Exact powered-denominator lifting and factorized EMR assignment

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2026-09-17 · 869a01c107a795a6b758b2c94134360e2d0a92d8

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This document describes the production design used when a completed local 4D term is projected to a causal-flow representation (CFF) after UV Taylor operations have raised one or more propagators. It expands two tightly coupled requirements:

1. carry each original source_edge through raw UV Taylor recursion, then create a separate canonical projection view whose denominator classes retain an exact component binding and a physical source-incidence witness; and
2. rewrite the still-factorized numerator over those exact occurrences, then select the fewest native generated source-map rows among at most three certified proposals, with rank supplying proposal order and count tie-breaks.

This exact-source reconstruction and bounded-dispatch machinery is exclusive to the projected local-4D route. Both direct local-3D representations instead complete the loop-energy integration first and apply every UV Taylor operator to the complete/global CFF expression. Their bodies are identical; the explicit-sum form only omits the localized form’s orientation selectors.

The canonical algebra boundary is implemented. The policy below specifies its invariants. Owner-provenance regression examples below still exercise raw physical inputs. Their original-owner expectations remain valid at that boundary; the completed class projection adds an explicit certificate before transferring ownership and does not change those tests’ inputs.

The two requirements cannot be solved independently. A denominator power first creates the occurrence-local energy namespace in which the generalized residue problem is posed. The numerator then has to use exactly that namespace, and the same factor-to-occurrence assignment must control both CFF generation and numerical numerator evaluation.

This is GammaLoop production machinery. It is not a requirement that the diagnostic 3Drep CLI prepare its inputs or expressions in the same way. It is also deliberately independent of LTD; no LMB variable is ever used as an energy-identity or energy-capacity index here.

Equality is defined for the complete fixed-cut functional

The direct local-3D and projected local-4D constructions need not agree on an individual generalized residue-map entry or an individual raised-order slot. Their authoritative equality is

for each fixed physical Cutkosky cut $𝐶$, after both sides have summed all generalized residue maps and combined all derivative pieces from every raised order required by that cut. The number, labeling, and individual values of the intermediate entries may differ. They are coordinates of a residue decomposition, not observables.

For example, a double-pole result can be represented schematically by

For any regular $ℎ$, the change

(with the second $ℎ$ evaluated at the expansion point) changes the two stored pieces but leaves ${\mathrm{ℛ︀}}_{𝐶}$ exactly invariant. Higher raised powers have the corresponding triangular freedom between adjacent derivative slots. A test which pairs lu_cut_order values across two construction routes therefore imposes a nonphysical canonicalization unless that particular fixture independently proves the stronger identity. The safe comparison sums first, with the Cutkosky cut held fixed.

This does not weaken orientation-local UV subtraction in the direct-3D route. Each direct residue-key selector must still give its required local UV behavior. It only means that projected local-4D, which is defined after an explicit residue sum, has no canonical one-to-one key or raised-order pairing with that selector-local decomposition.

Factorized staged map composition

Projection integrates disconnected or nested Taylor components in stages. A state is retained as a CFF carrier times a still-factorized numerator, rather than as one expanded polynomial. At component $𝑎$, its exact source map acts only on the numerator factors owned by $𝑎$; the resulting carrier/numerator pair is passed unchanged to component $𝑎+1$. After all UV components have been integrated, the outer CFF map acts once on the still-unmapped soft and cograph factors. Symbolically, the implementation realizes products of mapped factors and sums of CFF carriers without expanding the full numerator.

Genuinely independent unsummed component residue states form the mathematical Cartesian product of their sums. A child coefficient which has already summed its source-local residues is stored once, together with its frozen localizing factor. Different Taylor topologies may have child maps of different lengths; those maps are consumed inside their own sums and are never paired with outer maps. Each actual outer CFF branch maps the completed coefficient times the remaining numerator once, then multiplies the frozen factor and contributes to the checked cut-indexed sum. No artificial child production hosts or selector copies are constructed. This distinction preserves both the complete fixed-cut functional and the factorized numerator representation.

Executive summary

For a term

the production projection performs the following operations:

1. A negative integer power den(...)^-r becomes r denominator occurrences. The numerator is not expanded.
2. Every occurrence retains its source_edge. Contracted cograph and UV source minors use those IDs to provide the already known skeleton, topology domain, and external attachment. Matching momentum signatures never solve an inverse incidence or Kirchhoff problem.
3. The r occurrences of a rewritten denominator retain a source-backed occurrence graph. A raised source wrapper is realized by serial subdivision, with r-1 auxiliary two-valent vertices; a powered self-loop becomes an r-edge cycle. Equal channels may share a powered quotient only when their serial-path or pure-cycle incidence is certified. Nonadjacent equal channels retain their original incidence while sharing one algebraic class.
4. Exact signatures, masses, full denominator polynomials and component domains are canonicalized into algebraic channels. This is the safe D(Q)=D(-Q) equivalence used by repeated-channel algebra. Source minors seed the construction, but relabeling algebraically identical occurrences with another compatible physical owner cannot change the CFF residue or loop rank.
5. Physical owner IDs survive separately for occurrence-energy provenance, physical-surface projection, diagnostics, and mapping back to the parent graph. Cut support is the union of all physical owners which instantiate an algebraic channel, with raised-line representatives added without discarding that union.
6. The completed hard numerator is analyzed in certified class variables; non-UV factors retain their physical EMR variables. Neither is expanded. Addition takes a maximum degree, multiplication and multilinear slots add degrees, and a nonnegative integer power repeats a factorized base.
7. Exact denominator occurrences in one certified signed class form a candidate set. Distinct masses, pole families and component domains keep separate sets.
8. Completed hard roles zero and one, including original numerator factors, may use their class’s candidate set. Physical and soft roles retain their owner restrictions. Cyclic/greedy allocation proposes a baseline assignment and at most two deterministic alternatives. The proposal producing fewer native CFF map rows wins; descending degree envelopes and stable proposal order break count ties.
9. One immutable assignment plan owns both the bounds passed to generalized CFF and the substitutions used later to evaluate the numerator.
10. If an occurrence has momentum -Q rather than Q, that literal algebraic sign is retained by the numerator mapper. It is composed with any later coherent reversal of the complete rational-routing component by comparing the original exact signature with the final parsed occurrence signature. The resulting sign converts the occurrence energy back to the physical numerator convention; denominator evenness never causes the numerator to be treated as even.
11. Once incidence, power subdivisions, and non-vacuum source-crown hedges are complete, Graphica canonically relabels the skeleton with internal propagators treated as undirected and external crown incidences left directed. Connected rational-routing components then receive one deterministic coherent direction. This supplies deterministic equality and cache keys without inferring topology.
12. Each additive Taylor term prepares its proposals once. Counts and generated payloads use the same canonical topology/options/per-occurrence-capacity key. Only winning payloads enter the full cache; a losing contender leaves a count-only memo. Each term retains its selected immutable numerator plan.

The resulting graph is therefore a source-backed occurrence representation of the rewritten rational function. Provenance is not discarded: it supplies the source-minor scaffold and all maps back to the physical graph. It is also not an extra argument of the rational function. Once algebraically identical occurrences have been lifted, changing only their compatible owner labels must not change their residue, loop rank, or owner-free CFF cache identity.

Terminology and identity layers

Several IDs coexist in this path. Conflating signature equivalence with physical incidence, or occurrence-local capacity with physical EMR ownership, produces incorrect topology or numerator sampling.

An occurrence-local exact energy is not a new physical momentum. If a double propagator is represented by exact energies E_a and E_b, both can be certified representations of the same physical energy E_5. They remain distinct variables only because a generalized residue calculation needs one energy slot per denominator occurrence.

Why UV processing creates this problem

The unexpanded graph normally has one denominator wrapper per propagator. UV Taylor operations differentiate and expand rational factors. Even if the input topology contains only simple propagators, a completed local 4D term can contain

den(owner, P, m2, D(P))^-2

or a product such as

den(owner_a, P,  m2, D(P))^-1
den(owner_b, -P, m2, D(P))^-2.

At that point, using the original graph unchanged is insufficient: the energy integral has a double or triple pole, whereas an ordinary graph edge encodes only one denominator occurrence. It is wrong to discard the source edge and try to reconstruct a graph from the rewritten signatures: the UV/cograph minors and their attachment to the unshrunk graph are already known from the original graph. It is equally wrong to promote that retained label into algebra. Two occurrences with the same domain, mass, and denominator D(Q)=D(-Q) represent the same rational channel even when they came from different physical edges. A compatible owner relabeling may change physical provenance, but cannot change the contour residue or the number of active loops.

The numerator poses a second problem. UV differentiation can simultaneously produce higher powers of temporal momenta. For example, the rational term may contain

After creating two occurrence energies, the numerator can be evaluated through either occurrence. Choosing just the first is algebraically possible on the physical diagonal, but it supplies unnecessarily high generalized-CFF bounds and can make CFF generation much more expensive. Worse, advertising balanced bounds while evaluating all factors through the first occurrence would be incorrect. This is why structural lifting and numerator assignment are one coupled operation.

Production data flow

raw local 4D sectors, retaining physical owners for Taylor recursion
  |
  | group compatible sectors; exact cancellation returns typed zero
  v
canonical projection view: factorized numerator + powered denominator classes
  |
  | source-minor lookup plus independent signed-channel certificate
  v
known UV/cograph skeleton and occurrence channels keyed by
  (component domain, signature up to sign, mass^2, full polynomial)
  |
  | source-backed attachment + owner-independent occurrence lift
  v
source-backed occurrence graph
  |
  | explicit source-crown completion for non-vacuum boundaries
  | post-construction Graphica canonical relabeling
  v
canonical temporary exact ParsedGraph
  |                         physical factorized numerator
  |                                      |
  |                                      | EMR degree analysis
  |                                      v
  |                         physical edge -> physical degree
  |                                      |
  | exact literal +/-Q candidates        |
  +--------------------+-----------------+
                       |
                       | cyclic/greedy proposals, K <= 3
                       v
            exact occurrence -> energy bound
            factor -> exact occurrence assignment
                       |
                       | generate/lookup actual native map counts
                       v
             chosen payload + matching immutable plan
                       |
             +---------+---------+
             |                   |
             v                   v
      selected CFF          exact numerator mapper
      payload               used by every orientation,
                            residue, and contact sector

Each candidate plan is immutable before its generalized CFF request. Selection retains the winning generated payload with that same plan and exact-source numerator mapper. This is the certification boundary: the expression eventually sampled is the expression whose bounds were supplied to CFF.

Projection prepares each canonical exact source with its initial immutable plans; selection derives the final placement challenger from the best native score. It reuses an expression only when canonical topology, occurrence-local capacity and generation options match, and maps each term with its selected plan. Equal physical energies do not allow the cache to redistribute their individual bounds. Independent requests never combine into a larger Cartesian capacity, so no preliminary registration pass is needed.

Non-negotiable common-LMB commutative reconstruction invariant

The projected local-4D route is not required to preserve an individual CFF orientation. It is therefore free to choose any convenient EMR spelling of a completed Taylor numerator on the reconstructed factorized UV graph. That freedom does not make the rewrite heuristic. Every chosen spelling must pass the following exact commutative check before its generalized-CFF image is trusted.

Let the Taylor operator be defined in the compatible hard sub-LMB $\ell =({\ell }_1,\dots ,{\ell }_{𝐿})$. Provenance attached before differentiation retains, for every original edge $𝑒$, both its immutable owner and its hard momentum

After taking the requested Taylor coefficient, let its still-factorized numerator be ${𝑁}_{𝑇}(\left\{{𝐻}_{𝑒}\right\})$. Keep this raw sector unchanged for outer Taylor recursion. Its projection copy replaces completed hard roles zero and one with Q(uv::class(id), index) and an explicit routing sign. Each class binds one formal momentum channel; a channel such as ${𝐾}_1-{𝐾}_0$ is not expanded into two independent allocation families. Denominator polynomials alone may be expanded to certify equality in the common frame.

Rational-shell multiset buckets sum numerators without distributing their factors. Positive den factors remain indivisible numerator blocks with their full polynomial. A bucket retains one deterministic physical incidence witness; merging equal buckets never concatenates their denominator lists. A hard factor without an unambiguous surviving class stays a fixed affine-frame carrier. Sectors combine only when their actual component and frozen bindings agree. Grouping precedes the empty-result check: same-frame sectors +N and -N produce the existing typed zero, including every final cut order, without requiring energy maps. Cancellation cannot remove a frozen localizer or a remaining component request from a different binding. The raw recursive sectors remain available unchanged for an enclosing Taylor operation.

Build the UV skeleton from the retained witness; never infer its incidence from a signature matrix. A differentiated denominator may add serial copies of its own line. Any further quotient contraction requires a serial-path or pure-cycle incidence certificate; nonserial equal classes preserve incidence. Choose a deterministic LMB $𝜅=({𝜅}_1,\dots ,{𝜅}_{𝐿})$ for this reconstructed graph and write each of its EMRs as

For an MSbar vacuum counterterm the fixed-external part is absent, ${𝐶}_{𝑎}=0$, in a centered hard chart. A compatible chart built from retained physical edge momenta may instead represent the same vacuum integral by a balanced affine loop translation. Their signed incidence sum must vanish at every UV vertex; they do not represent momentum entering the vacuum graph. The exact source certificate retains them on both sides rather than rejecting each shifted denominator separately. A future on-shell counterterm may retain a fixed insertion such as $(𝑚,0,0,0)$; it is simply included on both sides of the same check.

For nested Taylor operators, the enclosing UV node fixes the hard/soft split. If a compatible carrier $𝑄$ has soft part $𝑆$ in that node’s reference LMB, its rescaling in the projected route’s inverse-hard scale is $𝑄\mapsto (𝑄-𝑆)/𝑡+𝑆$. The same affine transport applies to a retained child’s hard-momentum payload; its owner and provenance role remain fixed. Literal external-coordinate carriers stay soft even when their out-of-domain LMB rows vanish. At the vacuum-integration boundary these external coordinates must also stay fixed when solving for canonical loop momenta. Finite MUV integration retains one Laurent term beyond the maximal pole order; the legacy forest owner requests wood.max_loops + 1, matching its finite-term extraction contract.

Vacuum masses within the component under Taylor expansion retain the hard rescaling mUV -> mUV/t. Direct local-3D Taylor expansion applies the equivalent forward scaling mUV -> lambda*mUV to an integrated coefficient only when the current subgraph contains its owner. Disjoint owners remain fixed; a later enclosing operation can scale them. The direct route tags each connected coefficient’s mass as transient mUV(owner) before multiplying coefficients, using the existing symbol and subgraph encoding. Output copies restore physical mUV, while stored sectors retain ownership for later Taylor operations. The separate normalized localization kernel stays frozen throughout.

Four-dimensional and IntegratedCts runtime behavior is unchanged. IntegratedCts projects the coefficient as C(mUV) * s^(4*r), where s = uvIntegratedLoopScale and r counts consumed loops. The four-dimensional s -> s*t transformation restores only the consumed-measure power; it does not cancel the coefficient’s mass rescaling in an enclosing operation. Integration and physical projection set s = 1. The active denominator rewrite introduces the enclosing vacuum mass independently. See Signs and Integrated Projections for the integrated-owner and frozen-kernel distinction.

The mapper proposes a factorized UV-graph numerator $\widehat{𝑁}(\left\{{\widehat{𝑄}}_{𝑎}\right\})$. Correctness means

as an exact symbolic identity in one common set of formal loop four-vectors, masses, and fixed external data. Tests may erase provenance tags for this comparison, but must preserve the graph numerator’s factors, including in diagnostic copies. Use exact factor cancellation and normalize scalar CFF denominators before inserting the numerator. The separate analytic UV-subgraph expansion boundary is described in UV renormalization.

This check has four important consequences:

1. Raw recursive sectors retain every original numerator owner and hard payload. The projection copy transfers completed hard factors to a class only after exact signed-frame normalization, preserving the numerator’s factorization.
2. All eligible hard factors of one certified class may use its occurrences. Physical-source factors and new soft Taylor factors do not acquire this freedom; distinct masses, pole families and domains stay separate.
3. Such an assignment is correct before it is optimal. A deterministic first valid copy is sufficient; cost-driven redistribution is allowed only after the boxed identity passes and only among occurrences of the certified class.
4. Canonicalizing $𝐷(𝑄)$ as $𝐷(-𝑄)$ cannot discard an odd-numerator sign. The complete signed coordinate map from the tagged hard momentum to the selected UV occurrence participates in the boxed equality. Denominator evenness changes the denominator channel, never the numerator transformation.

New soft Taylor factors are outside this child-contour assignment. Their separate, exactly certified cograph routing occurs at the outer CFF boundary described in Stage 6; it never moves an original or child hard factor.

The LMB has a deliberately limited role here. It is a common coordinate chart used to prove that two EMR expressions denote the same function. It is not an energy identity, an occurrence owner, a CFF bound label, or a fallback when provenance is missing. Once the exact identity is certified, only the candidate UV-graph EMR expression and its physical/occurrence metadata are passed to generalized CFF.

The denominator side has a parallel certificate: after substituting the same common coordinates, the reconstructed occurrence multiset must reproduce the Taylor term’s denominator momenta, masses, powers, and topology domains exactly (up to the safe even relation $𝐷(𝑄)=𝐷(-𝑄)$). Numerator equality and denominator equality are checked separately so denominator evenness cannot hide a numerator sign error.

This commutative identity is the first diagnostic boundary for every projected local-4D mismatch. If it fails, the defect is in provenance preservation, owner-backed skeleton construction, occurrence assignment, or signed EMR rewriting. If it passes but the residue-summed result differs from direct local-3D UV, the defect is downstream—in CFF input normalization, prefactor or component composition, or residue aggregation. Rank-envelope optimization must never be used to repair a failed identity.

Worked diagnostic: GL04 temporal-square T0 sector

The minimal GL04 acceptance probe localizes the original numerator on edge 5,

With integrated and threshold counterterms disabled, the first differing forest contains the Taylor-vacuum components labelled 1zs and 8Ok. Certifying its derivative-free outer 8Ok/T0 sector first cleanly separates identity reconstruction from the derivative-bearing 1zs/T2 sector. The typed local-4D T0 sector entering projection reports exactly

component 1zs:
  owner 5, momentum  Q5
  owner 6, momentum -Q5

component 8Ok:
  owner 2, momentum  Q2
  owner 1, momentum -Q2
  owner 7, momentum -Q2

numerator:
  i*g^5*Q(provenance(owner=5, role=fixed, hard=Q5), 0)^2

Every occurrence has the Taylor-vacuum mass and no external shift. Introduce common formal four-vector carriers

Erasing only the test copy of the provenance metadata, the authoritative post-T rational function becomes

The owner-built reconstructed UV graph has the same five occurrences and the same deterministic component LMBs. Because T0 generated no new numerator factor, the only admissible candidate is the unchanged fixed-owner spelling

Substituting the reconstructed graph LMB therefore gives

exactly. There is no minimax choice in this comparison: the original fixed factor stays on owner 5, and all five denominator powers are one. At the inner CFF residue, the actual source map sends

so both ordinary pole branches map the numerator to ${𝐸}_5^2$, while the zero-sampling contact maps it to zero. The outer component leaves that already sampled factor untouched.

An independent diagnostic using the production generalized-CFF engine then compared the disconnected five-denominator source with the product of its separately embedded three- and two-denominator components. The combined source had $6\times 5=30$ branches—six scalar three-denominator orientations times the five quadratic two-denominator key families contact:+, contact:-, contact:0, remainder:+, and remainder:-. Its total agreed with the separately embedded component product at the tested points to approximately ${10}^{-16}$.

This worked sector therefore passes the pre-CFF common-LMB certificate, the generalized-CFF component-product diagnostic, and a static trace of the GammaLoop host/prefactor/outer-factor coefficient. It is not the first unequal boundary in the enclosing forest. The isolated 1zs/T2 comparison below locates the remaining discrepancy. Changing this fixed numerator assignment would violate the exact certificate and cannot be a valid repair.

Worked live reproducer: GL04 temporal-square 1zs/T2

The full-graph generation LMB used by the minimal reproducer contains

For the 1zs Taylor component, set

The frozen reconstruction fixture represents the complete truncation T0+T1+T2, rather than the isolated T2 coefficient. Its exact denominator occurrence multiset is

owner 5: +q, +q       (production occurrence ids 9 and 12)
owner 6: -q, -q, -q   (production occurrence ids 10, 11, and 13)

and the still-factorized numerator is algebraically

This is the authoritative full-truncation left-hand side. It is obtained before any CFF call, and no numerator/denominator cancellation is performed or needed. The original factor $({𝑞}^0{)}^2$ retains TaylorFixed(owner=5, hard=+q) provenance. The occurrences beyond the first copy of each owner and every factor generated by differentiating a denominator retain DenominatorDerived provenance.

To perform the actual reconstruction certificate, introduce neutral proof coordinates rather than treating occurrence ids as independent physical energies. Let every parsed occurrence have a formal momentum ${𝑃}_{𝑗}$. The source records two signs for each selected occurrence:

where $𝐻$ is the immutable hard momentum, $𝑅$ is the raw rewritten denominator momentum, and ${𝑃}_{𝑗}$ is the canonical parsed momentum. In this reproducer the selected owner-5 occurrences have $ℎ=𝑟=+1$, while the owner-6 occurrences have $𝐻=𝑅=-𝑞$, ${𝑃}_{𝑗}=-𝑅=+𝑞$, hence $ℎ=+1,𝑟=-1$. Consequently the production mapper must give

$$ \begin{aligned} (Q_5^0)^2_{\rm fixed} &\longmapsto (P_9^0)^2,\\ D_5(q) &\longmapsto (P_{12}^0)^2-\boldsymbol P_{12}^{2}-U,\\ D_6(-q) &\longmapsto (P_j^0)^2-\boldsymbol P_j^{2}-U, &&j\in\{10,11,13\},\\ p\mathbin{\cdot}(-q) &\longmapsto p\mathbin{\cdot}(-P_j) =-p^0P_j^0+\boldsymbol p\mathbin{\cdot}\boldsymbol P_j. \end{aligned} $$

Only now impose the common reconstructed-graph LMB chart,

Every mapped ${𝐷}_5$ becomes $𝐴$, every mapped ${𝐷}_6$ becomes $𝐵$, every mapped dot becomes $𝑥$, and the candidate numerator becomes

That last zero-difference identity is the required numerator certificate. The parallel denominator certificate is the multiset ${𝐴}^2{𝐵}^3$, with the common Taylor-vacuum mass $𝑈$, the same component domain, and momentum equality using only $𝐷(𝑞)=𝐷(-𝑞)$. The LMB is used only to establish these identities; it does not define occurrence ownership or CFF rank capacity.

The actual immutable production plan has now been enumerated term by term. Set

It makes the following deterministic choices:

original fixed Q5^0 * Q5^0:                   9, 9
every positive owner-5 D5 wrapper:            12
the D6*D6 factors in the cubic term:           10, 11
the single D6 in the (U+p^2) term:             10
both p·(-Q5) factors in the quadratic-dot term: 10, 10
the p·(-Q5), D5, D6 factors in the linear term: 10, 12, 11

Thus the exact expression that GammaLoop asks the mapper to sample is

After applying the common-LMB diagonal ${𝑃}_9={𝑃}_{10}={𝑃}_{11}={𝑃}_{12}=𝑞$,

and therefore

exactly. This is the real post-plan certificate: it includes every selected occurrence and every signed hard/raw/parsed conversion, rather than comparing an earlier pre-plan expression. The parallel denominator certificate is also unchanged and exact. Values of the formal (P_j) away from the common-LMB diagonal are not an alternative physical routing and are not used to judge the reconstruction identity.

The executable regression gl00_gl04_planned_lifts_match_post_t_numerators_in_common_loop_coordinates runs both reconstruction proofs on the production source. It builds the actual owner-5/owner-6 denominator records, asks the production EnergyPowerAnalyzer for an immutable plan, and applies it with ExactSourceEnergyMapper. Distinct occurrence witnesses first check that original factors stay on their base owner and derivative-created factors stay within that owner’s copies; the test does not prescribe the particular allocation shown in this recorded trace. It then imposes the common chart and proves the complete factorized numerator identity. The denominator certificate compares the retained source multiset D(+q)^2 D(-q)^3 with the reconstructed occurrence product, including mass and UV-domain ownership. The GL04 case certifies the full T0+T1+T2 truncation; isolated coefficient extraction and the complete three-route comparison require the additional checks below.

Isolated coefficient and complete-source certificate

Writing $𝐷={𝑞}^2-{𝑀}^2$ and stripping the common numerator coupling, the actual second Taylor coefficient of the child is

This equality is a diagnostic expression for the coefficient, not a production cancellation or expansion. The production source retains its factorized numerator and owner-backed occurrences. The recorded 2026-09-05 diagnostic extraction separately certified the child and the complete child-plus-outer source after applying all immutable occurrence assignments and the signed $𝐻=ℎ𝑅$, $𝑃=𝑟𝑅$ maps. Both numerator differences vanished exactly in independent neutral four-vectors, and both denominator multisets matched. The complete source has physical occurrence counts

{1:1, 2:1, 3:1, 4:1, 5:2, 6:3, 7:1, 8:1}.

The proof must resolve every EMR component, retain independent crown momenta, and never install a replacement for emr_vec_index(edge,cind(0)): that spatial temporal component is already literal zero. Replacing it would silently turn every zero in the diagnostic into an energy variable.

An independent child contour uses $𝐸=\sqrt{|𝒒{|}^2+{𝑀}^2}$ and $𝑎=𝒒\cdot 𝒑$. Its positive-pole residue sums are

Each real-axis energy integral contributes $-𝑖{𝑅}_{𝑘}$ with the usual Below closure. Expanding the exact energy-integrated two-propagator child gives the same coefficients. For the complete cut [1,2], the additional moving child pole in the outer contour begins at order ${𝑡}^8$ before the ${𝑡}^{-3}$ spatial measure, so it cannot change these three coefficients.

At the rational sample

with $𝑀=0$, physical masses one on edges 0 and 8 and zero on every other edge, and all graph numerator factors one except $({𝑄}_5^0{)}^2$, the complete isolated T2 contour was recorded in the former production phase convention as

Before the scalar-base correction, the complete one-call exact CFF agreed with projected local-4D at approximately 300 digits, giving $-2.79328040709378476003397681050696066768826159163\times {10}^{-6}𝑖$. The independent analytic contour agreed with direct local-3D. Thus the reconstruction and factorized child/outer composition passed before the shared generalized scalar contact was changed. In that recorded post-correction diagnostic, direct, projected, and one-call results agree with the analytic value: one-call minus direct is numerically zero at 1024-bit precision, and projected minus direct is about $-1.25\times {10}^{-306}𝑖$. The unchanged production GL04 q5_temporal_square acceptance also passed with the complete Taylor truncation, integrated UV counterterms, and thresholds enabled.

GL00 odd-numerator child certificate

The same regression includes the complete post-Taylor child source for the physical GL00 edge-5 temporal probe used by scalar_3l_cross_section_gl00_q5_temporal_component_inspects_match. Its one-loop UV child has physical owners 4 and 5, with ${𝑄}_4=𝑞$, ${𝑄}_3=𝑝$ and ${𝑄}_5=-𝑞-𝑝$. The original ${𝑄}_5^0$ stays on owner 5; $𝑝$ is soft boundary data. Both child vertices contribute the retained factor ${𝑔}^2$, where g = UFO::SCALAR_COUPLING.

Write ${𝐷}_{𝑣}(𝑞)={𝑞}^2-{𝑀}_{𝑣}^2$ with the complete Minkowski square. For a soft Taylor parameter $𝜆$, the source numerator is $-{𝑞}^0-𝜆{𝑝}^0$, while its denominators are ${𝐷}_{𝑣}+𝑂({𝜆}^2)$ and ${𝐷}_{𝑣}+2𝜆(𝑝\cdot 𝑞)+𝑂({𝜆}^2)$. The independent complete T0+T1 oracle is therefore

Before normalization, production stores distinct mass fields: the full propagator expression uses ${𝑀}_{𝑣}^2$ (mUV), while exact-occurrence metadata uses ${𝑀}_{𝑒}^2$ (mUVexp). The regression preserves this distinction. Its numerator proof compares the complete factorized tagged payload ${𝑔}^2[{𝑑}_5({𝐻}_{5,\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}^0-{𝑝}^0)+2(𝑝\cdot {𝐻}_{5,\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d}}){𝐻}_{5,\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}^0]$, with ${𝐻}_5=-𝑞$, to the numerator above. Separately, its denominator proof checks the full product ${𝐷}_{𝑒}(+𝑞){𝐷}_{𝑒}(-𝑞{)}^2$, ${𝐷}_{𝑒}(𝑞)={𝑞}^2-{𝑀}_{𝑒}^2$, including physical owner multiplicities {4:1, 5:2}, masses and the child UV domain. The original factor stays on the base occurrence; the denominator derivative uses only owner 5′s new occurrence. The odd terms retain their sign independently of the denominator’s evenness.

local_4d.rs introduces the two mass fields, and final integrand simplification identifies mUVexp = mUV after exact mapping. Applying that specified substitution yields the normalized rational oracle above. The GL04 fixture also preserves these fields separately. Neither numerator is distributed into a polynomial for the comparison.

This is a complete child-sector reconstruction certificate. It does not certify the full cut sum or every GL00 forest. The GL00 numerical regression complements it with complete physical-route evaluations including local and integrated UV and threshold counterterms.

One source convention across recursive scalar bases

Reconstruction correctness ends before CFF normalization. Let ${𝐺}_{𝑐}$ denote a raw independently generated component, including positive local $1/(2{𝐸}_{𝑗})$ factors. Its target source-frame coefficient is

For generalized and ordinary public components, the signed contour is ${𝐽}_{𝑐}={𝛽}_{𝑐}{𝐺}_{𝑐}$, using $𝑑{𝑞}^0/(2𝜋𝑖)$ for each loop. The physical Minkowski amplitude uses ${𝑖}^{{𝐿}_{𝑐}}{𝐽}_{𝑐}/(2𝜋{)}^{3{𝐿}_{𝑐}}$, including the vertex and propagator numerator factors of its model. The former $(-𝑖{)}^{{𝐿}_{𝑐}}$ production factor is superseded; the real source-frame conversion described here is unchanged. Internal pure-CFF catalogues retain their native orientation normalization, distinguished below. Here ${𝑁}_{𝑐}$ counts denominator occurrences. The single source_frame field stores ${𝐵}_{𝑐}$, the numerator-bound-independent connected-component frame, where ${𝑟}_{𝑐}$ is the surviving denominator rank. The generalized core sign is already encoded in its Laurent functional; multiplying it again would let an unused degree allowance change a scalar integral. In the ordinary GammaLoop path, surface conversion removes the positive half-edge factors and restores $1/{\prod }_{𝑗}(-2{𝐸}_{𝑗})$, which already supplies $(-1{)}^{{𝑁}_{𝑐}}$. A generalized expression retains its positive factors on each variant, so its typed adapter supplies that parity. More precisely, the actual GammaLoop adapter is product_c [(-1)^(rho*N_c) * frame_c]: rho is one when the whole expression retains variant-local positive factors, and frame_c is that component’s single source_frame, independent of energy-factor ownership. Thus mixed expressions also apply this parity to their ordinary components. Fresh exact sources and stored production roots obey the same rule; they do not require separate fresh-versus-stored formulas.

The same convention must survive denominator deletion inside a generalized functional. BoundedCffBuilder and KnownFactorCffBuilder carry the original source prefactor through every remainder, contact, and recursive reentry. LowerSectorCffBuilder converts a scalar base by ${𝛽}_{\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}/{𝛽}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}$ once, at its embedding boundary. It derives the native factor from the base that was actually built:

Every connected pure component with $𝑛$ denominator occurrences and surviving rank $𝑟$ has native factor $(-1{)}^{𝑛+𝑟-1}$ in every medium. Repeated channels require no separate parity convention. Scalar bases use the same recursion in vacuum and thermal media. Free tadpole pruning averages the two contour closures: each vacuum orientation has weight one half, so a free one-line pure component evaluates to $1/(2𝐸)$ and its native factor is $-1$. Contacts retain the parent contour instead. Ordered inherited rows and structural cuts select terminal poles; an embedded complete residue basis uses the original ambient contour order to determine that selection. These are contour metadata on the actual graph, not projected graph factors. Occurrence interpolation preserves this scalar-base normalization: its coefficients reconstruct the numerator and introduce no additional terminal multiplicity.

Thermal tadpoles retain both energy orientations s = +/-1. Their weight is c_s + n_B(E - s mu) for bosons and c_s - n_F(E - s mu) for fermions. For a free tadpole, c_s = 1/2; for a parent contour selecting s0, c_s = (1 + s*s0)/2. Edge-energy numerators are evaluated at s E; loop-energy numerators use the complete affine routing map at that pole. Ordinary contraction numerators keep their signed thermal distribution differences. At zero temperature and zero density these weights reduce to the corresponding vacuum prescription; a zero-temperature Fermi sea retains its occupation contribution.

Component products use their actual native factors and their assembly sign, not metadata inferred from an unrelated fresh child source. Odd free-tadpole numerators follow the symmetric prescription, while odd contact numerators retain their inherited contour value. Convergent complete source integrals remain the comparison boundary for cancellations among auxiliary contacts. Public ordinary sources require their denominator rows to span every loop energy; initial-state cut aliases cannot supply missing denominator rank. Formal lower-sector contacts keep their internal rank-reduction path.

The GL04 outer graph exposes the missing conversion particularly clearly. Edges 3 and 8 carry the same momentum with different masses, so they remain distinct denominator channels. The complete physical cut obeys both

Deleting edge 3 changes the denominator-count parity while leaving the nonterminal core metadata unchanged. Before the fix, ordinary scalar and both lower-topology contours passed independently, but the generalized quadratic differed by minus twice the physical edge-3 contact. The coefficient of $({𝑝}^0{)}^2$ in ${𝑅}_2$ is $-1/(16{𝐸}^3)$, giving exactly the full live T2 discrepancy without fitting a sign to the UV result.

The permanent regressions gl04_outer_unequal_mass_quadratic_obeys_complete_cut_identity, mixed_carrier_contact_matches_independent_convergent_contour, and attached_tadpole_repeated_bubble_matches_independent_convergent_contour compare complete convergent sources against literal pole residues. Their scalar, lowered, and terminal controls distinguish a source-frame error from an incorrect contour oracle. Complete physical-cut sums are the comparison boundary; branch counts and individual generalized keys are not route invariants. The recorded initial-cut repeated-spectator contour check also passed unchanged, checking the source-frame sign needed by the original root contact.

Stage 1: retain denominator multiplicity without expanding the numerator

The local-4D term projector traverses the Symbolica expression structurally. For a negative integer power of a denominator wrapper, it repeats the FourDDenominator record abs(power) times. Non-denominator powers remain in the numerator. The first argument of every typed GS.den wrapper is the original EdgeIndex; the UV Taylor operator differentiates and rewrites the other arguments while carrying that owner unchanged. Consequently, the completed Taylor coefficient still has a direct lookup into the original graph and never has to recover an edge from the rewritten momentum.

Taylor collection can also expose a positive typed GS.den factor inside a numerator multiplying a negative occurrence of the same wrapper. Projection does not cancel those factors or distribute that nested numerator sum. It splits only an outer additive Taylor result whose addends need separate denominator topologies. For example,

((A+B)*den_0 + den_1) / (den_0*den_1)

is projected with numerator (A+B)*den_0 + den_1 and denominator occurrences [den_0,den_1]. The numerator remains one factorized atom, and generalized CFF performs any resulting pinches internally. By contrast, a completed outer sum T_0/den_0 + T_1/den_1 is parsed as two terms because its addends genuinely carry different denominator topologies. The focused regression term_projection_preserves_complete_factorized_values protects that distinction.

For example,

first^-2 * second^-1 * N

becomes

numerator:    N
denominators: [first, first, second]

The regression term_projection_preserves_complete_factorized_values rebuilds the complete expression from every projected term. Its cases include wrappers with the same physical edge and momentum but distinct full_expr provenance, repeated powers, positive numerator denominators, and distinct outer topologies. It compares exact factorized values instead of prescribing storage counts. A factorized expression such as

survives term projection exactly as a factorized atom.

This step supports arbitrary negative integer powers. It does not contain a special case for a double pole or for a self-energy graph.

Stage 2: recover the source skeleton and classify algebraic channels

Two keys are retained here for different purposes. The physical source_edge locates an occurrence in the already known cograph or UV source minor. It tells GammaLoop which original component, attachment, and topology domain produced the factor, so no graph has to be inferred from a matrix of rewritten momentum signatures. Separately, GammaLoop computes an exact momentum signature in the temporary source coordinates and canonicalizes it up to an overall sign. The algebraic-channel key is

(
    topology_domain,
    canonical_momentum_signature_up_to_sign,
    canonical_mass_squared,
)

The retained full_expr payload, input order, and physical owner label do not change this channel. The owner remains the authoritative link back to the source skeleton and physical graph, but it is not a variable of the rewritten rational function.

This gives three separate invariants:

• every exact occurrence can be attached to the appropriate original source minor without an algebraic graph-reconstruction search;
• repeated copies of one source wrapper form a powered line only when they represent the same rewritten denominator up to D(Q)=D(-Q); and
• after that source-backed construction, the rational residue and loop rank of an algebraic channel are invariant under a compatible relabeling of its physical owners.

A compatible relabeling changes only the source_edge provenance among occurrences which instantiate the same normalized denominator in the same topology domain and for which the source-backed lift remains valid. It does not move a factor between UV and cograph domains, alter a mass or affine external shift, or change a non-vacuum boundary. Those changes describe a different rational problem rather than a provenance relabeling.

If one source edge appears with incompatible normalized signatures or masses in one additive 4D term, generation fails instead of choosing an incidence by heuristic. UV-vacuum and cograph denominators are constructed in disjoint source minor domains even when their algebraic data happen to look alike.

Provenance is retained, not discarded

Owner provenance remains on every FourDDenominator. Across construction and the later projection back from the owner-free ParsedGraph, it is used to:

• select the source-minor component and attachment which seed the exact skeleton;
• recognize repeated copies produced by raising one physical wrapper;
• project occurrence-local energy IDs back to physical edge IDs;
• retain distinct occurrence-energy provenance;
• build Cutkosky and LU cut support as the union of physical owners in each algebraic channel;
• recognize which exact surfaces are original physical surfaces;
• report and diagnose the source of a UV-expanded factor;
• recover parent-graph spatial momenta and affine external shifts.

The rule is therefore:

Use the original source graph and retained owners to recover the known skeleton and all physical maps. Add only derivative-created serial copies of those same lines unless a further serial-path or pure-cycle contraction is certified. Normalized denominator algebra classifies channels on that scaffold; it never supplies incidence. Completed hard numerator ownership transfers only at the exact canonical-class certificate boundary.

Stage 3: instantiate the source-minor scaffold and lift powers

The original graph is never reconstructed. Instead, each occurrence first receives a deterministic attachment in the appropriate source minor.

For cograph denominators, a disjoint-set union over the original graph nodes contracts source edges absent from the additive exact term. Each surviving owner identifies the endpoints of its original EdgeIndex in that quotient. UV denominators use a separate disjoint-set union for the UV source minor: absent UV edges are contracted there, and every active UV owner inherits its endpoints from that minor. The two node domains remain distinct.

There is no Kirchhoff solver, signature-matroid reconstruction, or signature-to-endpoint fallback. A rewritten momentum such as k1+k2 still has a source_edge; its source-minor attachment identifies where that factor came from. Exact signatures are subsequently checked against the scaffold. If the source-backed construction and supplied momenta cannot realize a valid exact occurrence graph, generation fails explicitly.

For a raised source wrapper, once its base attachment is known, multiplicity r is subdivided as

tail --occ_0-- x_1 --occ_1-- ... --occ_(r-2)-- x_(r-1) --occ_(r-1)-- head

This introduces precisely r-1 auxiliary vertices. If tail == head, the result is an r-edge cycle. Every occurrence keeps its own exact energy ID. Its ParsedGraph signature is normalized up to sign, and every segment is directed coherently.

That source-backed subdivision is a construction device, not an owner-sensitive definition of the integral. Equal algebraic occurrence multisets are subsequently canonicalized without owner labels. Thus two copies of D(Q) must yield the same one-loop, double-pole CFF whether their provenance is (edge 0, edge 1) or (edge 0, edge 0), provided both owner assignments are compatible with the same source skeleton and topology domain. The source IDs avoid a combinatorial endpoint search; they do not grant two spellings of the same rational function different residues.

Here, “owner-independent” does not mean that GammaLoop throws away the source graph and reconstructs another graph from canonical signatures. The only topological operations are contraction of known source-minor edges, local subdivision for denominator multiplicity, and canonical relabeling of the result. Owner invariance is a contract on that deterministic lift. There is no enumeration of candidate incidences, Kirchhoff-matrix inversion, or signature-matroid search anywhere in this path.

The UV Taylor operator can remove a soft/cograph shift from a denominator. For example, an original source edge may carry Q_hard+Q_soft, while its rewritten UV denominator carries only Q_hard. Its scaffold attachment still comes directly from the original edge. To determine only whether that attachment is used forward or backward, GammaLoop tests both signs modulo the span of the known opposite topology domain:

exact_row - sign * source_row in span(opposite_domain_rows), sign in {+1,-1}.

Exactly one sign must pass. This rank calculation is a routing-coordinate validation after both graph minors are already fixed; it never chooses nodes, constructs a graph, or repairs momentum balance. The focused hard-plus-soft triangle regression checks that the correct sign leaves only a soft residual and that the opposite sign raises the quotient-space rank.

Serial subdivision is the graph-theoretic representation of a dotted line: the new two-valent vertices do not introduce independent momentum constraints. Raw occurrence signs and physical owners are retained separately by the exact numerator/provenance maps rather than encoded as alternating segment directions.

Non-vacuum boundaries use explicit source-crown hedges

After internal incidence is fixed, a non-vacuum exact signature can leave an external-momentum imbalance at a source-backed node. GammaLoop represents that boundary explicitly with source-crown hedges carrying the corresponding external coefficients. This completes the already known source topology; it does not use external or loop signatures to infer internal endpoints. Contracted components containing no exact denominator or cut carrier are multiplicative identities and do not manufacture causal surfaces.

The implemented non-vacuum path currently wires pure-external source crowns: the relevant parent crown hedge has no parent-loop coordinate and carries its known external coefficients into the exact source. This is already sufficient for ordinary non-vacuum amplitude boundaries. A future on-shell UV scheme may instead leave a two-point exact topology with a fixed insertion such as (m,0,0,0). That extension should add the fixed four-vector as an explicit boundary payload and wire it to the same source-derived endpoints. It does not change the internal disjoint-set construction and does not require an incidence or Kirchhoff reconstruction.

Determinism

Occurrence input is first put in a stable order so raised-wrapper subdivisions are reproducible. After all source attachments, auxiliary power vertices, cut carriers, and crown hedges exist, Graphica canonically relabels the edge-coloured graph. Internal propagator incidences are undirected during this relabeling because D(Q)=D(-Q); external crown incidences remain directed. In the canonical node namespace, exact occurrences are grouped only for routing by connected component and the key

(UV/cograph domain, canonical signature up to sign, mass, power marker).

For each such component, GammaLoop compares the sorted multiset of directed (tail,head,signature,...) edge keys with the multiset obtained by reversing every incidence and negating every complete momentum signature. It retains the lexicographically smaller whole-component spelling. Reversing the component as one unit is essential: a same-owner dotted chain, or equal rational denominators carried by successive distinct source edges, must not acquire alternating segment directions. Disconnected equal denominators remain separate routing components, and owner IDs remain provenance rather than an algebraic routing key.

Exact edges are then sorted in the canonical node namespace and the occurrence-to-original-factor map is reordered with them. Source-node and power-node names survive as aliases of the canonical nodes; cache keys deliberately clear those names and edge labels. Reversing the order of Symbolica factors therefore does not change the parsed exact graph or its CFF cache entry. Crucially, this pass only relabels and orients an already source-constructed graph: it never derives an endpoint from a momentum signature.

Rational incidence and physical support cross the boundary separately

The owner-free exact CFF and the maps back to the physical graph deliberately cross separate interfaces:

• Energy provenance is occurrence-local. Each canonical exact energy maps back to the physical owner carried by its original denominator occurrence. Distinct owners therefore remain distinguishable to numerator evaluation and diagnostics even when their denominators are algebraically identical.
• Cut support is channel-wide. For each algebraic denominator channel, GammaLoop collects the sorted, deduplicated union of every physical source_edge which instantiated it. Any exact occurrence used as residue support is projected to that full union. This prevents an arbitrary canonical occurrence representative from erasing a valid physical Cutkosky carrier.
• Raised-line compatibility is additive. Downstream LU selection may name a canonical representative of a physical raised-edge group. GammaLoop adds those representative aliases to the already unioned physical support and merges equal support-sign entries. It does not replace the original owner set. Thus existing raised-cut matching continues to work without losing provenance.

The channel used for this neutral cut-support union includes the topology domain, mass, and exact signature canonicalized up to sign. Consequently D(Q) and D(-Q) share support, but equal-looking UV and cograph factors, or factors of different mass, do not. None of these support operations modifies the rational incidence, exact energy map, or numerator routing sign.

Stage 4: analyze the factorized numerator before occurrence substitution

The numerator is analyzed in physical Q(edge, index) variables and, for completed hard UV factors, certified Q(uv_class(id), index) variables. These references remain distinct from the exact occurrence IDs allocated later.

For each active physical edge or UV class, the analyzer computes a certified polynomial degree bound using the following composition rules:

These rules avoid algebraic expansion. In particular,

[ (Q^0+c)r

]

retains a repeated factorized base. Allocation reuses repeated assignment cycles where possible instead of eagerly copying the base r times; it never expands the power into r+1 monomials. Additive branches reuse capacity because the degree of a sum is a maximum, not a sum. Inert tensor projectors keep their heads and argument structure; recognizing their multilinearity here does not change their global Symbolica attributes.

Opaque nonlinear functions of EMR energies and negative energy-dependent powers are rejected. Production does not guess a bound for an expression it cannot certify.

Stage 5: certify equivalent exact occurrence energies

For a physical EMR input which has nonzero numerator degree, the exact source mapper finds denominator occurrences whose rewritten momentum is literally

The source constructor certifies each owner’s same-momentum, same-mass serial occurrences. The mapper exposes the retained base as the fixed-factor candidate and derivative-created copies as the new-hard-factor candidates. This uses source incidence and provenance, not a search over equal on-shell expressions or a largest equality class. Candidate order is canonical and deterministic.

Candidate sets for different physical edges must be disjoint. A missing, empty, duplicated, or overlapping candidate set is an error. In particular, GammaLoop does not treat an arbitrary shifted momentum Q_e+p as an alias for Q_e, and it does not use an LMB coordinate as a fallback.

The literal +/-Q_e restriction is important for sign safety. It supplies both an algebraic energy identity and an exact routing sign. Equality of positive on-shell energies alone would not be sufficient to map an odd numerator.

Completed canonical UV input uses typed classes instead: the mapper certifies every retained occurrence against the class’s exact signed component-frame binding and denominator polynomial. This permits a class to span several physical owners while preserving the incidence witness. It does not equate channels using additional momentum-conservation identities. A reconstructible hard factor without a surviving pole is a fixed affine carrier, whose temporal dependence still requires valid source-coordinate bounds and sampled mapping; it is not an empty class or an invented occurrence.

Stage 6: propose certified assignments and select by actual map count

For raw physical input, original factors retain one owner occurrence and newly derived hard factors use that owner’s certified copies. Completed canonical UV input permits every eligible hard factor to use its class’s certified pool. The factorized expression supplies structural degree bounds: sums take componentwise maxima, while products and multilinear slots add loads. The baseline allocates freely assignable class-owned unit factors by cyclic occurrence offsets. Product and multilinear children advance the offset; sum branches share their starting offset. Fixed physical factors retain their occurrence, while other indivisible factors use deterministic least-loaded placement within their certified pool. A raw original quartic stays fixed; a class-owned quartic may use a factorized lift under its exact certificate. The former Cartesian/Pareto hard frontier is not enumerated. Actual native map count selects among the bounded proposals; the descending degree envelope and stable proposal order break count ties.

Cyclic and greedy hard proposals

For the special case of d freely assignable unit factors and n equivalent occurrences with no fixed load, any valid assignment has nonnegative loads

The smallest possible maximum load is

For this unit-factor rank objective, cyclic allocation attains the quotient/remainder distribution:

and assigns load q+1 to the first r canonical candidates and q to the rest. This is whole-envelope optimal in the stated special case and deterministic; it does not predict native map count. Sums preserve their branch structure and share offsets. Fixed loads and indivisible positive-denominator blocks require the actual structural allocation; the unit-factor formula does not promise an optimal envelope for those cases.

Cyclic-baseline examples without fixed loads are:

The planner assigns individual admissible factor slots, not merely total degrees. For three completed class-owned unit factors,

[ (Q^0+c_1)(Q0+c_2)(Q^0+c_3)

]

over two equivalent occurrences a,b, the canonical plan is conceptually

[ (E_a+c_1)(E_b+c_2)(E_a+c_3),

]

with bounds (2,1). Another balanced assignment would be algebraically valid, but canonical ordering makes generation reproducible.

For

[ (Q^0+c)2,

]

the internal planned representation repeats the unexpanded base and maps it as

[ (E_a+c)(E_b+c).

]

The original expanded polynomial is never constructed. Raw physical owner restrictions remain in force; cross-owner reuse requires the completed hard factor’s certified canonical class.

Native map-row selection

An affine baseline needs one proposal. For nonlinear input, the planner chooses one eligible occurrence pool deterministically by its largest baseline load, then its degree beyond the pool size, then stable family order. A UV-class pool can supply a second proposal which packs unavoidable excess after its initial pass. After scoring the initial proposals, selection asks the current winner for a reversed placement in that same pool; a one-step rotation replaces a duplicate. Ordered capacity duplicates are skipped, leaving at most three certified proposals. No Cartesian product across pools is constructed. Equal sorted envelopes at different occurrence positions remain distinct capacities.

For each proposal, the ordinary exact source generator receives the same source and options with that plan’s occurrence bounds. The score is the actual stored row count generated.expression.orientations.len(). This is the generator’s operational output after its normal complete-map emission/coalescing; retained zero rows count too. It is neither the sum of key lengths nor a degree-product formula, and it has no extra physical-orientation multiplier. Different Taylor terms keep their own source maps even when key lengths differ.

The smaller count wins; an equal count prefers the smaller descending degree envelope, then the earlier admitted proposal. For example, (4,2,2) breaks a count tie ahead of (4,3,1) despite equal maximal degree. The same generated payload, assignment and bound report move forward together. The cyclic/greedy proposal family assumes neither monotonicity of map count under rank reduction nor independence between occurrence pools. K<=3 bounds the number of candidates, not each generation’s time or memory, and does not promise a global minimum. A generation error remains an error rather than silently discarding an unsupported contender. Stage 8 describes how the hard cache avoids rebuilding or retaining losing trial payloads.

The small sunset regression checks the complete contour instead of fixing this cost model’s row count. Its source has channels k0, k1-k0 and five serial occurrences of k1, with energies (2,3,7) and numerator k0*k1^p; D0, D1, D2 denote the three corresponding quadratic denominators. With both measures dk/(2*pi*i) closed below, integrating k0/(D0*D1) gives -k1/[2*E1*(k1^2-(E0+E1)^2)]. Multiplying by k1^p/D2^5 and taking the negative sum of the remaining simple and fifth-order pole residues gives 1213/16387080192 for p=5 and -365/47775744 for p=7. The test covers all 126 degree-five load distributions and the bounded allocator’s degree-five/seven proposals. These independent values certify signs and complete sampling; they establish neither a preferred allocation layout nor a globally minimal native map.

Soft Taylor energies at the outer CFF boundary

Denominator differentiation also creates soft factors S^0. Their DenominatorDerivedSoft provenance records the actual crown carrier and literal payload. A later enclosing Taylor operator consumes a newly hard part on that carrier, while any remaining soft part keeps its provenance. This is separate from assigning new hard factors among serial copies of a UV denominator.

At final projected assembly, the completed child coefficient is multiplied by the untouched outer numerator before choosing soft routings. For each tagged soft carrier, the active cograph momentum rows provide exact rational basis alternatives. Every candidate must reproduce all loop coefficients, with the fixed external shift explicitly restored. No on-shell energy equality or change of original numerator ownership supplies this certificate. Concrete spatial components retain their original routing and contribute zero energy rank.

The same factorized sum/product rules combine candidate loads into a Pareto frontier. Up to three proposals with distinct ordered active-capacity vectors, ranked by descending-envelope/deterministic order, enter the raw outer-source generator. Selection uses the same native row count as the hard path, before surface conversion, cut selection or hosting; proposal order breaks ties. Final selected (host, source map) branches can have a different count and are not the objective of this raw-row search.

Only the incumbent and one challenger are retained at a time. The loser is dropped before any persistent graph surface, bound-report or cut/hosting effect. The winning raw payload is converted once and reused with its matching factorized Atom for numerator mapping. This soft boundary has no persistent count memo. When an empty contraction reuses an existing production root expression, the rank baseline follows that existing path without regenerating or changing the root’s established capacity.

Child contour maps are already consumed and play no role in this routing. A loop-dependent carrier outside the active span is an error, not a constant in the rank analysis. The search is best-of-three within the existing certified basis/Pareto proposal class, not a global count optimum. Basis enumeration and frontier construction can still grow combinatorially; the three-candidate generation budget does not bound that preparation work. Neither preparation nor scoring expands the numerator or searches for cancellation by expansion. The direct-3D and LTD paths retain their existing behavior.

Stage 7: use one immutable plan for generation and evaluation

The assignment plan contains both:

• exact occurrence -> certified degree bound; and
• factor-local physical edge or certified UV class -> chosen exact occurrence.

The first map is passed to the shared generalized-CFF generator. The second map is retained by the exact-source numerator evaluator. Each orientation, LU residue, threshold residue, and lower-sector contact term maps the numerator using the same assignments.

This prevents a subtle but serious inconsistency. It would be wrong to balance the advertised bound as (1,1) but later replace every physical Q^0 by E_a. The CFF expression would then have been generated for a lower rank than the expression it actually samples. Because one immutable plan owns both operations, that state is not representable.

Sampled directions also survive lower-sector pinching. If a contact sector assigns a zero sample to the occurrence which owns a factor, that exact zero map remains authoritative even if an equivalent dotted occurrence survives. The numerator is not silently reassigned to the surviving occurrence after the pinch.

Stage 8: cache exact topology and per-request capacity

One nonserialized Local4dProjectionContext owns reuse for one graph’s complete UV computation in either forest backend. Canonical sectors, owned source analysis and immutable numerator templates share a 48 MiB/4,096-entry retention budget. Winning CFF payloads and count-only contenders use 64 MiB/4,096 keys; component-local mapped subtrees use 16 MiB/16,384 entries. The accounting includes owned keys, containers and symbolic payloads; oversized entries bypass retention. These are retained-payload limits, not limits on whole-process RSS. A context must not cross the graph boundary: both orchestrators keep the parent graph fixed throughout the context lifetime, and keys describe requests within it.

An immutable Arc keeps the certified assignment, prepared numerator and signed mapping context together. Row memoization remains separate mutable state and is cleared after each component wave. Cold, warm, disabled and eviction paths must return the same complete coefficient. Tests retain the configured byte ceiling as a resource contract without fixing cache occupancy, hit counts or preparation counts.

Each completed Taylor coefficient prepares its source-backed exact graph and initial immutable plans together. The final placement challenger depends on native scores and is constructed during selection. Counts and generated expressions use the same complete canonical ParsedGraph, energy-edge map and generation-options key, including the unchanged per-occurrence bounds. Every term is evaluated with its selected plan.

ExactCffGenerationCache keeps a small count-only memo for each scored key and admits a full generated payload only when that key wins a request. Known contenders are compared by count without cloning their payloads. A key which previously lost can later win a different comparison; if its payload was not retained, it is generated only after selection. Previously winning cache entries remain reusable. Cold comparisons retain only an incumbent and one challenger, dropping a losing trial immediately. A request makes at most three core calls, including any recovery of a count-only winner, and never regenerates a freshly generated winner solely to convert or map it.

No maximum is taken across independent requests: such a join can manufacture combinations of high ranks on different channels which no individual numerator needs. Nor does the cache sum or redistribute bounds between occurrences of equal physical energies. Only requests with identical occurrence capacities share a generated expression. This keeps cache reuse independent of the factor-to-occurrence assignment and avoids repeating source construction and rank planning in a separate registration pass.

Provisioning one outer CFF for independently evaluated residue branches is a different boundary. That single source requires the per-edge maximum over the factorized branch products, including their common remaining numerator. The existing physical-EMR analyzer consumes these products one at a time and merges only degree maps, with initial-cut and tree-edge energies excluded. Equal bodies can share an analysis, but opposite bodies cannot cancel across branches. No tagged symbolic sum or expanded numerator is constructed, and the original branch expressions, residue keys and numerator maps remain unchanged.

The production-shaped regression dod_one_triangle_keeps_separate_denominator_topologies applies the real degree-one UV Taylor operator to the three-edge UV triangle in the double-triangle skeleton. The coefficient retains its natural denominator topologies with owner multiplicities

(1,1,1), (2,1,1), (1,1,2).

The base and singly dotted terms are separate exact sources. No common denominator is imposed across them, so denominator clearing does not manufacture positive typed factors in their numerators. The parser preserves each original factorized numerator and can group terms only when their typed denominators already agree. Genuine positive typed denominator factors remain supported; their CFF pinches are distinct from artificial factors introduced by regrouping. The fixed cograph component is part of each canonical key, so cache reuse does not conflate different surrounding graphs or future non-vacuum boundaries.

Why D(Q)=D(-Q) cannot introduce numerator sign mistakes

This deserves a separate proof because the denominator and numerator have different parity properties.

Let the literal exact denominator carry momentum

For a scalar propagator,

It is therefore correct to give +Q and -Q the same normalized denominator signature and algebraic channel. If they are repeated occurrences of one source wrapper, they validate as copies of that powered line. If they belong to distinct source edges, those owners remain in physical provenance and unioned cut support, while the rational occurrence graph remains owner-invariant. Neither case reconstructs topology from the sign.

It does not replace Q by -Q in the numerator. After owner-free graph canonicalization, a complete rational-routing component may also have been reversed coherently. Write its final parsed occurrence momentum as

The mapper determines s_route by comparing the final parsed signature with the original exact signature; it must not reuse only the earlier canonical_up_to_sign result because the later component canonicalization may have reversed that direction again. The occurrence-local CFF energy map is the temporal component p_parsed^0, so a physical numerator written in terms of Q is reconstructed as

because both signs are their own inverses. The certified occurrence sign stored by the mapper is therefore the product s_literal * s_route; an explicit negation is performed exactly when that product is -1.

Consequently,

An odd numerator therefore retains its sign. A vector numerator is handled in the same way: spatial components remain in the physical parent-graph EMR basis, while the corrected temporal energy is inserted with the temporal unit vector. The mapper never performs a blanket substitution Q -> -Q on the physical numerator.

There are thus two separate routing layers:

1. The parsed graph uses one coherent canonical direction for each completed rational-routing component. Its initial attachment came from the source minor, but owner labels are absent from the owner-free rational identity.
2. The numerator mapper retains the literal sign, derives the final component routing sign from the parsed occurrence, and composes both with the occurrence-local temporal map to recover the physical Q convention.

The first layer is even denominator algebra on a source-backed skeleton; the second is signed physical-numerator mapping. Denominator evenness affects only signature equivalence and never silently turns an odd numerator into an even one. Likewise, unioning +Q and -Q cut support is a neutral provenance operation and never rewrites a numerator factor.

A focused sign oracle

The test exact_energy_bounds_keep_original_on_canonical_occurrence_across_routing_signs uses three equivalent occurrences with sign patterns (+,-,-) and (+,+,-). An original degree-two numerator stays on its retained base occurrence and maps to E^2; changing the other routing signs cannot authorize dispatch to a serial copy.

The complementary test denominator_derived_dispatch_composes_each_occurrence_routing_sign dispatches two derivative-created temporal factors over derived copies. It varies the immutable hard sign h, the literal occurrence signs, and the parsed routing sign. After the signed conversion, the complete mapped numerator is

The independent shifts retain odd powers of E, so a lost numerator-side sign cannot be hidden by an even final power. The test also forces an abstract vector factor onto each eligible derived occurrence and checks the same physical spatial and temporal hard-momentum convention.

The end-to-end exact-CFF identity test also retains an explicitly odd factor Q^0+c while cancelling one denominator. It verifies at two rational phase- space points that

[ =

]

matches both the lower exact source and the ordinary CFF source. This protects the odd numerator mapping independently of a purely even Q^2 probe.

Complete example 1: a same-owner cubic powered line

The unit fixture exact_source_preserves_the_complete_cubic_propagator_contour starts from the physical graph

        edge 0
    a ----------> b -> c -> a
                         |
                         +----> d

and constructs three exact denominator records for physical edge 0. Two records carry one retained full_expr label and the third another, but all have the same rewritten momentum and mass.

The source represents the repeated propagator by serial occurrences on its contracted physical incidence. The regression checks valid momentum balance, retained physical ownership, and the complete signed contour against the independent cubic-pole residue $-3/(16{𝐸}^5)$. That oracle detects lost or duplicated powers without fixing auxiliary-node counts or canonical edge ordering.

This is the simplest complete realization of

[ D(Q)^{-3}

]

as a graph accepted by the shared CFF recursion.

Complete example 2: owner relabeling preserves the rational CFF

The regression exact_source_owner_relabeling_preserves_residue_rank_and_cut_provenance uses a physical graph with two parallel edges:

        edge 0
    a ----------> b
        edge 1
    a ----------> b

It constructs two spellings of the same rewritten denominator multiset:

distinct provenance: (owner 0, owner 1) -> D(Q)^-2
relabeled provenance: (owner 0, owner 0) -> D(Q)^-2

The source graph and retained IDs make both constructions immediate: no algorithm searches momentum signatures for a two-node bubble or solves a Kirchhoff incidence problem. Their normalized signatures and masses identify the same algebraic double channel. The owner-free exact graphs are both valid, have one topological loop, and give the same CFF residue at exact rational on-shell points after the neutral physical-energy aliases are identified.

What changes is only the physical projection. The distinct spelling retains energy provenance {0,1} and cut support {0,1}. The relabeled spelling retains energy provenance {0} and cut support {0}. In the first case, each exact occurrence projects to the union {0,1} for cut eligibility, even though its occurrence-local energy still maps to the one owner which produced it.

This fixture protects the precise boundary: source IDs recover the known skeleton and survive as provenance, but owner identity cannot control the exact rational incidence, loop rank, or contour residue. It also demonstrates why cut support must be unioned after canonicalization: choosing one canonical occurrence must not hide the other physical owner.

Complete example 3: opposite routing within one powered line

The fixture exact_source_normalizes_opposite_spelling_inside_one_power_chain uses two occurrences of one rational denominator with momenta +Q(0) and -Q(0), plus a balancing denominator of a different mass.

The +Q and -Q occurrences enter the same owner-local power group because their canonical signatures and masses agree. Their parsed signatures are both normalized, and both serial segments are attached to the source-edge-0 scaffold before receiving one coherent canonical direction. The raw opposite signs remain in the exact numerator mapper. The separate physical edge completes the known source scaffold, so the whole exact graph is momentum-balanced without a signature-derived sign bridge or an owner-sensitive rational residue.

A separate end-to-end test, exact_cff_keeps_opposite_source_routing_without_a_sign_bridge, uses two distinct physical source edges whose exact signatures normalize to one double-pole channel. Their physical provenance remains separate while the owner-free occurrence graph represents the common rational channel. The test finds two explicit orientations, keeps the original physical edges undirected, and verifies the complete orientation sum

[

]

in GammaLoop’s physical production normalization, including the typed source-frame bridge. No signature-derived incidence bridge is needed: source routing and numerator sign restoration are handled at their proper layers.

Complete example 4: the production-shaped cubic UV rewrite

The most representative scalar fixture is scalar_cubic_exact_uv_rewrite.dot. Its full graph is

incoming -> v5

v0 -> v2   [edge 1, outer LMB carrier]
v3 -> v0   [edge 2]
v0 -> v5   [edge 3]

v2 -> v1   [edge 4, numerator Q(4)^0 + m]
v1 -> v3   [edge 5, inner LMB carrier, numerator Q(5)^0 - 2m]
v2 -> v3   [edge 7]

v1 -> v4 -> outgoing

Edges 4, 5, and 7 are the only UV-active cycle. In its leading UV routing they become

Their owner edges have genuinely different physical parent-graph momentum signatures. After the local 4D UV expansion, all three UV denominators are the same massive UV propagator up to routing sign. Each owner nevertheless retains its edge as physical provenance, and the UV source minor supplies the known triangle scaffold without reconstruction. The shared CFF layer keeps all three occurrences and their numerator energies independent, even though their physical denominators coincide. Canonical graph labels do not merge their rank bounds or numerator samples.

The integration fixture’s numerator probe is deliberately factorized:

[ (Q_4^0+m)(Q_50-2m).

]

The physical analyzer retains these as two separate linear edge dependencies; it does not first expand their product or replace them by an LMB polynomial.

The analytic companion test then uses the sharper single-channel probe

All three exact cubic occurrences share the relevant algebraic energy, but this quadratic probe is an original numerator factor. It retains its canonical source occurrence and supplies one degree-two bound, as the companion test explicitly requires. Only newly denominator-derived factors may enter the bounded dispatch search. No expanded quadratic polynomial is constructed.

The unit test exact_cff_cubic_uv_rewrite_matches_production_convention uses the source-compatible routing (4:+Q,5:+Q,7:-Q) and permutes the input factor order. For every order it verifies:

• a valid momentum-balanced exact graph;
• one active topological loop;
• no spurious causal surface from the zero-denominator cograph component;
• complete signed CFF values independent of input factor order;
• the analytic cubic-pole production contour $-3𝑖/(128{𝜋}^3{𝐸}^5)$;
• complete contours for original and derivative-created quadratic factors, retaining their distinct ownership domains;
• the correct lower-sector parity for a q_0^2 numerator, $+𝑖/(128{𝜋}^3{𝐸}^3)$; and
• the temporal/spatial cancellation of a full q^2 numerator.

The mirrored fixture scalar_mirrored_cubic_exact_uv_rewrite.dot reverses edge 5 and changes the inner carrier so that the leading routing is

[ Q,-Q,-Q.

]

Both fixtures are exercised through three production routes:

1. orientation-local direct local 3D UV;
2. direct local 3D UV with an explicit orientation sum; and
3. local 4D UV followed by exact CFF projection and an explicit orientation sum.

The first two routes are parity comparators and do not use the lifting described in this document: they apply Taylor operators after complete/global CFF energy integration and differ only by selector omission. Only route 3 reconstructs exact source occurrences and performs bounded EMR dispatch.

They are compared at several small and large momentum points. The strongest comparison evaluates the explicit-local-3D and projected-local-4D routes directly as 1000-bit Arb values and compares the Arb numbers without conversion through f64. This makes the fixture sensitive both to structural errors and to precision loss in the precision-escalation path. Because the two real graph numerator factors are odd affine functions of Q(4)^0 and Q(5)^0, while the original and mirrored UV cycles contain Q/Q/-Q and Q/-Q/-Q respectively, their passing three-route comparison is also an end-to-end sign oracle—not only a denominator-evenness test.

Complete example 5: UV and cograph topology remain separate

The fixture exact_cff_separates_uv_topology_from_the_cograph uses two parallel bubbles joined by a bridge:

    edges 0,1              edge 4              edges 2,3
 a ========== b -------------------------- c ========== d

Edges 0 and 1 are represented with the UV expansion mass; edges 2 and 3 use their physical masses. Even if canonical momentum signatures coincide after a contraction, the UV and cograph occurrences live in distinct topology domains.

The exact source therefore preserves both factors rather than accidentally joining them into one owner- or signature-based power chain. The related exact_cff_lu_residue_factorizes_from_quadratic_cubic_spectator fixture selects an LU residue on the cograph bubble and combines it with a cubic UV spectator whose routings are (+,+,-) and whose numerator is quadratic. It verifies that the combined LU residue equals the cograph LU factor times the independent UV factor, including the production prefactor bridge.

This is important for nested and disconnected UV structures: source-minor attachment keeps the domains structurally separate. Any later signature equivalence is repeated-channel algebra on the completed graph and cannot merge UV with cograph topology merely because their momenta look alike.

Complete example 6: numerator/denominator cancellation, including LU

The exact-CFF test exact_cff_uncancelled_powered_denominator_matches_lower_source constructs

[ .

]

The positive D(Q) remains a provenance-wrapped factor in the numerator. Its polynomial value is analyzed, while its owner/momentum/mass metadata are not double-counted as numerator powers. The exact powered source is compared with

[

]

and with the ordinary unpowered production CFF. The equality is checked at two exact rational points. This simultaneously tests:

• a raised denominator;
• a numerator factor which algebraically cancels one copy;
• an odd temporal factor Q^0+c;
• exact occurrence mapping; and
• normalization against the ordinary source.

The companion test exact_cff_uncancelled_powered_denominator_matches_lower_lu_residues performs the same comparison after selecting a physical LU raised residue. It checks the powered exact source, the lower exact source, and the ordinary CFF after assembling the complete fixed-cut functional: all generalized residue-map entries and all raised-order derivative pieces are summed before comparison. Any stronger per-order agreement observed in this fixture is diagnostic only.

At acceptance level, raised_cut_numerator_cancellation.dot contains two LU cross-section graphs:

• powered_cancel, whose edge numerator is ${𝑄}_1^{𝜇}{𝑄}_{1𝜇}-{𝑚}^2=𝐷({𝑄}_1)$; and
• lower_bubble, the corresponding graph with the cancelled propagator removed and the compensating graph sign.

The test raised_cut_numerator_cancels_one_propagator_in_both_orientation_modes evaluates two nontrivial momentum points in all three local-UV routes listed above. It requires the powered and lower graphs to cancel to relative 1e-12, verifies the combined process sum, explicitly sums orientations, and uses OSE-weighted, explicitly summed LMB channels. Integrated UV terms are disabled so the oracle isolates the local raised-cut and CFF machinery.

This is the direct LU analogue of the self-energy mechanism in which a factorized p_slash from the integrated UV result combines with a neighboring propagator numerator to form p^2 and cancel a denominator.

Complete example 7: the epem_a_ddx GL0 self-energy source

The scalar fixtures isolate the mechanism, but the motivating production case is the NLO epem_a_ddx self-energy graph GL0.

Structured generation traces show two different stages:

Thus the only genuinely raised physical EMR dependency is the quadratic bound on edge 3 in the contracted GL0 source. The two propagator edges adjacent to the self-energy insertion are physically related, but after the vakint result and its external EMR shift are remapped, edge 3 deterministically owns the remaining quadratic dependency; edge 5 remains linear. This is a physical-EMR provenance decision, not an LMB choice.

The physical degree-two bound is not itself permission to redistribute its factors. Raw source factors retain their owner occurrences. A completed local-4D projection must first certify their class binding; eligible hard factors with two class occurrences can then use a (1,1) proposal. The native map-row comparison selects among the bounded proposals. The physical-source trace above neither records that selected occurrence plan nor proves its map count.

GL2 supplies a useful contrast. Both of its contracted integrated-UV sources remain linear, as expected for the triangle correction which factorizes the Born graph. The GL0/GL2 comparison therefore tests that the generalized energy machinery is activated only where the self-energy numerator actually requires it.

State::generation_summary(...) exposes contracted-source diagnostics in each graph report’s stats.cff_energy_degree_bound_reports field. The DDx acceptance reads those reports and distinguishes the physical parent bounds from exact-source assigned bounds. They are generation diagnostics, separate from the ordinary graph retained by ProcessIntegrand; acceptance checks should use this summary rather than infer transient bounds from the final evaluator.

Complete example 8: multi-loop and rank-deficient exact sources

The approach is not restricted to one-loop bubbles.

The sunrise_pow4.dot shared-core fixture exercises a multi-loop repeated topology with degree-five numerator support on one occurrence. It verifies that generalized CFF constructs nonempty known-factor completion sectors and retains only energy surfaces in denominator trees. Other fixtures exercise a single-quadratic lower sector and high-contact completions on multi-loop penta topologies.

The GammaLoop fixture exact_rank_deficient_source_keeps_the_complete_active_direction begins with a two-loop graph formed by two triangles sharing a vertex. Its exact term retains only one denominator with momentum equal to the sum of the two parent loop carriers. The exact source correctly identifies one inactive loop direction and one complete active direction. It does not invent a physical owner by selecting an arbitrary LMB coordinate.

The companion initial-cut fixture verifies that a cut carrier remains a literal external alias even when a stored noncanonical signature contains a component outside the active exact-source span.

These tests show why LMB data and EMR ownership have distinct responsibilities: LMB coordinates express and reduce the exact signature space, but physical EMR provenance determines numerator identity and bounds.

Generalized-CFF behavior after the lift

Once the exact parsed graph and occurrence bounds have been produced, the shared three-dimensional-reps crate handles generalized residues. Its focused fixtures cover:

• a repeated box with branching denominator trees;
• one-loop quadratic, cubic, and quartic completions;
• box_pow3 with repeated quadratic recursive contact and remainder sectors;
• repeated high-power channels with a common auxiliary sampling scale M;
• multi-loop sunrise powers and penta contact sectors;
• several choices of which Q/Q/-Q occurrence owns an equivalent bound; and
• retention of distinct affine maps even when two terms have the same coarse orientation.

In particular, the shared-core fixture cff_repeated_occurrence_bounds_preserve_equivalent_physical_numerators compares the complete signed scalar and quadratic functions requested by

(3,2), (4,2), (5,2),
(3,1)+(4,1), (3,1)+(5,1), (4,1)+(5,1)

on a repeated Q/Q/-Q denominator at two deterministic points. The fixture reverses both endpoints and the full momentum signature, verifies the graph, and includes the reversed numerator sign. Each representation acts on the factor-local polynomial class specified by its bounds; equivalent physical numerators must yield the same complete function. Separate analytic quadratic and quartic contours retain independent mathematical oracles. GammaLoop’s exact source normalizes denominator signatures and restores the raw sign in its separate numerator mapper.

Higher-degree reconstruction interpolates one occurrence at a time and divides each interpolation basis polynomial only by that occurrence’s denominator. Contracting that edge passes an exact, factorized quotient to the existing lower-sector CFF completion. It does not replace several equal denominators by a single powered channel or impose a common sampling node on their numerators.

When a bound exceeds the direct interpolation range, generalized CFF may sample an EMR energy as a signed integer multiple a*M of the common auxiliary scale. This remains an occurrence-local EMR substitution. It is never an LMB rewrite, and the final physical expression is required to be independent of the nonzero value chosen for M.

Why the construction is generic

The confidence claim is deliberately scoped: the construction is generic for the factorized polynomial EMR numerators and rational denominator powers produced by GammaLoop’s local UV machinery. It is not a claim that every arbitrary symbolic function can be projected without certification.

Within that production class, genericity follows from the following properties.

1. Denominator multiplicity is arbitrary

Any representable negative integer power becomes the corresponding number of occurrences of its rewritten denominator. The serial-chain constructor accepts an arbitrary raised-wrapper multiplicity and adds exactly one fewer auxiliary vertices. Algebraically coincident occurrences retain separate bounds and samples regardless of how many distinct owners supplied them. No power-two or GL0 branch exists in the implementation.

2. Skeleton recovery is source-backed and rational identity is owner-free

The source edge and its cograph/UV minor provide the original skeleton, component domain, and attachment for every process and UV-spinney shape. This sidesteps the generic inverse problem of deriving graph incidence from momentum signatures. Within one raised source wrapper, domain, exact momentum signature up to sign, and mass must agree. Across owners, the same normalized signature and mass identify the same rational channel. Relabeling compatible owners can change physical provenance, but not the channel’s residue or loop rank.

3. Canonicalization is topology-preserving

Cograph and UV graph minors are constructed directly from the physical source, including contraction of absent edges. Explicit crown hedges retain non-vacuum external boundaries. Only after this scaffold exists does the owner-blind occurrence lift and Graphica relabeling establish deterministic rational equality and caching. No signature matroid, Kirchhoff system, or incidence search is part of exact-source topology construction.

The rank solve at source-topology validation is the two-candidate quotient-space test for the routing sign of an already attached owner: exact_row -/+ source_row must lie in the span of the opposite source domain. It returns one unique +1 or -1, or fails. It cannot propose endpoints, connect components, or synthesize an edge to repair vertex balance.

4. Numerator degree composition is structural

The analyzer implements polynomial composition rules over sums, products, powers, dot products, and multilinear functions. It does not recognize a particular numerator string or expand it into process-specific monomials.

5. Actual native map rows select among certified bounded proposals

The hard allocator preserves fixed owner loads and factorized sum/product rules using cyclic unit-factor assignment, compressed repetitions and greedy indivisible-block placement. Its nonlinear alternatives pack excess and reverse the best scored placement in one eligible pool. Production compares actual native source-map rows among K<=3 certified proposals, with descending degree envelopes and deterministic order breaking count ties. Quotient/remainder balancing is rank-optimal for freely assignable unit factors without fixed loads; this does not imply globally minimal map count. The separate soft-routing boundary retains its exact-basis/Pareto proposal construction.

6. Bound and evaluation cannot diverge

One assignment object drives both generalized-CFF bounds and later numerator substitutions. Generic future terms cannot accidentally take a different mapping path after their bounds have been certified.

Batching does not weaken this statement. Reuse requires equal canonical topology and identical per-occurrence capacity, while each term still evaluates through its own factor-local plan. Canonical topology identity includes external edges and affine boundary data, so reuse remains valid for disconnected and non-vacuum exact sources as well as vacuum UV factors.

7. Routing signs are first-class data

Canonicalization up to sign normalizes the denominator signature stored in the parsed graph. A rational-routing component then receives one coherent canonical direction, which can reverse that intermediate spelling. The numerator mapper composes the separate literal occurrence sign with the final parsed-routing sign and restores the physical numerator convention. This works for odd, even, scalar, and tensor numerator factors.

8. Provenance projection is channel-complete

Owner relabeling does not erase physical information. Occurrence-energy maps retain the particular source owner, while neutral cut support is the set union of all distinct owners in the algebraic channel. The union automatically deduplicates repeated copies of one owner and extends to any channel multiplicity. Adding raised-group representatives preserves compatibility with physical cut selection without replacing that owner set. None of these rules depends on a bubble, triangle, GL0, or a particular number of raised edges.

9. Unsupported cases fail closed

The implementation returns contextual errors for:

• a non-integer or unsupported energy-dependent denominator power;
• an opaque nonlinear energy function;
• a physical EMR degree with no certified literal exact occurrence;
• duplicate or overlapping candidate ownership;
• one source edge instantiating incompatible normalized denominators in an additive term;
• a source-backed exact topology which violates loop-momentum conservation; and
• a rank which has no valid source-carrier basis.

Failing instead of selecting a convenient owner or LMB coordinate is part of the genericity guarantee: adding a more complicated nested self-energy cannot silently fall through to a test-specific approximation.

Minimality and runtime consequences

The design minimizes work at three levels.

First, the graph lift is minimal: a power r requires exactly r occurrences, and raising one source wrapper requires r-1 auxiliary subdivision vertices. It does not clone the surrounding graph or algebraically cancel numerator/denominator pairs upstream.

Second, the numerator lift compares actual native map rows for a small set of cyclic/greedy proposals. Assigning four eligible class-owned unit factors as (2,2) instead of (4,0) can reduce reconstruction/contact work, but topology and complete-map coalescing determine the actual count. The bounded search may prefer a less balanced proposal when it produces fewer rows. Raw original factors remain fixed; completed class-owned hard factors use certified class pools, and soft routing uses its separate exact off-shell certificate. The additional contender has a generation cost of its own; a net runtime or memory benefit requires measurements rather than an envelope-only argument. All proposal and selection work preserves the factorized numerator.

Third, retaining natural Taylor denominator topologies avoids artificially raised propagator powers and compensating numerator factors. Canonical batching reuses expressions only for equal topology and identical per-occurrence capacity, avoiding rank inflation across independent terms. The real degree-one triangle therefore presents its base and singly dotted terms as separate source topologies, each with its own factorized numerator and capacity.

For genuine numerator factors, upstream algebraic cancellation such as D(Q)/D(Q)^2 -> 1/D(Q) is deliberately not performed. Supplying the proper occurrence bounds lets generalized CFF perform the required pinches and lower-sector reconstruction internally. Preserving natural Taylor topologies does not require such cancellation: it avoids creating the compensating factors in the first place, without making source incidence depend on process-specific numerator simplification.

Maintained invariants

Future changes to this path should preserve all of the following:

1. A denominator occurrence is never dropped merely because another occurrence has the same owner.
2. The original EdgeIndex survives the Taylor operator as source_edge.
3. Source owners and the original graph determine the contracted cograph/UV scaffold and physical attachments; signatures never reconstruct that skeleton or choose original endpoints.
4. Compatible owner-relabeling comparisons require valid source-backed lifts on both sides, with the same topology domain, mass and D(Q)=D(-Q) channel. Their equal rational residue is not permission to infer incidence from denominators. Moving a completed original hard factor to a class occurrence requires the independent exact numerator and signed-binding certificates.
5. Raising one source wrapper produces the requested occurrence multiplicity through minimal serial subdivision; the owner’s label is not subsequently part of rational CFF identity.
6. UV and cograph denominator node domains never merge.
7. Repeated occurrences of one owner must have the same normalized signature and mass, or generation fails.
8. +Q and -Q share a normalized denominator signature, while the literal sign and any later coherent component reversal are composed separately for physical-numerator mapping.
9. The quotient-space rank calculation can choose only the unique routing sign on fixed source endpoints; it cannot reconstruct or repair incidence.
10. Occurrence-energy projection retains the particular physical owner which supplied each occurrence, independently of owner-free rational incidence.
11. Cut support for an exact algebraic channel is the sorted, deduplicated union of all distinct physical owners which instantiate that channel.
12. Raised-edge representative aliases are added to unioned cut support for LU compatibility; they never replace or erase the original physical owners.
13. A physical numerator Q is reconstructed with the physical sign even when assigned to a -Q occurrence.
14. Numerator energy degrees use physical EMR variables or explicitly certified completed-UV class variables; those identifier types remain distinct.
15. LMB coordinates never own an energy bound or serve as an identity fallback.
16. The numerator remains factorized through analysis and mapping.
17. The same immutable per-term plan owns bounds and substitutions.
18. CFF cache reuse requires equal retained topology, semantic options and identical ordered per-occurrence bounds. Bounds of degenerate occurrences are never summed or redistributed; independent terms retain their own plans.
19. Additive branches reuse capacity; multiplicative and multilinear slots consume capacity.
20. Lower-sector pinching does not reassign a factor away from its certified occurrence.
21. Input factor order does not alter the canonically relabeled exact source; changing only a compatible physical owner changes provenance, not the rational graph or residue.
22. Non-vacuum external boundaries remain explicit source-crown hedges.
23. No internal edge or external-balance edge is synthesized from momentum signatures; an inconsistent source-backed graph fails validation.
24. Unsupported mappings fail explicitly rather than choosing a convenient edge.
25. Public initial-state cut slots carry the canonical Default direction; their signed fixed energies remain in the exact energy maps. This label normalization occurs after fusion and preserves unfiltered residue IDs and their order; it cannot supply missing denominator rank or re-sign a numerator factor.

Code map

Local Taylor construction preserves spinor products, chains and traces after metric simplification. Their registered multilinearity supplies structural energy degrees without evaluating a trace or expanding its numerator. The integrated-CT owner performs analytic Dirac simplification on its own copy before Vakint; the local numerical route leaves finite tensor contraction to evaluator preprocessing after residue mapping. These placements preserve the same Taylor coefficients while avoiding repeated scalar trace expressions in the local residue sum.

The principal implementation sites are:

• crates/gammalooprs/src/uv/approx/local_4d.rs

  • factorized term projection;
  • negative-power extraction into repeated FourDDenominator records;
  • outer Taylor-sum splitting without canceling positive typed GS.den numerator factors against negative denominator occurrences.

• crates/gammalooprs/src/graph/three_d_source.rs

  • exact source coordinates and occurrence ordering;
  • disjoint-set cograph/UV source-minor contraction from original source_edge attachments;
  • minimal raised-wrapper subdivision and owner-independent rational canonicalization;
  • normalized denominator validation and repeated-channel signatures;
  • explicit source-crown boundary completion;
  • unique +/- routing validation on already fixed endpoints;
  • post-construction Graphica node/edge canonicalization;
  • literal +/-Q candidate certification;
  • occurrence-to-physical energy mapping and routing-sign restoration;
  • channel-wide physical cut-support projection.

• crates/gammalooprs/src/numerator/energy_degree.rs

  • factorized physical-EMR degree analysis;
  • equivalent-candidate validation;
  • deterministic cyclic/greedy hard and certified soft proposals (K<=3);
  • immutable factor-local mapping plan.

• crates/gammalooprs/src/cff/generation.rs

  • physical degree extraction;
  • physical-to-exact planning and raw native map-row selection;
  • count-only contender memoization and winning-payload reuse, with keys retaining exact topology, options and occurrence bounds;
  • raw ordinary-source generation before persistent surface conversion.

• crates/gammalooprs/src/uv/approx/projected_4d.rs

  • single-pass proposal preparation and cached generation of completed local-4D Taylor terms;
  • retention of each term’s selected exact numerator plan and matching payload.

• crates/gammalooprs/src/uv/approx/local_3d/residue_localizer.rs

  • projected soft-proposal selection before surface conversion;
  • shared cut/hosting conversion for the winning payload and ordinary paths.

• crates/gammalooprs/src/cff/mod.rs

  • exact-source CFF assembly;
  • retention and use of the planned exact-source numerator;
  • physical cut and surface projection;
  • unioned physical-owner support with additive raised-edge representatives.

• crates/three-dimensional-reps/src/generation.rs

  • generalized residue, finite-pole, and lower-sector CFF generation from the supplied exact occurrence bounds.

Validation map

The most relevant focused tests are:

Together these tests cover parsing, source-backed graph construction, canonical relabeling, sign handling, factorized rank planning, generalized-CFF contact sectors, exact analytic residues, local-3D/local-4D equivalence, and LU raised cuts. No single numerical process test is being used as a substitute for the structural invariants. The table identifies test contracts, not a claim that every selection has passed on the reconstructed stack.

Design conclusion

The essential distinction is:

Original source edges recover the known UV/cograph skeleton and retain physical provenance; normalized denominator classes pool completed UV algebra without supplying incidence. Numerator factors follow physical provenance or an explicitly certified completed hard-class binding, and one immutable plan owns their occurrence bounds and substitutions.

Canonical D(Q)=D(-Q) normalization is safe because owner labels remain in energy provenance and cut-support metadata rather than contaminating the rational residue, while the raw routing sign remains separate data used to reconstruct the physical numerator. Raised-wrapper subdivision supplies the correct number of residue variables. The physical cut projection unions all owners of an algebraic channel and then adds raised-line representatives, so LU compatibility does not cost provenance. Explicit crown hedges preserve non-vacuum boundaries, and post-construction Graphica relabeling supplies deterministic cache identity without any topology inference from Kirchhoff or incidence matrices. The bounded search keeps the numerator factorized and chooses fewer native map rows among its certified proposals. Because the same selected plan controls both CFF generation and evaluation, the construction extends to higher powers, nested self-energies, disconnected UV components, LU residues, and multi-loop sources without a process-specific patch.