Reciprocal Expression and Non-Expression: Toward a Minimal Mathematical Framework for Distinction, Possibility, and Recursive Conditioning

Reciprocal Expression and Non-Expression

Toward a Minimal Mathematical Framework for Distinction, Possibility, and Recursive Conditioning

John Swygert  |  November 12, 2026

DOI: To be assigned

Abstract

This paper develops a provisional mathematical framework for reciprocal expression and non-expression in TSTOEAO. It separates conceptual opposition, bidirectional transitions, causal conditioning, and recursive alteration of future accessibility. A conventional two-state baseline is extended by directional transition fluxes and an independently measured boundary variable. Non-expression is operationally divided into unrealized, inaccessible, and unobservable conditions. A preregistered accessibility rule and a conditional reciprocal-boundary hypothesis identify prospective tests, competing explanations, and quantitative failure criteria. These constructions are research proposals, not a derived or empirically established TSTOEAO physical law. The closed-loop candidate additionally yields a unique positive equilibrium and a locally stable response under its declared positive-parameter assumptions; these are consequences of the illustrative model, not a derivation of a universal TSTOEAO law.

1. The foundational question

The TSTOEAO Paradox: The Opposition of Non-Opposition asks whether expression and non-expression are merely conceptual contrasts or participate in a reciprocal process. Language as Mathematics in Disguise explores how structure and relationships may survive transformations of representation. Read together, these inquiries motivate a testable question: can realized outcomes alter the conditions governing what may be realized next, and can that relation be specified independently of the observation it seeks to explain?

The word “opposite” requires care. Logical negation, mathematical inverse, physical complement, and causal influence are different relations. A descriptive contrast alone does not demonstrate physical reciprocity.

2. Three claims that must not be conflated

First, distinction: a defined system can discriminate expression (X) from non-expression (N). Second, bidirectionality: transitions can occur in both directions under specified conditions. Third, mutual conditioning: the realization of one condition causally modifies the mechanism or constraints governing subsequent realization. Each claim is stronger than the preceding one and requires separate evidence.

3. A minimal two-state baseline

For a population of comparable systems, let p(t) denote the fraction in X. Let alpha and beta be the rates of N-to-X and X-to-N transitions. The standard continuous-time two-state model is:

dp/dt = alpha(1 - p) - beta p

For constant positive rates, p* = alpha/(alpha + beta), and p(t) = p* + [p(0) - p*] exp[-(alpha + beta)t]. Thus individual transitions may continue even as the population fraction approaches equilibrium. Bidirectional transition does not imply oscillation, a self-sustaining cycle, or an independent energy source. The model is conventional mathematics and is not itself evidence for a new TSTOEAO law.

4. Mutual conditioning requires an additional mechanism

A stronger candidate introduces an independently measurable conditioning variable y, such as an accessible boundary configuration, available pathway, or material state that influences transitions:

dp/dt = alpha(y)(1 - p) - beta(y)p

dy/dt = h(p, y, u)

Here u denotes controlled external inputs. This occupancy-based feedback model is useful but insufficient to distinguish effects of being expressed from effects of transitions themselves: two populations can have the same p and different turnover. A transition-sensitive extension is required.

Define directional fluxes J_NX = alpha(y)(1 - p) and J_XN = beta(y)p, so dp/dt = J_NX - J_XN. Replace the boundary law with dy/dt = H(y, p, J_NX, J_XN, u). This allows boundary changes to depend separately on occupancy, expression events, and loss-of-expression events. Fluxes must be measured or estimated with specified time units and uncertainty.

At p = 0.5, alpha = beta = 0.01 yields total transition flux 0.01 per unit time; alpha = beta = 1 yields flux 1. Both have dp/dt = 0, yet the second has 100 times the turnover. If transitions consume energy or leave material records, the boundary response may differ despite identical occupancy. This example is illustrative, not empirical evidence.

Causal identification requires two intervention directions: change y while controlling common causes and measure subsequent fluxes; separately change the transition process and measure y, then future fluxes. Correlation or nonzero derivatives alone do not establish reciprocal causation. A comparator with externally driven rates and no transition-to-boundary feedback must be included.

5. Beyond a binary state: not-yet-expression

Non-expression must not be treated as a single measured state without specifying the mechanism. (i) Unrealized: a state is presently permitted but has not occurred. (ii) Inaccessible: a state is excluded by currently measured constraints. (iii) Unobservable: a state is realized but inaccessible to a specified detector or receiver. Nonexistence is a different claim again. These categories can overlap in ordinary language but must remain distinct in experimental definitions.

Let S be a prespecified state space, b a vector of independently measured boundary conditions, and c(s,b) a registered constraint function. Define accessibility prospectively as A(b) = {s in S : c(s,b) <= 0}. Realized states x(t) must obey the specified evolution or transition law; membership in A(b) permits realization but does not guarantee it.

x(t) in A[b(t)] ;   db/dt = H(b, x, J_NX, J_XN, u)

An outcome may change b without immediately changing the realized state, and a transition may occur without changing A(b). Both possibilities must be allowed as potential counterexamples. Accessibility must be predicted from measured constraints before the subsequent outcome, not reconstructed afterward to fit it. Detector access requires a separate observation model, rather than being conflated with physical accessibility.

6. Relational invariance across representations

The language paper motivates asking whether a relation survives changes of representation. For a mapping F between two systems and corresponding transformations T_A and T_B, one candidate preservation condition is:

F(T_A(x)) = T_B(F(x))

This commutation condition is familiar in established mathematics. Its usefulness here is methodological: identify the relation in advance, test whether it survives transformations, and record where it fails. Similar diagrams across language, geometry, sound, or physical systems are not sufficient; a specified invariant and an independently tested consequence are required.

7. What would count as evidence?

A credible test preregisters the state space, expression category, independently measured boundary b, accessibility constraint c, directional transition fluxes, energetic inputs and costs, comparator model, expected effect directions, and minimum meaningful magnitudes. Accessibility and transition propensity are separate outcomes; the primary outcome and any joint decision rule must be fixed before observation.

Two-direction causal identification requires interventions on b to estimate changes in transition flux, and interventions on turnover to estimate subsequent changes in b and later transition propensity. When feasible, compare matched occupancy and measured energy expenditure at different turnover rates; otherwise explicitly model and measure energy, temperature, common forcing, and other potential confounders. A manipulation of turnover alone is not necessarily an isolated intervention on flux.

8. Conditions of failure and limitation

The bidirectional claim is weakened when prespecified transition rates are bounded below a meaningful threshold. The mutual-conditioning claim is weakened when adequately precise intervention estimates exclude the preregistered minimum effect in either required direction. A proposed recursive-boundary mechanism fails in its declared regime if transition activity produces no predicted boundary change, or the predicted accessibility change does not occur.

For each effect define a smallest meaningful magnitude delta > 0 and a confidence or credible interval procedure in advance. A nonsignificant result alone is not proof of no effect; an interval excluding effects at least as large as delta in the predicted direction is more informative. The model must also accept a stable-boundary counterexample (dy/dt = 0) as a failure of its stronger claim within that regime, rather than inventing a new unmeasured boundary afterward.

9. The unresolved derivation

Candidate conditional reciprocal-boundary proposition (not a canonical TSTOEAO law): In a preregistered system and boundary regime, an intervention changes directional turnover by at least δJ under specified controls. Within a registered time window τ, the independently measured boundary changes by at least δb in the predicted direction. Subsequently, a prespecified primary outcome changes: either (a) accessibility, measured by a declared set distance d_A and threshold δA, or (b) a declared transition propensity with threshold δr. These alternatives are not interchangeable after observation. This is a conditional, falsifiable feedback hypothesis, not a law derived uniquely from TSTOEAO.

10. Measuring accessibility rather than assuming a scalar

An accessible set A(b) is not itself a number. Before using an accessibility-change threshold, specify a metric appropriate to the state space. For finite S, one candidate is d_A(A1,A2) = |A1 △ A2|, the number of states belonging to exactly one set. For example, {a,b,c} and {a,b,d} have equal cardinality but symmetric-difference distance 2. A weighted symmetric difference can distinguish consequential states; continuous domains may instead require a specified measure, Hausdorff distance when applicable, or a task-specific functional. None is universally preferred.

The testable statement becomes d_A(A(b_after),A(b_before)) ≥ δA, with metric and threshold fixed before data collection. A boundary can change without changing accessible membership; conversely, accessibility can change without an immediate realized transition. The theory must not claim an accessibility result from a transition-rate result unless its decision rule explicitly allows that composite outcome.

11. Worked computational example: a boundary with and without turnover sensitivity

Open-loop illustration (not a closed reciprocal cycle). Consider a hypothetical two-state ensemble with p = 0.5. Let α = β = k. Directional fluxes are J_NX = k(1−p) and J_XN = kp, and total turnover J = J_NX + J_XN = k. At k = 0.01, J = 0.01; at k = 1, J = 1. Both cases have dp/dt = 0. This is an illustrative calculation, not empirical data.

Define an independently measurable boundary variable b, with candidate law db/dt = γJ − λb, where γ and λ are preregistered positive parameters and external forcing is held fixed. With b(0) = 0, γ = 0.2 and λ = 1 (consistent time units), b(t) = 0.2k(1−e^(−t)). At t = 1, the predicted values are approximately 0.00126 for k = 0.01 and 0.12642 for k = 1. The difference is approximately 0.12516 despite identical occupancy.

Now define a countermodel db/dt = −λb, with b(0) = 0, so b(t) = 0 regardless of turnover. This countermodel retains two-way transitions yet rejects transition-driven boundary updating. An experiment that precisely supports the countermodel under the prespecified regime contradicts the candidate turnover-sensitive proposition for that regime. Neither model is uniquely entailed by TSTOEAO.

To test accessibility as a distinct outcome, let S = {s₁,s₂,s₃,s₄} and A(b) = {s₁,s₂,s₃} when b < 0.10, but A(b) = {s₁,s₂,s₄} when b ≥ 0.10. At t = 1, the low-turnover model predicts the first set, and the high-turnover model predicts the second; their symmetric-difference distance is 2. This deliberately simplified threshold rule illustrates a preregistered prediction, not a known physical boundary law.

The toy example also exposes a limitation: increasing k may require extra energy or change temperature. Unless these are measured and controlled, a difference in b cannot be attributed uniquely to turnover. A valid study would require a feasible manipulation, matched or modeled energetic costs, independent boundary measurements, uncertainty estimates, and a stronger conventional comparator. The accessibility threshold is stipulated for illustration; a physical test must independently demonstrate that crossing it changes realizable pathways, rather than merely relabeling states. Because α and β are held fixed, the analytic boundary solution does not establish boundary-to-transition feedback.

12. A closed-loop candidate and its failure conditions

The previous example demonstrates turnover-to-boundary response while holding transition rates constant. To model dynamic reciprocity, we must additionally specify how the independently measured boundary alters subsequent transitions. The following is an illustrative conventional feedback system, not a law derived from TSTOEAO.

J_NX = α(b)(1 − p);   J_XN = β(b)p;   J = J_NX + J_XN

dp/dt = J_NX − J_XN;   db/dt = γJ − λb

Choose, before observing data, α(b) = k exp(−θb) and β(b) = k exp(θb), with k, γ, λ > 0 and θ > 0. Here increasing b suppresses N→X and enhances X→N. The exponential forms are assumptions chosen to make the feedback explicit; no TSTOEAO premise presently requires these functions or their signs.

At p = 1/2 and b = 0, J = k, dp/dt = 0 and db/dt = γk. As b becomes positive, the same occupancy would yield dp/dt = −k sinh(θb) < 0. Thus a turnover-induced boundary shift can subsequently bias transitions toward N. Unlike the open-loop illustration, this model contains both causal links. Once b affects α and β, J generally changes over time; the earlier solution b(t) = (γk/λ)(1 − exp(−λt)) no longer applies globally.

A direct boundary intervention at matched p and controlled u predicts a measurable change in the rate ratio: log[β(b)/α(b)] = 2θb. An independent turnover intervention predicts a boundary response after accounting for energy input, temperature, occupancy and common forcing. Both links must be identified experimentally; fitting trajectories alone cannot establish them.

Prespecified failure: if sufficiently precise interventions show θ below a declared meaningful threshold, the proposed boundary-to-rate link is unsupported in the tested regime. If the measured turnover-to-boundary response excludes the predicted γ effect, the other link fails. If both links exist but follow different functions, this particular candidate is contradicted even though generic feedback remains possible. A stable-boundary model (γ = 0) and a boundary-insensitive-rate model (θ = 0) are explicit nested countermodels.

Recursive accessibility and dynamic reciprocity remain distinct outcomes. A registered accessibility experiment tests d_A(A_after,A_before) against δ_A; a registered rate-feedback experiment tests a specified change in α or β (or their ratio) against δ_rate. A study may prespecify a joint rule, but cannot switch its primary outcome after observing results.

13. Equilibrium and stability of the closed-loop candidate

For the illustrative closed-loop model in Section 12, equilibrium requires α(b*)(1−p*) = β(b*)p* and γJ* = λb*. With α(b)=k exp(−θb) and β(b)=k exp(θb), these imply:

p* = 1/[1 + exp(2θb*)];    J* = k/cosh(θb*);    λb* = γk/cosh(θb*).

For k, γ, λ, θ > 0 and b ≥ 0, the final equation has exactly one positive solution: λb increases strictly with b while γk/cosh(θb) decreases for b > 0. Consequently b* > 0 and 0 < p* < 1/2. These are model-specific predictions, not established universal properties of expression.

The Jacobian at the equilibrium, in coordinates (p,b), is:

M* = [ −(α*+β*)       −θJ* ;   γ(β*−α*)       −λ ].

Its trace is −(α*+β*+λ) < 0. Its determinant is λ(α*+β*) + γθJ*(β*−α*) > 0 because b* > 0 implies β* > α*. Therefore both eigenvalues have negative real parts: the equilibrium is locally asymptotically stable. The eigenvalues may be real or a complex-conjugate pair, permitting monotonic or damped-oscillatory relaxation. This local result does not, by itself, establish global stability or perpetual cycling.

The feedback signs are illustrative: increasing b suppresses N→X and promotes X→N in this particular candidate. Reversing signs or choosing different rate laws can change the equilibrium and stability conclusions. Any experimental application must register those signs and functional forms prospectively.

Dimensional consistency. If time has units T and b has units B, then k and λ have units T⁻¹, θ has units B⁻¹, α and β have units T⁻¹, p is dimensionless, and J has units T⁻¹ per modeled system. Accordingly γ has units B per transition (equivalently B when J is expressed as transitions per system per unit time), so γJ and λb both have units B·T⁻¹. The exponent θb is dimensionless. An empirical study must define the physical units of b and the normalization of transition counts.

A test of the exponential rate law may reject its specific prediction log[β(b)/α(b)] = 2θb without rejecting every possible boundary-dependent feedback law. Likewise, finding the predicted stable equilibrium would corroborate this candidate under appropriate controls, not uniquely validate TSTOEAO.

14. Derivation challenge and explicit limits

A distinctive TSTOEAO prediction would require a boundary-update restriction derived from independently stated premises, rather than choosing γ, λ, the threshold, or an invariant after examining results. One possible research question is whether a prespecified relational invariant I(b_after) = I(b_before) must coexist with a predicted accessibility change. This is a question for derivation, not a conclusion of the present paper. Ordinary feedback models may also satisfy such a condition.

A decisive preregistered study must declare which regimes count as in scope, what negative evidence would exclude the predicted effects, and whether accessibility or transition propensity is primary. Stable-boundary systems cannot be reclassified after the fact as hidden reciprocity by introducing an unmeasured boundary. Failure of this conditional physical hypothesis would not by itself disprove the broader philosophical distinction between expression and non-expression.

Conclusion

Reciprocal expression is more fruitfully investigated as a conditional process in which realized transitions may reshape independently specified constraints on future realization, rather than as a required perpetual alternation of two states. Flux-sensitive boundary equations, an operational taxonomy of non-expression, prospective accessibility constraints, and two-direction interventions sharpen this hypothesis. Whether TSTOEAO entails any distinctive quantitative law beyond established feedback theory remains unresolved. The closed-loop example specifies a possible full causal sequence, while leaving the decisive challenge unresolved: deriving a distinctive quantitative restriction from independently stated TSTOEAO premises and testing it against strong conventional alternatives. The closed-loop example admits a locally stable nontrivial equilibrium, demonstrating that reciprocal feedback need not entail sustained oscillation. Its rate laws remain illustrative and require independent derivation and testing.

References and source note

Swygert, J. (2026, November 12). Language as Mathematics in Disguise: Relational Structure, Context, Encoding, and the Transformation of Meaning. Updated manuscript.

Swygert, J. (2026, November 12). The TSTOEAO Paradox: The Opposition of Non-Opposition. Manuscript.

The following works provide established mathematical and causal-inference context; none is presented as independently validating TSTOEAO. The proposed flux-sensitive boundary and accessibility hypotheses are reconstructions for future testing.

Norris, J. R. (1997). Markov Chains. Cambridge University Press.

Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos (2nd ed.). Westview Press.

Pearl, J. (2009). Causality: Models, Reasoning, and Inference (2nd ed.). Cambridge University Press.

Aubin, J.-P. (1991). Viability Theory. Birkhäuser.