IX-C-ST: A supported partial product for one sharp ordered word Sequential local Grade S adaptations of IX-C, IX-Free, and IX-Free-2 Actual-functional descent, boundary-charge separation, and finite certificate transport on a prescribed cone

We adapt the sequential IX-C–IX-Free–IX-Free-2 programme to a causally saturated regular laboratory on the prescribed I-B conical spacetime. The three gates are retained in order: first-stage products and exact descent criteria; canonical two-edge outputs and degree-two second-stage maps; and degree-four descent with simultaneous overlap agreement. The carrier is the already adapted IX-A-ST space of actual off-shell polynomial functionals. The covariance and the VII-D triangle finite part are supplied chart data.

For the selected word $B_f,W_g,B_{f^{\prime }}$, we prove injectivity of both degree-two and both degree-four presentation maps. A chart-local advanced-trace witness separates every nonzero sharp generator from the partial contractions and the old microcausal algebra. An anchored tensor-domain argument then eliminates the old overlap of the entire new-domain sum, not merely its pairwise intersections. Consequently every fixed real common parent parameter passes both scalar descent and gluing: ${\mathcal{C}}_{\mathrm{desc}}={\mathcal{C}}_{\mathrm{glue}}=\mathbb{R}$. The prescribed coefficients define one truncated partial product modulo ${\hslash }^4$, with presentation independence, reversal, and coherence of both associations of this word through order ${\hslash }^3$.

The potential value of the result is a controlled example in which actual-functional factorization is achieved without assuming continuity of unchanged sharp approximants or altering the old product. The algebraic findings are adapted from the cited Cheung manuscripts; the added construction is their supported conical transfer and a complete finite certificate interface, including destructive-clock readout and reference-preserving causal transport. No global product, actual-stage continuity, causal lift, interacting Ward identity, or spacetime emergence follows. The inherited ultraviolet analysis remains an explicit premise and is not re-proved.

and exact release boundary{ The explicit request for one combined .tex file is followed. Parts I–III retain the ordered stages IX-C-ST, IX-Free-ST, and IX-Free-2-ST. A later part uses only results already established in the earlier part and the shared stated inputs.

Part I ends with an unresolved factorization gate. Part II advances that gate but still releases neither complete association. Part III constructs the declared truncated partial product. These intermediate verdicts are not contradictory and are not retrospectively erased.

This is not IX-D-ST, IX-E-ST, or a unified IX-ST. No registry modification is asserted.}

The selected programme concerns the exact off-shell insertion $B_f$, a quadratic metric observable $W_g$, and one ordered three-factor word. Its source papers have distinct releases: IX-C gives criteria; IX-Free constructs the degree-two blocks; IX-Free-2 completes the declared descent and gluing problem IXC,IXFree,IXFreeTwo. A numerical three-test pairing is not, by itself, a binary map on actual intermediate functionals. That distinction determines the order of the proofs below.

The operational background is Enter's distinction between a consumed quantum input name, a classical readout, and an unmeasured residual EnterArticle,EnterI. The philosophical essay emphasizes that an output record is not the return of an unknown quantum state and that a finite interface does not identify its mathematical labels with a gravitational object EnterPhilosophy. We use finite Born instruments with explicitly proved complete channel identities; no new measurement-uniqueness claim is made.

Spacetime Enter adds locations, calibration, complete channels, and declared causal routes to a prescribed spacetime. The clock is a finite encoding. Its destructive readout does not remove the geometric time coordinate or derive the metric IB,En09. The finite operational layer in Section sec:operational is independent of the distributional factorization proofs.

A local Grade S package consists here of:

1. a prescribed spacetime and a supported regular laboratory;

2. mathematical objects with stated domains and explicit inputs;

3. complete finite channels, including arbitrary untouched references;

4. one destructive clock with a declared calibration;

5. computational support bounds and a separate physical causal-containment certificate;

6. one nonempty finite transport realization with resources and durations accounted for.

Grade E, meaning a derivation of spacetime or its metric from records or quantum data, is not constructed.

discipline. The carrier, covariance, old product, and parent finite part are the explicit inputs in Section sec:inputs. Their cited proofs are not independently certified here. In contrast, the presentation injectivity, graph-specific leaf arguments, local charge witness, and algebraic gluing are proved on the supported chart below. A documentary separation is not a theorem premise.

boundary. The main algebraic mechanism is attributed to Cheung's IX-C, IX-Free, and IX-Free-2. This manuscript does not claim priority for those results. Its contribution is a sequential, supported adaptation with a common host contract and a complete certificate interface. The latter does not strengthen a partial product into an interacting field theory.

Fix a deficit $0<{\delta }_{\mathrm{c}}<2\pi$, set

and fix a conversion speed $c_{\mathrm{light}}>0$. The regular host is the prescribed product

with $\phi$ periodic of period $2\pi$. The stipulated completion has defect

It is a worldsheet, not a worldline. The lapse in $T$ is one; static proper time is $T$. These data are prescribed, as in I-B IB.

The regular product (eq:host) is not globally hyperbolic.

Proof. Fix a regular spatial point of radius $r_{\ast }>0$. Take two events there separated in $T$ by $4r_{\ast }/c_{\mathrm{light}}$. At their middle time, points on the same radial ray with radii tending to zero lie in the causal diamond between the two events. They have no convergent subsequence in the regular product. The diamond is not compact.

Choose a product-adapted isometric developing embedding

with developed metric $\eta =\mathrm{diag}(-1,1,1,1)$. Choose

Observation tests belong to

For each such test $f$, the whole truncated developed past

is included in $K_{\ast }$. The compact laboratory contains a neighbourhood of $\iota (K_{\ast })$, and its closure is disjoint from $\mathcal{W}$.

The response has zero value and zero first $t$-derivative at $t=0$. No additional incoming homogeneous or lateral boundary data are supplied. All compact test families have a positive smoothing margin inside the supported presentation.

The spacetime host is never identified with ${\mathbb{R}}^{1+3}$. An auxiliary flat extension can be used to write comparison formulae, but only the supported data and pairings are transported to the host.

The IX-A-ST zero-Cauchy response on $V$ is independent of auxiliary field extensions that agree near $K_{\ast }$. The same applies to its fixed smooth derivatives and compact observation pairings.

For a smooth vector field $v$ compactly supported in $V$, the supported prescription satisfies

This identity constructs no global conical inverse.

Proof. The difference of two extensions has zero source on the complete retained past and zero initial response data. The local uniqueness and finite-propagation argument of IX-A-ST therefore gives zero response on $V$ IXAST ({:saturation}). For (eq:compactidentity), $v$ itself is a zero-data solution with source $\square v$. The same local uniqueness identifies it with the prescribed response. Only a supported Cauchy statement is used; the global Green-operator hypotheses are different BGP.

Changing the regulator label, subtraction scale, stage label, or parent parameter $c$ changes a description register. It leaves the host, lapse, initial surface, worldsheet, and installed causal relation unchanged.

Proof. Those geometric objects are fixed inputs of Definition def:support. No metric-update map is an output of a catalogue command.

This is the fixed-host separation used in VIII-ST STInterfaces.

Configurations $\Psi =(h,\varphi )$ and configuration directions are smooth and off shell on the supported presentation. Metric configurations are symmetric covariant tensors. All compact coefficient kernels are understood with their actual tensor symmetries and retained boundary terms.

Set

The notation ${\mathsf{E}}_0$ refers only to the supplied supported comparison response.

Write

Distributional pairings are complex linear in their test. Complex tests give complex-valued functionals on real configurations; complexified directions are equivalent by polarization.

The carrier is the already adapted $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$ of IX-A-ST IXAST, not a reconstruction of the original global Minkowski carrier. It has the following properties.

1. Its elements are actual finite polynomial functionals with compact symmetric distributional coefficient kernels. The admission condition is

   $$({\mathrm{D}}_{\mathbf{\eta }}F{)}_h^{\prime }(\Psi )\in {\mathcal{S}}_1={\mathcal{M}}_1+{\mathcal{Q}}_{\Sigma }$$

   for every smooth configuration and every finite list of smooth directions.

2. The sharp linear kernels and intrinsic trace are

   $$q_a=L^{\ast }a=C^{\ast }(\theta {G_{\mathrm{A}}}^{\mathrm{chart}}a),{\mathcal{Q}}_{\Sigma }=\{q_a:a\in \mathcal{D}(V;T^{\ast }U{)}_{\mathbb{C}}\},$$
   $$\mathfrak{b}(m+q_a)=({G_{\mathrm{A}}}^{\mathrm{chart}}a){|}_{\Sigma }.$$

   Here ${G_{\mathrm{A}}}^{\mathrm{chart}}$ is the supplied advanced chart comparison prescription restricted to the retained support. Elements of ${\mathcal{M}}_1$ have no nonzero nonspacelike wavefront covectors; compact smooth kernels belong to ${\mathcal{M}}_1$. The trace is well defined on actual linear kernels.

3. The sharp adjoint retains

   $$C^{\ast }(\theta b)=\theta C^{\ast }b+\delta (t)\mathcal{B}(b{|}_{\Sigma }).$$

   No surface term is dropped.

4. Classical pointwise multiplication, conjugation, and fixed smooth directional differentiation preserve ${\mathcal{P}}_{\Sigma }$. Classical multiplication is separately continuous. The Hausdorff topology ${\tau }_{\Sigma }$ records strong compact coefficient kernels and all intrinsic trace coordinates.

5. The generating maps $f\mapsto B_f$, $g\mapsto W_g$ are fixed-support continuous, and $B_f$, $W_g$, the selected mixed coefficient below, and the selected scalar-contracted self coefficient belong to ${\mathcal{P}}_{\Sigma }$. Their specified fixed-support families and trace coordinates are continuous.

6. For every nonzero real $f$, the unchanged regular approximants satisfy

   $$B_{f,\epsilon }/⟶B_f\text{in }{\tau }_{\Sigma }.$$

No quantum product on all of ${\mathcal{P}}_{\Sigma }$, no joint classical continuity, and no quantum hypocontinuity is included.

The intrinsic trace is called a charge below. It is not an electric charge, an energy, or a stress tensor.

The comparison covariance is

Use

and

These are supplied chart formulae, not covariances derived from the conical metric.

Retain the VII-C tensor identities

The nonzero mixed component with $h_{00}$ is

The reverse ordered mixed component is $\overline{K(x,y)}$; the contraction selects vector index $0$. The displayed infrared powers are locally integrable in three momentum dimensions. High-frequency integrals are distributional.

The full old compact polynomial microcausal algebra ${\mathcal{P}}_{\mathrm{mc}}$ is included faithfully in ${\mathcal{P}}_{\Sigma }$. Its unchanged coefficients are

with ${C_0}^H(F,G)=FG$, supplied by

The old convention is Hermitian:

The microcausal wavefront condition excludes joint covectors entirely in either closed causal cone, including zero individual covectors in those cones.

These inputs retain the VII-C and IX-A-ST domain distinctions VIIC,IXAST; the functional microcausal convention is the one reviewed in FR. In particular $H_{ww}=0$ does not imply $K=0$.

The selected already admitted comparison outputs are

Their admission is the explicit selected-output input in Assumption ass:carrier, not closure under an unspecified new contraction.

Fix the radial regulator $\rho$, its prescribed placement on the two original field slots of each outer insertion, and $\mu >0$. The middle $W_g$ is unsmoothed. For that placement,

where $D_{\epsilon }$ is the corresponding smoothed $S_{00}$ comparison block. There are ${u_{\mu }}^{\rho },r\in {\mathcal{D}}^{\prime }(X)$ with

and, for every compact $A⋐X$,

The estimate holds on arbitrary joint tests supported in $A$, including those crossing $\mathcal{R}\cap \{x=x^{\prime }\}$. The allowed positive widths are also restricted by the supported smoothing margin.

The residue is the positive parameter measure

restricted to $X$, where

Its support is the reflected observation relation $\mathcal{R}$, not an observation equality diagonal. For allowed regulator changes,

with no extra normal derivative, nonconstant weight, or independent intersection term in that parent conversion. Also

The parent distributions have dagger symmetry under conjugation and exchange of $x,x^{\prime }$.

This is the stated VII-D input, as used by IX-B VIID,IXB. The supplied IX-B-ST draft is the preferred supported presentation of the same calibration, but no public registration of that draft is assumed. Neither its existence nor this paper re-proves the ultraviolet ledger.

Fix one real number $c$ for an entire mathematical construction and put

All logarithms use fixed units, so their arguments and scale ratios are dimensionless. One uses the same $c$ for all presentations, both associations, and declared numerical observation descendants.

For a fixed smooth multiplier $a$ and a differential operator $D$ of order $d$ on $X$, the parent finite part transfers to $aD{u}_{\mu }^{\rho }$, with residue $aDr$ and test order $64+d$. Conversion of $U_c$ becomes $caDr$, with the same $c$.

Proof. Transpose $aD$ onto the test in (eq:parentestimate). For a common compact support,

The constant is bounded by a finite polynomial in the $C^{64+d}$-seminorms of $a$ and the coefficients of $D$. The resulting bound tends to zero uniformly on bounded joint-test sets. The conversion assertion is linearity.

This is numerical observation differentiation, not configuration differentiation, exposed functional-slot differentiation, host evolution, or a binary-product theorem.

The construction in this section may be instantiated with any one of the three stage tags

One finite experiment has one clock factor. A stage tag is classical programme data, not an additional geometric time coordinate.

Let $C\cong {\mathbb{C}}^{2^m}$, $m\ge 1$, have calibration basis $\{|{\tau }_k⟩\}$. Let $S$ contain the surviving finite registers, and define

The channel

is CPTP and satisfies

All identities hold with every untouched reference. The quantum clock names are consumed.

Proof. The recorded Kraus operators are $|k{⟩}_R\otimes D_k$. Their squared moduli sum to $I_{CS}$. Expansion in the calibration basis identifies the $k$-th output block with the $kk$-input block and proves both formulae and the record-discard identity. Tensoring those identities proves the reference extension. There is no quantum $C$-output.

The construction agrees with the I-B and En09-A interface IB,En09. Consumption concerns the clock encoding, not $T$, $t$, the lapse, or the prescribed cone.

If distinct calibrated values $t_k$ must all equal the time of one fixed event whenever their outcomes occur, at most one outcome has positive probability. The clock channel has no left inverse on all clock states.

Proof. Distinctness proves the first assertion. For the second, the clock superpositions $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with identical surviving input, give the same output in (eq:clock). They cannot both be recovered by a left inverse.

A nontrivial clock catalogue is therefore a programme/history calibration unless a separate time-observable contract is supplied.

A finite catalogue entry records:

Only finitely many entries are installed in one experiment. For an arbitrary real $c$, a finite entry may store its symbolic specification or a declared finite approximation. There is no finite-bit exact encoding of all arbitrary real numbers or all smooth functions.

Let $i$ be a classical catalogue index and let $P$ be an arbitrary finite payload. Pinch the catalogue in its designated basis and write

For installed values $v_{ki}\in [-1,1]$, set

and let $r(k,i)$ be the fixed report word. Define

The map (eq:tablechannel) is CPTP, reference preserving on its surviving factors, and has conditional sign mean $v_{ki}$. For the same labels, pinching, report words, and payload interface,

Proof. Kraus operators are

Completeness follows by summing first over $y$. This also proves the complete surviving-block statement.

For the distance, include a reference and use orthogonality of all recorded flags. The trace norm of the output difference is at most

The last inequality is trace-norm contractivity of pinching, which is an average of unitary conjugations, on arbitrary operators. A basis input at a maximizing index attains the bound.

For a supplied complex coefficient evaluation $z_{ki}$, one may install

If $|z_{ki}-{\hat{z}}_{ki}|\le E_{ki}$, then

For the parent numerical coefficient, an enclosure can use

when $e_T,e_r$ are supplied enclosures of the regular pairing and residue pairing. This is a triangle-inequality consequence of Lemma lem:transfer. No executed quadrature, field acquisition, charge measurement, or residue measurement is asserted.

For finite instruments $\{{\mathcal{A}}_w\}$, $\{{\mathcal{B}}_w\}$, record equality on all inputs is equivalent to ${\mathcal{A}}_w^{\ast }(I)={\mathcal{B}}_w^{\ast }(I)$. Equality on all recorded residual observables requires the stronger identities ${\mathcal{A}}_w^{\ast }(B)={\mathcal{B}}_w^{\ast }(B)$ for every declared residual observable $B$.

Equal records can coexist with residual trace-distance difference $1/2$.

Proof. Pull each output effect back to the input and use separation by density matrices. For the example, append the same one-outcome record to identity and computational dephasing. On $|+⟩⟨+|$, their residuals are $|+⟩⟨+|$ and $I/2$. Their difference has eigenvalues $1/2,-1/2$.

The complete-instrument distinction agrees with the VII-E, VII-F, and VIII comparison records STInterfaces. A finite table does not determine a residual source functional.

For a finite layered network of complete block channels, the backwards ancestor set $N_{s,t}(Y)$ of a joint output $Y$ satisfies

An intervention outside that set cannot affect the joint output. Reference-assisted comparison uses the joint ancestor–reference reduction.

If every dependency port is placed in the regular laboratory and every dependency edge has a supplied future-causal route, then scheduled ancestry is contained in the prescribed physical causal past.

Proof. A block adjoint is unital. A block disjoint from the current observable support acts on identity and changes nothing; a meeting block enlarges support only within that block. Reverse induction proves the inclusion, and testing output observables proves the intervention statement. Carry the untouched reference algebra throughout. Concatenating the supplied future-causal routes proves the last assertion.

Thus actual influence is bounded by the complete scheduled ancestry; equality is not assumed. A distributional contraction graph, or a parenthesization schedule, is not itself a physical port network En05,STInterfaces.

The regular laboratory permits declared finite preparations, local unitaries, classical processing and storage, destructive readout, and exact finite-register carriage along specified future-causal paths. These are ideal finite-channel permissions, not derived field dynamics or a microscopic Hamiltonian implementation.

Choose a developed spatial segment of length $L_{\ast }>0$, with a compact regular tube in the laboratory. Install $n+1$ stations along it, each with a $p$-qubit register, $p\ge 1$. Initialize all but the sender in a fixed blank. Read the clock and produce the complete finite report at the sender. Retain every outcome and wait for a positive buffer interval. Forward the classical certificate on explicit causal wires. At each hop exchange neighbouring payload contents by opposite carriage paths taking developed time $2L_{\ast }/n$, not by an instantaneous remote gate.

The route transmits the unknown finite payload exactly, including every untouched reference, and forwards its complete classical certificate history. Every non-output payload register ends in a payload-independent blank. All communication dependencies are causal in the supplied metric. Orthogonal sender payloads give a receiver trace-distance contrast one.

Proof. After hop $j$, the sender payload factor and all its reference correlations occupy station $j$; every other payload register is blank. Tensor-factor exchange proves this by induction. Copying an already classical history preserves its joint correlations and is not cloning an unknown quantum state. Each carriage speed is $1/2$ in developed units; local processing and buffering are timelike. The receiver controls are used only after their classical inputs arrive. For the contrast use $|0⟩⟨0|,|1⟩⟨1|$ on one payload qubit.

The transported payload is a finite register, not the continuum field. The unit signalling witness is an installed transport operation, not entanglement generated by the cone IB,En08.

If the report has $b$ bits, the catalogue $g$ bits, and lookup uses $w$ work bits, a conservative reserve is

basis-state qubit carriers, plus separately declared external reference storage. There are $n$ exchanges, $n$ history transmissions, at most $nb$ elementary history-copy operations, and the declared clock/probe readouts. All local processing has positive charged duration. Carriage takes $2L_{\ast }$; preprocessing, copying, and buffering are added to it before fixing the laboratory time window. No reset or thermodynamic cost theorem is claimed.

For compatible implementations on the same complete interfaces, telescoping gives

This follows by replacing one channel factor at a time and using diamond-norm contraction Watrous. It is not a norm bound for the continuum partial product.

-C-ST: first stages and exact criteria

The maps $f\mapsto B_f$, $g\mapsto W_g$ are injective.

Proof. Sharp-test injectivity is included in the adapted carrier input; it also follows from the stronger local separation proved in Lemma lem:restricted below. For $W_g$, polarization of $\int g{v}^2=0$ gives $\int guv=0$. Choose $v=1$ near $\mathrm{supp}g$; arbitrary compact $u$ then gives $g=0$.

The reverse first-stage coefficient is

It belongs to ${\mathcal{P}}_{\Sigma }$, and

Proof. The locked reverse covariance is the conjugate mixed kernel. The middle observable has two metric slots, giving factor $2$; the single sharp insertion supplies one minus sign. Conjugation of (eq:Q) gives the displayed reversal. Admission follows from the carrier's conjugation closure.

This retains IX-C's reverse normalization IXC ({:reverse}).

Define actual-functional subspaces

Let

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

The prescriptions

define bilinear maps on $\mathcal{B}\times \mathcal{W}$ and $\mathcal{W}\times \mathcal{B}$, with values in

They satisfy classical normalization, locked first-stage normalization, and reversal.

Proof. Injective generating tests make the displayed bilinear assignments unambiguous on the actual argument spaces. Classical closure and the admitted mixed outputs place their coefficients in ${\mathcal{P}}_{\Sigma }$. Proposition prop:reverse proves reversal.

On fixed compact observation supports, $(f,g)\mapsto Q_{f,g}$ and its reverse are jointly continuous into the strong coefficient topology. Their metric-derivative trace coordinates vanish, also after all required directional insertions. The first-stage generating families are continuous into $({{\mathcal{P}}_{\Sigma }}^{[3]},{\tau }_{\Sigma })$.

Proof. The coefficient of $Q$ is obtained by multiplying the fixed distribution $K$ by $f\otimes g$ and taking one fixed scalar-slot derivative. On common supports, finite-order distribution estimates bound this operation by finitely many smooth seminorms of $f,g$, uniformly on bounded smooth test sets.

Contracting its scalar slot with a smooth configuration leaves a compact smooth metric kernel, by the selected output property in the carrier input. Replacing the scalar configuration by a fixed scalar direction has the same property. A metric insertion removes the only metric slot. All higher derivatives vanish. Thus every relevant trace coordinate is zero. The classical first-stage term has the inherited generating test-family continuity. Combining its coordinates gives the last assertion.

This is continuity along specified test families, not hypocontinuity on arbitrary carrier stages IXC ({:firstcontinuity}).

For the selected tests set

All three outputs in (eq:oneedges) exist in ${\mathcal{P}}_{\Sigma }$, with all required directional descendants. They have field degree four.

Proof. Every factor is already admitted. Their products are classical pointwise products, so the carrier's classical closure applies. No new multiplication in singular variables is used.

At this stage the desired two-edge placements are only labels:

where $V_3=f\otimes g\otimes f^{\prime }$.

The three two-edge placements have insertion sign $+1$, assignment multiplicity $2$, and residual degree two. Their fully contracted completion has the parent coefficient ${T_{\epsilon }}^{\rho }$.

Proof. The only available nonzero edges are metric edges $12,23$ and scalar edge $13$. Each two-edge choice removes four of the six slots. Both outer insertion signs occur, giving $+1$. For $12+23$, the two middle slots have two assignments. For $12+13$, either middle slot is used, again giving two; the third case is its reverse. The tensor identity fixes index $0$ on each mixed metric edge. Full contraction adds the remaining edge and gives the supplied triangle.

This counts slots; it does not yet prove the products in (eq:twolab) exist IXC.

All spans and tensor products below are algebraic over $\mathbb{C}$. Let

and define

Let $v:E\to A$ be surjective and let $\lambda :E\otimes Z\to Y$ be linear. There is a unique bilinear map $\ell :A\times Z\to Y$ with

if and only if

For two presented arguments, the condition is vanishing on

Proof. Necessity follows by inserting a zero argument. For sufficiency, two presentations of the same argument differ by an element of the indicated kernel; the condition makes their assigned values equal. Linearity gives bilinearity and surjectivity gives uniqueness. Apply the argument to both slots for the last assertion.

This is the IX-C actual-functional criterion IXC ({:descent}). It is not a topological quotient theorem.

Define the scalar relation space

After the natural ordering of its three test factors, let $\alpha ,\beta$ be its pairings with ${u_{\mu }}^{\rho },r$.

The calibrated scalar assignments

descend exactly when

The permitted real set is empty, a singleton, or all of $\mathbb{R}$.

Proof. The descent criterion gives precisely those linear relation equations, using injectivity of the remaining sharp argument. Each equation is affine in $c$. If two distinct real values solve every equation, their difference forces $\beta =0$, and then $\alpha =0$; all real values solve. If there are not two solutions, the solution set is empty or a singleton.

Part I does not evaluate this test. Part II does.

Let ${\mathsf{L}}_j,{\mathsf{R}}_j$ denote proposed second-stage coefficient maps on the left and right intermediate spaces, with order zero classical. Write $P=B_f{W}_g$, $Q=Q_{f,g}$, $P^{\prime }=W_g{B}_{f^{\prime }}$, $Q^{\prime }={Q^R}_{g,f^{\prime }}$.

Assuming that the displayed second-stage maps exist on actual arguments and that the scalar coefficient is supplied only by the degree-two block, the two associations agree modulo ${\hslash }^4$ exactly when

Proof. Expand the first stage $P+\hslash Q$ and then its second multiplication, and do the same for $P^{\prime }+\hslash Q^{\prime }$. The order-zero equality is classical associativity. Equating the next three coefficients gives exactly (eq:assoc1)–(eq:assoc3).

The intended prescriptions are

Until existence, admission, descent, and overlap agreement are proved, these remain prescription equations IXC ({:coherence}).

If a coefficient prescription descends through every actual presentation kernel and glues consistently on its actual tensor domain, its value is independent of presentation. No such conclusion follows from agreement of formal labels alone.

Proof. The difference of two presentations is a relation annihilated by descent. Any further change of domain decomposition is annihilated by the gluing condition.

Relative to the shared inputs, Part I constructs both admitted first stages, their fixed-support continuity, the one-edge outputs, and exact descent, scalar, and coherence criteria. With Section sec:operational, it forms a local Grade S criteria package.

At this stage neither sharp second multiplication is constructed. Its factorization release is Unresolved, not a universal obstruction.

Proof. The claims are Theorem thm:first, Propositions prop:firstcontinuity, prop:oneedges, prop:placementcount, and the criteria of this section. The common operational layer does not use an unconstructed product. The graph products and relation-kernel discharges needed for a second multiplication have not been used in Part I.

-Free-ST: presentation kernels and degree-two maps

For $v={\sum }_i{a}_i{f}_i\otimes g_i\in E_L$, use the faithful algebraic embedding

Faithfulness follows by choosing linearly independent first factors and applying dual linear functionals on their finite span. No completed tensor product is substituted.

The exact relation equations are

Proof. Observation-variable integration by parts gives

There is no boundary term because $v$ is compactly supported inside $V^2$. The independent scalar and metric slots recover the coefficient on all product tests. Product-test density identifies the joint distribution. The reverse calculation is identical.

Let ${J^{\mu }}_{\alpha \beta }(x,z)$ be the supported kernel of $L$, and let ${e_{00}}^{\alpha \beta }{h}_{\alpha \beta }=h_{00}$. Define

Symmetrization is averaging over the three metric slots.

The kernel $\mathsf{P}[v]$ is defined and compact in its field variables, and

Consequently

The right relation is the exchanged equation with $v(x,y)=v_R(y,x)$.

Proof. Before smooth multiplication, $J(x,z_1)$ and $\delta (z_2-z_3)$ are tensor factors in disjoint variables. Observation integration by parts gives the positive derivative in (eq:Prelationkernel). Saturated finite propagation gives compact field support. One scalar derivative and three metric derivatives recover the symmetric kernel with factor $3!$. Polarization and product-test density prove the equivalence.

These are the concrete IX-Free relations IXFree ({:relations}, {:prelations}). Neither equation has yet been solved for $p_L,p_R$.

The restriction of (eq:K) to $V^2$ has

Proof. For $a=t-s$, distributional differentiation of (eq:K) gives

The terms without the explicit factor $t$ solve the homogeneous wave equation. The $t$-term yields the displayed derivative.

Away from $a=0$ and $|\mathbf{x}-\mathbf{y}{|}^2=a^2$, Abel damping of the radial Fourier integral gives

That set is dense in every open product chart. Thus ${\square }_{x}K$ has full support, and a differential operator cannot enlarge support. Hence $K$ has full support as well.

This is chart support, not a theorem about the defect IXFree ({:fullsupport}).

If $I$ is a connected open interval and $T\in {\mathcal{E}}^{\prime }(I\times N)$ satisfies ${\partial }_{t}T=0$, then $T=0$.

Proof. A compact test of integral zero on $I$ has a compact primitive. Thus $T$ depends on a time test only through its integral. Choose an integral-one test disjoint from the compact time projection of $\mathrm{supp}T$. Its pairing is zero, so every such constant-in-time pairing is zero. Product-test density gives the result.

Both $q_L:E_L\to {\mathcal{L}}_2$ and $q_R:E_R\to {\mathcal{R}}_2$ are algebraic isomorphisms onto their images.

Proof. Equation (eq:qrelation) and Lemma lem:primitive give $vK=0$. If $v$ were nonzero at a point, it would have a smooth reciprocal near that point, making $K=0$ on a nonempty open set. Lemma lem:fullsupport excludes this. Hence $v=0$, including as an algebraic tensor. Conjugation and variable exchange give the right case.

For this sector,

For every fixed real $c$, both calibrated scalar maps in Theorem thm:scalarfactor exist uniquely as algebraic maps on the actual degree-two intermediate blocks.

Proof. Both relation kernels vanish by Theorem thm:qinjective. The actual sharp argument has its unique observation test. Thus the parent pairing descends, and constants belong to ${\mathcal{P}}_{\Sigma }$.

This evaluates descent only. The overlap problem remains separate IXFree ({:cdesc}).

Every wavefront covector of $K,\overline{K}$, or $S_{\mu \nu }$ has nonzero covectors in both endpoint slots. Their coordinate Euclidean norms are equal.

Proof. Split each Fourier integral into bounded and unbounded momenta. The bounded-momentum part is smooth, since every coordinate derivative remains integrable there.

The two high-frequency mixed phases have endpoint covectors