Elimination Geometry

Mian Huang; Xueqin Wang

cs.LGAug 18, 2026

Elimination Geometry

Authors: Mian Huang, Xueqin Wang

Abstract

This monograph develops elimination geometry (EG), a typed, native-loss, audit-oriented framework for studying when locally optimal objects can be realized by a shared deployment rule. Elimination and compression may erase distinctions required by prediction, inference, control, or representation. EG asks which distinctions are lost, whether the induced defect is visible to the declared task, and whether changing information, architecture, action space, or deployment domain can repair it. EG separates local solvability, global realizability, and finite-sample certifiability. It derives native defects from the original objective and distinguishes architecture obstruction from model approximation, generalization, and implementation error. The monograph synthesizes tools from geometry, optimization, information theory, statistics, and machine learning into interfaces for integrability, representation admissibility, resource constraints, observational overlap, and common deployment. Formal results address regular, coordination, singular, compositional, and resource-limited mechanisms with explicit antecedents and claim boundaries. Applications include sparse model selection, distribution-free prediction, observational treatment policies, routed expert and retrieval systems, and learned score fields. Obstruction-Aware Learning and Inference links structural diagnosis to finite-data authorization, mechanism-matched intervention, and independent validation. Reproducible synthetic and real-data studies illustrate how certificates can guide architecture repair while recording failed gates and unresolved cases. The framework requires the deployment contract, native endpoint, competing explanations, information and compute budgets, and validation rule to be fixed before a persistent performance floor is attributed to architecture.

Figures & tables

Symbol	Meaning
Core elimination and architecture
$\mathsf{X}$	instance, covariate, history, or problem-index space
$P$	population law on $\mathsf{X}$ or on observed biological/statistical units
$\mathsf{A}_{x}$	auxiliary fiber at instance $x$
$H_{x}(a)$	lifted objective before elimination
$J(x)$	eliminated objective $\inf_{a\in\mathsf{A}_{x}}H_{x}(a)$

Table 1

Panel of Figure 1 — Figure 1.1 : Three distinct questions. Local optimization, global deployment, and finite-data certification require different assumptions and different remedies.

Panel of Figure 3 — Figure 3.1 : Reference risks in the four-component decomposition. The first two gaps are population-level structural terms; generalization and implementation control the distance from the class optimum to the learned rule.

Component	Diagnostic evidence
Model approximation	stronger local oracle family, oracle refit, simulation truth, or local-family comparison
Architecture obstruction	transport or monodromy certificate, coordination decomposition, resource lower bound, or zero-obstruction witness
Generalization	independent-unit split, confidence sequence, stability, or data-dependent complexity bound
Implementation	primal–dual gap, KKT residual, multiple restarts, or certified inner-solver tolerance

Table 3.1 : Evidence associated with the four components of risk.

Deployment contract	Step budget $k$	Residual factor	Exact minimum (decimal)
Affine one pass	$0$	$1$	$1/9\;(0.111111)$
Affine plus local refinement	$1$	$1/4$	$1/36\;(0.027778)$
Affine plus local refinement	$2$	$1/16$	$1/144\;(0.006944)$
Affine plus local refinement	$4$	$1/256$	$1/2304\;(0.000434)$
Quadratic-feature one pass	$0$	—	$0\;(0.000000)$

Table 4.1 : Exact one-pass and local-refinement ledger for the three-point quadratic laboratory.

Laboratory	Oracle and native defect	Contract and obstruction	Repair and evidence
Coarse-graining	conditional oracle/fiber law; carrier loss/fiber KL	carrier plus decoder/back-map; lost oracle distinctions	refine carrier/faithful back-map; marginal and transfer risk
Graph distributions	vertexwise CDF; integrated squared-CDF loss	graph filter plus validity interface; validity barrier	nonlinear projection; CRPS/minimax rate
Sequential decisions	historywise kernel; conditional KL or Bellman defect	shared network/memory; coordination/memory tax	rectangular hull/memory refinement; held-out policy value
One-pass refinement	instancewise local optimum; excess local objective	one-pass class plus step budget; architecture floor	representation/local refinement; exact frontier and held-out audit

Table 4.2 : The four laboratories under one structural vocabulary.

Mode	Acts on	Held fixed	Returned object	Certified consequence
P	local oracle choice	original lift and target	$A^{\star}$	zero local defect
G	deployed field	$H_{0}$ and $J$	$SA^{\star}$	exact insertion defect
X	frozen surrogate	original target $J$	$Q_{G}-T_{k}d$	target jet restored
V	joint objective	declared feasible class	optimizer of $H_{0}+\lambda R$	target generally changes
C	joint law	every marginal $\mu_{t}$	$\gamma\in\Gamma(\mu_{t})$	additive means fixed

Table 6.1 : The five operational contracts.

Panel of Figure 9 — Figure 9.1 : An elementary elimination square. Nonzero circulation records order dependence. On a general complex, zero face curvature must still be supplemented by a period audit.

Panel of Figure 10 — Figure 10.1 : The admissibility chain. Exact elimination alone is insufficient: a lift must be tied to target slack, stripped of slack-invisible gauge, and then connected to a deployment state whose distinctions are measured by native defect.

Panel of Figure 13 — Figure 13.1 : Rectangularization preserves the locally available kernels but removes the cross-history constraint that forces them to share one parameterization or memory state.

Panel of Figure 14 — Figure 14.1 : Monodromy depends on the output contract. A point-valued representative can fail to return to itself after a loop, while a quotient-valued object such as a projector or unordered root set can remain globally well defined.

Panel of Figure 17 — Figure 17.1 : Contextual observability. Two internal kernels are operationally equivalent only if no legal precontext, postcontext, and terminal task can distinguish them.

Panel of Figure 19 — Figure 19.1 : Finite-data certification transports a covered set of population worlds through a declared query before producing a terminal label.

Declared item	What must be auditable
coverage regime	Fixed-record, simultaneous, or anytime validity; pointwise, uniform, or asymptotic scope.
protected scope	The times, queries, subgroups, model components, and selection rules covered by one event.
population content	The nuisance, regularity, sampling, and identification assumptions encoded in a complete world.
set status	Whether the reported set is exact or an inner or outer approximation, and how that approximation changes honesty or decisiveness.
empty-set rule	$C_{n}=\varnothing$ is reported as model conflict $M$ ; it is not used for vacuous inference.

Table 19.1: The confidence-world construction contract.

State	Frontier condition	Meaning and authorized next step
common-deployable	$V_{\rm cd}\leq\eta$	The feasible sets have a common point; exhibit and deploy one common witness.
uniformly impossible	$V_{\rm lo}>\eta$	Every compatible world is infeasible; report impossibility or revise the budget or contract.
truth-mixed unresolved	$V_{\rm lo}\leq\eta<V_{\rm pw}$	Some worlds are feasible and others are not; collect data that separates them.
common-witness conflict	$V_{\rm pw}\leq\eta<V_{\rm cd}$	Every world is individually feasible but the feasible-set intersection is empty; shrink $C$ or enlarge the legal routing or robust-deployment grammar.

Table 19.2: Four confidence-world deployment states under attainment.

Object	Typical form	Quantifier, guarantee, and proper use
Bayesian belief	posterior, Bayes risk, expected task information	Prior-average; evaluates expected task value and helps select the next experiment.
Certificate evidence	worldwise likelihood, max–min KL, characteristic time	Worst binding opposite-label world under fixed confidence; authorizes stopping and a terminal claim.
Recursive state	colored belief quotient, labeled update kernels, $N_{h}$	Closed under every future observation context; determines which states may be safely merged.

Table 20.1: Three statistical objects that must not be conflated.

Dept.	$n_{M}$	$n_{F}$	Estimate	Simultaneous interval
A	825	108	$-20.35$	$[-34.32,-2.96]$
B	560	25	$-4.96$	$[-33.14,31.09]$
C	325	593	$2.86$	$[-10.47,16.35]$
D	417	375	$-1.84$	$[-15.67,11.99]$
E	191	393	$3.83$	$[-11.57,19.83]$
F	373	341	$-1.14$	$[-8.95,6.58]$

Table 20.2 : Department admission-rate differences and 95% simultaneous exact intervals, in percentage points.

Panel of Figure 21 — Figure 21.1 : The three architecture chains after finite-information closure. Each has its own domain, exactness obligation, and reporting scale; they meet only at a common certified action.

Panel of Figure 22 — Figure 22.1 : The proposed OALI workflow. Arrows denote audit stages rather than theorem implications. A certificate motivates a repair candidate but does not establish saturation, mechanism exclusivity, or held-out improvement.

Diagnosed mechanism	Candidate structural intervention
excess oracle variation	additional charts, local experts, or test-time refinement
label monodromy	atlas, quotient, set-valued, or randomized output
missing branches	partial transport, abstention, or variable-cardinality output
coordination obstruction	rectangularization, memory refinement, or revised sharing
carrier information loss	richer carrier or task-relative sufficient state
decoder nonsaturation	stronger decoder with the carrier audit retained

Table 22.1 : Structural mechanisms and corresponding repair families.

Figure 23.1 : Synthetic multi-cycle graph illustrating partial transports, cycle-space obstruction, and atlas coverage.

Check	Audit units	Result
degree certificate	240 loops	all singleton sets recovered the generating degree; the certified adjacent-increment bound was $0.02056<\pi$
fixed two-chart common witness	240 loops	maximum recorded native defect $1.79\times 10^{-30}$
full-root-set common witness	240 loops	maximum recorded native defect $1.79\times 10^{-30}$
even-degree oracle identity	101 loops	maximum worldwise oracle-witness defect $5.55\times 10^{-31}$
odd-degree oracle identity	139 loops	maximum residual from the analytic $r^{2}$ identity $6.67\times 10^{-16}$
collapsed-root ablation	240 loops	averaging the two roots returns zero; maximum residual from the analytic $r^{2}$ identity $6.67\times 10^{-16}$

Table 23.1: Radical-loop common-witness deterministic code-path audit.

Stage and location	Certified object	Handoff and non-negotiable boundary
1. Native defect Parts I–IV; Chs. 1 , 5 , and 14	With continuous single-valued point output and uniform loss frozen, $D_{\theta}(a)=\|a^{k}-z(\theta)\|^{2}$ is the objective-native loss scale and $\inf_{a}D_{\theta}(a)=0$ .	The same defect prices constructive upper witnesses and converse lower certificates. A convenient parameter distance cannot replace it.
2. Lift Parts III–IV; Chs. 10 and 14	The radical question is whether $z:S^{1}\to\mathbb{C}^{\star}$ lifts through $p_{k}(a)=a^{k}$ ; a lift exists exactly when $k\mid\deg z$ .	This covering lift is not permission for a target-calling auxiliary coordinate. Any carrier that exposes $u=z(\theta)$ must be declared and pass the lift-admissibility audit.
3. Architecture obstruction Part IV; Ch. 14	The failed lift becomes an exact native floor: for the round loop, Theorem 14.1 gives $0$ when $k\mid\ell$ and $r^{2}$ otherwise.	The claim is for the declared continuous, one-chart, point-valued class under uniform risk. Average risk, measurable output, an atlas, or a quotient is a different contract.
4. Visibility Parts IV–V; Chs. 15 and 17	A labeled root exposes monodromy; the full root set or an appropriate quotient can erase it. Legal contexts decide whether that distinction remains task-observable.	Visibility is a semantic gate, not another scalar tax. A change of output meaning or context class must be declared before certification.
5. Confidence world Part VI; Ch. 19	Finite data determine a compatible degree set and hence a feasible–impossible–unresolved certificate; regularity and coverage can collapse that set to one degree.	Hidden winding may not be guessed. Without adequate regularity or coverage, unresolved is the honest result, and every conclusion must hold throughout $C_{t}$ .
6. Common witness Part VII; Ch. 21	When $C_{t}$ stays away from zero and deployment exposes $u$ , the fixed rule $W_{k}(u)$ , or the fixed two-chart atlas with retained chart label, belongs to every world’s repaired feasible set.	Worldwise divisibility alone does not produce one common $\theta\mapsto a(\theta)$ . If $u$ is hidden or a global labeled root is required, these witnesses are inadmissible.

Table 23.2 : The radical loop as a seven-step certification chain.

Audit component	Frozen settings	Held-out result	Interpretation
Training data	10,000 / 30,000 / 55,000 images	$5.084/3.016/2.642$ nat	continued improvement; no saturation claim
Training effort	4 / 12 / 24 epochs	$4.296/3.053/2.642$ nat	continued improvement; no saturation claim
Encoder width	64 / 256 / 512 hidden units	$3.778/2.642/2.527$ nat	continued improvement; no saturation claim
Selected one-pass gap	validation-selected finite library	$2.493$ ( $0.126$ ) nat; lower bound $2.373$	passes $>0.5$ -nat gate
Eight-step residual ratio	uniform $T=8$ refinement	$0.354$ ( $0.027$ ); upper bound $0.380$	passes $<0.80$ in $5/5$ runs

Table 24.1 : Prospectively frozen MNIST audit. The first three rows report mean test amortization gaps in nats across five independently trained decoders. The final two rows are the seed-level primary endpoints; parentheses contain the seed standard deviation and confidence bounds are one-sided 95% Student

t

bounds.

Link	Frozen gate	Confirmatory result	Decision role
Protocol integrity (3)	protocol ID, source hashes, and confirmatory completeness	all three passed	the contract predates scoring
Operational gap (3)	gap lower bound $>0.004$ , oracle dominance, proximal residual near zero	gap 0.00831; lower 0.00768	positive fitted-baseline defect; not a class infimum
Structural saturation (1)	strict-KKT Jacobian conflict $>0.05$	conflict 0.869	one fixed pass cannot hit both certified neighborhoods
Targeted repair (1)	residual-ratio upper bound $<0.25$	ratio 0.134; upper 0.141	two-step mechanism-matched reduction
Equal-information/equal-compute control (1)	repair/generic upper bound $<0.50$	ratio 0.218; upper 0.233	rules out extra native information or computation alone
Negative control (1)	permuted-gradient lower bound $>1.05$	ratio 3.049; lower 2.887	destroying the mechanism reverses the gain

Table 24.2 : Digits R2 equal-information, equal-compute control upgrade. Bounds are one-sided 95% paired bootstrap bounds over 360 confirmatory images. The table groups all 12 frozen gates; every gate passed.

Link	Frozen gate	Confirmatory result	Architecture role
Protocol and numerical quality (5)	ID, ten hashes, 89 rows, oracle residual, oracle dominance	all five passed	excludes post-hoc refitting and a numerical pseudo-oracle
Operational gap (1)	baseline-gap lower bound $>10^{-4}$	gap 0.000478; lower 0.000304	positive native defect of the fitted one-pass rule
Structural saturation (1)	strict-KKT conflict $>0.02$	89 points; conflict 0.601	one thresholded-affine pass cannot hit both certified neighborhoods
Targeted repair (1)	residual-ratio upper bound $<0.30$	ratio 0.100; upper 0.149	large reduction in native defect
Equal-information/equal-compute control (1)	target/generic upper bound $<0.50$	ratio 0.118; upper 0.187	carrier and MAC alone do not explain the gain
Mechanism negative control (1)	target/permuted upper bound $<0.50$	ratio 0.0456; upper 0.0670	correct gradient coordinates matter

Table 24.3 : Frozen patient-level diabetes confirmation. One-sided 95% paired bootstrap bounds treat the patient as the independent unit. All 12 frozen gates passed.

Mechanism	Target contract	Coupling price and certificate	Reproducible evidence
Uncorrected shared Gibbs	perturbed unless $\epsilon_{t}=0$	stationary defect $\epsilon_{t}/(1-\gamma_{t})$ ; conditional and kernel discrepancy	exact binary slopes: 1.006 one-sided, 2.006 centered; all 18 evaluated perturbation bounds pass
MH-corrected coupled move	exact under the declared ratio	rejection and mixing variance; acceptance and autocorrelation	comparator only; no numerical claim in this laboratory
Exact-ratio particle bank	population base change is exact; finite- $N$ self-normalization remains	$\chi^{2}$ overlap, ratio evaluations, memory, and ESS-triggered refresh	smooth-family MSE $0.221\times 10^{-3}$ versus $0.658\times 10^{-3}$ , with 2,016 versus 3,936 fresh draws

Table 24.4 : Three diagnostic scales for posterior-family Monte Carlo reuse. The final column reports only the fixed analytic pilots described above.

Endpoint	Estimate	95% interval	Unit and method
Sample-level macro-F1	0.9304	(0.9171, 0.9443)	10 samples; bootstrap
Rare-population recall	0.4683	(0.3419, 0.5709)	10 samples; bootstrap
Unresolved-cell rate	0.0129	(0.0104, 0.0155)	10 samples; bootstrap
Edge-match precision	0.9959	(0.9851, 0.9989)	486 matches; Wilson
Edge-match recall	1.0000	(0.9921, 1.0000)	484 shared branches; Wilson
Missing-branch detection recall	0.9623	(0.9070, 0.9852)	106 missing branches; Wilson

Table 24.5 : Synthetic cytometry smoke test with 95% intervals. Sample-level endpoints use 5,000 bootstrap resamples of the ten samples; discrete edge and cycle endpoints use Wilson intervals.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix


Term	Working definition and boundary	Gateway
A–C
Architecture class	A declared set $\mathfrak{A}$ of deployable fields satisfying the output, measurability, sharing, regularity, and resource contracts. It is not the training algorithm or merely the subset that one optimizer happens to reach.	Ch. 2
Architecture grammar	The rules that generate legal representations, fields, compositions, and resource budgets. In finite-data work the grammar itself may be uncertain; it is not automatically fixed by a model name or parameter count.	Ch. 2
Architecture obstruction	The infimum of native defect over a declared architecture class under a stated risk aggregation rule. It is a class-level realizability floor, not the implementation gap of one trained deployment.	Ch. 11
Atlas	A cover of the instance space by domains admitting legal local sections or experts, together with an admissible routing rule. Adding an atlas changes the deployment contract; it is not merely widening one global model.	Ch. 15

Appendix

Table F.1: Controlled glossary of book-level terms.

Source	Primary book locations
Elimination Geometry I	Chapters 2 and 5–8; native defect, P/G/X/V/C, exactification, graph-CDF/CRPS, and model-specific examples
Elimination Geometry II	Chapters 7–9 and boundary material in Chapter 26; exactification converse, probability-validity gates, integrability, and representation limits
Elimination Geometry III	Chapter 18; soft conditional chains, pushforward, base change, conditioning, and zero temperature
Foundations of Elimination Geometry	Chapters 2, 6, and 18; defect fibrations, obstruction towers, and interchange coherence
Elimination Obstruction Transfer	Chapters 11–12; second elimination, regular obstruction transfer, flow duality, and population thickening
COT	Chapters 6, 13, and 18; rectangularity, coordination tax, architecture base change, and dequantization

Appendix

Table 30

No.	Formal result	Class	Antecedent and audited increment
1	Proposition 3.1 : exact four-component risk identity	B	Algebraic telescoping over the classical approximation, estimation, and optimization ledger. The book-level increment is the contract-relative split of approximation into local-model and shared- architecture accounts; compare amortization-gap work ( 44 ; 136 ) .
2	Theorem 3.2 : certified learning bound	S	Standard uniform convergence plus approximate empirical-risk minimization. The specialization preserves the model/architecture floor; it adds no new concentration inequality.
3	Theorem 5.1 : conjugate defect identity	C	Fenchel–Young/Bregman identity ( 31 ; 162 ) . The book uses it to fix the native defect and its direction.
4	Proposition 6.1 : tower associativity	C	Classical infimal-convolution/Bellman associativity ( 18 ; 7 ) ; the typed tax interpretation is the book’s organizational layer.
5	Theorem 7.1 : jet exactification	S	Elementary Taylor subtraction, adjacent to first-order surrogate touching ( 133 ) . The specialization uses the exact elimination defect rather than an arbitrary correction.
6	Theorem 7.2 : converse exactification normal form	P	Within the declared frozen-lift class, target-jet preservation forces defect-jet subtraction modulo a flat term. The source claims this normal form, not tangent-surrogate methodology in general.

Appendix

Table 31

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