OMNIBAL: FROM ONE DIMENSION TO ANY DIMENSION Omnibal - Omnidimensional Midpoint balance framework
OMNIBAL: FROM ONE DIMENSION TO ANY DIMENSION
Formulas, compact data structures, irregular sequences, and budget-aware coding agents
Imagine a line of numbers becoming a square, a cube, and then a higher-dimensional arrangement. Now imagine choosing squares, cubes, fourth powers, or another nonnegative integer power independently of that arrangement.
That is the starting point for my OMNIBAL framework: identify the rule behind a structured sequence, represent that rule compactly, and choose an aggregation algorithm that uses the structure.
The opportunity is practical. When values follow an established progression, a generating formula can replace an explicit list, and an exact aggregate can sometimes replace a term-by-term loop. When the data is irregular, the system must retain the irregular information and account for the work required to inspect it.
The current Python package connects AP, GP, HP, polynomial differences, geometry, OmniFit-style segmentation, residual corrections, and agent-budget allocation through a common set of interfaces.
Caption: A conceptual journey from a line to higher-dimensional projections. The displayed powers are examples;
DIMENSION AND POWER ARE TWO DIFFERENT CHOICES
Dimension describes the arrangement. Power describes the operation applied to each value.
A one-dimensional sequence can contain fourth powers. A three-dimensional arrangement can use first powers. A 100-dimensional shape can use squares.
For a shape with D side lengths:
shape = (n_1, n_2, ..., n_D)
X = product(n_d, d=1,...,D)
For equal sides:
X = n**D
Here X is a discrete number of positions, not a physical volume.
The AP power-sum engine supports p = 0, 1, 2, 3, ... . This does not mean that negative or fractional powers automatically share the same finite polynomial formula. Those cases require appropriate domains and different methods. HP handles positive reciprocal powers explicitly.
For the bases 1, 2, and 3:
p = 0: 1, 1, 1
p = 1: 1, 2, 3
p = 2: 1, 4, 9
p = 3: 1, 8, 27
p = 4: 1, 16, 81
These categories are independent of the dimensional layout.
Caption: Columns change the shape; rows change the exponent. The number labels illustrate powers of three example bases rather than listing every value in the depicted grids.
FROM AN ORDINARY ARRAY TO A GENERATING DATA STRUCTURE
An ordinary array stores every element. A known progression can instead store its generating parameters.
For an AP:
a_i = F + h*i
i = 0,...,X-1
The representation needs F, h, X, and optionally the dimensional shape. The shape occupies O(D) stored numbers, while the AP generator uses a fixed number of parameters.
Coordinates map to a row-major index:
index = 0
for coordinate, side in dimension order:
index = index*side + coordinate
value = F + h*index
Coordinate mapping takes O(D) arithmetic. If a consumer requests all X values, producing those values still requires at least O(X) work. Compact representation is most useful when the consumer can operate directly on the generating rule.
In the package:
shape = Shape((2, 2, 2))
sequence = AP(3, 2, shape.count)
sequence.at_coordinates((1, 1, 1), shape) # 17
AP: THE MIDPOINT AND EVEN-MOMENT ENGINE
The AP formulation measures every term from the midpoint:
M = F + h*(X-1)/2
For nonnegative integer p:
S_p = X * sum(
C(p,2*m) * M**(p-2*m) * h**(2*m) * mu_(2*m)(X),
m=0,...,floor(p/2)
)
C(p,k) is a binomial coefficient. The normalized centered moment is:
mu_(2*m)(X) = (1/X) * sum(
(i-(X-1)/2)**(2*m), i=0,...,X-1
)
This definition assumes X > 0. An empty sum returns zero before normalized moments are calculated.
The first reusable blocks are:
mu_0 = 1
mu_2 = (X**2-1)/12
mu_4 = (3*X**4-10*X**2+7)/240
mu_6 = (3*X**6-21*X**4+49*X**2-31)/1344
Odd centered contributions cancel by symmetry. The surviving blocks give:
S_0 = X
S_1 = X*M
S_2 = X*(M**2 + h**2*mu_2)
S_3 = X*(M**3 + 3*M*h**2*mu_2)
S_4 = X*(M**4 + 6*M**2*h**2*mu_2 + h**4*mu_4)
Consider the eight values 3, 5, 7, 9, 11, 13, 15, 17 arranged in a 2-by-2-by-2 shape:
X = 8
F = 3
h = 2
M = 10
mu_2 = 21/4
mu_4 = 777/16
S_4 = 8*(10000 + 12600 + 777)
= 187016
This is exact rational/integer arithmetic.
Higher powers can generate moments through a raw-power recurrence:
P_0(X) = X
P_k(X) = (
X**(k+1) - sum(C(k+1,j)*P_j(X), j=0,...,k-1)
)/(k+1)
c = (X-1)/2
T_k(X) = sum(C(k,j)*(-c)**(k-j)*P_j(X), j=0,...,k)
mu_k(X) = T_k(X)/X
For fixed powers 0 through 4, the implementation takes a fixed number of arithmetic operations after calculating X. Its general-power implementation uses O(p² + p log p) arithmetic operations and O(p) stored numbers, excluding binomial-generation and rational-normalization costs.
GP: SUMS AND PRODUCTS OF MULTIPLICATIVE SEQUENCES
A geometric progression is:
a_i = a*r**i
Let q = r**p. Then:
If q != 1:
sum(a_i**p) = a**p*(q**X-1)/(q-1)
If q == 1:
sum(a_i**p) = X*a**p
The product of nonnegative integer powers is:
product(a_i**w) = a**(w*X)*r**(w*X*(X-1)/2)
For 3, 6, 12, 24, 48, 96:
sum = 189
product = 23887872
GP representations are useful for exponential schedules, bounded retry backoff, and decaying historical weights. Fixed-power queries use O(log X) exponentiation multiplications, while exact operand and result sizes can grow substantially.
HP: RECIPROCAL POWERS WITH AN EXPLICIT ACCURACY POLICY
For harmonic progression values:
a_i = 1/(F+h*i)
H_p = sum((F+h*i)**(-p), i=0,...,X-1)
The exact implementation sums rational values directly. It rejects zero denominators. For positive denominators, a midpoint approximation reuses the centered moments:
M = F+h*(X-1)/2
T_k = X*mu_k
H_approx = sum(
rising(p,2*m)/(2*m)!
* h**(2*m)*T_(2*m)/M**(p+2*m),
m=0,...,K-1
)
Here rising(p,k) = p*(p+1)*...*(p+k-1), with rising(p,0)=1.
Let:
rho = abs(h)*(X-1)/(2*M)
For positive denominators, positive integer p and rho < 1, the package supplies the conservative bound:
abs(H_exact-H_approx) <= E
E = X/M**p * rising(p,2*K)/(2*K)!
* rho**(2*K)/(1-rho)**(p+2*K)
The truncated expansion is approximate. Exact special-function identities also exist. For F>0, h>0 and z=F/h:
H_1 = (digamma(z+X)-digamma(z))/h
For integer p > 1:
H_p = (hurwitz_zeta(p,z)-hurwitz_zeta(p,z+X))/h**p
The package exposes these identities as reference strings, not numerical special-function evaluations. Exact HP sums generally take O(X) arithmetic. Its K-block approximation constructs moments in O(K² + K log K) arithmetic, excluding coefficient and rational-normalization costs.
IRREGULAR DIFFERENCES AND RATIOS: KEEP THE INFORMATION
Real data often has no common difference or common ratio. The representation must reflect that.
For arbitrary differences d_j:
a_i = F + sum(d_j, j=0,...,i-1)
sum(a_i) = N*F + sum((N-1-j)*d_j, j=0,...,N-2)
For arbitrary multipliers r_j:
a_i = a_0*product(r_j, j=0,...,i-1)
product(a_i) = a_0**N
* product(r_j**(N-1-j), j=0,...,N-2)
Examples:
First value: 10
Differences: 2, 5, -3, 0, 11
Values: 10, 12, 17, 14, 14, 25
First value: 2
Multipliers: 3, 2, 1/2, 5
Values: 2, 6, 12, 6, 30
The VariableSteps and VariableRatios adapters build ordinary prefix indexes. Preprocessing and storage are O(N); subsequent range sums and products use O(1) arithmetic queries. Separate zero counts make product queries safe around zeros.
These are classical data-structure techniques incorporated into a common interface. Arbitrary ratios still carry N-1 pieces of input information.
Caption: AP, GP and HP rules remain distinct. OmniFit preserves exact structured segments and irregular exceptions instead of discarding observations to manufacture a pattern.
OMNIFIT: EXACT SEGMENTS, RESIDUALS, AND PAD–SOLVE–CLIP
The Python OmniFit adapter scans the original order and identifies exact contiguous AP, GP and HP runs. Unmatched values remain raw segments. It preserves order, duplicates, signs and outliers.
If there are S retained segments, descriptor storage is O(S). Detection reads all N observations. A noisy input can require S=N, so compression is conditional on structure.
A second approach stores a baseline with sparse corrections:
x_i = baseline_i + e_i
sum(x_i**p) = sum(baseline_i**p)
+ sum((baseline_i+e_i)**p-baseline_i**p, i in residuals)
When every baseline value is nonzero:
product(x_i**p) = product(baseline_i**p)
* product(((baseline_i+e_i)/baseline_i)**p, i in residuals)
Zero baseline products trigger direct exact multiplication. R residuals occupy O(R) storage, but building the fit still inspects all observations and R may equal N.
For a required padded shape, a valid AP tail can be removed analytically:
AP_sum(F,h,N,p)
= AP_sum(F,h,X_pad,p)
- AP_sum(F+h*N,h,X_pad-N,p)
For a GP product with nonzero tail product:
actual_product = padded_product/tail_product
Synthetic tail values are computational bookkeeping, not newly observed data. Direct evaluation at N is usually cheaper if padding serves no other purpose.
HYBRID AP AND POLYNOMIAL DIFFERENCE HEADS
For a polynomial f of known degree q:
Delta f(i) = f(i+1)-f(i)
d_k = Delta**k f(0)
f(i) = sum(d_k*C(i,k), k=0,...,q)
sum(f(i), i=0,...,X-1)
= sum(d_k*C(X,k+1), k=0,...,q)
The sequence 1, 3, 7, 14, 25 has difference heads (1, 2, 2, 1). Under that cubic model, the next terms are 41 and 63.
Any finite list can be reconstructed with a sufficiently high-degree polynomial. That fact does not establish a meaningful future pattern. Building all differences from arbitrary N observations takes O(N²) arithmetic in this implementation.
AP, GP and HP can also form an explicit mixed schedule:
f(i) = alpha*AP(i) + beta*GP(i) + gamma*HP(i)
sum(f(i)) = alpha*sum(AP)
+ beta*sum(GP) + gamma*sum(HP)
Linearity applies to the first-power sum. Higher powers introduce cross terms, which the implementation handles through exact enumeration.
GEOMETRY: SHAPES, MEASURES, AND DISTANCES
The geometry family describes continuous shapes and feature spaces. It is distinct from the discrete count of entries in a tensor.
For a hyperrectangle with positive side lengths s_j:
V = product(s_j)
Boundary_measure = 2*sum(V/s_j)
For a D-dimensional hypercube:
Number_of_k_faces = C(D,k)*2**(D-k)
For p >= 1, the Lp distance is:
distance_p(x,y) = (sum(abs(x_j-y_j)**p))**(1/p)
distance_infinity(x,y) = max(abs(x_j-y_j))
For a radius-R Lp ball, p > 0:
V(D,R,p) = (2*Gamma(1+1/p))**D
* R**D/Gamma(1+D/p)
For p<1 this set is not a norm ball. The Euclidean case is p=2:
V(D,R,2) = pi**(D/2)*R**D/Gamma(1+D/2)
Sphere_boundary(D,R) = D*V(D,R,2)/R, for R>0
For an ellipsoid:
V = V(D,1,2)*product(semiaxes)
For a generalized superellipsoid defined by
sum((abs(x_j)/a_j)**p_j) <= 1:
V = product(2*a_j*Gamma(1+1/p_j))
/Gamma(1+sum(1/p_j))
For a general D-simplex:
V = abs(det(edge_matrix))/D!
Boxes and simplexes can use exact rational inputs. Gamma-based volumes use floating-point arithmetic in this package. A distance calculation costs O(D); the implemented general simplex calculation costs O(D³) arithmetic and O(D²) working storage.
CODING AGENTS: SCORING, EVIDENCE, AND CONSERVED BUDGETS
An agent can use positive, normalized relevance signals for symbols, paths, callers and verification evidence. A weighted power mean provides one explicit aggregation choice:
M_p = (sum(w_i*x_i**p))**(1/p), for p != 0
M_0 = exp(sum(w_i*log(x_i)))
Weights are nonnegative and normalized to sum to one. The familiar arithmetic, geometric and harmonic means correspond to p=1, p=0 and p=-1.
This creates a transparent scoring rule, not proof that the ranking improves correctness. The signals and weights require evaluation.
For integer time, token or action allowances, choose positive AP, GP or HP weights:
AP: w_i = F+h*i
GP: w_i = a*r**i
HP: w_i = 1/(F+h*i)
A(k) = sum(w_i, i=0,...,k-1)
b_i = floor(B*A(i+1)/A(C))
- floor(B*A(i)/A(C))
sum(b_i, i=0,...,C-1) = B
For AP weights:
A(k) = k*(2*F+(k-1)*h)/2
One planning example is 480 seconds per issue, with 60 seconds reserved for final review. Distributing the remaining 420 seconds over 28 equal-weight calls gives a target of 15 seconds per call.
Actual tool and inference latency will vary. A controller must measure elapsed time and check:
remaining = total-elapsed-final_reserve
action_timeout <= remaining
The local BudgetController checks calls and remaining time before dispatch and passes a timeout to subprocess execution. The notebook demonstrates a real temporary Python source repair and verification. It does not invoke Gemma, enforce LLM token generation, or modify the Kaggle evaluation harness.
Known AP, fixed supported power:
O(1) aggregation arithmetic after O(D) shape reduction.
General AP power p in the current implementation:
O(p²+p log p) arithmetic, excluding comb/gcd costs.
O(p) stored numbers.
GP sum/product, fixed power:
O(log X) exponentiation multiplications.
Exact HP sum:
Generally O(X) arithmetic.
Arbitrary differences/ratios with prefix indexing:
O(N) build and storage; O(1) range-query arithmetic.
Emit C budget allowances:
O(C) time and output storage.
These counts describe arithmetic operations and stored numbers. Exact integers and fractions become more expensive as their bit lengths grow. Writing a large answer also takes time.
Reading arbitrary source files, sorting arbitrary values, searching an unindexed repository, or running an LLM retains its own computational costs.
FROM A FORMULA LENS TO EXECUTABLE SOFTWARE
The notebooks write importable Python files, execute the package, and demonstrate each family with its usage and complexity. A separate usage notebook loads the existing package from:
Omnidimensional Data Structure and Algorithms
The package's 17 unittest methods passed locally, covering many formula, boundary, irregular-data, geometry and budget cases. The usage notebook's 16 code cells also executed locally against the package without changing its source files.
Those are local engineering checks. No Kaggle leaderboard gain or overall model speedup has been measured, and the Kaggle runtime itself has not been exercised here.
Potential applications include analytical fixtures, structured simulation components, regular layout calculations, schedule design, and repeated aggregates over known progressions. The next useful experiment is to choose a real workload, include preprocessing and output costs, and compare it with an efficient baseline.
My OMNIBAL research presents these families through a common language of dimensions, powers, midpoint balance and fit/correction. The classical identities remain their mathematical foundation. The engineering opportunity is to make valid structured computation easy to recognize, reuse and test.
The question for a developer is concrete: which part of your workload follows a rule that you can represent and verify before you enumerate it?
REFERENCES AND ATTRIBUTION
Kumaran Kanniappan — Random Omnidimensional Hybrid AP GP HP
Kumaran Kanniappan — Omnidimensional Formula usecase: Claude API token and Cost Saving
https://www.linkedin.com/pulse/omnidimensional-formula-usecase-claude-api-token-kumaran-pk1zf
Andrometocs — Omnidimensional AP Series Formula Framework
Andrometocs — Nth Power AP Series presentation
Andrometocs on LinkedIn
https://in.linkedin.com/company/andrometocs-global
kumarandatascientist on Kaggle
https://www.kaggle.com/kumarandatascientist
NIST Digital Library of Mathematical Functions, Section 5.19 — Mathematical Applications
#OMNIBAL #Omnidimensional #DataStructures #Algorithms #AppliedMathematics #Python #CodingAgents #Andrometocs
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