Architecture and system characteristics

Use this register for construction contracts and assumptions. For conceptual explanations, see Theory; for verification outcomes and exclusions, see Results.

This register separates the recursive relational software architecture, specified mathematical constructions, conditional dynamics, and proposed physical interpretations. Grouped-record promotion is an architectural specification; Gram restriction, geometric assembly, and quantum models have separate contracts. Their counts, metrics, and verification claims cannot be transferred without an adapter.

Status: checked means executable verification of the stated construction; conditional identifies required model choices; open identifies missing construction or physical acceptance; excluded rejects the specified branch. Kernel proofs, written analytic arguments, exact finite checks, and measured physical evidence are distinguished. This register summarizes linked research, not a new proof certificate. Checked mathematics does not establish a physical interpretation.

Current synthesis status

Track	Positive result	Boundary or exclusion	Source
Operational analytic completion	Research reports safe Agda closure for unit norm, metric regularity, unit-circle flow, and the assembled half-turn.	Standard-pi identification is an explicitly accepted written proof, not a kernel ordinary-real interpretation. Operational whole-return assembly remains unresolved; the standard-complex whole-return theorem is a separate written result.	Half-turn, completion net, standard return
Fixed-carrier Gram tower	Orthogonal restrictions compose and conserve retained-plus-record norm; rung spectra are r+10 and 10.	Probe selection, metric-isotropic preparation, Hamiltonian, and physical units are additional obligations. Rebuilt carriers are a different tower.	Restriction construction
Source-controlled geometry	One orbit/facet/cone rule reconstructs the declared tetrahedron, six-panel belt, and pentachoron with marked interfaces.	Metrics, generators, seeds, and interface roles are inputs. Matching geometry does not select a coefficient protocol or identify particles.	Common constructor
Conditional spinor quantization	Reduced and ideal endpoint-parent Hamiltonians are unitarily equivalent. Four conserved-flux blocks give an exact invariant 16-dimensional simulation sector.	No selected physical platform, measured calibration, or particle identification. Structural endpoint flatness is not exact finite-stiffness pinning of independent links.	Acceptance contract
Absolute mass readout	The common charging/displacement model has an explicit elementary denominator and a general splitting bound.	Matching the proton/electron reference bounds splitting by about 42.5 eV, not the approximately 1.293 MeV target. Common coupling and clock rescaling cannot repair this branch.	Charging-law exclusion
Typed constituent ports	Tensor-lifted belt rims retain both spin components; compatible pair operations can be specified.	The current common-vacuum graph gives repulsion or cancellation in breakup comparisons, not nuclear attraction. A sourced cross-carrier interaction is missing.	Rim and breakup test
Joint compatibility	Canonical quantum-state completion preserves the original annotations and all probabilities.	It adds zero prediction constraints in the tested class; intermediate locality is an extra selector. Distinct spatial/flavor/gravity source maps remain unspecified.	Falsified sufficiency claim
Braid composition	Preserving one attachment restricts endpoint channels to two or three components.	A separate unitary model leaves amplitudes free; 372:1 is an input constraint. The braid representation and sourced amplitude law remain open.	Composition model
Storage resources	Exact error-transparent interaction expansion and sine-profile midpoint optimization expose concrete resource demands.	The 769-term pulse is not directly a one-/two-body operator on the same qubits. The finite-work optimum is objective-specific; preparation, clock, and bath resources remain.	Branch comparison
Element/isotope catalogue	Observations and candidate construction records have explicitly separate provenance.	Coverage is non-exhaustive hydrogen-1 and hydrogen-2, not an explanation or catalogue of all stable isotopes.	Catalogue

These exclusions reject their stated models or sufficiency claims, not every possible carrier construction. They do not erase the positive identities and conditional constructions used in the tests. Theory synthesis · results and exclusions.

Retained comparison architecture

Organizing specification. The middle triple carries9:12->6,8:11->5 and 7:10->4. Vertical moves retain the label/source/target presentations; rung4 is the physical-reference locus. Candidate retained transports satisfy both squares and both complete routes to that reference. The intended horizontal generator and physical reference value/metric remain to be realized.

Source-backed structure. The comparison constructor retains its supplied pointed equivalence, both boundary packages and composition parents. Endpoints alone do not determine a witness; flatness is not imposed on the comparison type. The fixed-domain higher witnesses are distinct from a carrier connection. Actual carrier-edge operations and their attachment to rung4 still need an adapter.

Checked components. Shared paths supply109 mixed comparison relations. Reference changes and higher fillers have exact coherence tests. A marked joint probe adapter reconstructs all137 matrix responses. A separate witness/response packet preserves finite equivalence data without identifying it with the supplied amplitudes. Full mean/fluctuation transport accounts for averaging defects, and promoted-value composition distinguishes endpoint policies that share a mean.

Conditional dynamics. A preparation increment B and common witness transport have an associative retained update law. Updating the30 shared primitive legs preserves the137-slot factorization. One explicit feedback candidate computes B from retained incoming-family disagreement: it preserves family means, contracts active disagreement in a declared control metric, and retains enough data to reconstruct old members. It drives no change on family-constant fields.

Prior-source reuse and current gate. The earlier reversible carrier/record exchange already supplies an endogenous mismatch update. Its faithful packet retains16 conserved coordinates and137 mismatches; its averaged contraction transfers mean-square budget into covariance. A fresh adapter test rejects identifying that mean law with the newer32-family mean-preserving relaxation. Keep the latter as a distinct candidate, not a derivation of the older dynamics. A conditional source-labelled experiment now maps actual native matrix responses to declared coherent preparations, validates witness boundaries and carries the faithful packet through exchange and detector readout. Independent memory, trace gain and physical controls are explicit inputs. Shared-leg state closure and strict operator factorization fail; the adapter preserves event composition, not that stronger native descent. Native instrument selection, physical standards and absolute coupling remain open. Another generic preparation law or normalization by137 does not close them.

Model boundaries. Family promotion is the selected record architecture. Independent products, Gram restrictions, biclique fillers and probe realizations are separately specified constructions until an adapter identifies them. Their ranks and costs must not be transferred implicitly. The1,2,4 product-homology result remains conditional; no new physical constant or gauge identification has been derived. Probe/context choice, measure and metric are still inputs.

Public synthesis: theory and results. Detailed current gate: retained source comparisons, amplitude source audit, preparation and transport, and family feedback. The sections below retain the individual constructions, assumptions and checks.

Structural characteristics

Characteristic	Precise statement	Assumptions	Status	Verification / source	Open question
Carrier probe realization	On S12, f_i indicates permutations fixing i. Counting overlaps gives diagonal 11! and off-diagonal 10!; dividing once by 10! gives G12=10I+J.	Stabilizer-indicator probes; counting inner product; fixed normalization.	Checked	Gram tower, checker	What selects these probes physically?
Nested tower	V_r is the span of the first r probes, r=4,...,12. The induced Gram is G_r=10I_r+J_r.	One ambient S12 carrier; chosen witness order.	Checked	Gram tower and checker above	Derive the witness order and the physical significance of the floor.
Gram spectrum	Common coefficient direction has eigenvalue r+10; r-1 contrast directions have eigenvalue 10.	Fixed-carrier Gram realization.	Checked	Gram tower checker	These norm eigenvalues need a dynamical operator before they represent excitation energies.
Restriction map	From r to r-1, surviving coefficients become y_i=x_i+x_r/(r+9), using one-based indices. Nested orthogonal restrictions compose.	Gram-orthogonal projection; remove the last probe.	Checked	Gram tower checker	Identify the actual witness operation with this projection.
Retained-plus-record budget	B_r=B_(r-1)+D_r, with D_r=10(r+10)x_r^2/(r+9). The signed residual permits reconstruction.	Quadratic Gram norm; orthogonal restriction; records retained.	Checked	Gram tower checker	Derive a physical energy observable and its units.
Mean linear descent	Mean retained fraction is r/12 for coefficient second moment C=G12^-1.	Excitation isotropic in the Gram metric; fixed budget normalization.	Conditional; checked	Gram tower checker	What prepares this excitation distribution?
State-dependent descent	The removed fraction depends on the state. Rank reduction alone does not fix an energy fraction.	Specified state, restriction, and metric.	Checked	Trial descent, checker	Determine the system's state or ensemble.
Two pointed carriers	Two four-point carriers identified at one reference have seven labels and six independent scalar potentials after the reference is pinned.	Shared-point identification; scalar potential model.	Checked construction	Shared-state hypothesis, observation construction	Select their realization in the ambient carrier. Seven labels alone do not establish a tower rung.
Comparison domain	144 arrow-pair slots plus nine state-pair slots give 153. Twelve outer labels give 1836 comparison attempts.	All ordered nonidentity arrows; shared reference; specified labelled programme.	Checked	Shared-state hypothesis; programme checker	Select outer-label-dependent dynamics and the physical resource readout.
Tetrahedral projector geometry	Twelve three-axis fibers give dimension 36; ranks of L,R,N are 12,3,1 and LR=RL=N. The specified three-stage supports are 108,432,1296.	Natural axis basis, distinct stage ports and dense collective feedback; not an invariant minimum.	Exact construction	Geometry	Supply a physical Hamiltonian rather than identifying the arrow count with mass.
Binary tetrahedral bridge	The cover 2T to A4 restricts to Q8 to V4; End(C^2)=1+3 recovers the vector action by Pauli conjugation.	State space and operator space remain distinct; scalar and vector operator sectors are central-even.	Exact representation comparison	Bridge and checker	Explain the physical selection of an extra spinor factor.
Native spinor-selection test	Native selected-line transport composes through endpoints; all six half-turn commutators return +1. A projection-only triangle can return -13/3375 without the required spinorial commutator.	Original geometry/filter operations only; no spinor input or physical apparatus supplied.	Native-source conjecture rejected in this model	Native-loop audit	Physical branch deferred until an independent rotation interaction and coherent readout are supplied.
Potential observation	For carrier coefficients c, q_i=<f_i-f0,psi>=10(c_i-c0). These are readouts, distinct from expansion coefficients in a contrast basis.	Seven selected probes; carriers {0,1,2,3} and {0,4,5,6}.	Checked	Observation square, checker	Derive the observation map from the intended physical comparison.
Readout metric	For n surviving readouts, the minimum compatible carrier norm is q^T M_n^-1 q, with M_n=10(I_n+J_n). Orthogonal hidden state adds further norm.	Linear readout map; least-norm representative.	Checked	Observation square and checker	Specify hidden state and its dynamics.
Observation commutes with descent	Observing after carrier restriction equals deleting the missing readouts after observation. From 12 through 7, all six readouts survive unchanged.	Fixed probe embedding; highest-label-first deletion.	Checked	Observation square checker	Test alternative witness orders and carrier realizations.
Comparison survival	At target rungs 7,6,5,4, respectively 153,78,27,0 original slots are evaluable from live readouts alone.	Same embedding and deletion order; original 153 functionals.	Checked	Observation square checker	Characterize survival across all admissible orders.
Consistent surviving updates	At 7->6, all 78 descending comparison projections commute with readout deletion when using the induced inverse metric.	Metric-orthogonal equality updates; rows independent of the deleted readout.	Checked	Observation square; matrix identity and seed checks	Construct record-dependent updates for the other 75 slots.
Minimal deleted readout record	At six-to-five readouts, B6=B5+(3/35)(q6-s/6)^2, where s=q1+...+q5. Recording q6 also reconstructs all original comparison values.	Least-norm readout budget.	Checked	Observation square checker	Integrate raw comparison records and orthogonal budget records in one state representation.

System as a software architecture

The system is a recursive relational index with retained history. Within each triple, the same records have three presentations: individual labelled records, records indexed by from-label, and records indexed by to-label. Every group retains all matching records.

Between triples, summarize the grouped records and assign each a fresh unique label. The resulting interface is fresh label ↔ grouped record (1:1). The many:many structure remains inside the record through retained members or links. This is family promotion. The next level organizes families of relations, then families of families.

Conceptual pseudocode for promotion:

groups = summarize(current_records, grouping_keys, retain_members=True)
next_records = addcolumn(groups, "label", unique_index)

retain_members=True states the preservation requirement; ordinary aggregation or deduplication alone does not preserve all member records. Grouping keys, next-level endpoint fields, and label lifetime are part of the implementation contract. A fresh index identifies a group within its declared namespace; its stability across later updates requires an identity policy.

Tower presentations

The gauge column records the proposed correspondence. Relational squaring of the full four-label off-diagonal relation now gives a candidate1,2,4 quantity: the degree2 Betti number of its Dowker complex. Identifying that operation with the tower's family promotion remains open; the137-slot missing-arrow complex has no positive-degree rational homology.

Rung	Gauge assignment	Record presentation	Records describe
12	U(1)	Individual labelled paths	Original relations
11	SU(2)	Paths indexed by from-label	Original relations
10	SU(3)	Paths indexed by to-label	Original relations
9	U(1)^2	Individual freshly labelled family-records	Families of original relations
8	SU(2)^2	Family-records indexed by from-label	Families of original relations
7	SU(3)^2	Family-records indexed by to-label	Families of original relations
6	U(1)^4	Individual freshly labelled family-of-family records	Families of families
5	SU(2)^4	Those records indexed by from-label	Families of families
4	SU(3)^4	Those records indexed by to-label	Families of families

The operator-confirmed transport roles are 9:12->6, 8:11->5, 7:10->4. The middle triple presents transport between the outer triples:

12 --9--> 6
 |        |
11 --8--> 5
 |        |
10 --7--> 4

Vertical arrows retain the label/source/target presentations. Both squares must commute, so the routes12->6->5->4 and12->11->10->4 give the same rung4-relative observation. Rung4 is the designated physical-reference locus. Local reanchoring must retain its offset to that reference. Family promotion introduces identities for retained groups; it does not by itself specify the horizontal transport.

Diagram realization and checker verify both squares for two explicit retained-grouping candidates. All137 leaves and their masses reconstruct, and both routes give the same fixed-reference residual. The candidates still differ at the target (32 versus1 records), so presentation coherence does not select the generator. Reference value and metric at rung4 remain to be realized; the checker supplies a fixture value.

Anchored normalization gate

Fixing the reference locus and its unit norm does not select a metric. Three conditional retained Gram-chart models G_r=aI+bJ satisfy the clarified squares, orthogonal budgets and the same unit rung4 reference while giving different contrast costs and transports. The existing S12 stabilizer-indicator probes with counting measure select a=10,b=1, hence a normalized bottom contrast coefficient 5/7; that is conditional geometry, not a physical coupling prediction.

Anchored Gram audit and checker verify the alternatives. An adapter from the Gram charts/probes to retained label/from/to family records is required before this Gram construction can be identified with the intended horizontal generator or rung4 physical metric.

Concrete carrier-probe adapter

Label the twelve S12 fixed-label probes by the actual directed arrows and restrict along the induced S4 action. This equivariant adapter has rank6: opposite-arrow probes coincide, with six antisymmetric coefficient records restoring the source. Four state probes add independent directions; a3-cycle shows that arrow probes alone cannot reconstruct the state branch.

Probe adapter and checker compute the transported visible Gram5I+J versus uniform-carrier GramI+J; no scalar normalization identifies them. On the121+16 slots, fixed-label probes have rank52, while transition probes have rank88. Uniform product counting gives unequal arrow/state slot norms and nonzero cross-block overlaps. A measure that matches the source arrow metric still leaves3-cycle weight and state metric free. The adapter therefore requires an explicit state/readout branch before identification with the rung4 physical normalization or full rung transport.

Joint probe-to-response adapter

The joint carrier field now carries the actual slot residual matrices. Its factorized arrow/state construction reproduces the137-slot assembly under an explicit identity-pair readout. State responses and arrow reversal-class sums have a direct decoder;85 within-class coefficient records restore every named slot. A faithful alternative uses a marked reference edge and an outside state basepoint, giving a rank137 adapter with exact matrix decoding.

Joint response adapter and checker verify both. Uniform carrier averaging differs from equal-slot averaging. The endpoint-fixing feature envelope closes as52->113->121. For the faithful marked adapter, a valid disjoint-context example shows decoded averaging fails to commute with pointwise composition. Independent slot pairing restores the mixed term exactly; retaining the multiplicativity defect is another possible coherence policy. Context choice and the physical rung4 observation/normalization remain inputs, and feature ranks are not physical degree-of-freedom counts.

Mean/fluctuation packet transport

For declared same-slot reference composition, retaining the mean and every centered slot response is an invertible change of coordinates. The exact composition law generates an ordered covariance correction rather than adding an independent coefficient. It is associative and unital. Within-family plus between-family covariance makes both source- and target-indexed grouping routes agree with the full retained response, for uniform and nonuniform weights.

Packet transport and checker prove the algebraic identities and test the137-slot matrix fixtures. A normalized mean cannot preserve all same-context products unless it selects a single slot. A valid shared-leg sign-reflection control also has equal means and second moments but different third iterates. A mean plus one binary correction is not therefore a generally closed successor state.

Independent-context pairing is also associative and generally changes the response. The next source constructor is the intended generator's joint context incidence and transported weights. This packet result does not identify slot multiplication with marked-probe pointwise multiplication, supply the horizontal rung maps, or select a physical metric. Its correction identity is not a claim that nonzero matrix defects are null-homotopic in the earlier graded model.

Joint incidence from promoted-value composition

Using the actual endpoint keys of the rung fixture, composing each promoted family mean through the reference expands into pairs of its retained leaves. The coefficient of a pair i,j in family F is m_im_j/(MM_F). Direct and staged expansion agree; both marginals retain the original leaf masses. This is an algebraic consequence of composing means, not physical statistical independence.

Context incidence and generator separation and checker show that the inherited-key candidate has32 final families and977 supported leaf pairs, whereas the common-target candidate has one family and18769 pairs. Their first-moment responses agree, but the promoted-value composition readouts differ by nonzero between-family covariance. Memberwise composition adds the within-family correction. These counts describe probe support, not physical fields or promotion costs.

The context law is determined once the operation and endpoints are specified. Selecting which family ports are composed, and whether the operation uses promoted values or retained member packets, remains the source-constructor gate. This result uses the rung fixture's32 endpoint families, not the earlier packet checker's15 leg-index families.

Record-port composition and reference-return lift

Literal target-to-source composition is not already supplied by the two endpoint fixtures. It admits916 leaf pairs but cannot preserve both uniform marginals: endpoint ('a',0,0) has nine incoming versus four outgoing records. Both promoted policies have disjoint source/target namespaces and no literal composable pairs. Matrix composability alone therefore does not complete their record ports.

Endpoint gate and typed lift and checker construct a conditional repair: attach a parallel reference d_a and return r_a to each record, then compose br_ba when a and b share a target. The result retains a's source/reference, and triple compositions agree. This incoming-family rule reproduces the977 versus18769 expanded leaf-pair supports of the two horizontal candidates without changing leaf masses or the137-slot count.

Reference attachments and next-level endpoint selection remain source inputs. The checked lift does not identify a physical generator or derive its metric; restoring excluded records or changing weights are controls, not adopted repairs.

Flat transport of the fixed reference

A declared invertible port frame P_v now realizes reference attachments as d_a=P_tP_s^-1, with actual maps P_td4^-1C_aP_s^-1. Decoding recovers exactly C_a and the fixed d4. Common-target reference composition and both promotion rounds preserve the original matrix readouts, including when family targets differ. Independent passive frame changes leave decoded values unchanged.

Flat reference attachment and checker verify this conditional realization. An invertible reference twist on a genuine carrier cycle produces nonidentity holonomy that no frame change erases. Thus local reference inverses do not prove global path-independent attachment. Source flatness or retained path-indexed transport remains required, alongside endpoint-generator selection and the physical metric. Reference flatness here is not a claim that the actual comparison responses have zero curvature.

Source audit of reference flatness

The reference-witness audit and checker distinguish three types: canonical anchor-coordinate changes telescope exactly on171 anchors;28 shared-leg witness routes agree strictly while109 have nonzero differences filled by existing degree-two cells; carrier reference transport would require a separate edge/route assignment. The degree-one A->B comparison witnesses are not themselves degree-zero transports on the32 carrier endpoints.

Neither flat nor curved carrier reference transport is selected by those witnesses. The missing source adapter must declare elementary edge operations, port spaces, degrees, ordered composition and the route-preserving rung4 observation. This returns to the existing route-calibration programme's source instantiation gate; another freely chosen connection would not resolve it.

Conditional permutation-edge classification

If an arrow i->j acts by an S4 permutation carrying i to j, and the assignment is equivariant under every carrier relabelling, there are exactly two lifts: swap i,j while fixing the other labels, or swap both complementary pairs. The first has nonidentity triangle holonomy; the second is globally flat. The distinction survives deleting the counted reference arrow. A loop's basepoint alone cannot detect the first lift's spectator permutation.

Equivariant lifts and checker exhaust the finite possibilities. The source discriminator is the action on the two nonparticipating labels. Active permutation semantics itself remains an assumption, and neither these permutation groups nor their loop data are yet identified with a physical gauge group or the shared-leg DG witnesses.

Source-backed swap sector and its boundary

The existing Agda fourQ swap has images(0,2,1,3): it fixes the distinguished value0 and the other spectator3. Its conjugates under pointed relabellings give the three unmarked-state transpositions, with nonidentity triangle loops. The flat double swap moves0 and is not an endofiller of this fixed package.

Source swap audit and checker verify this from the existing source definition and image certificate. They also check the scope hostile: between differently pointed packages, both lifts are admissible; the general Filler signature does not select an involution. The pointed swap sector therefore supplies a source-backed nonflat candidate, not an identification of all12 carrier arrows or the physical rung4 reference. The next identification is fixed-package operation versus boundary-changing comparison, with the actual filler retained rather than its existence alone.

Source comparison retains its witness

The actual compare-rule accepts both complete boundary packages and a supplied pointed equivalence. Its output retains that equivalence; composition retains both parent packages. It permits fixed-package and moving-boundary comparisons rather than selecting the fourQ swap universally.

Finite source groupoid and checker realize four differently pointed copies of the carrier:96 comparison payloads, six witnesses per endpoint pair. Relative to root frames, the retained witness is an S3 stabilizer element; its composition and frame-change laws reconstruct the exact permutation. A nonidentity loop cannot be removed by changing frames. Setting every witness to identity is a special flat section, not a generally faithful coordinate change. These are finite witness counts, not replacements for137 slots or a physical gauge identification.

Thus flatness need not be imposed on the source comparison type. Retain supplied witnesses and route parents, and test the actual loop data. The next adapter must map these typed source comparisons into the137-slot matrix model without silently forgetting their isotropy witnesses.

Witness-preserving matrix packet

A two-channel adapter now retains the actual matrix C and a faithful S3 witness matrix rho(h) as separate blocks. Reference-bridged composition preserves both C2d4^-1C1 and h2*h1. All137 original responses and their mixed rectangles remain unchanged, and a zero response does not erase its witness. Sample witness assignments are explicit inputs; the block dimensions are not physical field counts.

The checker also rules out identifying the existing matrices with a functor from the finite pointed comparison groupoid: its based loops have sixth power identity, whereas the fixture's base response normalized by d4 is I/2. Witness absorption into an unrestricted response matrix and witness averaging both lose information. The next source constructor must explain response amplitudes and their relation to retained equivalence witnesses; the direct block encoding supplies no physical interaction or normalization by itself.

Amplitude preparation remains a source input

The amplitude source audit and checker locate the numerical inputs: the shared-leg matrices assign their coefficients, and the retained comparison series transports a supplied initial field. The separate scalar scattering benchmark supplies field-theory weights explicitly; it does not derive these matrix amplitudes from the carrier count.

A centered determinant-one shear family preserves the137-slot topology, formal witnesses, fixed reference and total mean. Sign reflection preserves a declared scalar Frobenius cost but reverses nonzero mixed responses; normalized slot traces distinguish the fixtures even under passive frame transport. Thus mean, reference and this scalar cost do not select amplitude preparation. The next physical input is a source preparation/evolution law and its action with the retained witnesses. Matrix values remain fixtures until that identification.

Boundary preparation and witness-dependent update law

A conditional local law now separates Prepare(B):A->A+B from Transport(h):A->rho(h)Arho(h)^-1. Their ordered composition is (h2*h1,B2+Transport(h2)(B1)). The checker verifies associativity, inverses, retained parents, port-frame covariance and fixed-reference decoding. Applying a common witness action with explicit preparations to the30 primitive legs preserves the137-slot factorization and all rectangle identities; the slot cross terms follow from multiplying the updated legs rather than an independently fitted coupling.

Witness/endpoint-only coherent affine kicks on this finite rational groupoid are boundary-potential differences, by incoming-arrow averaging. Genuine preparation history therefore cannot simply be compressed into the finite witness: two prepared transposition operations can have identity witness and nonzero accumulated amplitude. Preparation magnitudes and schedules remain source inputs; this conditional algebra is not yet physical dynamics, arbitrary slot evolution or the full intended rung transport.

State-derived family preparation candidate

A family-disagreement law now computes B=eta*(P*T_h(A)-T_h(A)), where P is the retained incoming-family mean on primitive legs. This explicitly postulates relaxation toward family agreement; it is not derived from promotion alone. The checker verifies30 shared legs,137 slots and the actual32 target families. All family and global means survive, active leg disagreement scales by(1-eta)^2, and slot fluctuation cost splits into quadratic and quartic scaling. Retained kicks reconstruct old members even at full projection.

The rule respects passive carrier relabelling and common witness transport. It prepares no change on family-constant data and does not choose initial amplitudes or their means. Relaxation, gain and the invariant control metric still require source/physical justification; no energy-dissipation or numerical constant claim follows from the finite contraction law.

Prior exchange reduction and failed family-feedback identification

The exchange adapter test reuses the earlier16-carrier-plus137-record reversible source. Its retained coordinates a=q+U^Tw and delta=w-Uq reconstruct the full state. Every exchange fixes a. Under the declared lazy uniform schedule the mismatch mean evolves by I-(I+U*U^T)/274, while the exact covariance channel accounts for all mean-square budget decrease. No physical energy loss follows from averaging.

The checker reruns the old closed-record fixture and verifies reconstruction, budgets, moments and a separating singleton-family example. The mismatch mean has no nonzero fixed directions; the newer slot-family projector fixes32. A linear intertwiner from mismatch means cannot preserve nonzero family means. This is not an exclusion of nonlinear, history- or covariance-enriched adapters.

The old model already supplies an endogenous B-like response. Its source-derived kernel, fixed information and covariance must be retained. The family-feedback candidate remains separate; the old stationary1/137 result cannot be transferred to it. Identifying exchange variables with the matrix-leg/rung model and physical charge/current observations remains the adapter task.

Conditional source-to-exchange experiment: ten-iteration audit

The consolidated deliverable provides a reusable boundary-validated experiment compiler, actual native matrix-response seeding,16 conserved anchors plus137 mismatches, covariance, source provenance and ordered histories. Declared oscillator pulses and a loss/gain/noise/phase detector model give signed-echo and rectangle falsifiers. The final checker freshly runs the resource/end-to-end audits and eight supporting checks.

The scope is conditional: a shared-leg trace preparation is not exchange-closed, native boundary-response witnesses are not writable record memory, and all109 rooted rectangles reject invertible shared-leg operator factorization on the same state space. The compiler attaches independent instrument memory and preserves event concatenation rather than asserting that rejected factorization.

Normalization and resources are separately checked. Vacuum already gives equal raw1/137 mismatch shares; unknown preparation/loss and phase jumps obstruct uncalibrated interpretation. Record-only reset leaves carrier excitation. Full swap-reset transfers state and energy to an environment; apparatus overhead remains unmodelled. Ideal excitation conservation is not a complete energy law.

Conditional finite-sample Chebyshev designs are included, not experimental results. The remaining constructor is an independently justified native action or calibrated apparatus contract selecting memory, preparation and controls, followed by a normalized physical current/field action. The stronger physical objective remains explicitly open.

Optical plant and endpoint-phase synthesis

The optical/endpoint verdict combines the prior four-port cavity plant with the137-feature endpoint-phase adapter. The frozen lossy plant is not exchange (retained singular values1 and .48). Its ideal lossless swap boundary realizes all137 addressed exchanges with transported carrier supermodes and retained record/environment ports. Extending the prior zero-phase fixture requires the return phase in both branches of the subsequent loss coupler; the combined model checks this correction explicitly.

Independent complex port tomography, phase-sensitive router bounds and timed reference records give a prospective error certificate. The synthetic four-event bound.04391744 passes the predeclared tolerance.08. Delayed maps use distinct input/output frames; omitting the connection or silently resetting environment memory fails. An uncertified upper bound is inconclusive, not itself a rejection.

The final checker freshly reruns the synthesis and original checkers. The bounded mathematical combination is complete; measured hardware realization remains unverified. Required input is independently acquired port/router/clock calibration, an explicit storage/environment contract and held-out apparatus observations. Synthetic evidence and the1/137 share cannot supply those physical controls.

Repeating the promotion rule

Repeating incoming-family promotion with the current child-endpoint footprint policy predicts137 ->32 ->32 ->32 records. A family's child-target footprint is the singleton of its grouping key, so distinct families have distinct promoted targets. Every later incoming family is a singleton wrapper. This is a general obstruction for promoted targets that injectively encode their incoming key.

A control using a common promoted target instead gives137 ->32 ->1 ->1. Both policies preserve retained records, weighted means and direct/staged returns. Thus those coherence tests do not select the next-level relation or derive the 1,2,4 progression. The earlier source-then-target transaction fixture uses a different schedule from the tower's repeated incoming promotion.

Generator obstruction and prediction and checker. The next construction must specify relations between promoted families that supply shared next-level endpoints and genuine non-singleton higher families.

Family incidence and cycle records

The source/target family incidence graph has64 vertices,137 retained leaf edges and17 components: one32-vertex arrow component and16 state pairs. Its signed incidence matrix differs from the joint mean map only by invertible row scaling. Therefore the readout rank is64-17=47 and its hidden edit space has dimension90.

Compatible simultaneous mean increments have equal member-count-weighted source and target totals in each component. A spanning-forest solver realizes every such request. The90 fundamental graph cycles provide explicit independent edits invisible to all family means. They reconstruct the residual of any leaf edit after a compatible forest lift; the minimum-change lift chooses a different, orthogonal representative of the same readout.

Incidence derivation and cycle records and checker. Promoting connected components yields17 groups with only self-relations and then stabilizes. The next recursive candidate must specify how retained comparison cycles become higher cells, beyond this stationary component quotient.

Higher comparison cells with explicit boundaries

The retained incidence graph now generates local comparison cells. Every complete K(2,2) block gives a four-edge rectangle boundary:672 cells span the90-dimensional hidden edit space. Complete K(2,3) and K(3,2) blocks give1304 three-rectangle boundaries of rank582, filling all582 relations among rectangle boundaries. The next kernel has dimension722. Boundary-of-boundary checks pass exactly.

A common local rule assigns dimension p+q-2 to each complete K(p,q) block for p,q>=2, with alternating deletion faces. The1074 dimension4 cells now have verified boundary rank722. Continuing the same rule gives416 dimension5 cells of boundary rank352 and64 dimension6 cells of boundary rank64. All positive-degree rational homology vanishes; degree0 retains17 component classes. A spanning-forest control attaches only90 independent2-cells and terminates immediately, showing why an independent-kernel basis alone does not generate further layers.

Higher comparison-cell construction and checker. Complete finite hierarchy and full-complex checker verify every consecutive boundary composite, exact rational ranks, termination at dimension6, and an orientation-sign negative control. These cells record comparisons of retained edits. The cell counts and boundary layers have no established identification with the tower's1,2,4 superscripts.

Dowker geometry and a candidate doubling invariant

The higher comparison complex has the homology of the source Dowker complex: its simplices are source families sharing a target neighbor. Independent exact simplicial computations match the biclique ranks on both fixtures and three small controls. The full four-label off-diagonal primitive relation gives the tetrahedral boundary S^2; removing arrow(0,1) removes one triangular facet and leaves a disk. Arrow pairing gives the product homotopy type.

Restoring the arrow changes137 slots to160 and produces Betti numbers b0=17,b2=2,b4=1, with arrow component S^2 x S^2. The original arrow component has type D^2 x D^2. Rectangular closure of Dowker faces explains the product. Iterated relational squaring of the full primitive relation gives (S^2)^(2^k), hence b2=1,2,4 and Poincare polynomial (1+t^2)^(2^k).

Dowker identification and recurrence and checker. The remaining architectural bridge is to derive relational squaring from family promotion and determine how the omitted-arrow/reference choice treats the primitive sphere. The gauge correspondence remains separate.

Independence test for the promotion step

Pulling the primitive sphere generator back along the two pairing projections gives rank2 for independent arrow pairs, rank1 for synchronized/equal-source/ equal-target pairs, and rank0 for composable pairs. Explicit fixed-coordinate sphere cycles witness both independent classes in the144-pair construction. Sharing a fixed reference tag preserves independence; equating variable endpoints changes it.

Retained incoming promotion sends12 primitive arrows to four family labels and reconstructs the same12-arrow relation with b2=1. In the existing fixture the two factors enter earlier, in the explicit product-of-arrows constructor. Grouping that record set has not supplied another independently variable operand.

Promotion independence audit and checker. An architectural doubling step requires an explicit new comparison constructor, its two projections and admissibility/reference rules. Independent all-pairs comparison passes the projection-class test; endpoint-constrained alternatives measured here do not preserve both classes.

Recursive comparison constructor

An explicit independent all-pairs constructor now follows retained-family promotion. For incoming fibres F_v, F_w, its members are F_v x F_w, and (q x q)^-1(v,w)=F_v x F_w makes grouping-then-comparison agree exactly with comparison-then-grouping. Applying the same rule twice gives operand arities1,2,4 and incoming-family counts4,16,256. Full primitive arrows give record counts 12,144,20736; the omitted-arrow relation gives11,121,14641.

Recursive family-comparison construction and checker verify both route equalities, complete tuple coverage and parent provenance. Explicit coordinate sphere cycles have independent projection evaluations at all three full-relation stages. Together with the product theorem this gives b2=1,2,4 and four-factor Poincare polynomial (1+t^2)^4.

All-pairs comparison is a declared architectural operation. Omitted-arrow disk factors describe the counted subrelation; the documented direct reference is retained separately. Auxiliary state-pair slots and scalar comparison responses have no new recursive rule in this constructor.

Reference semantics: retained data and selected slots

The earlier theory/assembly specification retains the direct reference d:A->B and excludes it from counted comparison slots. The matrix checker verifies that a reference-only change alters the residual while all137 slots stay fixed. The11-arrow construction therefore describes a selected comparison view; interpreting it as deletion from the full retained carrier adds an assumption.

An explicit marked-carrier check retains12 primitive arrows and selects11. Full pairing has144 entries, of which121 are counted and23 touch a marked reference. The16 state slots give137 counted entries. Changing the mark changes selection without altering retained incidence. Full primitive Dowker geometry remains S^2; the selected subrelation is D^2.

Reference-semantics audit and checker. The next adapter must relate the assembly's direct reference d to local marks q_A,q_B and comparison-leg maps. Composition through a fixed type U permits all independent leg pairs; equating variable primitive endpoints is a different rule.

Equivariant reference selection

A stabilizer audit finds no local-arrow pair fixed by the symmetries of bare four-element carriers with one cross-carrier endpoint reference, a carrier bijection, or a rooted bijection. Two ordered anchors per carrier plus an orientation rule suffice.

Retaining the documented Bool x Bool square structure gives another route: a rooted square has a unique antipode. The outward root-to-antipode arrow is equivariant under all square symmetries and leaves11 other arrows. Choosing an adjacent arrow requires an axis choice. Under standard binary labels the antipodal mark is(0,3), while the earlier(0,1) mark is adjacent.

Reference-selection audit and checker. The carrier definition in theory section1 explicitly retains only a finite set and S4 symmetry, so the square selector would add structure. Once endpoint roots and an outward-arrow convention are supplied, the allowed selectors form a nine-context family with no equivariant preferred member.

Reference-choice family audit and checker verify covariance under all576 independent relabellings. Every context has137 slots; the1233 context-tagged records cover160 underlying slots. Selecting one context reduces its stabilizer from36 to4. The family can be retained without selecting or averaging; optional uniform context mixing gives nonuniform underlying slot weights.

Context descent and orientation transport

Mean readouts and uniform returns commute with relabelling of both context and member data;324 exact cases pass. Each context change has four witnesses. The45 stabilizer-orbit means form a witness-independent interface, with induced member-count return costs. Full member residuals retain witness dependence.

At fixed member data the nine context means are independent functionals. Their return gains are1,126/137 or116/137 according to whether zero, one or two local marks change. A reference-context choice can therefore affect a readout.

On the full product's H2, context stabilizers independently reverse the two sphere orientations. The class space retains rank2 but its common invariant vector subspace is zero. This gives an explicit orientation representation over reference contexts; rank and sign-invariant readouts can descend without a preferred signed generator.

Context descent and orientation and checker. Next extend this action through recursive comparison and classify compatible readouts/metrics. Metric normalization is additional data.

Recursive orientation readouts and return metrics

The rank1,2,4 class spaces now carry compositional independent-sign actions. Allowing operand interchange at each binary comparison node gives signed tree groups of orders2,8,128. Exact invariant-space calculations give no nonzero invariant linear scalar and one common-scale quadratic metric at each rank. With ordered operands and signs alone there are respectively1,2,4 diagonal metric parameters. An additional additive-budget law carries one scale across ranks. Least-change returns commute with transport when their covectors move too.

At rank4, tree-preserving symmetry permits three quartic invariants: a sum of fourth powers, a sibling-pair term, and a cross-block term. Full leaf-permutation symmetry leaves only two. The higher readout therefore detects whether pairing history is retained, even though the quadratic norm does not.

Orientation/readout classification and checker. These metrics live on oriented homology coordinates; the relation to leaf-edit cost and a physical readout is a further construction. A positive-norm return from zero requires directional request data.

Rebracketing and quartic descent

The independent comparison constructor now passes103680 named-member reconstructions across the five ordered four-factor bracketings. All120 routes including factor exchange preserve content while retaining their parent histories. A readout that descends across those routes equates the sibling and cross-block quartic coefficients, leaving alphasum(x_i^4)+betasum_(i<j)(x_i^2*x_j^2).

The interface law (Q,F)(R,G)=(Q+R,F+G+betaQR) is associative and gives the same readout and gradient on every route. The coordinate F-betaQ^2/2 is additive. Depth-dependent coupling is a tested history-sensitive counterexample. Differential minimum-change returns agree under the chosen Euclidean coordinate metric; finite nonlinear target solving is separate.

Route/readout coherence and checker: 9720 exact readout/gradient cases. Next induce this coordinate metric from the retained member or comparison-cell edit cost.

Orientation metric induced from comparison-cell costs

Pulling edge-edit cost back along the boundary gives zero on every closed2-chain. A positive orientation-class cost therefore needs costed higher-cell records or another specified member coupling. Unit Euclidean costs on Dowker triangle records induce M_n=(4/64^(n-1))*I_n: scales4,1/16,1/65536 at ranks1,2,4. Explicit harmonic representatives minimize that cost; their differential returns agree with the earlier scalar-metric coordinate returns.

Induced cell-cost metric and checker verify full exact matrices at ranks1,2 and the product-count formula with local identities at rank4. An additive class budget of scale4 requires triangle weights64^(n-1). The primitive sphere has class cost6 with unit rectangle weights and4 with unit triangle weights, so metric transport between presentations is an additional requirement beyond homology agreement.

Weighted chain comparison with retained closure constraints

An explicit rectangle-to-triangle chain map now preserves both boundary squares. For a rank4 primitive map, each six-cell source vector reconstructs from four target coordinates plus a two-dimensional orthogonal residual. All729 tested source vectors preserve their exact edit cost. The sphere budget is6=4+2. The unconstrained minimum target lift has a nonzero source boundary; its retained residual restores closure. On closed classes the transported cost remains6.

Weighted chain-transport study and checker verify all81 source-anchor assignments. Their maps have ranks2,3,4, but every complete closed-class budget remains6. Promotion must carry content, complementary cost and admissibility constraints. The two-factor chain transport is next.

Two-factor weighted chain transport

Both independent product orientation classes now pass explicit rectangle-to- triangle transport. Unit rectangle costs induce (1/14)*I2. For the tested product anchor map, the unconstrained quotient and retained residual Grams are respectively (2029/35280)*I2 and (491/35280)*I2; their sum restores the closed source cost. The unconstrained lifts are not closed; the retained residuals restore closure.

Two-factor metric transport and checker verify both boundary squares, periods, harmonic representatives and rational normal-equation certificates. The closed-class rectangle/triangle cost ratio is 8/7, versus3/2 at the primitive level. Local assignment counts84 and64 give the product-scale formulas6/84^(n-1) and4/64^(n-1). The rank4 rectangle scale1/98784 is a product-count prediction; the full rank4 map is not enumerated.

Costed higher-cell records and additive comparison

A harmonic-sector store now implements the chosen additive class budget Q=6*sum(h_i^2). Rectangle-cell weights are1 at rank1 and84 at rank2, as required by the induced metric scales. All six/1152 source cells, target views, closure residuals, versions and event histories are retained. Reconstruction, closure and cost decomposition are checked after every update.

Costed-record contract and checker verify direct rank2 returns against primitive edits followed by snapshot comparison. Source cells, views, residuals and total edit costs agree exactly. Stale/foreign requests preserve the full store; compensation restores content with fresh revisions. The additive budget is explicit policy. Parent links bind snapshots; live propagation and nonharmonic closed modes remain extensions.

Live parents and internal closed-cell modes

The two-factor store now retains the full closed-cell state x=Zh+b, with b closed and orthogonal to the two harmonic classes. Its1037-dimensional internal boundary-mode space is generated by2496 local3-boundaries; every generator passes admissibility checks. The exact budget is6||h||^2+84*||b||^2. A three-cell boundary mode can add cost252 while leaving both class coordinates unchanged.

Live costed-cell contract and checker verify parent-version propagation, retained internal modes, class extraction, source closure, target/residual reconstruction and weighted cost. Stale requests, nonclosed modes and disguised class edits preserve the full store on rejection. Live and full rebuilds agree; compensation restores content with fresh versions. The fixture has fixed topology, two primitive parents and one derived node.

Complete observation of the closed-cell state

The product-anchor triangle view has rank279 on the1039-dimensional closed source space: it detects both orientation classes and277 internal directions, leaving760 hidden closed modes. There are1328 local3-boundaries that individually vanish in this view. A nonzero b and -b can have the same classes, target view and budget252 while remaining distinct retained cell states.

Observability/completion study and checker construct a minimal linear completion using760 selected source-cell coordinates. The augmented rank is1039; exact decoding reconstructs a state using every basis coordinate. Coordinate selection is label-dependent. Complete residual retention is an alternative. Partial-view returns still require a residual-selection law.

Closure-constrained partial and complete returns

The partial triangle interface now has an exact minimum-cost closed return: T2dx=dy and d2dx=0 give392 independent constraints and760 free closed modes. Rational normal equations select an update orthogonal to those modes, preserving the pre-existing hidden component. The complete760-coordinate augmentation decodes a unique source update. Its excess over the partial optimum is hidden, with an exact Pythagorean cost split.

Closed-interface return contract and checker verify both interfaces, zero/linear replacement laws, closed but unrealizable target rejection, live commits and compensation. The exact solver also backs up live reconstruction when rational recovery from numerical proposals is insufficient. A target cycle must lie in the actual view image to be admissible.

Presentation-independent closed returns

Nine product-anchor maps have closed-view ranks279,207,247,207,138,187,247,187,215, but identical pulled-back class functionals. Exact class-return solves through different maps give the same harmonic source update. Complete target/residual packages roundtrip the same closed source state and preserve its weighted cost. Invertible coordinate rescaling also preserves the return when the request moves with it.

Cross-presentation return law and checker include a zero-class mode hidden by one map and visible in another: the two partial requests then require different minimum updates. Return coherence requires equal admissible source constraints and transported cost; equality of homology classes alone is weaker.

Successor-signature audit

Adding assembly, retained reference and residual data still leaves the successor underdetermined. Two endpoint policies satisfy full recovery, transported weights and C=d+rho but iterate as137 ->32 ->1 or137 ->32 ->32. A computed residual adds no free parameter to(C,d). A new filler with that residual as boundary would require a separate witness type and constructor.

Successor-signature audit and checker verify these countermodels. Literal promotion gives32 family records; independent family pairing gives1024 ordered pair objects. The prior product/fibration law establishes compatibility of that added comparison operation, not its derivation from one-label-per-family promotion. The next-level identity and endpoint types must be fixed before identifying its iteration with the tower.

Typed reference-cone successor

Literal promotion supplies32 retained family records. Interpreting each reference comparison as a boundary sigma_F:C_F=>d exposes a higher-cell typing condition: common target d is insufficient for parallel endpoints. In the generic fixture, all992 off-diagonal pairs require a source connector C_F=>C_G before a prospective 3-cell boundary can be formed.

Typed reference-cone audit and checker construct all1024 aligned boundary types. Additive connectors C_F-C_G make every scalar discrepancy telescope to zero. The1089 connectors among32 assemblies and the reference have rank32, so they remain dependent on the original spokes. An actual higher filler, its admissible kernel modes and cost require an explicit constructor. Family grouping alone does not supply that type-ascending witness.

Aligned filler recurrence and obstructions

The existing biclique chain complex now supplies explicit filler records: d_k*K=u-v, with retained endpoint chains, coefficient choice and cost. Exact labelled sections give two degree-ascending steps. A fixed3-boundary has1459 filler directions; every difference between3-fillers has a4-filler, with733 choices at fixed boundary. Fresh zero discrepancy selects zero; retaining a closed filler preserves its kernel history.

Higher-filler construction and checker solve all1459 degree3 kernel generators. Both H2 sphere classes obstruct3-fillers; one H4 class obstructs a subsequent5-filler. Support restrictions can introduce additional failures. The section is a declared selector with recorded unit coefficient costs. A carrier adapter must still map named reference comparisons into these chain endpoints and their allowed support.

Actual path-boundary adapter and mixed defect

Retaining the typed assembly's shared legs gives4 endpoint objects,31 primitive/ reference legs and137 path-reference comparison2-cells. Their boundary rank is28, leaving109 closed rectangle relations:100 arrow-block and9 state-block modes. Keeping explicit composite names and composition witnesses yields the same kernel by an exact chain retraction. Treating composites as independent arrows would erase these relations.

Typed assembly adapter and checker verify that a rectangle's finite matrix readout is (y_j-y_0)*(x_i-x_0), with the analogous state formula. It vanishes at first order but can be nonzero for simultaneous leg perturbations; direct-reference changes cancel. Thus the actual assembly supplies factored mixed-defect data. The earlier biclique filler dimensions belong to a distinct complex and require an explicit path/response-preserving adapter before reuse.

Recursive comparison-record closure

Reference-decorated maps admit one associative composition law: (C2,d2) after (C1,d1) gives (C2C1,d2d1), with residual rho_new=d2rho1+rho2d1+rho2*rho1. The same constructor passes two recursive applications and all five four-input bracketings, retaining child histories. Agreement and intermediate basis covariance are preserved.

Closure experiment and checker expose its inputs: a composable network and, for primitive-leg reconstruction, a factorization of the direct reference. Current common-Hom(A,B) comparisons cannot compose without an endpoint adapter. Endomorphism self-reuse doubles occurrences while retaining one independent seed. This law preserves cell degree; it supplies a compositional constraint for the family successor, not its higher-cell constructor. An additive norm budget also requires further structure.

Graded higher witness and cube coherence

An explicit differential-graded model now realizes delta(h)=C-d. The two composite homotopies h2C1+d2h1 and C2h1+h2d1 share the exact composite residual boundary. Their difference is delta(h2h1), giving a degree-two comparison witness. The degree-three product h3h2*h1 fills the next signed six-face cube. Higher records retain both endpoints, fillers and parents.

Graded witness construction and checker verify the Leibniz rule, both degree ascents and boundary-of-boundary zero. The finite test uses an explicitly chosen contractible complex. Canonical lifts give zero higher products; closed history changes give nonzero higher witnesses without changing lower residuals. A carrier differential, admissible lifts, endpoint transports and cost/readout must be supplied for the actual assembly. This is a candidate higher realization, not a derivation of family promotion.

Shared-leg realization and base-change coherence

The actual A->U/V->B assembly now supports a free DG witness construction. Relative leg witnesses have boundaries x_i-x_0 and y_j-y_0; two base-path witnesses retain y_0x_0-d, preserving the independent direct reference. All137 comparisons have two compositional witness routes. Their discrepancies are filled by109 products k_jh_i whose matrix readouts are the mixed defects.

Shared-leg realization and checker verify both blocks and two root contexts. Changing base legs and returning preserves the reference response while shifting its witness by -delta(K): the same rectangle product supplies the higher comparison. Hom(A,B) dimensions are 138,246,109,0 in degrees0..3, with boundary ranks137 and109. The original two-leg network has no nontrivial next cube; further ascent needs an explicit network- extension or generator rule. Root choices and free witness adjunction remain model inputs, with metric and physical readout identification open.

Recursive extension through the direct-reference inverse

In a declared reversible-reference sector, r=d^-1:B->A closes existing A->B records under Q star P=QrP, with unit d. The same rule passes two applications and gives rho_new=rho1+rho2+rho2rrho1. Its graded extension supplies a nonzero cube and degree2,4,8 products of an original rectangle witness. The last product reuses the same two witness labels; occurrence growth adds no independent inputs.

Reference-bridge extension and checker verify formal inverse reductions, graded boundaries, basis covariance and singular-reference rejection. Both composition orders are now typed, giving rho2rrho1-rho1rrho2 as their commutator. Strict invertibility is an added sector assumption; a weaker witnessed return needs its own unit coherences. The rule supplies compositional closure, with family promotion, word selection, metric and physical coupling identification still to be determined.

Witnessed reference returns and pentagon coherence

The reconstruction inverse for indexed rows does not itself prove the direct matrix reference invertible. A weak DG return instead retains delta(u)=rd-1_A and delta(v)=dr-1_B. An exact singular-reference example with nonzero surviving homology satisfies both equations. Composition QrP then needs the retained correction e=du, with delta(e)=dr*d-d, to keep d as reference.

Witnessed-return construction and checker verify two constructor rounds, a nonzero degree-two associator and degree-three pentagon filler across all five bracketings. Unit-triangle coherence can require a closed-history edit; arbitrary fixed unit witnesses can obstruct that triangle. The weak construction retains reference drift and history changes explicitly. Carrier return selection, edit admissibility and physical readout/cost remain inputs; strict inverse closure is the zero-unit-witness special case.

Protected-history policy for coherent returns

Coherent return versions now preserve immutable parents and declared linear readouts, solve the unit/triangle equations, and record every permitted history edit and filler cost. Both unit histories are locked by default. A positive weighted least-change selector either returns an exact optimum or rejects an incompatible request. Reusing a coherent version is a zero-cost no-op.

History policy and checker verify a locked-history triangle in the singular-reference fixture and rejection of the fixed zero-differential obstruction. Explicit edit permission and weights 2:3 select u=v=2/5 at cost6/5; protecting the weighted total accepts, while protecting both individual readouts rejects. Accepted versions pass two recursive composition rounds and associator checks. Readout protection and metric weights are declared inputs; history selection leaves reference drift unchanged.

Return observability and history-metric obstruction

The existing slot, mixed and round-trip residual readouts depend on actual maps. At fixed maps they cannot distinguish independent closed unit-history changes. In the weak-return fixture,38 unit/triangle coordinates satisfy24 independent tangent constraints, leaving14 coherent-history directions invisible to the round-trip residual readout. Six triangle freedoms remain with units locked.

Observability audit and checker verify that every such direction has zero pulled-back response cost but positive coefficient cost. The policy's positive history metric therefore requires another input. Kind-preserving symmetry also leaves arrow/state metric weights free. A history-sensitive observable/dynamics, a separately justified record metric, or an observational quotient is needed before a physical normalization follows.

History equivalence under composition and active updates

At fixed actual maps, reference and return, closed witness changes stay closed under the recursive comparison law. Actual/residual observations therefore agree for every finite comparison word. All28 hidden-direction trials pass across five bracketings and a further comparison.

Operational-context audit and checker show that the full protected-update protocol has a stronger distinction: two coherent histories with equal current responses give accept versus reject under the same active reference-frame request. If edits are allowed, declared costs are0 versus1/5. Fully transported passive basis changes have no such obstruction. A response quotient is sound for fixed-frame composition; admission-sensitive history remains necessary for protected active updates. Its physical readout and the carrier's allowed active changes remain unspecified.

Consolidated operation contract and admission profiles

The operation contract separates source-defined family promotion from conditional comparison, return and active-update rules. For a protected active frame change g, full-support admission requires [g,delta(v)]=0 and [g,v]=delta(K_g). The triangle updates as W_new=gW+K_gd; certificates compose as K_(hg)=hK_g+K_h*g. Exact history changes preserve this admission profile. The singular-reference fixture has H1=0, so its14 hidden directions do not yield different full-support admission profiles; the preceding accept/reject counterexample used a different H1.

Weak-return checker and finite profile checker verify the rules. In a declared active-S4 diagonal-history model,256 histories reduce to15 exact permission profiles. Passive relabelling conjugates profiles; uniform family requests intersect them. Two retained pair-grouping rounds give stabilizer sizes6->2->1 with exact leaf recovery. This supplies a family permission field, not the missing next-level endpoints. Treating S4 relabellings as active physical requests and selecting a request schedule remain explicit choices.

Witnessed reference substitution on retained families

An existing comparison h_j with delta(h_j)=C_j-d supplies a reference-change witness. Keep actual slot maps fixed and set h_i'=h_i-h_j, with corresponding explicit updates to return-unit and triangle witnesses. Old records remain immutable parents. Two substitutions agree with a direct substitution; returning to the old reference restores all current fields. Thus retention does not by itself require frozen current witness coefficients.

Reference-substitution construction and checker verify all137 slots and32 freshly labelled family candidates, transported masses, round-trip restoration and all109 rectangle invariants. A retained total-mean reference makes the mean residual zero while preserving mixed defects; the exact mean/variance norm split passes. This is a fixed-domain reference operation, with reference selection, physical interpretation and any exclusion-mask update still to be specified. It supplies no complete next-level endpoint rule.

Architectural behaviour and verification

These are consequences of the stated architecture under the listed preservation conditions. The full recursive promotion/update implementation remains to be tested.

Characteristic	Behaviour	Condition / verification
Multiple access directions	Individual, outgoing-family, and incoming-family queries address the same relations.	Reconstruct the same labelled rows from both indexes, including empty groups where the domain requires them.
Higher-level addressability	A grouped family becomes an individually addressable record in the next triple.	Check unique labels within each namespace and label-to-group roundtrips.
Drill-down	A promoted record exposes its lower-level members and their provenance.	Retain member identity, endpoint fields, multiplicity and order where present; test recursive reconstruction.
Content-preserving reindexing	Switching source/target presentation preserves relational content.	Compare reconstructed records, not merely row counts or aggregate values.
Dependency propagation	A changed path can affect its containing families and their promoted records.	Retain membership links; compare incremental recomputation with full rebuild.
Compact interfaces	One label can address a rich many:many internal structure.	Verify dereferencing fidelity; measure storage separately from interface cardinality.
Measure transport	To preserve a weighted response, a promoted family carries the sum of its members' weights; its internal readout uses the conditional weights.	137-slot audit: exact grouping checks pass. Equal weighting of fresh labels generally changes the readout.
Reference symmetry and weighting	In the tested 137-slot domain, the independent directed-reference stabilizers give 45 slot orbits. Symmetry and normalization leave 44 orbit-mass parameters.	Symmetry checker: three distinct invariant measures pass. Primitive response derivatives are not yet feedback gains or physical couplings.
Identity over updates	A family may retain an identity across revisions or acquire a new content/version identity.	Choose identity and version policy; test membership changes, rebuild order, deletion, and index collisions.
Return actions	An operation on a family can act on its constituents.	Define the return map and conflict handling for overlapping families; test action/readout consistency.
Feedback	A return action can change lower-level records, which then update higher-level families.	Specify scheduling and a measured quantity before calculating gain or stability.

Reindexing supplies no intrinsic energy loss or settling law. Runtime costs, forgetting, record export, and state-changing return actions require their own accounting. The numerical damping and recurrence results below belong to their specified update prototypes, not automatically to this architecture.

Editable family means and derived return gain

A first return-contract prototype retains all137 member records and exposes arithmetic means of the32 source families and32 target families. Returning an edited mean makes the minimum Euclidean change to members. Editing family F by delta changes the mean of family T by |F intersection T|/|T| times delta. Thus the incidence data determine a cross-interface gain once the mean/readout and minimum-change contract are chosen.

The64 means have rank47, leaving90 internal member directions preserved by the return. Alternating source/target corrections have visible residual spectral radius0.290892665417 per cycle and converge to the joint least-change solution for compatible requests. A joint image test rejects conflicting summaries.

Family return study and checker. The readout, metric and schedule are trial choices; this does not yet implement multi-level promotion or identify the gain with a physical coupling.

Returning edits through promoted families

For disjoint family means y=A x, leaf Euclidean edit cost induces the family metric W=(A A^T)^-1=diag(member counts). Higher means must also retain member-count weights. With that metric, minimum-change return through families of families equals direct return to the same leaf readout. Source and target indexing routes agree when they implement that same readout.

The137-slot fixture checks32 families ->8 upper families ->1 global family. Direct/staged discrepancy is4.44e-16. Giving fresh labels a fresh Euclidean edit metric instead changes the update (norm difference0.428219) and increases its leaf cost. The overlapping64-family route requires the full induced metric (O O^T)^+ on its47-dimensional image; that route also passes direct-return checks.

Composition derivation and checker. These linear return tests do not yet implement persistent identity/version management or select the next-level endpoint fields.

Coherence of states and edits around the cycle

For fixed member identities, complete source/target presentations are invertible reindexings. Their cycle is identity on both states and edits. Minimum-change returns commute with those presentations when the member metric is transported. Mean-only views fail the cycle because they omit internal detail; adding the residual in the mean map's kernel restores reconstruction.

Fibration edit-cycle study and checker. Order-sensitive partial corrections are state-changing, noninvertible operations; their discrepancy alone does not establish transport holonomy. The verified case has fixed membership. Dependent edits that insert, delete, or regroup members require an identity/history and metric-transport law.

Changing membership and edit scope

A dynamic-index prototype now handles insertion, deletion, endpoint moves, and family-mean edits. Leaf IDs are stable; family IDs persist by direction/key; events retain old/new records. Incremental source/target indexes reconstruct the same live records as full rebuild after every operation. Current member counts determine the induced return metric; means and residuals are recomputed when membership changes.

A mean edit addresses an explicit member-ID snapshot. The prototype rejects stale membership before mutation. Insert-then-edit and edit-then-insert can legitimately differ because they address different members. Reverse event replay restores live state while leaving identity allocation/history retained.

Dynamic edit contract and checker: 250 scheduled operations,301 leaf events, exact rational view/cost checks. This is a single-threaded one-level prototype. Recursive promoted endpoints, full value-version/concurrency policy, and dependent-type proofs remain open.

Recursive endpoints and versioned return transactions

A two-level prototype now gives each promoted record endpoint IDs for the sets of its children's source/target endpoints. Full child membership is retained separately. First-level groups are source-indexed; second-level groups share target-footprint identities. Means use descendant leaf weights.

Stable group-key IDs carry changing versions, including child-version tuples. A return request names the family version and descendant leaf versions. The serialized commit validates all of them before mutation. Tests reject overlapping stale edits, moved-endpoint requests and ABA value restoration, while accepting an unaffected disjoint-family edit. Unchanged rebuilds preserve versions.

Recursive transaction contract and checker: 100 dynamic operations with exact rational means and recursive reconstruction. Endpoint footprints and version policy are explicit choices. This fixture uses full rebuild and simulated transaction interleavings; actual locking, distributed merging, and crash-safe storage are not implemented.

Recursive history and route coherence

The same Store now retains accepted old/new event batches and supports serialized LIFO compensation. Compensation restores content with fresh versions; it preserves the original history and rejects stale pre-compensation requests. Direct upper mean returns and staged child returns agree in leaf values before insertion, after insertion, and after deletion, when each route uses current snapshots. Their commit/version histories can differ.

Integrated history study and checker verify those routes, compensation, and a16-event rational JSON replay with a corrupt-precondition control. The JSON test is audit serialization, not a durable commit protocol or a replay of the entire identity registry.

Read/return laws and the membership-change obstruction

On fixed membership, uniform minimum-change mean replacement satisfies GetPut, PutGet and PutPut, preserves residuals/outside values, and commutes with complete ID reindexing. These are content laws, not full-store equalities: even a zero accepted shift advances the current prototype's versions and event history.

Insertion at a fixed value does not commute with extending a nonzero edit to the enlarged family. No alternative mean target repairs this: a whole-family return shifts every member equally, while edit-then-insert leaves the new member untouched. Original member scope must be retained, explicitly extended, or rejected for rebase. Fixed scoped increments commute even on overlaps; absolute mean replacements need not. Neither fact grants permission to use stale snapshots.

Algebraic laws and obstruction and checker: 1701 exact content-law cases,81 insertion cases, and a full-store metadata negative control. General statements have written algebraic proofs, not yet a machine-checked dependent-type formalization.

Composable member-scope transport

Membership transitions now have a pure semantic transport model: a partial injection records surviving IDs and constants supply new values. Edits follow the linear part, preserving survivor components and assigning zero to newcomers. The affine law T(x+delta)=T(x)+P(delta), identity, and composition hold exactly. Deleted edit components account for the exact lost Euclidean cost.

Transported old-family edits can be nonuniform in a newly enlarged family, so a mean-only return interface is not closed under this transport. Member vectors or mean-plus-residual edits are required. Delete/restore can recover content without recovering original edit scope: provenance cannot be inferred from initial/final ID sets alone.

Scoped transport laws and checker:3087 exact composition cases plus cardinality-change and invalid-provenance controls. The stable-ID insertion/deletion portion is now integrated with version-checked member-vector commits; pure renaming remains separate. A transported edit still requires an explicitly approved fresh rebase when its read versions are stale.

Version-checked transported vectors

The recursive Store now accepts immutable member-vector requests with leaf versions, store identity and an event cursor. Explicitly approved rebase follows actual retained events, drops deleted scope without resurrecting it on restore, and snapshots surviving versions. Commit validates the entire batch before mutation and retains request/provenance by accepted event index. Compensation uses the same event mechanism as mean returns.

Integrated vector-return study and checker verify untouched newcomers, pure/integrated transport agreement, fresh-version checks, full-store rejection invariance, compensation and composed rebases. Explicit leaf vectors and current-family mean requests have different scopes: insertion outside a vector's scope need not invalidate it.

Approval here is a trusted local policy switch, not authentication. The model is serialized; threaded locking and durable vector-provenance storage are open.

Record-type fibration cycle: supporting work

The existing table-fibration kernel provides dependent fibers and reconstruction of labelled records. The indexed-path study checks source/target presentations. Its path-composition successor was an alternative experiment; the current successor is grouped-record promotion.

Characteristic	Precise statement	Status	Verification / next step
Cyclic selected-field fibration	The existing kernel constructs the homotopy fiber over each selected field. A retained dependent total supplies a concrete successor record type.	Existing generic recovery theorem; new finite regression checked	Cycle construction, checker
Reconstruction	Each retained-total step reconstructs the preceding table with its fields. Construction history grows while table data remain equivalent.	Checked for the specified recurrence	147 tables, 1176 steps; retain selected-column provenance.
Section-valued field promotion (separate experiment)	Fiber the current selected field p:E->X, form S=Π_x Fiber(p,x), and use successor rows Σ_(sigma:S) X. This chooses one row per fiber, unlike the clarified intended grouping.	Explicit construction; 441 finite cases checked; not the intended rung operation	Section promotion, checker. Empty fibers can erase records; section contexts can duplicate them. Matching the intended endpoint schematic requires an explicit field-role/phase convention.
Three triples with multiplicities 1,2,4	Proposed gauge labels run U(1), SU(2), SU(3) at each multiplicity.	Conjectured correspondence	Define and count the recursively promoted endpoint structure. Retained-total row cardinality stays unchanged.
Presentation versus restriction	Fibration changes presentation; the Gram tower projects out state directions.	Distinct constructions	Supply the map from presentation data to live Gram state and records.
Path-composition alternative	Self-compose a path span by endpoint pullback, giving lengths 1,2,4; each length has flat, source-indexed, and target-indexed presentations.	Separate experiment; finite checks passed; not the selected promotion rule	Synthesis checker. The selected rule assigns identities to grouped families rather than extending path length.

Dynamical characteristics

Unless specified otherwise, the numerical results here concern the earlier Euclidean six-coordinate prototype. Its spectra and rates must be recomputed for the inverse-metric readout dynamics above.

Characteristic	Precise statement	Assumptions	Status	Verification / source	Open question
Reversible comparison memory	Exchanging the mismatch coordinate with a stored scalar reproduces projection for an empty record and is an involution. Carrier-plus-record squared budget is conserved.	Orthogonal equality projection; one record per comparison.	Checked	Memory lift checker, dynamics note	Select the record's physical realization.
Closed feedback	Reusing all records without export gives an orthogonal sweep, recurrence, and no attracting equilibrium.	Fixed schedule; no driving or reset.	Checked	Reused records checker, dynamics note	Recompute in the induced readout metric.
Closed sweep spectrum	The 159-coordinate Euclidean sweep has six +1 modes, one -1 mode, and 76 rotation pairs.	Existing comparison order and Euclidean metric.	Checked numerically	Spectrum checker	Schedule and metric dependence; physical mode identification.
Record retention	Returning amplitude rho*w exports (1-rho^2)w^2 for 0<=rho<=1.	Export port removes that budget from the modeled subsystem.	Checked	Retention study, checker	Derive a retention/export policy.
Damped settling	Every rho<1 gives spectral radius below one and decay to zero without input.	Full-rank comparison family; each record visited once per sweep; strict attenuation.	Checked; structural argument	Retention study	Add explicitly accounted driving or environmental feedback.
Driven floor	Independent tangent-plane noise sustains a periodic covariance; its magnitude scales with injection strength.	Euclidean metric, isotropic injection, fixed order.	Checked numerically	Back-action checker, shared-state hypothesis	Derive the source strength and its resource budget.
Schedule dependence	Forward and reverse projection orders yield different residuals for the same seed.	Irreversible projection programme.	Checked	Full programme checker	Determine the carrier-selected schedule.
Continuous interpolation	An extra auxiliary coordinate permits continuous norm-preserving half-turns through individual comparisons. Two-sweep maps also admit Hamiltonian embeddings.	Added coordinate/structure; selected phase branches and interpolation.	Conditional; checked numerically	Continuous-path checker, Hamiltonian checker	Derive an autonomous clock and physical symplectic structure.
Clock and phase freedom	Rescaling physical time leaves the discrete update map unchanged; additional full rotations also preserve sampled endpoints.	Discrete update data without a physical clock or interpolation law.	Established freedom; examples checked	Dynamics note and continuous-path checker	Supply a dimensional clock and phase-selection law.
Traversal resource versus mismatch budget	Attempts retain positive cost even when a mismatch is zero. Attempt count and dissipated squared mismatch are separate observables.	Unit cost per labelled attempt; projection dynamics.	Checked	Full programme checker and shared-state hypothesis	Identify which resource, if either, determines rest energy.

Equal comparison response: 137-slot operation test

With four probes from G12=10I+J, state features have squared norm11 and arrow features20. Tensor comparison slots therefore have raw norms121 and400. Raw feature storage does not give equal slot weights.

For normalized metric-orthogonal rank-one removal, an isotropic second moment C=sigma^2 K^-1 gives expected removed budget sigma^2 for every slot. This is an explicit sufficient condition for equal expected comparison contribution. It requires a common fresh ensemble: sequential projections change the state, and the initial16-dimensional budget is only16 sigma^2. A uniform selection probability1/137 still requires a scheduling rule and has no established identification with electromagnetic coupling.

Operation derivation and exact checker.

A stationary extension supplies the needed replenishment explicitly. In whitened coordinates, each comparison has Q=uu^T and P=I-Q. Independent refresh with covariance sigma^2 Q maintains C=sigma^2 I and equal removal sigma^2 per slot, while reopening the just-tested direction. Equality-preserving injection sigma^2 P/15 instead gives unequal periodic slot responses. Both inject and remove137 sigma^2 per stationary sweep; total balance alone does not select uniform slot weighting. The source strength and physical operation remain open.

Refresh study and stationarity checker.

Closed-record equilibrium and normalized exchange response

A closed alternative swaps each normalized carrier comparison coordinate with its stored record. Joint isotropic covariance is stationary without external injection. The joint space has16 common fixed directions and137 independent reflection directions. In this fixture the reflection normals' nonorthogonality graph is connected. A symmetric covariance invariant under every individual comparison is therefore scalar on their span.

The observable w_i-u_i^T q then has the same variance2a for every slot. For a>0 its share of the total exchange variance is1/137. Marginal stored budgets can still differ through arbitrary covariance on the fixed space. This conditional response result requires stationarity under every comparison, not merely one sweep. Its electromagnetic interpretation and physical preparation mechanism remain open.

Closed-record argument and checker.

Schedule mixing toward comparison-invariant equilibrium

For closed record exchanges H_i, choose each comparison with a positive, state-independent probability and include a half-probability idle step. The schedule-averaged covariance channel is a positive weighted average of orthogonal projectors on symmetric matrices. It converges to the common comparison-invariant covariance space.

If F projects onto the16-dimensional common fixed state space and A=I-F, the limit is F C0 F + tr(A C0) A/137. Positive active budget gives equal mismatch variances and normalized share1/137 regardless of the positive schedule probabilities. Individual histories remain reversible; this is ensemble equilibration, not deterministic state settling. Observation-event frequencies still affect frequency-weighted readouts.

Equilibration argument and checker. The physical scheduling mechanism and electromagnetic identification remain open.

Record endpoint current and local phase transport

Path records have an incidence boundary B j equal to target minus source. Family aggregation preserves this identity while retaining traversal history. A trial hopping action H=sum kappa_ab |psi_b-U_ab psi_a|^2 gives a concrete local U(1)-covariant transport operation and current J_ab=2 kappa_ab Im(conjugate(psi_b) U_ab psi_a). Canonical complex evolution satisfies dot(rho)+B J=0 for rho=|psi|^2.

Transport construction and checker. The hopping action, strengths, and canonical dynamics are explicit choices. That initial test treats links as backgrounds. The closed extension below supplies field dynamics; the137-comparison adapter, physical charge assignment, and coupling normalization remain open.

Closed matter/link feedback trial

A lattice gauge-inspired Hamiltonian adds conjugate link variables E, electric storage betaE^2/2, and triangular-loop storage mu(1-cos(C A)). Its equations are dot(A)=beta E and dot(E)=-J-mu C^T sin(C A). Matter current changes the links; their phases change subsequent matter transport. B C^T=0 and matter continuity preserve Gauss: B E=rho-rho_background. The fixture includes a static neutralizing background on the closed four-state graph.

Feedback construction and checker verify Gauss, matter norm, gauge covariance, reversibility, and second-order energy error. Coefficients, triangular cells, and canonical dynamics are trial inputs. This is a concrete feedback fixture, not yet a carrier-derived electromagnetic action or a determination of alpha.

Gauge adapter and complex lift

For the real comparison model, any skew generator commuting with every labelled reflection vanishes on all137 active comparison directions. A complex lift to C^153 supplies a global phase symmetry and canonical rank-one Hamiltonian pulses whose endpoints are the original exchanges. This is an added phase-space structure; pulse duration and physical action normalization remain free.

A simple product of two four-state amplitudes is not closed under the comparison updates: a valid projection takes its4-by4 tensor matrix from rank1 to rank2. The adapter to the matter/link model therefore needs a richer state representation.

Adapter audit and checker. A global phase is constructed in the extension. A subsequent local endpoint adapter retains the full tensor: x_ab transforms by exp(i(chi_B[b]-chi_A[a])), comparison features by the same diagonal D, and K by D K D^dagger. Records are neutral in this choice. All137 exchanges, their mismatch readouts and budgets transform consistently. Freezing K fails budget invariance.

Local phase adapter and checker. This transports representations; loop phases of the Gram remain unchanged. Charge identification and coupling dynamics remain open. A subsequent curved overlap extension allows independent edge phases in G_ab with diagonal11 and unit off-diagonal modulus. Positivity follows from the lower bound8; triangle phases can be nonzero. Five tested tensor metrics retain connected comparison normals and the conditional normalized mismatch share1/137.

Curved overlap study and checker. These phases extend the original stabilizer-probe model. Norm-compatible transport for a changing metric is available, but phase evolution, source backreaction, and physical field stiffness are not fixed.

Comparison-driven phase backreaction

A scalar-phase trial now couples the actual137 records to the curved tensor metric. Its Hamiltonian is H=(kappa/2) sum|w_i-u_i(theta)^dagger q|^2 + beta*p^2/2

• mu*(1-cos(theta)). The phase force is the negative derivative of that same mismatch storage, closing the comparison-state/overlap-phase loop. Global complex norm and H are conserved by the exact equations. Numerical checks confirm force, global phase covariance, and decreasing energy/norm errors with smaller steps.

Backreaction construction and checker. The action, frame, scalar reduction, and coefficients are explicit choices; full local gauge completion and physical normalization remain open. This is a different evolution from random scheduled reflections, so their stationary1/137 result requires a fresh test in this coupled model.

Physical identifications and normalization

Characteristic	Current statement	Assumptions / boundary	Status	Source	Next test
Energy dimension	E_r has dimensions Q_r P_r/T_r when Q,P,T are position, momentum, and time scales.	Dimensional consistency leaves the observable and its coefficient unspecified.	Dimensional identity	Energy normalization note	Construct physical action and time from the same carrier dynamics.
Quadratic budget as energy	The conserved norms supply candidate storage observables.	No selected physical Hamiltonian or GeV conversion.	Open	Energy normalization note	Derive the dynamics and energy readout together.
Full tower budget in GeV	No derived value. The 375 GeV example assumes the entire retained rung4 budget is 125 GeV and uses uniform descent.	Higgs excitation has not been identified with the whole rung4 budget.	Conditional example only	Calibration and audit, Higgs audit checker	Establish state occupations, energy reference, and dimensional normalization.
Proton/electron ratio	1836 is a comparison-programme count and a separate fixed-stage network support count, not a measured mass ratio.	The shared charging/displacement branch cannot match both the retained mass ratio and splitting; equal counts do not identify dynamics.	Physical identification open; common charging branch excluded	Geometry, mass closure	Supply a genuinely new source-supported charging or elementary interaction law and test absolute and relative energies together.
Physical spin	Adding S to V36 yields a 72-dimensional tensor carrier with a rank-two collective doublet; replacing the fibers yields dimension 24 and no invariant vector.	S is an additional input. Ordinary S4 doublets do not identify spatial spin or weak isospin.	Compatible extension, not physical selection	Spinor bridge	Independently source the rotation-path interaction; the native endpoint model does not select it.
Electric charge	I and N commute with the combined A4 action and have eigenvalue +1 on u; 2N-I has spectrum {-1,1}. The former universal no-go was false.	Algebraic eigenvalues are not electric charge. A scaled half-turn has expectation +1 but variance 8.	Counterexamples checked; physical identification open	Corrected charge scan	Supply independently specified charge sectors and electromagnetic coupling.
Electromagnetic coupling	Closed-record schedule mixing gives normalized mismatch variance1/137. A trial charge/field interface still has free strength g^2/kappa while that fraction stays fixed.	Relative fluctuation weight does not yet determine source coupling, Maxwell propagation, or a spatial Coulomb law.	Open physical hypothesis; interface freedom checked	Response bridge audit, checker	Derive a charge/current observable, its carrier action, and the normalized field response.

Corrections and scope boundaries

• Finite stabilizer covariance and cotangent lifts do not prove ADM field constraints, Lorentzian signature, Einstein dynamics, or a continuum limit. The legacy promotions are withdrawn.
• The half-turn's kernel operational laws and its written standard-pi identification have distinct verification levels. They do not automatically certify whole-return assembly.
• Traversal effort, stored occupation energy, excitation gaps, and rest energy must use explicit observable maps and a common reference; adjusting a clock cannot fix a ratio.
• Norm-charge is not electric charge. Changing the exterior reservoir between a composite and its breakup is not a binding-energy derivation.
• Counting permutations satisfying sigma(i)=j gives (N-1)! for every pair i,j. The off-diagonal (N-2)! belongs to intersections of two point stabilizers.
• Rebuilding probes on each S_r gives a different Gram family from restricting a single S12 probe space. Its factorial normalization changes are not a derived physical energy decay.
• The fixed-carrier Gram spectrum is r+10 and 10. It does not establish the older assignment of 12,11,4,10 as its eigenvalues.
• The earlier coefficient embedding uses M as its metric. Potential readouts use M^-1. Numerical results from either representation require the corresponding update rule.
• A state-space norm, a minimum readout norm, positive traversal cost, and a physical energy are distinct observables in the current constructions.
• The natural four-point S4 permutation representation is 1+3, not 1+1+2. The Pauli identity 2 tensor 2* = 1+3 concerns operator representations; it does not identify the separate tower/product-homology 1,2,4 counts with physical quantum numbers.

Next synthesis step

The detailed register above retains construction history. The current next steps are separate obligations, not one automatic unification or a claim that all prototype work is still missing:

1. Operational return: assemble the complex/tensor operators, metric interval, gluing, endpoints, piecewise generators, reconstruction/swap coherence, fixed-source faithfulness, and correspondence to the exported standard witness. Preserve the half-turn's mixed verification boundary. Completion frontier.
2. Comparison architecture: retain actual witnesses, members, parents, and versioned identity while specifying the intended horizontal generator and rung4 readout/metric. Candidate presentation squares and marked response decoders already pass their stated tests; they do not select physical normalization. Decide independent-context pairing versus explicit composition correction, passive relabelling versus active edits, and rejection/retry versus merging. Source comparison · response gate.
3. Physical adapters: choose a justified platform or an explicitly labelled digital simulation of the invariant spinor sector, then derive and test actual charging coefficients, transfers, preparation, physical units, and errors. Ideal algebra cannot certify those measurements. Acceptance tests.
4. Mass and binding: change the source-supported charging/elementary interaction law rather than retuning the excluded shared displacement model. Construct a bounded, number- and charge-preserving cross-carrier interaction and compare composite and breakup energies with identical field, boundary, and reservoir data. Mass exclusion · binding gate.
5. Additional selectors and resources: source a braid representation and amplitude law independently of the 372:1 constraint. For storage, address microscopic high-body interactions or justified ancillary approximations, and test packet objectives beyond the midpoint fixture. Neither compatibility alone nor an objective-specific optimum supplies the missing physical law.

Connecting record promotion to Gram restriction still requires explicit live-state, record, metric, and observation maps. Keep source occurrence counts, independent dependencies, cell degree, coefficient supports, and physical energies distinct. No new spectra, coupling, isotope stability, or gauge group follows by transferring numbers between prototypes.