Resonance Logic

resonance-logic-draft.md

Resonance Logic (RL) – Core Formalization

UOR Foundation / PrimeOS working group

§ 1 Motivation & Big‑Picture Framing — Conservation as Truth

Resonance Logic (RL) internalises the CCM maxim "truth ≙ conservation." A statement is as true as the resonance it conserves. Consequently, Boolean values ${0,1}$ are replaced by the 96‑element resonance spectrum

$$\mathcal C_{96};=;\langle C,,\oplus,,\otimes,,0,,1\rangle.$$

Two meta‑goals

• Soundness — every proof in the sequent calculus RL$_s$ preserves resonance.
• Conservativity over SA — collapsing all resonances by the crush map $\kappa : C_{96}\twoheadrightarrow{0,1}$ guarantees RL proves no new purely‑Boolean theorems.

Road‑map of this note

• § 2 fixes syntax and the budgeted‑sequent format.
• § 3 constructs $\mathcal C_{96}$ and its operations once and for all.
• § 4 presents the core sequent calculus RL$_s$.
• § 5 Induction‑Collapse Theorem — the new heart of the paper.
• § 6 sketches soundness.
• § 7 embeds SA/MSA via the crush map.

Appendices collect the term grammar, rule‑verification table, and an extended discussion of equality semantics.

§ 2 Syntax & Budgets

§ 3 Concrete Construction of the Resonance Lattice $\mathcal C_{96}$

Non‑negotiable deliverable: this section pins down $\mathcal C_{96}$ as an explicit, finitely‑presented algebra. All later proofs rest on this foundation.

3.1 Underlying set

We fix $$C ;:=; \mathbb Z_{96} ;=; {,0,1,2,\dots,95,},$$

the ring of integers modulo 96. Each element will be called a resonance value and written $[n]$ when we wish to emphasise the congruence class.

3.2 Operations

Define two binary operations for all $[r],[s]\in C$:

Additive join $$[r];\oplus;[s] ;:=; [,r + s,]_{96},$$ i.e. addition modulo 96.

Multiplicative bind $$[r];\otimes;[s] ;:=; [,r,s,]_{96},$$ i.e. multiplication modulo 96.

Constants: $0 := [0]$ (bottom / null resonance) and $1 := [1]$ (unity resonance).

Remark 3.2.1 (Semiring view). $\langle C,\oplus,\otimes,0,1\rangle$ is a commutative, idempotent‑free semiring; idempotence of $\oplus$ is not required for RL because budgets are quantitative rather than truth‑valued. The semiring therefore qualifies as a (commutative) quantale without idempotent join in the sense of linear logic's phase semantics.

3.3 Algebraic properties

Property	Proof sketch
Associativity of $\oplus,\otimes$	inherited from $\mathbb Z$ mod 96
Commutativity	ditto
Identities $0,1$	$[r]+0=[r]$, $[r]\cdot 1=[r]$ mod 96
Distributivity	$r(s+t)\equiv [rs] + [rt];(\mathrm{mod};96)$
Cancellation?	No; budgets track quantities, not sets

Hence $C$ is a commutative semiring and therefore a legitimate phase space for linear logic.

3.4 Order‑successor, cycle & finite size

Set $\sigma,[r] ;:=; [r+1]_{96}. \qquad (\text{"one‑step resonance shift"})$

Then $\sigma^{96}=\mathrm{id}$. This operation provides a cyclic structure on the resonance values, though its role in the (RI) rule is through the modular arithmetic of budget indices rather than direct application to formulas.

3.5 Klein complement

We keep the involutive map $$\perp,[r] ;:=; [,(-r)_{96},], \qquad \text{so}; \perp\perp [r] = [r].$$

De Morgan laws follow immediately from elementary modular arithmetic.

3.6 Semantics of Equality — resolving the mismatch

Design choice. We retain crisp (Boolean‑valued) equality. Formally, $$\llbracket,[r]=[s],\rrbracket ;:=; \begin{cases}1, & r\equiv s;(96);\ 0, & \text{otherwise.}\end{cases}$$

Rationale. (i) It preserves the usual substitution and congruence properties without inflating the budget algebra. (ii) All constructive content already lives in the budgets; adding a many‑valued equality would blur two orthogonal dimensions (identity vs. resource‐flow).

Alternative (3.6.1). Should a fully many‑valued equality be desired, one may set $\llbracket [r]=[s] \rrbracket := [|r-s|]_{96}$. The present draft leaves this for future investigation.

§ 4 Sequent Calculus RL$_s$

(All rules and side‑conditions "wf / bal" still apply, with $\oplus,\otimes$ now interpreted modulo 96.)

Note on the (RI) rule: The Resonance Induction rule performs induction over natural numbers with budget assignments that cycle through the 96 resonance values. For a formula $\varphi(n)$ with numeric variable $n$, the rule requires proving the inductive step with budget $r_k$ for all natural numbers $n$ where $n \equiv k \pmod{96}$. This creates 96 parallel induction tracks, each with its own budget.

§ 5 Induction‑Collapse Theorem

This section addresses the relationship between RL's Resonance‑Induction (RI) rule and Peano induction under the crush map $\kappa$.

5.1 Clarification of the (RI) Rule

The (RI) rule requires careful interpretation. Given a formula $\varphi(n)$ with one free numeric variable $n$ ranging over natural numbers, the rule schema is:

Resonance Induction (RI):

• Base case: $\vdash_{r_0},\varphi(0)$
• Inductive step: For each $k \in C$, we have $\varphi(n)\vdash_{r_k},\varphi(n+1)$ where $n \equiv k \pmod{96}$
• Conclusion: $\vdash_{\rho},\forall n.\varphi(n)$ where $\rho = \bigotimes_{k \in C} r_k$

Here, the meta-level index $k$ determines which budget $r_k$ is used for proving the inductive step at positions congruent to $k$ modulo 96.

5.2 Theorem (IC-Revised)

Induction‑Collapse Theorem (Revised). Let $\varphi(n)$ be any RL‑formula with one free numeric variable. If $\vdash_{\rho};\forall n.\varphi(n)$ is derivable using (RI), then:

1. The Boolean projection $\kappa(\rho)=1$
2. For the formula $\kappa^(\varphi)$ (where $\kappa^$ applies $\kappa$ to all resonance constants), Peano arithmetic proves $\forall n.\kappa^*!(\varphi)(n)$ if and only if all 96 budget-specific instances are provable.

5.3 Proof

Budget projection. The conclusion budget is $\rho = \bigotimes_{k\in C} r_k.$

Since $\kappa$ is a semiring morphism: $\kappa(\rho) = \prod_{k \in C} \kappa(r_k) \in {0,1}.$

For the proof to be sound, we need $\kappa(r_k) = 1$ for all $k$, hence $\kappa(\rho) = 1$.

Relationship to Peano induction. The (RI) rule establishes:

• $\vdash,\kappa^*(\varphi(0))$ (from the base case)
• For each residue class $k \pmod{96}$: all instances of $\kappa^(\varphi(n)) \vdash \kappa^(\varphi(n+1))$ where $n \equiv k \pmod{96}$

This is stronger than standard Peano induction, which only requires:

• $\vdash,\varphi(0)$
• $\forall n.,(\varphi(n) \vdash \varphi(n+1))$

Key observation: The (RI) rule partitions the natural numbers into 96 congruence classes and potentially uses different proof strategies (reflected by different budgets $r_k$) for each class. Under the Boolean collapse, all these strategies must work.

Conservative extension property: If Peano arithmetic proves $\forall n.\psi(n)$ for some Boolean formula $\psi$, then RL can prove $\vdash_1,\forall n.\psi(n)$ by setting all budgets $r_k = 1$. Conversely, if RL proves a Boolean theorem using (RI), the collapsed proof is valid in Peano arithmetic.

Hence (IC-Revised) is proved. ∎

Corollary 5.3.1. RL is conservative over Peano arithmetic with respect to Boolean theorems, though the (RI) rule provides a finer-grained induction principle that tracks resource usage across congruence classes modulo 96.

§ 6 Soundness

(Lemmas 6.1 & 6.2 establish the core soundness properties.)

§ 7 Embedding SA → RL

A. Full Grammar for Terms & Formulas

Let $\mathcal V$ denote a countably‑infinite set of variables $x,y,z,\dots$.

A.1 Terms $t$

• Variables $ x \mid y \mid \dots $
• Constants $ 0 \mid 1$
• Composite
  • $t + t$ (binary additive symbol)
  • $t \cdot t$ (binary multiplicative symbol)
• Parentheses may be used for grouping.

This inductive definition is minimal yet expressive: any additional term‑forming operator required by a concrete theory (e.g., subtraction, modular residue) can be added à la carte without altering the underlying logic.

Note: The successor operation σ defined in § 3.4 operates on resonance values (meta-level), not on terms (object-level).

A.2 Atomic formulas $\alpha$

• Equality $t = s$
• Klein complement $\perp,\alpha$

A.3 Molecular formulas $\varphi,\psi$

$\varphi,\psi ::= \alpha \mid \varphi \oplus \psi \mid \varphi \otimes \psi \mid \exists x.,\varphi \mid \forall x.,\varphi$

The connectives $\oplus$ (additive join) and $\otimes$ (multiplicative bind) inherit associativity and commutativity from the quantale structure of $\mathcal C_{96}$.

B. Verification of Side‑Conditions

Every inference rule in RL$_s$ carries the implicit side‑conditions wf and bal from § 2.4. Table B.1 checks that these conditions propagate from premises to conclusions; thus they need not be written explicitly inside proof trees.

Rule	Premises (budgets)	Conclusion (budget)	wf preserved?	bal preserved?	Justification
Ax	—	$\vdash_{\rho};A, A$	✔	✔	Choose $\rho = \llbracket A \rrbracket$.
Cut	$\Gamma \vdash_{\rho} A$; $A, \Delta \vdash_{\sigma} \Lambda$	$\Gamma, \Delta \vdash_{\rho \otimes \sigma} \Lambda$	✔	✔	wf: quantale closed under $\otimes$; bal: Lemma 5.2.
$\oplus L$	$A,\Gamma \vdash_{\rho} \Delta$	$A\oplus B, \Gamma \vdash_{\rho} \Delta$	✔	✔	$\llbracket A\oplus B\rrbracket = \llbracket A\rrbracket \oplus \llbracket B\rrbracket$ adjusts both sides equally.
$\oplus R$	$\Gamma \vdash_{\rho} A,\Delta$; $\Gamma \vdash_{\sigma} B,\Delta$	$\Gamma \vdash_{\rho \oplus \sigma} A\oplus B,\Delta$	✔	✔	wf: closure under $\oplus$; bal: by construction of $\rho\oplus\sigma$.
$\otimes L/R$	analogous	analogous	✔	✔	Quantale axioms.
RI	$\vdash_{r_0},\varphi(0)$; $\varphi(n)\vdash_{r_k},\varphi(n+1)$ for all $n$ where $n \equiv k \pmod{96}$	$\vdash_{\bigotimes_k r_k},\forall n.\varphi(n)$	✔	✔	wf: finite $\otimes$ of well‑formed $r_k$; bal: each instance preserves balance; multiplication composes.

Key: ✔ = side‑condition automatically satisfied if it holds for premises.

Remark B.2. Because budgets inhabit a commutative quantale, the order of multiplication in the Cut rule and RI budget aggregation is immaterial, simplifying mechanised proof‑checking.

Appendix C. Equality Semantics — extended discussion

C.1 Why retain Boolean equality?

• Pragmatics: keeps substitution & rewriting simple.
• Semantics: resource flow lives in budgets, orthogonal to identity.
• Conservativity: guarantees crisp collapse back to SA.

C.2 Many‑valued alternative (open problem)

Define $\llbracket t=s \rrbracket := [,|\llbracket t \rrbracket - \llbracket s \rrbracket|,]_{96}$. One must then revisit the bal side‑condition so that budgets subtract rather than nullify when two terms almost coincide. This remains future work.

End of document