Interior Observer Calculator

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.

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

• 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.

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.

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.

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

• 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.

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.

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.

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

• 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.

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.

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.

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

• 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.

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.

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.

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

• 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.

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.

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.

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

• 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.

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.

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.

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.

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

• 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.

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.

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.

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.

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.

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.

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.

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.

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.

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

• 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.

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.

Supporting references

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

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.

Supporting references

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

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.

Supporting references

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

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.

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.

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.

Supporting references

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

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.

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.

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

• 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.

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.

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.

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.

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

• 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.

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.

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

• 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.

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.

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

• 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.

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.

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.

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

• 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.

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.

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.

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.

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

• 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.

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.

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.

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

• 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.

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.

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

• 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.

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.

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.

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.

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

• 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.

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.

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.

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.

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.

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.

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.

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

• 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.

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.

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.

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.