For the purposes of this paper, C1 - C6 are adopted explicitly as standing axioms. The cited corpus supplies its upstream necessity derivations and provenance, but no unstated result from that corpus is used in the theorem proved here. Accordingly, the registry theorem is conditional on C1 - C6 exactly as written below.

C1.Quotient primacy and finite diagnostic closure. A physical distinction is retained only when it survives the complete predictive quotient; a primitive carrier may not contain a removable label, multiplicity copy, superselection spectator, or diagnostically silent direction [1][3][5].

C2.Minimal central phase. A nontrivial continuous faithful irreducible multiplicity-free real circle action has rank two, and its minimal central unital scalar closure is ℂ , uniquely up to conjugation [2].

C3.Complex operator selection. The Volume-I complex-selection tribunal places the relevant finite ordered objects in self-adjoint parts of complex C * -algebras on its theorem domain; the terminal Volume-III census then merges the surviving real, quaternionic, Jordan, restricted, finite, and singular charts into, or excludes their nonembeddable remainder from, one completion-enlarged complex operator envelope [3][5].

C4.One-world inheritance. Volume IV receives exactly that complex operator envelope. Representations, sectors, normal and singular domains, and finite or infinite completions are internal coordinates of one architecture rather than new primitive worlds [6].

C5.Typed process and interaction discipline. Input, output, source, probe, effect, and record roles may not be collapsed across sorts without a predictive-equivalence theorem. Interaction is defined relative to independently addressable source and probe roles, while a higher constituent number may be carried inside a composite source or probe rather than creating a new primitive sort [4][5].

C6.No silent surplus. A proposed primitive parent is certificate-irreducible: deleting, identifying, or factoring one of its roles must destroy a retained diagnostic. Conversely, a construction that factors through already admitted primitive modules is a descendant, not a new parent [3]-[5].

These statements do not contain the integers 4, 9, 16, 27. The dimensions enter only after the terminal operator cell and its typed response modules are classified. C1 - C6 are the complete architecture-level assumption set used in that classification; any rejection of one clause removes the theorem’s jurisdiction rather than being repaired by an implicit appeal to the external corpus.

Definition 1 (Operational diagnostic burden and equivalence).Let B be a finite-dimensional unital complex C * -factor,let τ B be its normalized trace,and put

Its diagnostic certificate is the Boolean triple b ( B ) = ( b Φ , b O , b R ) ∈ { 0 , 1 } 3 ,defined by the following independently checkable tests:

(D1) b Φ = 1 exactly when Z ( B ) = ℂ 1 B and the central action e i θ ↦ e i θ 1 B isfaithful;

(D2) b O = 1 exactly when 1 B is the distinguished Archimedean order unit and τ B ( 1 B ) = 1 fixes itsnormalization;

(D3) b R = 1 exactly when B sa 0 contains X , Y with [ X , Y ] ≠ 0 and the connected inner-automorphism orbit U ↦ U X U * is nonconstant.

Two candidate factors carry the same phase,order-unit,and reversible-distinction burdens precisely when both have certificate ( 1 , 1 , 1 ) under the same quotient and normalization conventions.This istheequivalence of the three primitive burdens,not an assertion that the two algebras have identical state spaces or every descendant prediction.A candidate is diagnostic-minimal when it hasacertificate ( 1 , 1 , 1 ) and no factor of smaller real dimension has that certificate.

Proposition 1 (Independent audit of diagnostic minimality). For a complex matrix factor M n ( ℂ ) ,

Since dim ℝ M n ( ℂ ) = 2 n 2 , the unique least-dimensional factor carrying all three burdens is M 2 ( ℂ ) .

Proof. Every M n ( ℂ ) has centre ℂ I n , a faithful central circle action, and the normalized order unit I n . For n = 1 the traceless self-adjoint space is zero, and (D3) fails. For n ≥ 2 , an embedded Pauli block supplies noncommuting traceless self-adjoint elements and a nonconstant connected unitary-conjugation orbit, so (D3) holds. The function 2 n 2 is strictly increasing for n ≥ 1 , which gives the stated minimum. □

Definition 2 (Primitive response cell). Let A be the terminal complex operator envelope.A primitive finite response cell is a finite-dimensional unital complex C * -algebra A 0 admitted as a local diagnostic cell of A such that:

(P1) A 0 is noncommutative,because a commutative cell carries no irreducible coherent regional distinction or nontrivial interactionalgebra;

(P2) Its central projections have already been quotiented into explicit superselection sectors,so the primitive cell is a factor;

(P3) Itis diagnostic-minimal in the operational sense of Definition1: no factor of smaller real dimension passes the same three diagnostic tests;

(P4) Nomultiplicity copy is counted as a new carrier type.

Theorem 1 (Minimal operator-cell theorem). Every primitive finite response cell is ∗ -isomorphic to

Proof. Every finite-dimensional complex C * -algebra is ∗ -isomorphic to a finite direct sum [7]

Primitive factor status removes m > 1 , because each central summand is an independently addressable superselection sector rather than one certificate-irreducible cell. Hence A 0 ≅ M n ( ℂ ) for some n . Noncommutativity excludes n = 1 . Proposition 1 supplies the operational audit: the three burdens are carried exactly for n ≥ 2 , and their least real-dimensional carrier is uniquely n = 2 . Any n > 2 is a larger representation inside the same complex operator architecture; it may support additional states, sectors, or composite records, but it cannot define a new primitive response type while the n = 2 cell already realizes the phase, order-unit, and noncommutative distinction burdens. □

Remark 1(Why the proof is not the Frobenius shortcut). Frobenius’theorem classifies finite-dimensional associative real division algebras as ℝ , ℂ ,and ℍ .That classification is not the correct terminal selector here.The corpus has already selected a central complex scalar line and a complex operator envelope;irreducible quaternionic scalar structure is not a second carried world.The relevant classification is therefore the finite-dimensional complex C * -algebra classification,followed by primitive factor and diagnostic-minimality reduction.This avoids treating ℍ simultaneously asascalar field, aregional carrier,andanobservable algebra.

Proposition 2(Representation size is not a primitive response type). A larger matrix algebra M n ( ℂ ) , n > 2 ,may describe a genuine higher-dimensional system,sector,or composite.Its matrix size does not by itself commission a new universal response parent.

Proof. The registry classifies certificate-irreducible response roles, not every representation dimension in which those roles may be realized. The terminal complex envelope already admits arbitrary finite and infinite representations as internal system coordinates. Promoting each n to a new parent would make the registry depend on the chosen system representation and would generate an unbounded catalogue rather than a finite universal grammar. Under quotient primacy and no-surplus closure, a new parent therefore requires a new irreducible typed role or a new invariant response module that does not factor through the existing phase, regional, complete, or interaction roles. Matrix size alone supplies neither certificate. □

Let τ ( X ) = 1 2 Tr   X be the normalized trace on M 2 ( ℂ ) . Define typed carrier copies

The copies are typed: V Φ is the scalar-amplitude carrier, whereas V O , V R , and V C are self-adjoint diagnostic carriers. Their common real scalar direction is not counted twice as a world degree of freedom.

Theorem 2 (Canonical carrier-rank theorem). The primitive response cell canonically supplies

Moreover, V R carries a canonical oriented Euclidean structure; it is not an arbitrarily postulated real three-space.

Proof. The center of M 2 ( ℂ ) is ℂ I 2 , which is two-dimensional over ℝ . Every self-adjoint 2 × 2 matrix has a unique Pauli decomposition

Therefore, ( A 0 ) sa has real dimension four, its order-unit line has dimension one, and the traceless complement has dimension three. The bilinear form

restricts to a positive Euclidean inner product on V R . Conjugation by S U ( 2 ) preserves the trace and this inner product. In the Pauli basis, its adjoint action is the standard double cover S U ( 2 ) → S O ( 3 ) [8]; connectedness fixes positive orientation. Thus, the Euclidean metric and orientation descend from the selected operator cell and its reversible action rather than being primitive spatial assumptions. □

Remark 2(No inference of three spatial dimensions). The rank-three carrier V R is the traceless self-adjoint contrast space of the minimal operator cell.It is not the dimension of physical space.Volume III’s theorem that the pregeometric premises do not force three spatial dimensions remains untouched.A later geometric realization may identify a regional contrast basis with three spatial directions only on an independently licensed branch.

Lemma 1(No independent order parent). The one-dimensional order-unit line does not generate a nontrivial primitive response parent.

Proof. A linear response T : V O → V O compatible with unital normalization satisfies T ( I 2 ) = I 2 . Since every element of V O is a I 2 , linearity gives T ( a I 2 ) = a I 2 . The response, therefore, contains no free coefficient direction. Allowing T ( I 2 ) ≠ I 2 changes normalization or the reference scale and belongs to representative choice, not to a quotient-stable physical response. Hence, the rank-one line has trivial determinant transport and supplies no positive parent. □

Definition 3(Primitive response jet). A primitive response jet is the lowest nonzero quotient-stable derivative of a typed response at its neutral configuration.A higher derivative is a descendant unless an independently commissioned symmetry or selection rule forces every lower derivative to vanish.

For a carrier V , a self-response is a typed map R : V in → V out . Its first nonzero jet has one input slot and one output slot, hence response arity r = 2 . An interaction is a response of an independently addressed probe to an independently varied source,

Its tangent coupling is the mixed derivative

It has two independently typed inputs and one output, hence r = 3 .

Theorem 3(Primitive arity theorem). Inside the frozen Natural Physics response signature,the only primitive complete-response arities are

Higher constituent number,ancillary extension,or nonlinear saturation does not create another primitive arity.

Proof. A process necessarily has an input and an output, so self-response begins with two typed slots. Interaction is defined only after an independently addressable source and probe have been commissioned, preserving their distinction, and the output gives three typed slots. Additional physical constituents can be included in a composite source or composite probe without adding another process sort. Sequential and ancillary constructions are compositions or extensions of the same typed maps. Higher Taylor jets describe nonlinear descendants of the primitive map and are already governed here by the finite-response constitutive law. A genuinely new primitive arity would therefore require a new irreducible role not factorable into source, probe, input, output, effect, or record. The terminal one-world signature contains no such uncommissioned role, and the corpus’s no-surplus rule forbids adding it merely to manufacture another parent. □

Remark 3(Why $27$ has regional rather than scalar output).A scalar experimental record is an effect applied after the interaction output;it is not the output carrier itself.Replacing V R out by a scalar record would collapse process and effect sorts,contrary to the regional and process typing theorems.Likewise,taking the full ordered-regional carrier in every source,probe,and output slot produces a reducible4³tensor containing fixed order-unit pieces,linear self-response blocks,and the genuine regional interaction block.Only the last is a new primitive interaction coefficient space.

Let the carrier-frame groups be

where the disconnected part of O ( V Φ ) implements the conjugation freedom already allowed by C2. For self-response, independent input and output redescriptions act by

and for interaction,

Proposition 3 (Frame redescription and complete coefficient retention). Under the independent product actions above,the phase module Hom ℝ ( V Φ in , V Φ out ) ,the regional module Hom ℝ ( V R in , V R out ) ,and the interaction module Hom ℝ ( V R src ⊗ V R prb , V R out ) are irreducible real representations of their respective product frame groups.The complete ordered-regional module has the exact invariant typed decomposition.

with dimensions 1 + 3 + 3 + 9 = 16 .Complete coefficient retention means retaining this entire irreducible module,or this entire typed direct sum,before any sector-specific constitutive restriction is imposed.

Proof. The standard real representation of O ( 2 ) and the standard real representation of S O ( 3 ) are absolutely irreducible. Their external tensor products are therefore irreducible representations of the corresponding direct-product groups, which proves the phase and regional statements; the same argument applied to three factors proves the interaction statement. For V C = V O ⊕ V R , the distinguished order line is not mixed with V R , so expanding input and output by this direct sum gives the four displayed invariant blocks and no others.

A symmetric, antisymmetric, trace-free, diagonal, isotropic, rank-restricted, or otherwise physically constrained coefficient space requires additional structure: for example, an identification of input and output frames, a constitutive tensor, a source constraint, or a declared symmetry. Such spaces may be valid descendants, but under C5 - C6, they cannot replace a universal primitive parent before that additional structure is commissioned. Thus, the exclusion is not that proper subspaces are mathematically impossible; it is that they are conditional sector reductions rather than primitive complete-response modules. □

The primitive modules are consequently

Theorem 4 (Primitive causal-response registry theorem). Relative to the terminal one-world complex operator envelope and the primitive response definitions above,the primitive complete-response signatures are unique up to typed real-linear isomorphism:

No alternative universal primitive response signature is admissible inside the frozen Natural Physics response jurisdiction.Representation-specific system dimensions remain allowed but are not registry parents by Proposition 2.

Proof. The minimal operator-cell theorem leaves one primitive cell, M 2 ( ℂ ) . Its center, traceless self-adjoint part, and complete self-adjoint part give exactly the nontrivial typed carrier ranks 2, 3, and 4; the normalized order-unit line is response-trivial by Lemma 1. The primitive arity theorem leaves only self-response with r = 2 and source-probe interaction with r = 3 . Self-response applies to each of the three nontrivial carriers, giving ( 2 , 2 ) , ( 3 , 2 ) , and ( 4 , 2 ) . The only new interaction block surviving sort decomposition is the regional source-probe-output map, giving ( 3 , 3 ) . Proposition 3 fixes the corresponding complete coefficient spaces and classifies every proper invariant or physically constrained subspace as a separately commissioned descendant.

It remains to exclude apparent alternatives. Direct sums and repeated blocks are a superselection or multiplicity structure and fail primitive irreducibility. A higher matrix size belongs to a larger representation of the same operator envelope and fails diagnostic minimality. Real, irreducibly quaternionic, exceptional Jordan, and nonassociative remainders have already been merged into the central-complex envelope, where predictively equivalent or excluded by the terminal census where not. Mixed carrier maps require an independently commissioned type-changing bridge; scalar multiplication, order-unit insertion, trace projection, and inclusion are fixed structural maps rather than new coefficient-complete parents. A phase bilinear product is already the fixed complex scalar multiplication. A full 4³ interaction tensor is reducible into order, self-response, observation, and regional-interaction blocks. Higher jets and higher constituent numbers are descendants or composites, not additional primitive sorts. Finally, higher determinant powers are harmonics of a parent character rather than new parent modules. The listed four signatures, therefore, exhaust the primitive response jurisdiction. □

Definition 4(Coefficient-transport group). Each real coefficient module E N ∈ { E Φ , E R , E C , E I } ,indexed by its real dimension N ∈ { 4 , 9 , 16 , 27 } ,carries the Hilbert-Schmidt Euclidean product induced from its typed carrier products.Define its complexified coefficient space

with the canonical Hermitian extension of that product.The carrier-frame groups in Proposition 3act through a subgroup of U ( ℋ N ) .The coefficient-transport group used by the global scalar readout is the full basis-neutral group

This is an observation-level premise:no preferred complex coefficient basis or proper complex coefficient subspace is included in the primitive data.It does not identify the physical carrier-frame group with U ( N ) .

Proposition 4(Least continuous coefficient character). Let χ : U N → U ( 1 ) be a continuous group character.Then for a unique m ∈ ℤ ,

On the central phase transport z I N ,one has χ ( z I N ) = z m N .If primitive observation applies the no-harmonic-surplus rule and selects the least positive nontrivial character,then m = 1 ,so the observable central winding is exactly N .

Proof. The commutator subgroup of U ( N ) is S U ( N ) , and every character into the Abelian group U ( 1 ) annihilates it. Hence, χ factors through the determinant quotient U ( N ) / S U ( N ) ≅ U ( 1 ) . The continuous characters of U ( 1 ) are exactly z ↦ z m , m ∈ ℤ . The least positive nontrivial choice is m = 1 ; choices m > 1 are determinant harmonics and are descendants rather than new primitive characters. □

Remark 4(Two distinct covariance statements). Carrier-frame covariance determines the tensorial coefficient module and the status of its proper subspaces.Coefficient-basis neutrality determines the unitary group on the complexified coefficient space used for scalar character transport.The determinant step relies on the latter and is therefore stated separately rather than inferred from the former.

Corollary 1 (Conditional primitive β-registry). If the coefficient module is complexified and transported according to Definition4,and if the observable scalar applies the least-character rule of Proposition4,then

Within this explicitly conditional observation jurisdiction,the resulting winding registry is not fitted to empirical data or selected from rival numerical lists.

Proof. Theorem 4 fixes the four real coefficient-space dimensions. Definition 4 supplies their N -dimensional Hermitian complexifications, and Proposition 4 proves that the least positive scalar character is the determinant with central winding N . Applying that conditional observation rule to the four dimensions gives the displayed set. □

Proposition 5 (Non-circularity).None of the selector premises contains the target integers or an equivalent numerical encoding.The numbers arise only after: 1)the complex operator cell is classified; 2)its center and self-adjoint decomposition are dimension-counted; 3)the typed response slots are counted;and4)the determinant character is applied.

Proof. The corpus premises concern quotient identity, central phase, complex operator closure, order unit, primitive factor status, typed source/probe/output roles, and response completeness. Their statements are invariant under basis changes and contain no target parent. The numerical values first appear in Equations (8) and (14) - (17). □

The effect of removing each load-bearing selector is summarized in Table 1.

Table 1. Ablation of the registry-uniqueness theorem.

Verdict 2(Exact foundational advance). The earlier registry theorem was conditional on choosing the ranks2, 3,and4and on declaring four primitive response modules.The present theorem removes that choice inside the terminal Natural Physics architecture.The ranks are the center,traceless self-adjoint contrast space,and complete self-adjoint diagnostic space of the unique primitive cell M 2 ( ℂ ) ;the arities are fixed by process and interaction sort structure.Accordingly,the primitive carrier-signature registry is a corpus-relative uniqueness theorem.The corresponding β -winding registry is its conditional observation corollary under the coefficient-transport and least-character premises;channel-level detection additionally requires nonzero generation,transfer,and observation.

By Theorem 4, the primitive carriers and arities are not free inputs. Their complete coefficient modules are

Set

Conditional Corollary 1 fixes the least positive central winding of each complete module to its dimension only under the coefficient-transport and least-character premises stated above. The earlier logarithmic-registry paper supplied the conditional determinant mechanism; the present corpus theorem removes the prior freedom in the primitive ranks and signatures [9]. Inside the present realization, this becomes the self-adjoint finite-spectrum operator

No continuous frequency parameter is fitted.

For a response state Q = ∑ N Q N ∈ ℰ and a dimensionless logarithmic comparison coordinate x , define

Then

The equation is kinematical in logarithmic comparison space; it does not introduce another time evolution. It states exactly which carriers may occur once a channel admits multiplicative comparison.

Theorem 5 (Support immutability). Let L j : ℰ ℂ → ℂ be a fixed bounded observation functional and T j ( N ^ ) a fixed bounded transfer operator.The observable

has Fourier support contained in { 4 , 9 , 16 , 27 } .A new nonzero frequency cannot be produced by changing amplitudes,phases,initial response data,or the convexconstitutive law;it requires changing the registry,observation coordinate,or transferoperator and therefore defines a new candidate.

Proof. Using the spectral projectors,

The stated support follows. None of the coefficients changes an eigenvalue of N ^ . □

Writing

recovers the selective-visibility rule of the registry paper: a parent can be structurally available and observationally absent because generation, projection, or transfer vanishes. Therefore, the empirical absence of 9, 16, or 27 in a channel does not contradict the existence of those modules; it contradicts the theory only when the channel map predicted nonzero detectability before inspection.

The empirical phase is not periodic in X but in the normalized multiplicative coordinate

Consequently − i ∂ x is the generator of scale translation and the response-number eigenvalue is

For a scalar projection, the phase, logarithmic wavelength, multiplicative recurrence ratio, and number of visible cycles are

Thus, a larger coherent or observed scale range does not by itself select a larger parent. It reveals more cycles of whichever parent is already generated and transmitted. Numerically,

These are wavelengths and recurrence ratios in scale space. They should not be confused with ordinary spatial wavelengths unless a route-specific observable map proves that identification.

Let a physical realization possess a coherent active domain with a causal diameter L coh , effective propagation speed v eff , and a collective carrier angular frequency ω c . Its causal crossing time is

A boundary condition contributes a dimensionless phase normalization κ bc ; for an elementary standing-wave convention, it is commonly π , while a full-wavelength convention gives 2 π . Define the primitive closure count

Realization Postulate 1 (Causal carrier realization). On a branch where the physical boundary-value problem identifies the number of independent carrier components with a closed causal mode count,require

This is an optional realization bridge.The response-registry theorem itself selects the ranks d = 2 , 3 , 4 structurally and does not depend on every sector possessing a literal cavity.

The distinction is essential. In a turbulent eddy, for example, the elementary turnover estimate ω L / v ~ 1 cannot explain N = 4 ; in a circular orbit, ω a / v = 1 identically. Equation (37) can therefore use only the collective coherent response frequency and domain licensed by the channel, not an arbitrarily chosen local kinematic frequency and length.

The numbers now have one source. The minimal central phase line V Φ = ℂ I 2 has real rank two; the traceless self-adjoint contrast space V R has rank three; and the complete self-adjoint diagnostic space V C = ℝ I 2 ⊕ V R has rank four. The first two-slot complete responses therefore carry 2², 3², and 4² coefficients, while the first certificate-irreducible source-probe-output interaction carries 3³ coefficients. Hence

The rank three in this table is internal operator contrast, not a proof that space has three dimensions. The rank four is one normalized ordering direction plus three regional contrasts, not four spatial wavelengths. The value 27 is not a shifted square: it is the complete coefficient count of the first genuinely typed interaction response. Larger Hilbert or matrix dimensions remain legitimate system representations, but Proposition 2 prevents representation size from being silently promoted into another universal parent.

Combining the optional causal realization with the primitive theorem gives

only where Equation (38) is independently established. The linear causal ratio realizes d ; response completeness fixes r ; the determinant character makes d r observable as logarithmic winding. None of these three operations may be substituted for another.

Let X = X ( t ) along a physical history. Differentiating (32) gives the ordinary instantaneous response frequency

Hence,

If the comparison variable is a coherent size X = L whose boundary evolves according to L ˙ = η v eff , then

This explains the apparent tension in the original intuition. At a fixed parent N , increasing L lowers the physical response frequency as L − 1 when η v eff is fixed. At a fixed physical frequency, a larger coherent domain permits a larger response number. The two comparisons hold different quantities fixed and are not contradictory.

A physical state need not choose one parent. It may carry

and a channel or population may therefore exhibit

where λ labels source history, formation channel, geometry, or other latent data. If relative phases are sufficiently decohered across a population, the cross terms vanish under averaging and

Thus, all parent powers can survive in different proportions even when no individual event displays a conspicuous multi-parent alignment. This is the natural operator interpretation of a heterogeneous compact-object population: the parents are simultaneous orthogonal response channels, while their observed proportions measure generation, projection, transfer, and decoherence.

Suppose a parent phase is observed with additive logarithmic-coordinate uncertainty δ x having zero mean and variance s . For Gaussian phase diffusion,

Therefore,

Higher parents are intrinsically more fragile under the same phase uncertainty.

A heterogeneous population need not have one variance. Let s follow a gamma distribution

Averaging (47) gives its Laplace transform,

At high parent number,

The earlier binary-black-hole study used the phenomenological hierarchy a ( N ) ∝ N − 3 / 2 . Equation (50) shows that this law is obtained asymptotically from a gamma mixture with

This is a candidate physical explanation, not a retrospective proof. A serious test must derive or freeze the variance distribution from formation modelling, measurement posteriors, and selection effects before evaluating the parent amplitudes.

The structural spectrum fixes parent support, but an observation generally sees a complex transfer envelope. Write one projected parent as

Its observed phase is N x + ψ j N ( x ) .

Proposition 6(Transfer-phase displacement).On any interval where H j N is nonzero and differentiable,the local logarithmic frequency inferred from phase is

A constant transfer phase changes only the fitted phase. A frequency displacement requires a nonconstant transfer phase.

Proof. Differentiate arg Z j N ( x ) = N x + ψ j N ( x ) with respect to x . □

Accordingly, a historical contact near 27.57 is not a fifth exact parent merely because it is numerically stable in a route. It can be a descendant of the exact interaction parent 27 only if an independently specified transfer supplies

over the fitted interval. This statement is falsifiable: once the route’s transfer phase is derived or measured and removed, the residual carrier must return to 27. If it does not, the shifted line remains an empirical candidate outside the exact registry.

Repeated detection of the same parent in two coordinates is weaker than demonstrating that the two phases are one physical phase viewed through two quotient maps. Let

Proposition 7 (Phase-locking scaling law). If Θ a − Θ b is constant on a common physical history,then

for a constant C > 0 . For equal parents, X b ∝ X a .

Proof. Constancy of the phase difference gives N a ln X a − N b ln X b = constant . Exponentiation yields the result. □

For the earthquake time-radius tribunal, equal 4 parents would imply the stronger prospective relation r ∝ τ only if the temporal and radial phases are demonstrably locked. Marginal power at 4 in both channels does not by itself establish that relation. This is a clean next-generation test because it distinguishes one causal response from two unrelated scale-space projections.

The exact parent registry is a statement of availability, not universal brightness. Factor the observed complex amplitude as

where G is generated response, P is quotient-level projection, T is physical and instrumental transfer, and D is coherence survival. The observable parent is absent when any load-bearing factor vanishes or when its product lies below the predeclared tribunal sensitivity.

A finite-memory fossil descendant makes the logic explicit. After a source switches off at t f , let

If propagation and averaging additionally accumulate Gaussian log-phase variance s N , the visible amplitude obeys

Thus, higher parents are the first to disappear under a common loss of coherence, and an old fossil may become β -silent without the parent registry ceasing to exist. This is the exact logical role of the BAO tribunal: it tests whether a predeclared fossil route remains silent rather than supplying arbitrary registry power in every smooth residual.

Verdict 3 (Simple physical meaning of β ).The primitive causal count d measures how many independent carrier components a coherent closure supports.The parent β = d r counts how many independent coefficients are required torecord the corresponding complete response.Logarithmic comparison turns that coefficient count into a scale-space winding.Observed parent proportions then measure how strongly each complete response is generated,projected,and preserved,not how many ordinary wavelengths fit across the visible object.