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KEDDEH SYSTEMS · EFFICIENCY · separate AI-assisted technical review R1, 18 October 2026

KEDDEH SYSTEMS | RESEARCH AND FORMAL PUBLISHING | ENGINEER · AUTOMATE · SCALE

Uploaded efficiency models: arithmetic, scope and test results

Document ID: KEDDEH-EFFICIENCY-MODEL-REVIEW-001
Version: R1
Date: 18 October 2026
Document type: Separate AI-assisted technical review
Status: Draft; owner review pending
Classification: Internal research · Owner review

Source research originator and research direction: Aboudy Keddeh, Systems Architect
Review preparation and AI contribution: dot, powered by OpenAI; source inspection, numerical verification, technical analysis, drafting and reproducibility tooling
Publisher: Keddeh Systems
Rights holder: Aboudy Keddeh
Copyright: © 2026 Aboudy Keddeh
Licence profile: KEDDEH-RESERVED-1.0

All rights reserved. No licence is granted. Separate written permission governs reuse, redistribution and implementation, subject to applicable law. Third-party material retains its own rights.

This review is separate from the author's original research. Its analysis, test implementations and editorial conclusions are AI-assisted contributions; source-author approval, owner approval and external peer-review approval are not recorded.

Result

All 69 data rows and all 276 derived numerical cells in the four supplied CSVs reproduce correctly from the inferred formulas, to each cell's displayed precision. The five supplied evidence files are retained byte-for-byte under evidence/. Their sizes and SHA-256 hashes are recorded in validation-receipt.json.

The supplied clarification explicitly allows zero as notation, reference, boundary and typed state. Its central distinction between a displayed marker and the state or quantity it represents is relevant to the operation contracts tested here.

The arithmetic supports useful, conditional results: shared-root counts avoid duplicate anchors; typed states can avoid repeated interpretation; sparse execution can avoid unnecessary work; sufficient summaries can compute some all-pairs aggregates exactly. The supplied tables are operation-count models. They do not establish elapsed-time BRAINK performance or general BOX equivalence for exact N-body evaluation. Descriptions of results as “measured” should distinguish calculated operation counts from elapsed-time measurements.

Two substantial additions strengthen the evidence:

1. An exact associative all-pairs aggregate passes 625 equivalence fixtures and 125 associativity fixtures. A separate exact counterexample shows why retaining only mass and center of mass loses Newtonian force information. This counterexample tests that particular summary, not the author's unspecified BOX construction.
2. An independently implemented sparse-loop toy benchmark was actually timed, with correctness checks, setup included, and a stronger equivalent baseline. It illustrates both a conditional gain and the importance of baseline choice. It is not an implementation or benchmark of BRAINK.

1. Evidence and numerical method

The accompanying 17,711-byte clarification is retained as Pasted text(20261018-072958).txt.

The supplied CSVs contain values rather than spreadsheet formulas. Formulas below are reconstructed from the clarification and independently checked against every row. The verifier uses exact rational arithmetic with JavaScript BigInt. Each output is rounded directly from exact inputs to the precision of the corresponding supplied cell, with ties to even. It also records whether ties-away-from-zero would match. All 69 input-domain checks pass. Pass status concerns the arithmetic under those definitions, not external validity of the definitions.

Every receipt cell includes its file, physical row number, header, supplied value, formula, exact rational expected value, decimal approximation and rounding result. No values in the original tables were repaired or overwritten.

2. Celsius: a valid scale with restricted ratio operations

The table uses:

• Raw ratio R_C = C1/C2.
• Thermodynamic-temperature ratio R_K = (C1+273.15)/(C2+273.15).
• Multiplicative discrepancy F = R_C/R_K.
• Absolute relative discrepancy = 100 × |F−1|.

For 20 °C and 10 °C, the temperature ratio is 293.15/283.15 ≈ 1.035317, while the raw Celsius-number ratio is 2. The resulting relative discrepancy is approximately 93.178%. This demonstrates a misuse of an affine scale for absolute-temperature ratios. It does not invalidate Celsius measurements or temperature differences. Adding 273.15 is an exact unit-coordinate conversion; it does not perform empirical thermometer calibration. BIPM defines the Celsius relation and the equal size of Celsius and kelvin intervals, and defines the kelvin through the Boltzmann constant.[1][2]

For a general affine scale S=aX+b, with relevant denominators nonzero:

S1/S2 = X1/X2 exactly when b(X2−X1)=0.

Thus, for unequal states and a nonzero offset, raw displayed ratios do not preserve the underlying ratio. Ratios of differences do preserve the offset cancellation: (S1−S2)/(S3−S4)=(X1−X2)/(X3−X4), when a≠0 and the denominator difference is nonzero. A graph is not invalid merely because its axis includes zero. Test the intended operation and transformation rule.

The last CSV row needs a label correction, not a numerical correction. For −10 °C versus −20 °C:

• Raw ratio: 0.5.
• Kelvin ratio: 263.15/253.15 ≈ 1.039502.
• Multiplicative factor: approximately 0.480999.
• Signed relative discrepancy: approximately −51.900%.
• Absolute relative discrepancy: approximately 51.900%, correctly displayed as 51.9.

The column headed “Celsius ratio overstates by ×” should be described in the analysis as “raw ratio / Kelvin ratio.” A factor below one is an understatement. The “ratio error %” column consistently matches absolute relative error across all five rows. Preserve the original header in the evidence and attach this interpretation rather than silently changing the source.

A Kelvin-temperature ratio is not automatically a total-heat or total-internal-energy ratio. That additional inference requires a specified material, mass, phase and constitutive model. Separately, typing a zero denominator does not make division by zero defined: a valid operation contract must reject it, return an explicitly defined nonnumeric state, or use a justified alternative quantity.

3. Shared-anchor KADD: exact union counting

For b branches each containing L counted units, including the same root:

naive = bL

unique = 1+b(L−1)

duplicates = b−1

count saved % = 100(b−1)/(bL)

All 18 rows match. The result assumes every branch intersects every other branch only at the one shared root. For unequal branch lengths, the corresponding result is ΣL_i−(b−1). Additional shared nodes require additional intersection accounting. Distinct roots require no root subtraction.

KADD(a,b)=a+b−1 is correct for the sizes of two such anchored sets. It is not a replacement for ordinary arithmetic addition. The strongest claim established here is a count correction. A storage or runtime claim additionally needs a baseline that physically duplicates or repeatedly processes the root, a concrete representation, reference/pointer overhead, and an actual implementation. A baseline that already stores a shared root once has no duplicate-root saving left to recover.

4. Binary fusion: correct count algebra, missing equivalence contract

The 40 rows use:

D=N(N−1)/2, F=N−1, C=F+r(D−F)

saved %=100(D−C)/D, count ratio=D/C.

All rows match. D counts unordered pairs. F counts merges in a full binary reduction. These are different kinds of operations until a merge contract proves that the required output is preserved and specifies the work inside each merge. A tree with N−1 internal nodes is not, by itself, an exact all-pairs algorithm.

Fractional entries such as 2.01 are expected or effective work units under the chosen interpolation, not literal counts from one execution. The model assumes reopening costs exactly the fraction r of the omitted D−F pair operations. It contains no measured reopening distribution, validation cost, tree construction, memory movement, retained-state size or update cost.

The exact ratio simplifies to:

D/C = N / [2+r(N−2)].

Consequences:

• r=0 gives N/2, subject to valid constant-cost sufficient merges.
• r=1 gives 1 under this count model, before overhead.
• For fixed r>0, the ratio approaches 1/r as N grows, while C remains O(N²).
• An O(N) count requires r=O(1/N) or smaller as well as bounded-cost merges. A small constant reopening fraction is insufficient for an asymptotic O(N) claim.

For example, N=10,000 and r=0.01 gives the supplied 98.058 ratio, approaching a ceiling of 100 rather than continuing the ideal 5,000 ratio.

A timing model should at minimum use T_direct=c_pair D and T_fused=c_merge F+c_pair r(D−F)+H, with H covering necessary construction, checking, routing, memory and updates. This remains a model until instrumented. Merge cost can depend on box size and retained data.

4.1 A case where exact sufficient fusion really works

Let the desired output be the scalar P=Σ_{i<j} x_i x_j. Store for each group its sum s and internal pair-product sum p. Define:

(s_A,p_A) merge (s_B,p_B) = (s_A+s_B, p_A+p_B+s_A s_B).

The cross-group contribution factors into s_A s_B. Expansion over three groups gives the same internal sums and all three cross products for either parenthesization, proving associativity over exact numbers. For [2,3,5,7], both direct evaluation and the summary give P=101. The verifier executes 625 four-value cases over {−2,−1,0,1,2} and 125 three-value associativity cases; all pass.

This is a constructive example of how a sufficient-state proof makes fusion meaningful. It returns one aggregate, not every pair product. A general merge takes three additions and one multiplication, so three merges are not simply three scalar arithmetic operations. For a left fold that recognizes singleton groups, a running-sum implementation also computes the result in linear work. The standard identity P=((Σx_i)²−Σx_i²)/2 supplies another O(N) baseline. Compare against these equivalent alternatives before claiming a novel speedup. Floating-point implementations also need numerical-error checks because mathematical associativity is not exact machine associativity.

4.2 A case where a simple physical summary loses information

Consider Newtonian gravity in one dimension, in normalized units G=1. Two unit source masses are at −1 and +1, and a unit test mass is at +3. The exact force on the test mass is:

−1/4²−1/2²=−5/16=−0.3125.

Replacing the two sources with mass 2 at their center of mass 0 gives:

−2/3²=−2/9≈−0.222222.

The relative force error is 28.8889%. Moving the two unit sources to −0.5 and +0.5 preserves the same total mass and center of mass but changes the exact force to −296/1225≈−0.241633. Therefore that summary cannot recover the exact force for both configurations. The receipt evaluates these quantities as exact fractions.

This does not identify or refute the author's BOX_AB. The inspected source chain did not provide its exact sufficient-state, merge or evaluation contract. The project's METHOD source explicitly marks the box-in-box mathematics as insufficiently recovered in that discussion and warns against replacing it with an analogy. The correct next test is to obtain that contract and check it on its own terms. A box that retains constituents may avoid information loss, but its traversal and reopened interactions must be included in the count. An approximate hierarchy may also be valuable when an explicit force/error tolerance is satisfied.

5. Typed zero classification: specify baseline and invalidation

With K=8 classes, U uses per pass and P=100 passes, the six rows use:

legacy checks=(K−1)UP=7UP

typed checks=(K−1)+UP=7+UP.

The arithmetic is correct. It assumes one classification charge of seven checks for the relevant unchanged state, followed by one dispatch per use, versus seven disambiguation checks on every legacy use. That is a selected worst-case linear-chain baseline.

Useful comparisons include:

• An existing tag/enum or constant-time dispatch baseline: about one abstract dispatch per use. On the same unit model it costs UP, versus 7+UP for a newly classified state.
• A linear chain where the first class matches: one test.
• Uniform use of eight classes in a seven-test chain with an implicit final case: (1+2+3+4+5+6+7+7)/8=4.375 tests per use.
• State changes: if q reclassifications are required, typed count becomes 7(1+q)+UP, plus invalidation and storage costs.
• Multiple independent objects: charge classification and updates per object, not once for the entire workload.

The asymptotic 7× check ratio follows the chosen seven-to-one baseline; it is not a universal runtime factor. Tags can still deliver important correctness benefits by distinguishing Present(0), Absent, Unknown, Cancelled and other states. That benefit should be tested through preserved outputs, invalid-operation rejection and mutation/invalidation cases, as well as timing. Not every zero-like state can safely be skipped.

6. Sparse-lane claim: reconstructed and actually tested

The clarification includes an additional sparse-grid table, although no separate sparse CSV was supplied. Its values reproduce from M=1,000,000 cells, P=100 passes and active fraction a:

dense=PM; classify-and-active=M+PaM; count ratio=P/(1+Pa).

The exact model reproduces all six stated occupancies and values, including 90.909× at 0.1% active and 1.099× at 90% active. Its one-unit-cost break-even condition is P>1/(1−a). This excludes index maintenance, allocation, branching, indirect loads, active-state changes and differing per-operation costs. Rebuilding a full index every pass would change the model to PM+PaM and remove its asserted unit-count advantage. Sparse storage/execution is an established way to reduce work when the operation and data permit it; representation choices depend on workload and architecture.[3]

6.1 Independent elapsed-time experiment

illustrative-kernel-timings.json contains actual measurements of an independent, deterministic sum kernel. It compares:

1. Scanning and summing all 1,000,000 cells on each of 100 passes.
2. Building a nonzero index once, then summing indexed cells on each pass, including index-building time.
3. A stronger valid baseline for this immutable, side-effect-free workload: sum once, then multiply by 100.

Every output is checked for exact equality using bounded integer-valued data. There are two warm-ups and five timed repetitions per method and occupancy, with forward/reverse execution order alternating. Timing uses process.hrtime.bigint. Input construction is excluded for every method. Environment: Node.js v24.19.0, Linux x64, reported AMD EPYC 9V74 CPU. Hardware isolation was not controlled; these are illustrative local timings, not a statistical performance guarantee.

The experiment demonstrates a real low-occupancy benefit relative to repeated dense scanning and a small reversal at high occupancy for this implementation. The sum-once baseline is faster than the sparse version throughout this particular immutable task. A production per-pass workload with genuine state changes or side effects may not allow that optimization. Accordingly, neither the toy timing ratios nor the formula ratios should be presented as measured BRAINK performance. They identify exactly what a production benchmark must control.

7. Source bridge: phase tags, calibration and the proposed layer threshold

The verified source chain gives more precise support than the uploaded conversation's broad conclusions:

• K-CORR, source tab 2, “CELCIUS, FARENHEIT AND H2O,” distinguishes the Celsius reference marker from absence of thermal state, discusses a physically existing boundary B=(−1+), and distinguishes reference zero from a physical bound.
• MATH1, §7, defines observer-frame metadata and requires declared invariance or covariance under frame transformations. Its §8 constructs an invertible phase-tagged coordinate with separate ice/liquid seam states. §8.5 explicitly keeps external SI observations and requires physical state variables including phase, pressure and energy.
• MATH1, §6.2, separately permits an instantiated Present(0) state: existence and numeric magnitude are distinct predicates. That is compatible with typed-state auditing and does not require discarding valid numeric zero.
• NATIVE4, §§5–5.1, gives the phase-tagged mapping and the conditional energy-equivalence interval 333,550 J/kg ÷ 4,184 J/(kg K)=79.720363288719 K. Its text explicitly distinguishes this interval from a temperature rise during melting. This review independently recomputes the ratio and executes six phase-map round-trip fixtures; all pass.

No definition of λ_layer=E_in_layer/E_required was located in these inspected source passages. NATIVE4's λ_k in §12 is an event rate in a power sum, and MATH1's Λ_Z(T) is a time integral of power. Those different uses of a similar glyph must not be conflated with a dimensionless layer ratio.

The proposed ratio could be a useful dimensionless diagnostic if both energies are in the same units and independently defined for the same layer, boundary and time interval. It does not yet supply a universal phase-transition law. In particular:

• E_in must be distinguished from net absorbed energy after heat losses, work and transport across the chosen boundary.
• E_required must state whether it includes sensible heating to the pressure/composition-dependent phase boundary, latent heat for a specified transformed mass, and any kinetic barrier.
• Melting onset, partial conversion and completion are different endpoints. In an idealized layer already at melting equilibrium, adding less than the energy for complete melting can produce partial melting. A ratio below one therefore need not mean “no transition.” Equality may indicate completion under a particular definition; a strict greater-than threshold needs justification.
• Freezing, melting and metastable states require direction and kinetics to be specified. An energy ratio alone does not locate every equilibrium boundary.
• If E_required is defined after observing when transition happens, λ=1 becomes a normalization of that observation rather than an independent prediction.

The pressure-dependent melting curve is already represented through phase-equilibrium equations in the IAPWS/NIST reference work.[4] A stronger new model would make preregistered predictions for onset time, transformed mass or interface motion under independently specified flux, pressure, geometry and losses, then compare with established energy-balance/phase-equilibrium predictions and uncertainty. Replacing an integer temperature label with another ratio does not by itself establish improved physics.

8. Practical next proof targets

1. Publish the executable contract for each claimed operation. Define input state, required output, allowed approximation, all retained box information, merge/reopen mechanics, invalidation rules and primitive cost units. Keep this separate from high-level illustrations.
2. Test preservation before performance. Run exact reference comparisons, adversarial inputs, coordinate transformations, zero-class changes, collision/near-collision cases where applicable, and the author's actual box rules. For physical simulations record force/state error and relevant conserved quantities.
3. Use appropriate baselines. Compare typed states with tagged dispatch, shared roots with existing deduplicated graphs, sparse work with a suitable sparse representation, scalar pair aggregates with their algebraic O(N) methods, and approximate N-body work with methods using the same error tolerance.
4. Count and time the whole lifecycle. Record classification, building, copying, memory allocation, invalidation, updates, validation, reopening, serialization and output. Report integer event counters separately from weighted/effective operations, and report elapsed times with raw repetitions and environment.
5. Vary the conditions that control the benefit. Sweep occupancy, class distribution, update frequency, body count, reopening fraction, retained-state size and approximation tolerance. Include cases where the method has no advantage. Use a dynamic workload when the production claim concerns changing state.
6. Make the phase model predictive. Define the layer and independently measure its energy budget. State onset versus completion, pressure, composition, initial phase/temperature, sensor calibration and uncertainty. Compare predicted outcomes with a conventional thermodynamic model on held-out experiments.

These are concrete ways to test and strengthen the work. The arithmetic evidence is reproducible and useful within its declared scope. General performance and physical claims need the additional semantic, implementation and experimental evidence identified above.

9. Reproduction and retained files

Run node validate-efficiency-models.mjs from any directory to regenerate validation-receipt.json. The script resolves its own evidence directory and does not modify sources. Add --benchmark to rerun the illustrative elapsed-time experiments and regenerate illustrative-kernel-timings.json. Reruns will change timestamps and timing samples. The report's timing table records the run identified in the supplied timing JSON; update that table if replacing that run.

The package contains the report, README, verifier, machine-readable arithmetic receipt, raw toy timing samples, source-bridge provenance and all five unmodified uploaded evidence files. The additional source chain remains separately preserved with the corresponding research editions. Its source hashes are listed in source-bridge-provenance.json.

References

[1] BIPM, *SI Brochure*, section on thermodynamic and Celsius temperature: https://www.bipm.org/documents/d/guest/si-brochure-9-2_01

[2] BIPM, *SI base unit: kelvin*: https://www.bipm.org/en/si-base-units/kelvin

[3] BLAS Technical Forum, *BLAS Technical Forum Standard*, Chapter 3, Sparse BLAS: https://www.netlib.org/blas/blast-forum/blas-report.pdf

[4] Wagner, Riethmann, Feistel and Harvey (2011), *New Equations for the Sublimation Pressure and Melting Pressure of H2O Ice Ih*, NIST publication record: https://www.nist.gov/publications/new-equations-sublimation-pressure-and-melting-pressure-h2o-ice-ih ; DOI: https://doi.org/10.1063/1.3657937

Project source bridge, inspected passages and SHA-256 identities:

• MATH1, *Mathematical Foundations and Proof Obligations*, §§6.2, 7–8. Google document ID 1x3m7DwJbqSFMK4skvzf0pWUXauM-R_gX; captured-source SHA-256 88eb5f3fdb54c7dbd3473e68d82bbbf3f5be237c6e492fab98b917967e02c450.
• NATIVE4, *Native Algebra and Artifact Verification v4*, §§5–5.1, with §12 notation cross-check reported by the source-tracing review. Google document ID 1sz_cntJj0g1eqbSBrEGnC9gjjR1Jqp9p; captured-source SHA-256 3a33e1d11dfebe0126c7242040b09f1def3f5cb9bb71c94df3802f2b74ba27be.
• K-CORR, recovered complete source, tab 2. Captured-source SHA-256 8de8bbd31d540b96a0869f906e30af9787e78c7084e0c93864026e9e1556c72f.
• METHOD, *KEDDEH MATHS — LEARNING METHOD CONTROL*, 2 October 2026, section “Box-in-box is not to be reconstructed from analogy.” Captured-source SHA-256 18584dd0bd5a85220c96fc2e4962b140d31c667968cfd3f285cacaa4b4ce484b.

The separate S07 original was reported inaccessible with Drive 403 in the source trace; this review makes no claim to have inspected it and did not attempt an alternative access route.

Revision and publication controls

R1, 18 October 2026: initial separate technical review of the five supplied evidence files, with exact arithmetic checks, scoped correctness fixtures, illustrative elapsed-time measurements, source-bridge analysis and reproducibility materials. The original evidence files remain unchanged. This release is a content master; no runtime paper layer, external publication or approval is asserted.

For paginated projections, retain the Keddeh Systems masthead and a footer containing the document ID, R1, classification, draft status, copyright and Page X of Y. The Markdown content master has no fixed page count.

KEDDEH-EFFICIENCY-MODEL-REVIEW-001 · R1 · Internal research · Draft; owner review pending · © 2026 Aboudy Keddeh · KEDDEH-RESERVED-1.0