorecard

Scope summary. Active repaired BBN deuterium scorecard carried by the calculator.

Depends on. premise.1, premise.2, paper12.baryon_dictionary_fraction

Premises

  • premise.1 and premise.2 fix the IO BBN setting.
  • paper12.baryon_dictionary_fraction repairs the old deuterium crisis by fixing the surviving baryon fraction route.

Proof outline

  • Carry the repaired IO deuterium prediction on the active BBN scorecard rather than the superseded low-baryon route.
  • Compare that fixed prediction against the current absorber compilation without re-fitting the value.
  • Retain the scorecard as a public carried output because the surviving route remains numerically clean against the precision sample.

Scope boundary

  • Active repaired deuterium scorecard only.
  • Does not claim a live calculator nuclear-network re-integration on demand.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper30_funrun_r_isw_deuterium_schur_report.txtInterior_Observer_Paper30_v1_3.docx
theoremPaper 30 Active Primordial Helium Scorecard
derived / scoped

Statement

The active repaired BBN scorecard and runtime package carry Y_p = 0.2477 on the public calculator surface.

Node id. paper30.primordial_helium_scorecard

Scope summary. Active primordial helium scorecard carried by the calculator.

Depends on. paper21.branch_assignment, paper30.active_branch_parameter_package

Premises

  • paper21.branch_assignment fixes the active runtime branch.
  • paper30.active_branch_parameter_package carries the reviewed active helium mass fraction on the public calculator surface.

Proof outline

  • Read the active helium mass fraction from the reviewed runtime package.
  • Carry that value unchanged into the live calculator constants and public bundle.
  • Cross-check the fixed value against the current primordial-helium data compilation without promoting the comparison itself to a new fit.

Scope boundary

  • Active carried helium scorecard only.
  • Does not claim a live calculator BBN network solve for Y_p.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper30_funrun_yp_21cm_nanograv_schur_report.txtInterior_Observer_Paper30_v1_3.docx
theoremPaper 29 Sound-speed Baryon-slot Selector
derived / scoped

Statement

The local sound-speed loading term uses R(z) = 3 rho_b(z) / [4 rho_gamma(z)] with the unique theorem-grade baryon slot omega_b,geom on the rebuilt reduced-stack scope.

Node id. paper29.sound_speed_selector

Scope summary. Local photon-baryon inertia coefficient for the active branch.

Depends on. premise.1, premise.2, paper21.branch_assignment

Premises

  • premise.1 and premise.2 fix the IO setting for the local pre-recombination plasma.
  • paper21.branch_assignment fixes the carried active branch used by the live calculator.

Proof outline

  • Identify R(z) as the local photon-baryon inertia coefficient rather than an observer-side readout scalar.
  • Use the rebuilt Paper 29 slot audit to rule out the late clustering branch for this local fluid coefficient.
  • Conclude that the unique theorem-grade slot for R(z) is omega_b,geom on the live calculator branch.

Scope boundary

  • Applies to the local sound-speed loading term R(z) on the active branch.
  • Does not by itself close the full drag-epoch or BAO standard-ruler theorem.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper29_sound_speed_baryon_selector_audit_report.md
theoremPaper 31 Geometric Pre-drag Ruler
derived / scoped

Statement

The active branch carries the published pre-drag ruler r_d = ∫ c_s(z) / H(z) dz with c_s(z) = c / sqrt(3[1 + R(z)]), yielding r_d = 144.013514253929 Mpc on the calculator surface.

Node id. paper31.geometric_pre_drag_ruler

Scope summary. Active-branch BAO ruler slot.

Depends on. paper21.branch_assignment, paper29.sound_speed_selector

Premises

  • paper21.branch_assignment fixes the active branch on which the ruler is read.
  • paper29.sound_speed_selector fixes R(z) to the theorem-grade slot omega_b,geom.
  • The published BAO surface uses one carried ruler slot rather than a per-query fit parameter.

Proof outline

  • Evaluate the active-branch sound-horizon / pre-drag integral using the carried closed background and the theorem-grade sound-speed slot.
  • Expose that carried value as the calculator's published r_d slot.
  • Use the same carried slot in D_M/r_d, D_H/r_d, and D_V/r_d outputs.

Scope boundary

  • Active-branch ruler slot only.
  • Does not claim a universal branch-independent drag-ruler theorem.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper31_full_recompute_legacy_branch_results.json
theoremPaper 31 Baryon Assignment
derived / scoped

Statement

The active branch exposes typed baryon slots and assigns omega_b,geom as the local chemistry density entering n_H(z), n_e(z) = x_e n_H(z), and kappa'_loc = a_loc n_e sigma_T in the live recombination primitives.

Node id. paper31.baryon_assignment

Scope summary. Typed baryon-slot split on the active branch.

Depends on. paper21.branch_assignment

Premises

  • paper21.branch_assignment fixes the active branch package.
  • The calculator distinguishes typed baryon slots for geometry, chemistry, and downstream operator use.

Proof outline

  • Separate the baryon inventory into typed calculator slots rather than one undifferentiated parameter.
  • Assign omega_b,geom as the chemistry density used by the live recombination primitives on the active branch.
  • Propagate that slot choice into the local background-state and opacity chains.

Scope boundary

  • Typed baryon assignment on the fixed active branch only.
  • Does not assert a theorem-grade completion of every possible chemistry closure.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper31_baryon_assignment_theorems.md
theoremPaper 31 Local Background State Map
derived / scoped

Statement

The calculator's local recombination state map (H_loc, T_R,loc, n_H,geom, u, a_loc) is a theorem-grade map on the active branch, with H_loc = (c / r_s) sqrt[(1-u)/u^3], T_R,loc = x^(-K_gauge) T_obs,0 (1+z), a_loc = u R_U, and n_H,geom = rho_b,geom / m_H.

Node id. paper31.local_background_state_map

Scope summary. Local recombination background-state primitives.

Depends on. paper21.branch_assignment, paper31.baryon_assignment

Premises

  • paper21.branch_assignment fixes the active branch package.
  • paper31.baryon_assignment fixes the local chemistry slot as omega_b,geom.

Proof outline

  • Start from the active-branch closed background and the typed chemistry slot.
  • Construct the local recombination state map H_loc, T_R,loc, n_H,geom, u, and a_loc at a supplied redshift.
  • Export those primitives as calculator-visible local-state quantities.

Scope boundary

  • Local recombination background-state primitives on the fixed active branch.
  • Does not by itself close the exact dynamic-network recombination problem.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper31_stage2_local_background_state_map_theorem.md
theoremPaper 31 Recombination Clock Transport
derived / scoped

Statement

The live primitive opacity chain derives kappa'_loc = a_loc n_e sigma_T, d tau_obs / dz = kappa'_loc c / ((1+z) H_loc), and Gamma_T / H_loc = n_e sigma_T c / H_loc once the local background state and omega_b,geom chemistry slot are fixed.

Node id. paper31.recombination_clock_transport

Scope summary. Primitive local opacity and LOS clock transport.

Depends on. paper31.local_background_state_map

Premises

  • paper31.local_background_state_map provides the local branch background state.
  • Chemistry-dependent outputs are derived only when x_e is supplied by the local Saha seed or another separately justified source.

Proof outline

  • Use the local background state and chemistry slot to build kappa'_loc, d tau_obs / dz, Gamma_T/H_loc, R_local,geom, and c_s,local.
  • Treat the local Saha seed as the default derived ionization input in the published calculator.
  • When a user overrides x_e, keep the local background state derived but mark chemistry-dependent rows as conditional.

Scope boundary

  • Primitive local opacity and LOS clock transport on the active branch.
  • Not a theorem-grade exact dynamic-network recombination closure.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper31_recombination_clock_transport_theorem.md
theoremLocal Saha Seed Theorem
derived / scoped

Statement

The default ionization seed is fixed by the local Saha law x_e^2/(1-x_e) = ((m_e k_B T_R,loc)/(2 pi hbar^2))^(3/2) exp(-chi_H/(k_B T_R,loc)) / n_H,geom.

Node id. local.saha_seed

Scope summary. Default local ionization seed used by the published recombination primitives.

Depends on. premise.2, paper31.local_background_state_map, paper31.baryon_assignment

Premises

  • premise.2 licenses standard hydrogen microphysics inside the horizon.
  • paper31.local_background_state_map fixes T_R,loc and n_H,geom on the active branch.
  • paper31.baryon_assignment fixes the chemistry slot as omega_b,geom.

Proof outline

  • Use the active-branch local radiation temperature and hydrogen number density as the thermodynamic inputs.
  • Apply the standard hydrogen Saha equilibrium equation to those local variables.
  • Solve the algebraic relation for the default local seed x_e used by the published primitive surface.

Scope boundary

  • Default local equilibrium seed only.
  • Does not claim an exact dynamic-network recombination history or visibility peak closure.

No published paper reference is used here; the theorem text is carried self-contained in the calculator dictionary.

theoremPaper 32 S^3 Solver Specification
derived / scoped

Statement

The IO perturbation/transfer geometry is organized on closed S^3 spatial sections rather than on a flat K=0 transfer basis.

Node id. paper32.closed_s3_solver_spec

Scope summary. Closed-space transfer geometry specification for the IO stack.

Depends on. premise.1, premise.2

Premises

  • premise.1 and premise.2 place the IO transfer problem on the interior black-hole branch with outside-equivalent microphysics.
  • Spatial sections in the accepted IO perturbation stack are closed rather than flat.

Proof outline

  • Specify the perturbation and transfer geometry on closed S^3 sections rather than a flat K=0 basis.
  • Use that specification to interpret acoustic and peak-location quantities in a closed-geometry transfer setting.
  • Treat flat-space extractions as comparison conventions rather than native calculator geometry.

Scope boundary

  • Geometry specification for the IO transfer stack.
  • Not a theorem-grade completion of the full TT/TE/EE solver by itself.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper32_s3_native_solver_specification_theorem.md
theoremPaper 32 Modular-DtN Field Transfer
derived / scoped

Statement

On the active scalar-source sector, the unique positive one-slot field transfer is T_field = exp[-(K_g tensor log(r_s Lambda_DtN^coex)) / (2x)], with quadratic descendant R_cov = T_field^* T_field, full source block P_src = B_+ o U_coex o T_field, repaired plus-branch window W_N^(+) = ((N+1) / (N_p+1))^(-K_gauge / x) on the affine odd-shell bridge image, and native amplitude A_s = (25/9) [gamma^2 / (1 + gamma^2)] [1 / sqrt(2)] [exp(4 pi sqrt(2)) - 1]^-1.

Node id. paper32.modular_dtn_field_transfer

Scope summary. Active scalar-source block and one-slot post-bridge field sector.

Depends on. premise.1, premise.2, paper32.hidden_identification_repair

Premises

  • premise.1 and premise.2 place the source/readout problem on the accepted IO interior-horizon branch.
  • The active source block is the one-slot modular-DtN field sector rather than the full perturbation hierarchy.

Proof outline

  • Use the coexact DtN spectrum to define the shell generator Y = log(r_s Lambda_DtN^coex).
  • Combine the reduced gauge modular weight with the accessible-line divisor 1/x to obtain the unique positive source transfer T_field.
  • Apply the plus bridge and pivot normalization to recover the source window and the native scalar amplitude on the active sector.

Scope boundary

  • Active scalar-source block only.
  • The exact source window is the repaired affine odd-shell law from the hidden-identification repair, not the older N / N_p shorthand.
  • Does not by itself close the exact Stage-2 history operator, the closed-S^3 perturbation hierarchy, or the peak/readout identification theorem.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper32_modular_dtn_field_transfer_theorem.mdpaper32_hidden_identification_repair_theorem.md
theoremPaper 32 Hidden-identification Repair
derived / scoped

Statement

The active source exponent lives on the boundary DtN shell variable s_ell = ell + 1, and under the accepted even-shell boundary lift plus the Paper 23 bridge rule N = n +- 1 it pushes forward to the affine odd-shell laws W_N^(+) = ((N+1)/(N_p+1))^(-beta) and W_N^(-) = ((N+3)/(N_p+3))^(-beta). The line-class descent is 1/x, not the legacy area factor 1/x^2.

Node id. paper32.hidden_identification_repair

Scope summary. Repaired shell relabeling and line-class descent for the Paper 32 source theorem.

Depends on. premise.1, premise.2, paper23.bridge_uniqueness

Premises

  • premise.1 and premise.2 keep the source/readout problem on the accepted IO branch.
  • paper23.bridge_uniqueness fixes the multiplicity-one adjacent-shell bridge grammar N = n - 1 or N = n + 1.

Proof outline

  • Separate the boundary DtN shell variable s_ell = ell + 1 from the scalar-shell bridge label N.
  • Push the DtN exponent through the accepted even-shell boundary lift n = 2 ell and the adjacent-shell bridge rule to obtain the affine odd-shell windows.
  • Use the one-slot line-class identification to keep the accessibility divisor at 1/x rather than importing the old area factor.

Scope boundary

  • Repairs the Paper 32 source/readout theorem on the active one-slot source sector only.
  • Does not by itself determine the branch coefficients B_(N,±) or the full perturbation/readout solver.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper32_hidden_identification_repair_theorem.md
theoremTyped Baryon-slot Specification
derived / scoped

Statement

The native solver preserves the typed baryon split: recombination chemistry omega_b,geom, primitive local opacity omega_b,geom, reduced visibility/readout omega_b,eff, while the perturbation block forbids any silent one-slot collapse on R. On the current calculator scope, the surviving closure is a typed local R operator built from primitive R_local,geom plus the coupled Thomson tuple, while the scalar metric-source slot still remains open. No silent one-slot collapse on R is licensed anywhere in the perturbation pipeline.

Node id. paper32.typed_baryon_slot_spec

Scope summary. Perturbation-era baryon typing on the closed S^3 solver grammar.

Depends on. paper32.closed_s3_solver_spec

Premises

  • paper32.closed_s3_solver_spec fixes the typed closed-S^3 solver grammar.
  • The perturbation block is not licensed to collapse to a one-slot baryon assignment.

Proof outline

  • Read the active typed baryon placements from Corollary 32.S3.1.
  • Assign theorem-grade slots only where the stack actually closes them.
  • Forbid any one-slot hierarchy-wide R reassignment while allowing only the later typed local R operator closure on its proper carrier.
  • Leave the scalar metric-source slot explicitly open instead of backfilling it from local helpers.
  • Reject any silent one-slot reassignment of the full perturbation R slot to a local helper quantity.

Scope boundary

  • Typed solver grammar only.
  • Does not derive a single theorem-grade baryon slot for the full observed hierarchy.
  • In particular, the full-hierarchy R slot is not licensed to collapse silently to one local scalar anywhere in the composed pipeline.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper32_s3_native_solver_specification_theorem.md
theoremPaper 31 Stage-2 Markov State
derived / scoped

Statement

The exact Stage-2 hydrogen radiative-transfer branch admits a lossless Markov closure on Y_rec = (x_e, T_m, D_-(q;z), L_-(z)), and no fixed finite-dimensional scalar moment vector replaces that field exactly.

Node id. paper31.stage2_markov_state

Scope summary. Exact Stage-2 history-state carrier.

Depends on. premise.2

Premises

  • premise.2 licenses the accepted exterior local atomic/radiative-transfer class inside the horizon.
  • Stage 2 belongs to the local bulk radiative-transfer sector rather than an observer-side readout patch.

Proof outline

  • Audit the exact FULL Stage-2 update chain and identify the outgoing characteristic distortion field plus line-handoff sector.
  • Show that the extended state (x_e, T_m, D_-(q;z), L_-(z)) is a lossless Markov closure of the exact branch.
  • Kill any fixed finite-dimensional scalar compression as an exact replacement.

Scope boundary

  • Exact Stage-2 state carrier only.
  • Does not by itself supply the final explicit dynamic-network renormalization operator.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

paper31_stage2_lossless_characteristic_markov_theorem.md
theoremInherited FULL Stage-2 Dynamic-history Builder Theorem
conditional / scoped

Statement

On the active IO local background with T_R,loc(z) = T_IO,0 (1+z) and H_loc(z) = (c / R_S) sqrt((1-u)/u^3) for u = 1 / [x (1+z)], the standalone inherited-FULL builder runs exact FULL HyRec history outside the forbidden pointwise wrapper and exports Y_rec(z) = (x_e(z), T_m(z), D_-(q;z), L_-(z)) with D_-(q;z) = interp_Dfnu(lna_0, dlna, Dfminus_hist(q;.), N_z, -ln(1+z)) and L_-(z) = interp_Dfnu(lna_0, dlna, Dfminus_Ly_hist(.), N_z, -ln(1+z)).

Node id. local.inherited_full_stage2_dynamic_history_builder

Scope summary. Conditional exact inherited-FULL Stage-2 history builder on the active IO local background.

Depends on. premise.2, paper31.local_background_state_map, paper31.stage2_markov_state

Premises

  • premise.2 licenses the inherited FULL atomic and radiative-transfer physics class on the local bulk branch.
  • paper31.local_background_state_map fixes the active local thermal and Hubble histories entering the standalone driver.
  • paper31.stage2_markov_state fixes the exact exported state grammar Y_rec = (x_e, T_m, D_-(q;z), L_-(z)).

Proof outline

  • Promote the existing standalone FULL-history route from the Paper 31 benchmark layer into the calculator instead of calling FULL HyRec through a pointwise (z, x_e, T_m) wrapper.
  • Run FULL HyRec on the active IO local background arrays with the explicit history-grid fix used by the benchmark route.
  • Export x_e, T_m, Dfminus_hist, and Dfminus_Ly_hist on the requested observer-redshift grid without silently collapsing the characteristic field to one preferred scalar.

Scope boundary

  • Conditional exact inherited-FULL history builder only.
  • Uses inherited FULL atomic and radiative-transfer physics under Premise 2 rather than a new universal IO-native renormalization theorem.
  • Does not pick a preferred one-dimensional compression of D_-(q;z) unless the caller chooses one explicitly.

The theorem text is self-contained here; there is no published paper reference for this post-Paper-32 local completion step.

theoremThomson-history Realization Theorem
derived / scoped

Statement

Any exact IO-native closure on the surviving acoustic branch must be realized on the coupled tuple (thomson_drag_rate, thomson_hierarchy_rate, tau_c, dtau_c, slip, shear) with tau_c = 1 / thomson_drag_rate and dtau_c = - d(thomson_drag_rate) * tau_c^2.

Node id. calculator.thomson_history_realization

Scope summary. Coupled Thomson-history carrier for the perturbation hierarchy.

Depends on. paper32.closed_s3_solver_spec, paper31.stage2_markov_state

Premises

  • paper32.closed_s3_solver_spec fixes the perturbation block as part of the typed closed-S^3 solver grammar.
  • paper31.stage2_markov_state fixes the exact history-state carrier feeding the perturbation seam.
  • The tight-coupling system depends on distinct drag and hierarchy rates together with tau_c, dtau_c, slip, and shear.

Proof outline

  • Localize the surviving acoustic perturbation seam to the broad Thomson-history leg rather than a narrow visibility patch.
  • Read the carrier formulas for thomson_drag_rate, thomson_hierarchy_rate, tau_c, and dtau_c from the accepted local hierarchy code path.
  • Promote the coupled tuple to the exact admissible carrier and kill one-site downstream rescalings as fake closures.

Scope boundary

  • Tuple carrier only.
  • Does not yet derive the exact IO-native operator acting on that tuple.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_thomson_history_realization_theorem_report.md
theoremFull Typed R Hierarchy Operator Theorem
derived / scoped as maps

Statement

On the accepted equal-rate scoped branch thomson_hierarchy_rate = thomson_drag_rate, the primitive acoustic loading object stays R_local,geom(z) = 3 rho_b,geom(z) / [4 rho_gamma(z)], and the full oscillator-site hierarchy maps are Gamma_gammab = thomson_drag_rate, Gamma_bgamma = R_local,geom thomson_drag_rate = R_local,geom / tau_c, c_bgamma^2 = 1 / [3 (1 + R_local,geom)], M_bgamma = 1 + R_local,geom, L_odd/even = R_local,geom / (1 + R_local,geom), F_tca = tau_c / (1 + R_local,geom) with tau_c = 1 / thomson_drag_rate, and the split Silk operator D_heat = R_local,geom^2 / [6 (1+R_local,geom)^2 thomson_drag_rate], D_visc = 16 / [90 (1+R_local,geom) thomson_hierarchy_rate], D_silk = D_heat + D_visc. Dynamic odd/even modulation is generated inside the oscillator by c_bgamma^2 and L_odd/even; the observed peak-height ratio is the downstream transfer/readout functional of that evolved hierarchy and not a separate baryon-slot assignment.

Node id. local.typed_r_operator

Scope summary. Equal-rate scoped branch of the full typed R hierarchy operator on the closed scalar photon-baryon oscillator.

Depends on. paper29.sound_speed_selector, paper32.typed_baryon_slot_spec, calculator.thomson_history_realization

Premises

  • paper29.sound_speed_selector fixes the primitive local enthalpy ratio R_local on omega_b,geom.
  • paper32.typed_baryon_slot_spec forbids any one-slot full-hierarchy reassignment of R and therefore forces a typed operator closure instead.
  • calculator.thomson_history_realization fixes the exact drag/hierarchy tuple consumed by the local hierarchy.

Proof outline

  • Keep the primitive enthalpy ratio itself on the inventory branch omega_b,geom instead of back-propagating observer-side omega_b,eff into the local plasma leg.
  • Read the full hierarchy site map from the closed scalar oscillator: momentum exchange depends on (thomson_drag_rate, R_local), pressure restoration on 1/[3(1+R_local)], inertia on 1+R_local, and dynamic odd/even loading on R_local/(1+R_local).
  • Split the standard Silk integrand into the heat-conduction term carried by baryon-photon slip and the viscosity term carried by the photon hierarchy, then bind those two pieces to thomson_drag_rate and thomson_hierarchy_rate respectively.
  • Conclude that the hierarchy requires a site-wise typed operator rather than a slot swap, and that the final observed odd/even peak pattern is downstream transfer/readout of this evolved oscillator rather than a new primitive R slot.

Scope boundary

  • Closes the full site-wise hierarchy operator carried by the primitive local R leg and the Thomson tuple on the accepted equal-rate scoped branch thomson_hierarchy_rate = thomson_drag_rate.
  • Does not claim a one-slot closure of the full observed peak/readout hierarchy.
  • Observer-side omega_b,eff remains downstream readout packaging and is not back-propagated into the primitive local R leg.
  • Does not yet claim a theorem-grade nontrivial drag-vs-hierarchy deformation.

The theorem text is self-contained here; the linked report is supplementary supporting material only.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_typed_r_operator_theorem_report.md
theoremTyped R Site-uniqueness Theorem
derived / scoped

Statement

On the accepted equal-rate scoped branch thomson_hierarchy_rate = thomson_drag_rate, the site-wise hierarchy placements carried by local.typed_r_operator are the unique admissible typed operators on the current IO stack: pressure/restoring force must use c_bgamma^2 = 1 / [3 (1 + R_local,geom)], inertia and dynamic odd/even loading must use M_bgamma = 1 + R_local,geom and L_odd/even = R_local,geom / (1 + R_local,geom), momentum exchange must use the pair Gamma_gammab = thomson_drag_rate, Gamma_bgamma = R_local,geom thomson_drag_rate, and Silk damping must split uniquely as D_heat = R_local,geom^2 / [6 (1+R_local,geom)^2 thomson_drag_rate] and D_visc = 16 / [90 (1+R_local,geom) thomson_hierarchy_rate]. No alternate site placement is compatible with the primitive local R theorem, the no-single-slot theorem, the Thomson-history tuple theorem, and the lower-triangular non-backpropagation boundary.

Node id. local.typed_r_site_uniqueness

Scope summary. Uniqueness-by-elimination for the four actual R insertion sites in the closed scalar hierarchy on the equal-rate scoped branch.

Depends on. paper29.sound_speed_selector, paper31.baryon_assignment, paper32.typed_baryon_slot_spec, calculator.thomson_history_realization, local.typed_r_operator

Premises

  • paper29.sound_speed_selector fixes the primitive local enthalpy ratio on omega_b,geom for the local acoustic leg.
  • paper31.baryon_assignment proves photon-baryon coupling is composite in (kappa', R) and that diffusion is not a one-slot baryon observable.
  • paper32.typed_baryon_slot_spec and its lower-triangular boundary forbid observer/readout back-propagation and any hierarchy-wide one-slot collapse of R.
  • calculator.thomson_history_realization fixes the exact drag/hierarchy tuple consumed by the local hierarchy.
  • local.typed_r_operator provides the candidate site map to test for uniqueness.

Proof outline

  • Kill every alternative that moves pressure or inertia off the primitive local enthalpy object; doing so would require a drag-vs-sound branch separation or observer-side back-propagation that the current stack explicitly forbids.
  • Use the local hierarchy convention tau_c = 1 / thomson_drag_rate to fix the momentum-exchange pair uniquely as photon-side 1/tau_c and baryon-side R_local / tau_c; the inverse-R alternative is a convention mismatch on this carrier rather than a second admissible site.
  • Identify the diffusion heat term with the slip/TCA sector and the viscosity term with the photon hierarchy sector; any swapped or collapsed rate placement breaks the explicit local generator even though the equal-rate limit hides that distinction numerically.
  • Conclude by elimination that the carried site map is the unique admissible typed hierarchy operator at the four actual insertion sites.

Scope boundary

  • Uniqueness theorem for the local closed scalar hierarchy sites on the accepted equal-rate scoped branch only.
  • Does not claim that the final observed peak-height pattern is itself a primitive one-slot baryon observable.
  • Does not validate the late-time TT driver handoff numerically.

The theorem text is self-contained here; the linked report is supplementary supporting material only.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_typed_r_site_uniqueness_theorem_report.md
theoremTyped Split Thomson-history Realization Theorem
derived / scoped as maps

Statement

The live conformal Thomson tuple is built through the typed split history path kappa'_loc(z) = a n_e(z) sigma_T on the local chemistry/electron-inventory branch, then the observer-side optical packet d tau_obs / dz, tau_obs, g_obs = exp(-tau_obs) d tau_obs / dz, and finally the accepted equal-rate scoped conformal tuple thomson_drag_rate = |(d tau_obs / dz) / (dC / dz)|, thomson_hierarchy_rate = thomson_drag_rate, tau_c = 1 / thomson_drag_rate, dtau_c = - d(thomson_drag_rate) tau_c^2. So the implementation realizes the tuple from typed local opacity plus typed visibility/readout history, not from one undifferentiated opacity scalar.

Node id. local.typed_thomson_split_history_realization

Scope summary. Implementation-level typed path from Stage-2 chemistry history to the accepted equal-rate scoped conformal Thomson tuple.

Depends on. paper31.baryon_assignment, calculator.thomson_history_realization

Premises

  • paper31.baryon_assignment fixes primitive local opacity on the inventory branch and reduced visibility/readout as a distinct downstream layer.
  • calculator.thomson_history_realization fixes the exact tuple grammar consumed by the hierarchy.

Proof outline

  • Read the local chemistry/electron history x_e(z) and form kappa'_loc = a n_e sigma_T explicitly on the local branch.
  • Build the observer-side visibility packet (d tau_obs / dz, tau_obs, g_obs) from that local history without collapsing the two layers.
  • Transport the observer-side packet onto the conformal clock and then package the Thomson tuple from the transported packet rather than from a raw opacity scalar.

Scope boundary

  • Implementation theorem for the scoped equal-rate tuple path used by the current TT driver.
  • Does not yet derive a nontrivial drag-vs-hierarchy deformation of the Thomson tuple.

The theorem text is self-contained here; the linked report is supplementary supporting material only.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_typed_thomson_split_history_implementation_audit_report.md
theoremClosed Scalar Metric-state Builder Theorem
derived / scoped as maps

Statement

Once the Einstein-normalized scalar stress summary (delta_rho, delta_p, rho_plus_p_theta, rho_plus_p_shear) is supplied on one closed scalar shell, the Newtonian metric state is fixed by psi = phi - 4.5 (a^2 / k_n^2) rho_plus_p_shear and phi' = -(a'/a) psi + 1.5 (a^2 / k_n^2) rho_plus_p_theta, while the synchronous metric state is fixed by h' = [k_n^2 s_2^2 eta + 1.5 a^2 delta_rho] / (0.5 a'/a), eta' = [1.5 a^2 rho_plus_p_theta + 0.5 K h'] / (k_n^2 s_2^2), alpha = (h' + 6 eta') / (2 k_n^2), and alpha' = -2 (a'/a) alpha + eta - 4.5 (a^2 / k_n^2) rho_plus_p_shear.

Node id. local.closed_scalar_metric_state_builder

Scope summary. Einstein-side scalar metric-state reconstruction on one explicit closed shell.

Depends on. paper23.closed_scalar_operator

Premises

  • paper23.closed_scalar_operator fixes the closed scalar shell geometry and the s_2^2 = 1 - 3K/k_n^2 factor.
  • The total scalar stress summary is supplied explicitly rather than guessed from a hidden matter model.

Proof outline

  • Write the scalar Einstein equations on one physical closed shell in Newtonian and synchronous gauge.
  • Solve algebraically for the metric variables and their first derivatives from the explicit stress summary.
  • Feed those metric variables into the acoustic quartet without introducing any extra closure factor.

Scope boundary

  • Metric-state builder from an explicit total scalar stress summary only.
  • Does not derive the stress summary from a full multi-species perturbation hierarchy by itself.

The theorem text is self-contained here; there is no published paper reference for this post-Paper-32 local completion step.

theoremClosed Scalar Adiabatic-seed Bridge Theorem
derived / scoped

Statement

On the active scalar-source branch, the source shell weight is C_N^src = A_s W_N^(+) = A_s ((N+1) / (N_p+1))^(-K_gauge / x) on the repaired affine odd-shell plus branch, and the leading radiation-era closed-S^3 adiabatic seed obeys delta_gamma^(S) = -(k_n tau)^2 s_2^2 R_n / 3, theta_gamma^(S) = -k_n (k_n tau)^3 s_2^2 R_n / 36, delta_b^(S) = 3 delta_gamma^(S) / 4, theta_b^(S) = theta_gamma^(S), eta^(S) = R_n [1 - (k_n tau)^2 / 36], followed by the explicit synchronous-to-Newtonian shift delta_gamma^(N) = delta_gamma^(S) - 4 (a'/a) alpha, theta_gamma^(N) = theta_gamma^(S) + k_n^2 alpha, delta_b^(N) = delta_b^(S) - 3 (a'/a) alpha, and theta_b^(N) = theta_b^(S) + k_n^2 alpha.

Node id. local.closed_scalar_adiabatic_seed_bridge

Scope summary. Active scalar-source shell weight plus leading closed-shell photon-baryon adiabatic seed.

Depends on. paper32.modular_dtn_field_transfer, paper32.hidden_identification_repair, paper23.closed_scalar_operator

Premises

  • paper32.modular_dtn_field_transfer fixes the active scalar-source shell weight A_s W_N^(+) on the repaired affine odd-shell plus branch.
  • paper23.closed_scalar_operator fixes the physical scalar-shell support and the closed-shell geometric factor s_2^2.

Proof outline

  • Take the active scalar-source shell covariance A_s W_N^(+) from the source block without adding a new primordial fit.
  • Write the leading radiation-era closed-shell photon-baryon adiabatic seed in synchronous gauge.
  • Transform that seed explicitly to Newtonian gauge on the same shell and package the resulting hierarchy state.

Scope boundary

  • Leading photon-baryon adiabatic seed on the active scalar-source branch only.
  • Does not derive low-shell puncture occupations, anomalous phase correlations, or full multi-species isocurvature families.

The theorem text is self-contained here; there is no published paper reference for this post-Paper-32 local completion step.

theoremClosed Scalar Transfer Projector Theorem
derived / scoped as maps

Statement

Given an explicit closed-shell scalar source history, the transparent LOS source law is S_T^(0) = exp(-kappa) phi' + g delta_gamma / 4, S_T^(1) = exp(-kappa) k_n psi + g theta_b / k_n in Newtonian gauge, S_T^(0) = -exp(-kappa) h' / 6 + g delta_gamma / 4, S_T^(1) = g theta_b / k_n, S_T^(2) = exp(-kappa) k_n^2 (2/3) s_2 alpha + g P in synchronous gauge, with S_T^(2) = g P and S_E = g P on the scalar polarization source, and the exact closed radial chain is Phi_0 = sin(beta chi) / [beta sin chi], Phi_l = [(2l-1) cot chi Phi_{l-1} - sqrt(beta^2-(l-1)^2) Phi_{l-2}] / sqrt(beta^2-l^2), dPhi_l/dchi = l cot chi Phi_l - sqrt(beta^2-(l+1)^2) Phi_{l+1}, d2Phi_l/dchi^2 = -2 cot chi dPhi_l/dchi + [l(l+1) csc^2 chi - beta^2 + 1] Phi_l, so that Delta_l^T(q) and Delta_l^E(q) are fixed by explicit LOS integration on chi = sqrt(K) (tau_0 - tau).

Node id. local.closed_scalar_transfer_projector

Scope summary. Closed scalar hierarchy-to-transfer projector on one explicit source history and one observer conformal time.

Depends on. paper22.spatial_mode_ladder, paper23.closed_scalar_operator, local.closed_scalar_metric_state_builder

Premises

  • paper22.spatial_mode_ladder fixes the closed spatial carrier and the scalar radial support on S^3.
  • paper23.closed_scalar_operator fixes the discrete scalar shell parameter beta = n+1 and the physical shell support n >= 2.
  • local.closed_scalar_metric_state_builder supplies the explicit scalar metric histories entering the transparent LOS source law.

Proof outline

  • Build the transparent scalar LOS sources directly from the hierarchy state, metric state, and observer-absolute visibility packet without rewriting them as hidden CLASS source patches.
  • Evaluate the exact closed scalar radial chain from the stable recurrence on beta = n+1 and the radial derivative identities.
  • Integrate the explicit source and radial kernels on the supplied conformal-time grid to obtain Delta_l^T(q) and Delta_l^E(q) on the closed support.

Scope boundary

  • Exact source law and exact closed radial chain on one explicit scalar history.
  • Numeric packet values still depend on explicit quadrature on the supplied conformal-time grid.
  • Does not derive the hierarchy history automatically from the full perturbation evolution problem.

The theorem text is self-contained here; there is no published paper reference for this post-Paper-32 local completion step.

theoremClosed Scalar Acoustic Generator Theorem
derived / scoped as maps

Statement

On one explicit physical closed-S^3 scalar shell n >= 2, the local photon-baryon acoustic generator is fixed by k_n^2 = n(n+2) / R_curv^2, q_n^2 = k_n^2 + K = (n+1)^2 / R_curv^2, s_l = sqrt(1 - K (l^2-1) / k_n^2), cot_K^gen(tau) = sqrt(K) / [k_n tan(sqrt(K) tau)], local baryon loading R = 3 rho_b / (4 rho_gamma) on omega_b,geom, and the explicit scalar RHS delta_gamma' = -(4/3)(theta_gamma + metric_continuity), delta_b' = -(theta_b + metric_continuity), theta_b' = -a'/a theta_b + metric_euler + k_n^2 c_b^2 delta_b + R * drag_rate * (theta_gamma-theta_b), theta_gamma' = k_n^2 (delta_gamma/4 - s_2^2 F_2) + metric_euler + drag_rate * (theta_b-theta_gamma), with gauge-to-quartet maps (metric_continuity, metric_euler, metric_shear, metric_shear_prime) = (-3 phi', k_n^2 psi, 0, 0) in Newtonian gauge and (h'/2, 0, k_n^2 alpha, k_n^2 alpha') in synchronous gauge, with higher multipoles and the reduced TCA contract driven by the coupled tuple (thomson_drag_rate, thomson_hierarchy_rate, tau_c, dtau_c, slip, shear). The local primitive loading R(z) used here is not a silent one-slot collapse of the full hierarchy-wide perturbation R slot.

Node id. local.closed_scalar_acoustic_generator

Scope summary. Local affine scalar acoustic generator on an explicit sampled closed-S^3 shell.

Depends on. paper23.closed_scalar_operator, paper29.sound_speed_selector, paper32.typed_baryon_slot_spec, calculator.thomson_history_realization, local.typed_r_site_uniqueness

Premises

  • paper23.closed_scalar_operator fixes the discrete scalar shell support and the shifted scalar operator on S^3.
  • paper29.sound_speed_selector fixes the theorem-grade local inertia loading R(z) on omega_b,geom for the primitive acoustic leg.
  • paper32.typed_baryon_slot_spec forbids collapsing the hierarchy to one baryon slot and keeps the metric-source leg explicit.
  • calculator.thomson_history_realization fixes the exact coupled Thomson tuple that an admissible local closure must consume.
  • local.typed_r_site_uniqueness proves that the carried site-wise R hierarchy operator is the unique admissible placement at the actual oscillator sites.

Proof outline

  • Insert the closed-S^3 scalar shell identities into the non-flat scalar hierarchy so the geometric recurrence factors are explicit on each discrete n shell.
  • Use the theorem-grade local history sample (x_e, T_m) to recover c_b^2 and its derivative, and use the theorem-grade primitive loading R(z) together with its typed Thomson-tuple composites for the local momentum-loading leg.
  • Feed the coupled Thomson tuple into the baryon drag, photon hierarchy damping, and tight-coupling contract equations to obtain the full local scalar photon-baryon RHS at explicit sample level.
  • Conclude that the local generator itself is fixed once the external metric-drive and Stage-2/Thomson sample builders are supplied explicitly.

Scope boundary

  • Local explicit-sample scalar generator only.
  • Does not derive the exact Stage-2 dynamic-network history builder or the total multi-species stress summary by itself.
  • Does not by itself integrate the full scalar history from initial conditions to transfer packets without those explicit upstream builders.
  • No theorem-grade hierarchy-wide one-slot collapse on R is claimed anywhere in this map.

The theorem text is self-contained here; the linked report is supplementary supporting material only.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_closed_scalar_acoustic_generator_theorem_report.md
theoremScoped Closed-scalar Pipeline Theorem
conditional / scoped

Statement

On the active scalar-source branch, the composed closed-S^3 scalar map from (P_src, C_N^src, ic_n, Y_rec, Thomson tuple, stress summary) to y^(n)(tau), then to Delta_l^X(q), and then to shell-summed C_l, is carried end-to-end once the exact Stage-2 history enters through the conditional inherited-FULL builder. The carrier laws, metric formulas, projector formulas, and LOS formulas remain derived / scoped as maps, while the full composition is only conditional / scoped because it inherits the Stage-2 builder status. No silent one-slot collapse on the hierarchy-wide perturbation R slot appears anywhere in this composition.

Node id. local.scoped_closed_scalar_pipeline

Scope summary. Composed active-branch closed-scalar pipeline from source shell to transfer packets and shell-summed spectra.

Depends on. paper32.modular_dtn_field_transfer, local.closed_scalar_adiabatic_seed_bridge, local.inherited_full_stage2_dynamic_history_builder, calculator.thomson_history_realization, local.closed_scalar_metric_state_builder, local.closed_scalar_acoustic_generator, local.closed_scalar_transfer_projector, paper28.closed_s3_shell_power, paper32.closed_s3_solver_spec

Premises

  • paper32.modular_dtn_field_transfer fixes the active scalar source block.
  • local.closed_scalar_adiabatic_seed_bridge fixes the active closed-shell source-to-initial-condition bridge.
  • local.inherited_full_stage2_dynamic_history_builder supplies the exact sampled Stage-2 history carrier at conditional inherited-FULL scope.
  • calculator.thomson_history_realization fixes the exact coupled Thomson tuple that the local hierarchy map must consume.
  • local.closed_scalar_metric_state_builder, local.closed_scalar_acoustic_generator, and local.closed_scalar_transfer_projector provide the explicit map-level formulas from shell data to transfer packets.
  • paper28.closed_s3_shell_power and paper32.closed_s3_solver_spec supply the shell weighting and closed-space LOS assembly grammar.

Proof outline

  • Compose the active scalar source block, closed-shell seed bridge, sampled Stage-2 history, Thomson tuple, metric-state builder, local acoustic generator, and scalar transfer projector without inserting any hidden flat-space or CLASS default.
  • Carry the resulting transfer packets into the explicit shell-weighted C_l assembly on the closed support.
  • Track the claim status through the composition: map-level formulas remain derived / scoped as maps, while the full end-to-end pipeline inherits conditional / scoped status from the Stage-2 history builder.

Scope boundary

  • Full composed active-branch scalar pipeline only.
  • Conditional on the inherited-FULL Stage-2 history builder rather than a universal IO-native Stage-2 renormalization theorem.
  • No theorem-grade hierarchy-wide one-slot collapse on R is licensed anywhere in this composition.
  • Not a universal automatic TT/TE/EE solver closure for arbitrary branches or sectors.

The theorem text is self-contained here; there is no published paper reference for this post-Paper-32 composed-pipeline step.

theoremScoped TT Driver Composition Theorem
conditional / scoped

Statement

On the active scalar-source branch, the executable TT carrier is the explicit composition Y_rec^scoped -> Thomson^conf -> metric/state history -> Delta_l^T(q) -> C_l^TT, with the Stage-2 segment supplied by the inherited-FULL builder on z <= z_exact_max, a thermal x_e = 1, T_m = T_R prehistory extension on z > z_exact_max, the repaired odd-shell source support, the explicit shell weight w(n) = ((n+1)^2 / (2 pi^2 R^3)) P_X(n), and one common early-time carrier for the whole run. The resulting C_l^TT array is a conditional/scoped executable spectrum rather than a theorem-grade validated full CMB closure.

Node id. local.scoped_tt_driver

Scope summary. Executable active-branch TT driver on the current repaired branch.

Depends on. local.inherited_full_stage2_dynamic_history_builder, local.typed_thomson_split_history_realization, local.closed_scalar_acoustic_generator, local.closed_scalar_transfer_projector, paper28.closed_s3_shell_power, local.scoped_closed_scalar_pipeline

Premises

  • local.inherited_full_stage2_dynamic_history_builder supplies the conditional Stage-2 history segment used by the executable branch.
  • local.typed_thomson_split_history_realization fixes the equal-rate typed Thomson-history path consumed by the local hierarchy carrier.
  • local.closed_scalar_acoustic_generator, local.closed_scalar_transfer_projector, and paper28.closed_s3_shell_power provide the map-level hierarchy, projector, and shell-sum laws.
  • local.scoped_closed_scalar_pipeline fixes the composed closed-scalar grammar and its status discipline.

Proof outline

  • Build the scoped history carrier from the inherited-FULL exact segment plus the explicit thermal prehistory extension.
  • Transport that history onto the conformal Thomson tuple, evolve the repaired closed-scalar hierarchy shell by shell, project the source histories to Delta_l^T(q), and assemble the shell-summed C_l^TT array.
  • Keep the runtime status honest: executable and reproducible, but only conditional/scoped because the history carrier inherits the inherited-FULL Stage-2 status and the high-shell source/phase frontier remains open.

Scope boundary

  • Executable TT carrier only.
  • Does not by itself prove that the returned spectrum is physically correct for arbitrary shell ceilings or arbitrary history-carrier choices.
  • Retains one common early-time carrier for the whole run; any shell-local alternative would require a new theorem-grade phase map.

The theorem text is self-contained here; there is no published paper reference for this post-Paper-32 executable TT composition step.

theoremScoped TT First-peak Support Theorem
Conditional/scoped/verified TT first-peak support on the repaired active-branch canonical carrier (n_max = 501), with inherited-FULL Stage-2 history and equal-rate typed Thomson specialization.

Statement

Conditional/scoped/verified TT first-peak support on the repaired active-branch canonical carrier (n_max = 501), with inherited-FULL Stage-2 history and equal-rate typed Thomson specialization. On that canonical carrier (exact_history_samples, prehistory_samples, n_max, shell_step) = (120, 40, 501, 1) with constraint_metric_source_only = True, constraint_consistent_seed = True, metric_baryon_momentum_slot = omega_b,eff, repaired odd-shell source support, and covariance shell weights, the executable TT spectrum lands in the physical first-peak family with ell_peak = 224, C_220 / C_peak = 0.9938104102565932, and C_2 / C_30 = 1148.794609154744. The neighboring ceiling n_max = 453 stays on the same family with ell_peak = 222 and C_220 / C_peak = 0.976859196443279. The surviving n_max >= 601 shell-ceiling drift remains open: on tested history carriers the peak drifts upward to ell_peak = 260 to 277.

Node id. local.scoped_tt_first_peak_support

Scope summary. Canonical repaired active-branch TT first-peak carrier only, with the surviving high-shell ceiling drift left explicit.

Depends on. local.scoped_tt_driver, calculator.peak_functional_separation

Premises

  • local.scoped_tt_driver provides the executable repaired TT carrier on which the first-peak audit is performed.
  • calculator.peak_functional_separation keeps the reported peak functional explicit rather than collapsing it to the background angle by fiat.

Proof outline

  • Run the canonical repaired TT carrier on the full odd-support ladder through n_max = 501 and record the resulting discrete TT peak functional.
  • Cross-check the neighboring ceiling n_max = 453 to show the same first-peak family survives below the canonical ceiling.
  • Check n_max = 601 on the same repaired branch and record the surviving upward drift as the exact remaining open boundary rather than hiding it.

Scope boundary

  • Canonical repaired first-peak carrier only.
  • This is not a theorem-grade full high-ell TT closure.
  • The surviving n_max >= 601 shell-ceiling drift remains open: on tested history carriers the peak drifts upward to ell_peak = 260 to 277.

The theorem text is self-contained here; the linked report is supplementary supporting material only.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_tt_first_peak_support_theorem_report.md
theoremNull-family Acoustic Readout Theorem
derived / scoped

Statement

The explicit null-family readout field omega_hat(eta) = (ev_eta tensor C_n) P_src(Phi) can be built from the theorem-grade source block, and the resulting background acoustic estimator class E_rs = integral_0^{eta_rec} c_s(eta) R_hist^ac(omega_hat(eta)) d eta is itself one-slot on the current scoped sector.

Node id. calculator.null_family_acoustic_readout

Scope summary. Explicit null-family acoustic readout field and estimator class.

Depends on. paper32.modular_dtn_field_transfer, paper29.sound_speed_selector

Premises

  • paper32.modular_dtn_field_transfer closes the one-slot source/readout block P_src.
  • paper29.sound_speed_selector fixes the theorem-grade local sound-speed slot entering the acoustic estimator kernel.
  • Gauge-neutral direction collection and history evaluation preserve one-slot degree on the current scoped sector.

Proof outline

  • Evaluate the one-slot source block at fixed history label and readout direction to obtain omega_hat(eta).
  • Apply the already-closed homogeneous acoustic history-reduction operator to that explicit field.
  • Integrate against the theorem-grade sound-speed kernel to obtain the background acoustic estimator class without creating a second gauge slot.

Scope boundary

  • Explicit readout field and estimator class only.
  • Does not yet derive the exact acoustic endpoint/phase scalar or the final peak-position angle.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_null_family_readout_theorem_report.md
theoremPeak-functional Separation Theorem
derived / scoped

Statement

The background ratio 100theta_s = 100 r_s(z_rec) / D_M(z_rec) is not, by itself, the exact physical peak-position readout theta_peak; theorem-grade numeric closure requires either the exact A_peak / closed-S^3 perturbation readout or a separate theorem identifying that peak functional with the background ratio on the relevant scope.

Node id. calculator.peak_functional_separation

Scope summary. Boundary between background acoustic ratios and physical peak-position readout.

Depends on. paper32.closed_s3_solver_spec

Premises

  • paper32.closed_s3_solver_spec fixes the full linear IO transfer as a typed map with a perturbation/readout side beyond the source block.
  • Peak-position observables live downstream of the primitive sky field and the quadratic power spectrum rather than at the raw background ratio alone.

Proof outline

  • Place theta_peak on the measurement chain primitive field -> harmonic coefficients -> C_l -> A_peak.
  • Use the typed-transfer theorem to separate the background ratio from the unresolved perturbation/readout block.
  • Conclude that no theorem may identify the numeric peak-position angle with 100theta_s alone without an additional peak/readout identification theorem.

Scope boundary

  • Boundary theorem only.
  • Does not itself derive the final A_peak -> theta_peak identification law.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_peak_functional_separation_theorem_report.md
theoremLate-time Baryon-counting Law
derived / scoped

Statement

On the late-time baryonic dust sector, n_b = rho_b / m_bar * [1 + O(k_B T / (m_p c^2))], so the surviving normalization ambiguity is the standard mean-baryon-mass convention rather than an open IO-side bridge.

Node id. paper35.late_baryon_counting_law

Scope summary. Late-time baryon-number conversion on the active branch.

Depends on. premise.2, paper21.branch_assignment

Premises

  • premise.2 licenses standard low-temperature baryonic matter inside the horizon.
  • paper21.branch_assignment fixes the active branch and its carried omega_b,geom slot.

Proof outline

  • Treat the late-time matter sector as nonrelativistic baryonic dust on the active branch.
  • Convert rest-mass density to baryon number density with the standard mean mass per baryon.
  • Keep the remaining normalization ambiguity explicit as a standard mass convention rather than an unresolved framework theorem gap.

Scope boundary

  • Late-time baryonic dust sector only.
  • Does not provide a primordial/source-era baryogenesis theorem.

No published paper reference is used here; the theorem text is carried self-contained in the calculator dictionary.

theoremLate-time eta_IO Closure Theorem
derived / scoped

Statement

The preferred late-time baryon-to-photon ratio eta_IO = n_b / n_gamma = 5.748778515174e-10 is closed on the active branch, equivalently eta_IO,late = C_eta(T_obs, m_bar) * omega_b,geom.

Node id. paper35.eta_io_late_closure

Scope summary. Late-time baryon-to-photon ratio convention used by the calculator.

Depends on. paper21.branch_assignment, paper35.late_baryon_counting_law

Premises

  • paper21.branch_assignment fixes the active branch package.
  • paper35.late_baryon_counting_law converts the carried mass density to baryon number density on the late-time dust sector.
  • The calculator needs one carried late-time eta_IO convention rather than multiple unresolved conventions.

Proof outline

  • Start from the active-branch physical-density slot omega_b,geom together with the late-time baryon-counting law.
  • Convert the active-branch baryon density and observed CMB temperature into the baryon-to-photon prefactor C_eta(T_obs, m_bar).
  • Fix the preferred exported eta_IO,late convention to that closed branch value.
  • Expose the convention directly through the calculator and bundle.

Scope boundary

  • Preferred late-time eta_IO convention on the active branch.
  • Does not claim that every alternative mass convention is closed to theorem grade.

No published paper reference is used here; the theorem text is carried self-contained in the calculator dictionary.

theoremPaper 24 Conditional Lithium Scorecard
conditional / scoped

Statement

Conditional on the Paper 22 BBN premise package and one empirical cluster-deformation input, the channel-resolved mass-7 route lands Li-7/H = 1.750087820365855e-10 with zero fitted parameters while preserving the repaired deuterium and helium scorecard.

Node id. paper24.conditional_lithium_scorecard

Scope summary. Conditional active BBN lithium repair scorecard.

Depends on. premise.1, premise.2, paper12.baryon_dictionary_fraction

Premises

  • premise.1 and premise.2 fix the IO BBN setting.
  • paper12.baryon_dictionary_fraction provides the repaired baryon-fraction route inherited by the active BBN scorecard.
  • The surviving lithium closure is conditional on the Paper 22 BBN premise package and one empirical cluster-deformation input.

Proof outline

  • Use the Paper 24 channel-resolved mass-7 route rather than uniform TT dressing or destruction-side fixes.
  • Condition that route on the surviving Paper 22 premise package plus the empirical cluster-deformation input.
  • Evaluate the resulting active lithium scorecard while keeping the non-mass-7 outputs on the repaired BBN scorecard.

Scope boundary

  • Conditional lithium-repair scorecard only.
  • Does not license an unconditional theorem-grade lithium closure independent of the Paper 22 premise package and cluster deformation input.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

Interior_Observer_Paper24_v2_1.docx
theoremPhase-equivalent Selector Theorem
derived / scoped

Statement

The strict-bare selector backbone solves theta_bare(z_sel) = r_s(z_sel) / D_M(z_sel) on the certified monotone interval and carries observer-side angle by theta_obs = J_theta theta_bare with fixed J_theta.

Node id. calculator.phase_equivalent_selector

Scope summary. Strict-bare phase-ruler selector backbone.

Depends on. premise.1, premise.2, paper21.branch_assignment

Premises

  • premise.1 and premise.2 fix the IO setting.
  • paper21.branch_assignment fixes the active branch package on which the selector is evaluated.
  • The strict-bare selector interval is certified monotone on its published domain.

Proof outline

  • Use the certified monotone interval to make the strict-bare selector exactly invertible on the published branch domain.
  • Map the selected bare phase-ruler leaf to observer-side theta with the fixed Jacobian J_theta.
  • Reduce theorem-grade numeric theta_* to identifying the physical selector leaf carried by the active branch.

Scope boundary

  • Strict-bare selector backbone on the certified interval.
  • Does not by itself identify which leaf is physical on the active branch.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_phase_equivalent_selector_theorem_report.md
theoremPacket Coefficient Fixing Theorem
derived / scoped

Statement

The outgoing-update and Ly-line coefficients are fixed on the surviving packet law by xr[1] = 3 x1s Dfplus_Ly[0] and xr[0] = x1s exp(E32 / TR) Dfplus_Ly[1], yielding the support-reduced packet carrier close to the final active endpoint.

Node id. calculator.packet_coefficient_fixing

Scope summary. Packet-law closure on the surviving active endpoint family.

Depends on. calculator.phase_equivalent_selector

Premises

  • calculator.phase_equivalent_selector reduces theta_* closure to fixing the active packet and leaf carrier.
  • The surviving outgoing-update packet law is constrained by the accepted Calculator boundary transport identities.

Proof outline

  • Use the outgoing-update and Ly-line boundary transport identities to fix the live packet-law coefficients rather than fitting them.
  • Construct the support-reduced packet carrier on the surviving active endpoint family with those fixed coefficients.
  • Show that the reduced carrier lands very close to the final active endpoint while keeping the coefficients theorem-governed.

Scope boundary

  • Packet-law closure on the surviving active endpoint family.
  • Does not yet prove that the surviving packet alone carries the physical selector leaf.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_effectc_packet_coefficient_fixing_theorem_report.mdcalculator_effectc_packet1500_support_promotion_theorem_report.md
theoremHigh-z Tail Slaving Theorem
derived / scoped

Statement

Support above z ~ 1500 is demoted to a slaved residual tail rather than an independent selector-bearing sector, with the reduced packet differing from the full endpoint family only by Delta_theta = +5.386582264233e-06 and Delta_ell = -4.390006880612e-04.

Node id. calculator.highz_tail_slaving

Scope summary. Endpoint-family support decomposition.

Depends on. calculator.packet_coefficient_fixing

Premises

  • calculator.packet_coefficient_fixing isolates the live packet carrier near the active endpoint.
  • Any support above z ~ 1500 still has to be checked for an independent selector-bearing branch.

Proof outline

  • Audit the continuation of the active endpoint family above the reduced packet support.
  • Show that the remaining high-z contribution is slaved to the packet carrier rather than introducing a second independent selector sector.
  • Demote the high-z continuation to a residual tail in the active-branch support decomposition.

Scope boundary

  • Support decomposition on the active endpoint family.
  • Does not by itself identify the physical selector leaf.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_effectc_highz_tail_slaving_theorem_report.md
auditPeak-window Tail Profile Audit
verified / scoped

Statement

After the best pure-amplitude rescale is removed, the residual first-peak parent-profile mismatch obeys RMS_rel[ell in [190,250]] = 2.950052007950388e-06, so the remaining tail is not a second active selector-bearing branch.

Node id. calculator.peak_window_tail_profile_audit

Scope summary. First-peak TT parent-profile audit on the active endpoint family.

Depends on. calculator.highz_tail_slaving

Premises

  • calculator.highz_tail_slaving demotes the high-z continuation to a residual tail candidate.
  • The physical relevance of that tail still has to be checked against the first-peak parent profile.

Proof outline

  • Compare the first-peak parent profile with and without the residual tail after removing the best pure-amplitude rescale.
  • Measure the remaining mismatch on the peak window and show that it is tiny.
  • Use that audit to rule out the residual tail as a second active selector-bearing branch.

Scope boundary

  • First-peak TT parent-profile audit on the active endpoint family.
  • Verified only on the relevant first-peak window, not as a full-spectrum theorem.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_effectc_peak_window_tail_profile_audit_report.md
theoremSelector-support Promotion Theorem
derived / scoped

Statement

The support-certified cumulative packet z_cross < 1500 carries the physical selector leaf on the active branch, fixing z_sel = 1092.267038673162.

Node id. calculator.selector_support_promotion

Scope summary. Carried active-branch physical selector leaf.

Depends on. calculator.packet_coefficient_fixing, calculator.highz_tail_slaving, calculator.peak_window_tail_profile_audit

Premises

  • calculator.packet_coefficient_fixing provides the support-reduced active carrier.
  • calculator.highz_tail_slaving and calculator.peak_window_tail_profile_audit remove the residual tail as an independent selector branch.

Proof outline

  • Promote the cumulative packet z_cross < 1500 from a reduced support object to the carried physical selector carrier on the active branch.
  • Collapse the old endpoint-family selector interval to the leaf transported by that certified support packet.
  • Feed the carried leaf back into the Calculator selector backbone as the physical active-branch leaf.

Scope boundary

  • Carried active-branch physical selector leaf only.
  • Does not establish a universal selector-promotion theorem off branch.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_effectc_selector_support_promotion_theorem_report.md
theoremActive-branch Theta-star Theorem
derived / scoped

Statement

Evaluating the exact strict-bare selector backbone at the carried active selector leaf gives theta_bare = 0.720492080259 deg, theta_obs = 0.600851617929 deg, and 100theta_* = 1.048683904879 on the fixed active branch package.

Node id. calculator.active_branch_theta_star

Scope summary. Fixed active Paper 10 legacy projected branch only.

Depends on. calculator.selector_support_promotion, calculator.phase_equivalent_selector

Premises

  • calculator.selector_support_promotion fixes the carried physical selector leaf on the active branch.
  • calculator.phase_equivalent_selector gives the exact observer-side map from that leaf to theta_*.

Proof outline

  • Evaluate the exact strict-bare selector backbone at the carried active selector leaf.
  • Obtain theta_bare, transport to theta_obs with the fixed Jacobian J_theta, and report 100theta_* for the active branch.
  • Check the same carried solution against the direct first TT peak observable so the derived number remains tied to the measured peak position.

Scope boundary

  • Fixed active Paper 10 legacy projected branch only.
  • Theorem-grade numeric closure on the carried selector leaf only.
  • Not a universal off-branch transfer theorem or a full TT/TE/EE solver closure.

Supporting references

Supporting references only. The theorem text carried on this node is the calculator's self-contained public dictionary entry.

calculator_active_branch_theta_star_theorem_report.md