Landmark IX-Free-2 Degree-four descent and overlap agreement for one sharp ordered word

We complete the algebraic descent and overlap tests left open in IX-Free for the ordered word $B_f,W_g,B_{f^{\prime }}$. The previously constructed graph outputs are used as inherited values.

A compact spatial-diagonal metric direction, obtained by applying the wave operator to a test tensor, proves sharp-generator separation even when the $00$-metric component is constrained to vanish. Slot isolation then proves injectivity of both degree-four presentation maps. Consequently all four prescribed degree-four coefficients descend to actual functionals.

An explicit advanced-trace witness proves that every nonzero sharp generator has nonzero intrinsic metric-derivative charge at some smooth configuration. The partial contractions have zero charge. The old microcausal algebra also has zero charge: a nonzero boundary coefficient would produce a timelike conormal singularity. These facts eliminate the live new-domain intersection and separate the sharp space from the old algebra. An algebraic anchored-domain lemma eliminates the old overlap of the entire new-domain sum, including possible multi-domain relations.

For every real common parent parameter, the prescriptions therefore glue to one truncated partial product preserving the locked first stages and the old product. The two associations of the selected word agree through order ${\hslash }^3$. Reversal, presentation independence, and fixed-support test-family continuity hold. Actual-stage continuity, associativity on unspecified triples, and the IX-D and G2 constructions are not asserted.

Figure fig:split-IX-Free-2 records one view of the same separation. Figure fig:chain-IX-Free-2 records one view of the same separation. Figure fig:pair-IX-Free-2 records one view of the same separation. Figure fig:gate-IX-Free-2 records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-IX-Free-2 records that separation. The drawing adds no hypothesis.

and scope{ Release: Constructed, relative to the inherited premises stated below. The result is one truncated partial product on an explicitly named tensor domain, together with coherence for the single ordered word $A=B_f,W=W_g,C=B_{f^{\prime }}$.

No graph-existence or triangle-calibration proof is reconstructed. No Einstein equation, time-ordered product, retarded product, global quantum algebra, causal-support lift, IX-D construction, or G2 opening is included.}

The release alternatives are kept distinct:

1. Constructed: the prescribed coefficients descend and glue with all declared existing coefficients, satisfying the copied interfaces on the named sector.

2. Canonical prescription obstructed: an explicit actual-presentation relation or overlap discrepancy contradicts the prescribed coefficients.

3. Contract proved obstructed: a contradiction excludes every coefficient system satisfying the copied interfaces.

4. Unresolved: a required proof remains unfinished.

A missing proof is not an obstruction. Failure of one prescription is not a universal contract obstruction.

Fix one real parent parameter $c$ at the outset. It is not changed by association, presentation, or observation descendant. The results below hold for every such fixed $c$.

Let

with metric

Metric configurations are smooth symmetric covariant tensors. Set

The operator ${\mathsf{E}}_0$ is the inverse of $\square$ with zero Cauchy value and zero first time derivative at $t=0$. The operator ${G_{\mathrm{A}}}^0$ is the componentwise advanced inverse of the same massless wave operator.

All configurations and directions are smooth and off shell. Compact directions supported in $M_{+}$ are allowed. We use the usual smooth mapping properties, Cauchy uniqueness, and finite propagation of these fixed linear operators.

With

the generators are

Complex tests define complex-valued functionals on real smooth configurations. Complexified directions can equivalently be used by multilinearity.

The supplied IX-Free source is used through the following ledger. Statements reported there from companion manuscripts remain inherited premises, not independently certified companion proofs.

The carrier ${\mathcal{P}}_{\Sigma }$ consists of actual finite polynomial functionals with compact symmetric distributional kernels, subject to

for every smooth configuration $\psi$ and every finite list of smooth configuration directions $\mathbf{\eta }$.

The linear sharp kernels and intrinsic charge are

The charge is well defined on actual linear kernels and vanishes on ${\mathcal{M}}_1$. Every element of ${\mathcal{M}}_1$ has no nonzero nonspacelike wavefront covector; compact smooth metric kernels belong to ${\mathcal{M}}_1$.

The sharp adjoint retains its surface term:

Classical multiplication, smooth directional differentiation, and conjugation preserve ${\mathcal{P}}_{\Sigma }$. Classical multiplication is separately continuous in the locally convex topology ${\tau }_{\Sigma }$, which records strong compact-distribution coefficients and all intrinsic trace coordinates.

The generating test maps are injective and have fixed-support test-family continuity. The inherited nonconvergence $B_{f,\epsilon }\nrightarrow B_f$ in ${\tau }_{\Sigma }$ for nonzero real $f$ is retained, but is not used as a charge witness.

The full old compact polynomial microcausal algebra ${\mathcal{P}}_{\mathrm{mc}}$ is included in ${\mathcal{P}}_{\Sigma }$. Its functional derivatives satisfy the standard microcausal wavefront condition: no nonzero joint covector is entirely future causal or entirely past causal, where zero individual covectors are allowed in those closed causal cones.

The inherited old product ${\star }_H$ is defined on ${\mathcal{P}}_{\mathrm{mc}}\otimes {\mathcal{P}}_{\mathrm{mc}}$, has classical order zero, and has its unchanged coefficients

For the reversal statement we use the inherited Hermitian old convention

No new continuity property of ${\star }_H$ is introduced.

The microcausal condition just specified is the usual functional-derivative condition; see FR. We will prove the zero-charge consequence needed here, rather than infer it from the informal word “bulk.”

The fixed mixed kernel $K$ has $\mathrm{supp}K=M_{+}^2$. The first stages are the actual maps

Their metric derivatives are compact smooth kernels, also after the required directional insertions.

Write

for the fixed scalar contraction kernel, and

The three one-edge outputs

exist and belong to ${\mathcal{P}}_{\Sigma }$.

For $V=f\otimes g\otimes f^{\prime }$, the three two-edge values are the already defined and admitted functionals

Their defining wavefront products, compact coefficient kernels, admission decompositions, and all required descendants have already been established in IX-Free. Their fixed-support test-family continuity is inherited.

The maps $q_L,q_R$ defined below are injective. The inherited degree-two second-stage maps are

They are maps on actual arguments, with inherited reversal and presentation independence.

Fix the inherited regulator and scale. The calibrated distributions ${u_{\mu }}^{\rho },r\in {\mathcal{D}}^{\prime }(X)$ exist under the analytical premise explicitly reported in IX-Free. For the one fixed real parameter $c$, set

The parent has dagger symmetry under conjugation and exchange of $x,x^{\prime }$, and its declared numerical observation operations use this same parameter. Algebraic scalar descent has already been evaluated:

The scalar family has inherited fixed-support continuity. No claim about arbitrary nonconstant contact terms follows from this scalar calibration.

The sign ledger is unchanged: each first-stage partial has factor $-2$; the complete word has two sharp insertion signs, whose product is $+1$; the middle-slot assignment factor is $2$, not $4$.

The construction must satisfy:

1. Every output is admitted, including the required metric derivatives and directional descendants.

2. Order zero is classical multiplication.

3. Every specified old-product overlap agrees with ${\star }_H$.

4. The first stages are locked to (eq:firstL)–(eq:firstR).

5. The constant kernel is $U_c$, using one real common parent parameter and its declared observation compatibility.

6. No subtraction changes a fixed old product.

7. Unchanged approximants need not converge sharply in ${\tau }_{\Sigma }$.

8. All finite relations between actual presentations are respected.

9. Every continuity claim has a named domain and proof; test-family continuity is not actual-stage hypocontinuity.

{}1.12

All spans and tensor products in this paper are algebraic over $\mathbb{C}$. Define

and

These last sums are direct by total field degree. The actual spaces ${\mathcal{L}}_4,{\mathcal{R}}_4$ coincide, but their ordered roles remain distinct.

Let

with ordered presentation maps

Inside ${\mathcal{P}}_{\Sigma }\otimes {\mathcal{P}}_{\Sigma }$, the declared domains are

Thus the full inherited old domain is used, not a smaller regular subdomain.

We will define coefficient maps

Their prescribed restrictions are displayed below. On the degree-four rows, $L_j,R_j$ initially denote the labelled prescriptions; their actual descent is proved in Section sec:descent.

{}1.2

On every row,

The explicit zero entries are part of the selected contraction prescription. They are not a classification excluding every alternative contact term.

For

use the inherited injective realization of this algebraic tensor product as smooth compact joint functions.

Let ${J^{\mu }}_{\alpha \beta }(x,z)$ be the distribution kernel of $L$, and let ${e_{00}}^{\alpha \beta }{h}_{\alpha \beta }=h_{00}$. The relation kernel is copied unchanged:

The symmetrization is the average over the three metric slots. IX-Free establishes

For $v_R(y,x^{\prime })$, set $\tilde{v}(x^{\prime },y)=v_R(y,x^{\prime })$. The right relation is

Full coefficient recovery uses one scalar derivative and three metric derivatives. The metric polarization factor is $3!$. Two metric derivatives do not recover this coefficient.

The prescribed values whose descent remains to be proved are

They descend precisely when

We now evaluate the kernels. No inference from injectivity of $q_L,q_R$ will be used.

The restricted separation lemma can be proved without a general range theorem for $C$.

Let $f\in \mathcal{T}$. Suppose

for every smooth symmetric metric direction $k$ with $k_{00}=0$ and every smooth scalar $\phi$. Then $f=0$. It is enough to use compact metric directions supported in $M_{+}$.

Proof. For $\chi \in {C_c}^{\infty }(M_{+})$, let $H$ have its only nonzero component

Then $k_{00}=0$. Because $C$ has constant coefficients,

The vector $CH$ is compactly supported away from $\Sigma$, so has zero Cauchy data there. Cauchy uniqueness gives

Directly from (eq:C),

Choose $\phi =t$. This is an allowed smooth configuration. Alternatively, use a compact smooth scalar agreeing with $t$ near $\mathrm{supp}f$. The hypothesis becomes

for every $\chi$. Hence ${\partial }_{t}f=0$. A smooth function independent of $t$ on $M_{+}$ and compactly supported there must vanish. Thus $f=0$.

Real $\chi$ already suffice for complex $f$, by applying the distributional identity to its real and imaginary parts.

The test direction $k=\square H$ is off shell. Equation (eq:compactinverse) is an identity for the fixed Cauchy inverse on compact zero-data tests, not an equation-of-motion restriction on configurations.

The maps

are linear isomorphisms onto their actual-functional images. In particular,

Proof. Suppose $p_L(v)=0$. Choose metric directions $k,a,b$ with

where $u,z\in {C_c}^{\infty }(M_{+})$; the other components of $a,b$ may be zero. Differentiate once in a scalar direction $\phi$ and three times in these metric directions at the zero configuration.

Using the unsymmetrized functional expression,

the only surviving terms place $k$ in the $Lh$-slot. Consequently the derivative is exactly

The factor $2$ is the surviving permutation count after $k_{00}=0$; it is consistent with the $3!$ recovery of the fully symmetric coefficient.

Define

It is a smooth compact test. Since (eq:isolation) vanishes for all $k,\phi$, Lemma lem:restricted gives $f_{u,z}=0$ for every $u,z$.

For any $d\in {C_c}^{\infty }(M_{+})$, take $u=d$ and choose $z$ equal to one near $\mathrm{supp}d$. Then

for every $x$. Thus every $y$-test contraction of $v$ vanishes, so $v=0$ as a joint smooth function and hence as an algebraic tensor.

The right map is the same classical product after $v_R\mapsto \tilde{v}$. Therefore its kernel is also zero. Surjectivity onto the named images is their definition.

For the algebraic presentation tensors of this paper,

The exchanged right equation has the same conclusion.

Proof. Combine (eq:Prelation) with Theorem thm:pinjective.

The conclusion concerns this particular polynomial product. It is not the false general principle that products of injective linear generating maps are automatically tensor-injective.

The four prescriptions (eq:lambdaL)–(eq:lambdaR) define unique algebraic bilinear maps

with the prescribed values.

Proof. Theorem thm:pinjective makes both relation kernels zero. Thus all four annihilation tests (eq:annihilation) hold.

More explicitly, an element of ${\mathcal{L}}_4$ has a unique presentation tensor in ${\mathcal{E}}_L$, and an element of $\mathcal{B}$ has a unique sharp test. The graph prescriptions are linear in that tensor and in the additional test. They therefore define unique bilinear maps on the actual images. The same argument applies on the right.

Their values are finite sums of inherited admitted graphs, so belong to ${\mathcal{P}}_{\Sigma }$, including the required admission descendants. Classical multiplication defines $L_0,R_0$.

No operation on an unproved “zero graph factor” is used. Linearity becomes sufficient only after the relevant actual presentation kernel has been evaluated as zero.

We first justify the zero-charge property of the old algebra using only its microcausal definition and the copied sharp boundary identity.

For the component convention in (eq:C), integration by parts gives

Its surface coefficient in (eq:boundary) is

In particular, $\mathcal{B}$ is injective on the boundary covector $b$.

Let $b$ be smooth near $\Sigma$. If $b{|}_{\Sigma }\ne 0$, then $C^{\ast }(\theta b)$ has a nonzero timelike wavefront covector.

Proof. By (eq:Bsurface), some component of $\beta =\mathcal{B}(b{|}_{\Sigma })$ is nonzero on a boundary patch. That component has local form

with $c$ smooth.

Choose compact localizers $\chi (t),\rho (\mathbf{x})$, with $\chi (0)=1$, so that $\int \rho (\mathbf{x})\beta (\mathbf{x}){\mathrm{d}}^3\mathbf{x}\ne 0$. The Fourier transform of the localized delta term at $(\tau ,0)$ is this nonzero constant. The transform of the localized Heaviside term is $O(|\tau {|}^{-1})$, by one-sided integration by parts. Thus the localized distribution is not rapidly decreasing along the normal rays as $|\tau |\to \infty$.

If the distribution had no timelike wavefront covectors on that compact patch, the local wavefront estimates, after a finite partition of unity, would give rapid decay in a cone around those normal rays. This contradicts the preceding asymptotic. Hence a timelike covector occurs.

If $\ell \in {\mathcal{S}}_1$ has no nonzero nonspacelike wavefront covector, then $\mathsf{b}(\ell )=0$.

Proof. Write $\ell =m+q_a$. The kernel $m$ has only spacelike wavefront covectors by Assumption ass:carrier. Therefore

also has no nonspacelike covector. The source $a$ is compact smooth, so $b={G_{\mathrm{A}}}^{0}a$ is smooth. Lemma lem:surfaceWF forces $b{|}_{\Sigma }=0$. Equation (eq:charge) now gives the conclusion.

This argument does not reverse the stated inclusion for ${\mathcal{M}}_1$: it does not assume that every spacelike linear kernel has already been identified with an element of ${\mathcal{M}}_1$.

At every smooth configuration and after every finite smooth directional insertion,

for

Proof. For the partial contractions, the inherited metric derivative is compact smooth. Inserting a scalar direction replaces the smooth scalar configuration by that direction. Inserting a metric direction removes the only metric slot, so subsequent metric differentiation is zero. Higher derivatives vanish by polynomial degree. All nonzero metric kernels in this calculation are therefore compact smooth elements of ${\mathcal{M}}_1$, with zero charge.

For $F\in {\mathcal{P}}_{\mathrm{mc}}$, contraction of its higher functional derivatives against smooth directions is a proper pushforward after inserting compact cutoffs equal to one on the coefficient supports. A wavefront covector of the remaining one-slot kernel can arise only from a joint covector with all integrated covectors zero.

A nonzero causal covector in the remaining slot, together with those zeros, would belong entirely to one of the excluded closed causal cones. Thus the remaining metric kernel has only spacelike wavefront covectors. It belongs to ${\mathcal{S}}_1$ by carrier admission, and Lemma lem:MCcharge proves its charge is zero.

The following trace formula fixes the sign for $\square =-{{\partial }_t}^2+\Delta$:

It is the advanced Kirchhoff formula; compare Ivrii. To see the normalization, the spatial Fourier transform of $\delta (s-|\mathbf{x}|)/(4\pi |\mathbf{x}|)$, for $s>0$, is $\sin (s|p|)/|p|$. The zero-future-data solution of

at $t=0$ is

Inverting the spatial transform gives (eq:advancedtrace).

For every nonzero $f\in \mathcal{T}$, there is a real smooth scalar configuration $\varphi$, which can be chosen compactly supported in $M_{+}$, such that

No directional descendant is needed.

Proof. The metric derivative is

Choose $x_0=(t_0,{\mathbf{x}}_0)$ with $f(x_0)\ne 0$. Select $\mathbf{z}$ with

Since $t_0>0$, a neighborhood $U⋐M_{+}$ of $x_0$ can be chosen away from the spatial point $\mathbf{z}$. Let

on that neighborhood. Choose $\chi \in {C_c}^{\infty }(U;\mathbb{R})$, nonnegative and positive at $x_0$, and form the complex scalar

Extended by zero outside $U$, it is compact smooth.

On the cone $t=r(\mathbf{x})$,

Using (eq:advancedtrace) and (eq:Bderivative),

The denominator is bounded away from zero on the relevant support. Positivity holds on an open spatial patch near ${\mathbf{x}}_0$.

Write ${\varphi }_{\mathbb{C}}={\varphi }_1+i{\varphi }_2$ with real compact smooth ${\varphi }_1,{\varphi }_2$. The trace is complex linear in the scalar configuration. Since (eq:positivewitness) is nonzero, at least one of the traces for ${\varphi }_1,{\varphi }_2$ is nonzero. Choose that real configuration. The metric configuration may be zero, since (eq:Bderivative) is independent of $h$.

This witness is independent of the inherited nonconvergence of regularized sharp generators.

The intersections are

Consequently,

Proof. Every element of $\mathcal{B}$ is $B_f$ for one test $f$, because the generating map is linear. A nonzero such element has $f\ne 0$. Lemma lem:sharpcharge gives a configuration at which its metric derivative has nonzero charge.

Every element of either partial-contraction space or the old algebra has zero charge at that configuration by Proposition prop:zerocharge. Equality of actual functionals would imply equality of their actual metric derivatives and intrinsic charges, a contradiction.

For algebraic tensor products,

Apply this identity to the displayed degree-two domains.

Metric–scalar homogeneity separates the locked first stages from the second stages. The species degrees are

Here the entries count metric and scalar slots.

The six new domains therefore have argument species degrees

The new domain is the algebraic direct sum

The prescribed coefficient maps therefore glue uniquely on $D_{\mathrm{new}}$.

Proof. The zero intersections involving the degree-four blocks are the inherited field-degree results of IX-Free. The locked first-stage domains have species degrees distinct from all second-stage domains and from each other.

All relations can be projected onto the distinct homogeneous argument-species components. The only component containing two listed domains is $((1,1);(1,1))$. Its two domains have zero intersection by Theorem thm:separation. Thus that sum is direct as well.

This proves a direct decomposition of the entire sum, not merely pairwise agreement. Each coefficient restriction is an actual linear map by the inherited first stages, inherited degree-two descent, and Theorem thm:degree4. Their sum is consequently a unique linear map on $D_{\mathrm{new}}$.

Let $A$ be a complex vector space and let $B,P\subset A$ satisfy $B\cap P=\{0\}$. Then

Proof. On $P\oplus B$, define $\pi (p+b)=p$. Extend this linear map algebraically to $A$, using a vector-space complement, with values in $P$. Then

The map $\pi \otimes \pi$ is the identity on $P\otimes P$ and vanishes on $A\otimes B+B\otimes A$. An element in their intersection must therefore be both unchanged and annihilated by this map, hence zero.

No continuity of $\pi$ is asserted or needed.

The full new-domain sum has zero old overlap:

Thus

For each fixed real $c$, all prescribed coefficients glue uniquely with the unchanged old coefficients on $D_{\mathrm{decl}}$.

Proof. Every new domain has a factor in $\mathcal{B}$ in at least one argument. Hence

Theorem thm:separation gives $\mathcal{B}\cap {\mathcal{P}}_{\mathrm{mc}}=0$. Apply Lemma lem:anchored with $A={\mathcal{P}}_{\Sigma }$, $B=\mathcal{B}$, and $P={\mathcal{P}}_{\mathrm{mc}}$. This proves vanishing of the old overlap of the entire new sum, including any potential sum-domain relation.

Proposition prop:newdirect already constructs each coefficient on $D_{\mathrm{new}}$. Combining it with ${C_j}^H$ on the disjoint old domain gives a unique map on $D_{\mathrm{decl}}$.

This proof does not subtract a regular coefficient, alter a fixed old product, or make a convergence assertion.

Define

For this declared sector,

Proof. The first equality is inherited. Every degree-four prescription descends independently of $c$. The degree-two scalar maps descend for every real $c$.

Proposition prop:newdirect and Theorem thm:oldgluing show that the complete declared domain has no overlap imposing an additional scalar equation. Thus gluing holds for every real $c$.

The parameter is free only within the inherited parent family. No conclusion is drawn about other normalizations or larger domains.

For $d\in D_{\mathrm{decl}}$, set

Extend this map $\mathbb{C}[\hslash ]/({\hslash }^4)$-linearly to

A pair of formal functionals is admissible when its formal tensor lies in ${\hat{D}}_{\mathrm{decl}}$. On such a pair, the partial product is obtained by this linearized map.

This formulation does not assert that every pair of elements of ${\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$ is admissible, or that the declared domain is closed under every intermediate multiplication.

The product of Definition def:product satisfies C-I1–C-I9 on its declared sector, with the continuity assertions proved in Section sec:continuity.

Proof. For C-I1, every new positive coefficient is a finite sum of inherited admitted graphs or an admitted constant. Classical closure handles $C_0$; old coefficients are inherited. All required admission descendants are included in those statements.

C-I2 is the definition of $C_0$. C-I3 follows from the unchanged restriction

the simultaneous-gluing proof shows this restriction is well defined. C-I4 is the first-stage coefficient assignment.

For C-I5, the only new scalar coefficients are the inherited maps with kernel $U_c$, using the one parameter fixed at the start. Their observation compatibility is recorded below. No new parent is selected during descent or gluing.

C-I6 holds because no old coefficient is changed. C-I7 is respected because no convergence of unchanged sharp approximants is invoked. C-I8 follows from descent to actual spaces and unique gluing on the full declared domain. C-I9 is satisfied by the precisely delimited fixed-support continuity theorem; no actual-stage continuity is asserted.

Write

Both formal intermediate tensors

belong to ${\hat{D}}_{\mathrm{decl}}$. Thus both associations now exist.

For every fixed real common parent $c$,

Their common coefficients are

Proof. The left association is

and the right association is