ω-Invariant Difficulty: A Standing Planetary Bounty on the Matrix-Multiplication Exponent

Research corpus codename: NOOSPHERE.

Date: 2026-07-10 Provenance: Compiled entirely from the frozen MindChain chapters (02-mathematics.md §4–§5, 04-experiments-and-falsifiers.md §3.2, README.md, 08-memetics-and-launch.md §3) and the Pearl dossiers (research/08-pearl-matmul-pow.md, noosphere/research/02-pearl-cupow-kernel-digest.md); this paper is THEORY end to end — no lab artifact exists for M-OMEGA and no measurement is reported here.

Label convention: every quantitative value carries [LABEL date] from {MEASURED, MODELED, SIMULATED, PAPER, THEORY, DREAM}. Design thresholds preregistered in the frozen chapters are THEORY 2026-07-10. Facts audited from Pearl's released code carry the dossier's research date MEASURED 2026-07-09. Claims from the arXiv literature are PAPER with the version date.

Abstract

Every proof-of-work system is a bet on the cost of some computation. Most systems bet on an algorithm. MindChain's useful-work lane bets on a relation: the difficulty puzzle constrains the committed output matrix $C$ and its keyed hash — never the algorithm that produced it. Validity is relation-based by definition, so a faster correct multiplication cannot invalidate a committed receipt; it can only make receipts arrive faster, and a per-block exponential retarget (M-PULSE) lowers the lottery probability in response — the same engineering principle by which Bitcoin-like retargeting absorbed the ASIC transition at the rate level without a validity fork. Strassen-like methods, the cited FFT-only $O(n^{2.89})$ direction PAPER 2026-07-10, pasted first-party article, ch02 §4.3, or any future improvement toward the true matrix-multiplication exponent $\omega$ are therefore absorbed, not survived: the protocol never requires a miner to use $O(n^3)$ arithmetic, and it never needs to know $\omega$. The inversion is deliberate: because credit is calibrated to the best known adversarial cost $c^\star(J)$, any undisclosed faster algorithm mines outsized credit until disclosed or detected — which makes MindChain a standing planetary bounty on the exponent $\omega$: every complexity theorist who attacks $\omega$ is, involuntarily, working on a MindChain issue. This paper states the mechanism, its single conjecture, its explicit non-claims, its seven named attack modes, and its preregistered kill thresholds. It is a theory paper: the smallest decisive experiment (E-FMM-01) is defined in the ledger but has not been run, and we say so plainly.

Claims

1. Validity invariant THEORY 2026-07-10. For an exact tensor relation $R(A,B,C)=1$, if two algorithms return the same canonical $C$ for the same committed job and satisfy the same proof relation, consensus validity is identical. Chapter 02 §4.2 derives this directly from the definition of relation-based validity; it is the one component of M-OMEGA that is definitional rather than conjectural, and it inherits the chapter's overall THEORY maturity label because the surrounding mechanism is unbuilt.

1. Conjecture M-OMEGA THEORY 2026-07-10. Quoted from ch02 §4.2: "Under the stationarity and competition assumptions, an ASERT-like controller restores the target success interval after a global multiplicative reduction in $c^\star$, without a protocol fork and without knowing the matrix-multiplication exponent $\omega$." The conjecture concerns aggregate rate only. Chapter 02 §15 lists it as conjecture 3: "An ASERT-like controller absorbs global fast-matmul improvements under stationary competitive load."

1. The ω-bounty framing THEORY 2026-07-10. Quoted from README.md novelty 2: "the puzzle constrains the committed output and its keyed hash, never the algorithm. Strassen, FFT-based O(n^2.89) multiplication, or any future fast-matmul breakthrough is absorbed by a per-block exponential retarget exactly as Bitcoin absorbed ASICs. Noosphere is a standing planetary bounty on the exponent ω." The bounty is structural, not escrowed: a discoverer of a materially faster allowed algorithm collects a transient credit advantage at old difficulty (ch02 §4.3: "Retargeting after deployment cannot retroactively erase the discoverer's transient advantage"), and the affected class's Loom credit is zeroed until recalibration.

1. Bounded influence THEORY 2026-07-10. The lane this mechanism credits starts powerless: at genesis work_loom_weight_cap = 0 THEORY 2026-07-10, ch08 §3 with a protocol maximum of 0.10 of consensus weight (denominator: total consensus weight) THEORY 2026-07-10, ch08 §3. Base security is a BLAKE3 Hashcash Ground (A-GROUND [BUILDABLE], ch04 §3.1) that "claims neither useful work, ASIC resistance, nor hardware egalitarianism." Ch02 §4.6: "M-OMEGA must never be cited to justify sole or uncapped finality."

1. Explicit non-claims THEORY 2026-07-10. From ch02 §4.2 and the §14 non-implication table, M-OMEGA does not claim: ASIC resistance; fair mining; demand existence; fair revenue distribution; hardware decentralization; or constant attack cost during the adjustment transient. A private method "yields temporary and possibly persistent monopoly rents" (ch02 §4.3). The ch04 ledger row states the same boundary: "It does not make private shortcuts fair, prove demand, or replace Ground."

1. No measurement exists THEORY 2026-07-10. No lab artifact for M-OMEGA exists as of this paper's date. E-FMM-01 and E-PEARL-01 are DEFINED in the ch04 ledger and have not been run. Every number in this paper is a design constant, a quoted threshold, or an audited fact about a competitor's released system — none is a MindChain measurement.

Construction

The ticket constrains the output relation

A useful-work receipt commits, before a randomness beacon, to the job's operand commitments, worker identity, and configuration (M-TENSOR, ch02 §1); after the beacon it commits the canonical output root and evidence. The winner predicate is a keyed hash of the committed output against an integer acceptance target $T_h$ — larger $T_h$ easier, predicate monotone in $T$ (ch02 §5.1). Nothing in the predicate names an algorithm, an instruction mix, an operation count, or a hardware class. Ch02 §4.1: "'ω-absorbing' means only this: a faster correct multiplication algorithm does not invalidate a committed output relation, and a rate controller can lower $p_h$ after aggregate throughput rises. The protocol never requires a miner to use $O(n^3)$ arithmetic."

The adversarial cost model

Credit is calibrated not to nominal FLOPs but to the conservative attempt cost over the public adversarial algorithm set $\mathcal{A}$ — "including naive GEMM, tiled kernels, Strassen-like methods, rectangular fast multiplication, reassociation, cached operands, sparsity, low-rank methods, custom silicon, and any newly disclosed method" (ch02 §4.1):

$$c^\star(J)=\inf_{a\in\mathcal A}\operatorname{Cost}_a(J)$$

where cost is "measured in the scarce resource named by the class, preferably joules plus binding latency and memory-bandwidth constraints — not advertised FLOPs." A receipt with lottery success probability $p_h$ has expected best-known cost per success:

$$D_h(J)=\frac{c^\star(J)}{p_h}$$

Assumptions carried by the rate claim (ch02 §4.1): stationary admitted task distribution over the controller window; no selective replacement by cheap structured instances; no receipt replay; enough observed competitive attempts; no private algorithm controlling a dominant fraction during calibration.

The absorber: M-PULSE, an anchored ASERT-like retarget

M-PULSE (ch02 §5, maturity [BUILDABLE], provenance here still THEORY 2026-07-10 — two-implementation conformance is defined but not run) fixes an anchor $(a, t_a, T_a)$, a desired interval $\Delta$, and a half-life $\tau_{1/2}$. The ideal next target is

$$T_{h+1}^{\star}=T_a\,2^{\frac{t_h-t_a-\Delta(h-a)}{\tau_{1/2}}}$$

and consensus computes the exact fixed-point form

$$T_{h+1}=\operatorname{clip}_{[T_{min},T_{max}]}\left(\operatorname{FixedExp2}(T_a,x_h)\right),\qquad x_h=\frac{t_h-t_a-\Delta(h-a)}{\tau_{1/2}}$$

with "one fully specified fixed-point approximation and overflow rule" and deterministic median-past/bounded-future timestamp validity. Derived algebraic facts (ch02 §5.2, not stochastic stability theorems): on-schedule blocks leave the target fixed; blocks late by exactly one half-life double the ideal target; early by one half-life halve it. A global speedup in $c^\star$ raises throughput, timestamps run early, and the exponent continuously tightens $T$ — no fork, no knowledge of $\omega$. Separate targets are required for economically different classes; "Ground proposal work must not share an uncontrolled retarget with optional external-demand receipts" (ch02 §5.1).

flowchart TD
    A["Faster allowed algorithm disclosed"] --> B["Conservative attempt cost c* falls"]
    B --> C["Aggregate receipt throughput rises"]
    C --> D["Block timestamps run early vs schedule"]
    D --> E["Pulse exponent x_h turns negative"]
    E --> F["FixedExp2 tightens target T - lottery p_h falls"]
    F --> G["Success interval returns to desired Delta"]
    G --> H["Speedup absorbed: no validity fork, omega never named"]
    subgraph PEARL["Contrast - Pearl prices the algorithm"]
        P1["Security object: prescribed tiled transcript"] --> P2["512-bit rolling state hashed per depth step"]
    end
    subgraph MIND["MindChain prices the output relation"]
        M1["Security object: keyed hash of committed output C"] --> M2["Any correct algorithm yields the same valid receipt"]
    end

Figure 1 — The ω-invariance loop: a disclosed speedup lowers the conservative attempt cost $c^\star$, throughput rises, timestamps run early, and the anchored exponential retarget tightens the target — the speedup is absorbed at the rate level with no validity fork and no knowledge of ω. Contrast boxes: Pearl's security object is the prescribed transcript; MindChain's is the output relation. THEORY 2026-07-10.

The disclosure engine: algorithm-transition auction

When a materially faster allowed algorithm is discovered, ch02 §4.3 prescribes: "Discovery of any materially faster allowed whole-attempt algorithm sets the affected class's old Loom credit to zero until M-TENSOR and M-OMEGA are recalibrated." Prospective transition uses an algorithm-transition auction THEORY 2026-07-10 (ch02 §4.3; ledger row P3/S-ALGORITHM-TRANSITION): challengers submit executable whole-attempt cost evidence; the current profile's optional credit is zero while the challenge is material; a successor profile activates only after "two independent implementations reproduce the cheapest allowed strategy" — the ledger's pass threshold requires reproduction "within 10%" THEORY 2026-07-10, ch04 §3.2 P3 with all losing artifacts public. The auction changes future calibration only; it cannot reinterpret past rewards, and "governance cannot exclude a cheaper algorithm merely because it harms incumbents." This is what makes the bounty credible: the fastest path to monetizing a fast-matmul breakthrough on MindChain is public disclosure through the auction, because private exploitation triggers recalibration-to-zero the moment it is detected, while disclosed evidence sets the new calibration the discoverer already leads.

flowchart TD
    A["Challenger submits executable whole-attempt cost evidence"] --> B{"Challenge material?"}
    B -->|"no"| I["Incumbent profile keeps its calibration"]
    B -->|"yes"| C["Affected class's old Loom credit set to zero for the challenge"]
    C --> D["Two independent implementations attempt reproduction"]
    D --> E{"Both within 10% of winning cost?"}
    E -->|"no"| G["Challenge fails - losing artifacts stay public"]
    E -->|"yes"| F["Successor profile activates prospectively"]
    F --> H["M-TENSOR and M-OMEGA recalibrated before nonzero credit"]

Figure 2 — The algorithm-transition auction (ch02 §4.3; ledger row P3/S-ALGORITHM-TRANSITION): executable cost evidence zeroes the affected class's credit while the challenge is material; a successor profile activates prospectively only after two independent implementations reproduce the winning cost within 10%, with all losing artifacts public. THEORY 2026-07-10.

Attack surface

The seven named failure modes (ch02 §4.4, quoted compactly): structured-instance selection (diagonal/low-rank jobs with $c^\star$ far below class calibration); cached-weight advantage (amortize one operand while calibration assumes two cold operands); private algorithm shock (mine at old difficulty, withhold the method); digest-only shortcut (compute only what the lottery predicate needs, not the claimed full output); demand starvation ("no threshold can manufacture demand"); verification bottleneck (solving accelerates but proof generation/availability does not); hardware discontinuity (fastest implementation proprietary to one architecture, centralizing proposal power).

Results

There is no measured table. No lab artifact exists for M-OMEGA as of 2026-07-10. This section therefore states exactly what the frozen chapters commit to — the design equations and preregistered bounds a future artifact will be judged against — and nothing else.

Design equations (all THEORY 2026-07-10, ch02 §4.1, §5.1; research/02 §8.3):

Preregistered quantitative bounds (all THEORY 2026-07-10; denominators stated with numerators):

Experiment status as of 2026-07-10: E-FMM-01 (algorithmic-breakthrough tournament feeding measured whole-attempt costs plus simulated 2×/10× private speed shocks into M-PULSE) — DEFINED, not run. E-PEARL-01 (digest-only / rolling-state attack tournament) — DEFINED, not run. The smallest decisive experiment ch02 §4.5 specifies — a benchmark adversary suite spanning dense naive/tiled, Strassen where dimensions permit, the FFT-only $O(n^{2.89})$ direction if code becomes available, sparse, diagonal, low-rank, cached-right-operand, output-only, and energy-optimized kernels, with recalibration covering proof generation, commitment hashing, delivery, and energy, not just GEMM runtime — has no artifact. When it does, its METRIC lines replace this paragraph.

Falsifiers & kill thresholds

Quoted verbatim where the frozen sources state them.

Kill condition, ch02 §4.6:

Set the class's Loom credit to zero if any allowed input family yields more than 1.05x credited work per joule relative to the public best calibration, if demand starvation makes the target rate unattainable for one full retarget half-life, or if a best-known strategy can satisfy the ticket without producing the deliverable. M-OMEGA must never be cited to justify sole or uncapped finality.

Ledger row, ch04 §3.2 (A-LOOM-CREDIT / M-OMEGA THEORY):

• Pass threshold (verbatim): "Best known adversarial accepted-receipt cost per joule is within 5% of the credited baseline at 99% confidence across admitted shapes; recalibration occurs before nonzero credit; admitted demand can sustain the target receipt rate."
• Kill threshold and rollback (verbatim): "Any reproducible >5% shortcut, stale calibration, satisfying the ticket without producing the deliverable, or demand starvation making the target receipt rate unattainable for one full retarget half-life. Set the affected class's credit and cap to zero; fee-paid jobs and Ground remain."
• Named adversaries and experiments (verbatim): "Digest-only shortcuts, FMM, cached weights, structured inputs, special hardware, demand starvation. E-PEARL-01, E-FMM-01, E-BLACKOUT-01."

The controller's own kill, ch02 §5.6 (M-PULSE, on which M-OMEGA depends):

Do not activate a parameter set if the two implementations disagree on one vector, if legal timestamp manipulation can change steady-state target by more than 10% over one half-life, or if a 10x shock fails to return median intervals to $[0.8\Delta,1.2\Delta]$ within two half-lives in the declared simulator envelope. Failure leaves the prior target rule active; it does not alter historical targets.

Standing discipline (E-FMM-01, ch04): "Pulse may absorb an honest global speedup, but it cannot cure a private shortcut that creates accepted security evidence more cheaply than credited work. Before any nonzero Loom credit and at every material kernel/hardware change, recompute measured work from the best known adversarial implementation. Stale calibration means weight zero."

A fired kill zeroes the useful-work class's credit and cap; the BLAKE3 Hashcash Ground and fee-paid jobs continue. The base chain does not depend on this mechanism surviving.

flowchart LR
    K1["Shortcut: over 1.05x credited work per joule vs public best calibration"] --> KILL["KILL: class Loom credit and cap set to zero"]
    K2["Demand starvation: target rate unattainable for one full retarget half-life"] --> KILL
    K3["Ticket satisfied without producing the deliverable"] --> KILL
    KILL --> S1["BLAKE3 Hashcash Ground continues"]
    KILL --> S2["Fee-paid jobs continue"]
    KILL --> S3["Base chain does not depend on the mechanism surviving"]

Figure 3 — The three preregistered kill triggers of ch02 §4.6 and what survives a fired kill: the useful-work class's credit and cap go to zero while the BLAKE3 Hashcash Ground and fee-paid jobs continue. THEORY 2026-07-10.

Reproduction

No lab artifact exists; what is reproducible today is the provenance of every claim and number above. Each command below prints the quoted source text (cwd shown; on Windows without grep, substitute findstr /n <pattern> <file>).

cwd: noosphere
grep -n "M-OMEGA" 02-mathematics.md            # §4 heading, §14 dependency row
sed -n '310,368p' 02-mathematics.md            # full M-OMEGA section incl. §4.6 kill
sed -n '370,423p' 02-mathematics.md            # M-PULSE equations and §5.6 kill
grep -n "A-LOOM-CREDIT / M-OMEGA" 04-experiments-and-falsifiers.md   # ledger row (line 148)
grep -n "S-ALGORITHM-TRANSITION" 04-experiments-and-falsifiers.md    # P3 auction row (line 165)
grep -n "E-FMM-01" 04-experiments-and-falsifiers.md                  # experiment definition (line 464)
grep -n "standing planetary bounty" README.md 08-memetics-and-launch.md
grep -n "work_loom_weight_cap" 08-memetics-and-launch.md

cwd: . (repo root)
grep -n "arXiv:2504.09971" research/08-pearl-matmul-pow.md
grep -n "194-second" research/08-pearl-matmul-pow.md
grep -n "8.2" noosphere/research/02-pearl-cupow-kernel-digest.md

When the E-FMM-01 tournament lab exists, its README will define the executable reproduction (adversary-suite runs plus 2×/10× shock injection into an M-PULSE simulator per ch02 §4.5) and this section will be superseded by exact commands against that artifact.

Related work

Pearl (the contrast). Komargodski and Weinstein, "Proofs of Useful Work from Arbitrary Matrix Multiplication," arXiv:2504.09971v4 (2025-11-13) PAPER 2025-11-13, audited in research/08-pearl-matmul-pow.md (research date 2026-07-09) and noosphere/research/02-pearl-cupow-kernel-digest.md. Pearl's security object is a prescribed tiled multiplication transcript: the released protocol hashes a per-output-tile 512-bit rolling state after each depth step, "precisely because output-only checking does not prove fresh work" (research/02 §8.2). That fixes the algorithm class — a miner must produce the prescribed transcript even if a faster output-only algorithm exists, and the dossier shows the whole-attempt overhead ratio is favorable only against a named baseline and rank regime: against an $O(n^\omega)$ output algorithm "the rank bound must scale more strictly" (ch02 §4.3; research/08 §4). Pearl's own difficulty machinery is a 194-second target interval with WTEMA retargeting MEASURED 2026-07-09, source audit, research/08 §5.3, and its released predicate carries a $k/r$-dependent work-normalization defect the dossier calls "a consensus-fairness defect, not a cosmetic unit mismatch" MEASURED 2026-07-09, research/08 §5.2. MindChain takes the opposite bet: it prices the output relation and treats the algorithm as adversarial, calibrating credit to $c^\star$ over the full public algorithm set. The boundary cuts both ways and ch02 §4.3 states it: "MindChain does not import Pearl's transcript-hardness conjecture by renaming it ω-invariance" — and retargeting alone "does not prove that the new puzzle remains fair or useful" (research/02 §8.2), which is why the calibration kill, not the retarget, is the load-bearing defense.

External references already collected in the source dossiers (research/08 §2.1; not independently re-audited here): Pass, "The Economics of Proof-of-Useful-Work," arXiv:2606.06700 PAPER — equilibrium model, not a Nakamoto-assumption proof; Basu, "The Usefulness Gap in Proof-of-Useful-Work: An Empirical Study of Pearl's cuPOW Protocol," arXiv:2606.04819 PAPER — independent empirical evidence; Bar-On, Komargodski, and Weinstein, "Proof of Work With External Utilities," arXiv:2505.21685 PAPER; Dotan and Tochner, "Proofs of Useless Work — Positive and Negative Results for Wasteless Mining Systems," arXiv:2007.01046 PAPER — the prior formal obstacle taxonomy; Buterin, "Hard Problems in Cryptocurrency: Five Years Later" (2019) §§3, 6–9, 16.

Bitcoin/ASERT precedent. Ch02 §4.3: "Bitcoin-like retargeting demonstrates the engineering principle that a threshold can absorb aggregate speed changes. It does not make the underlying puzzle 'ASIC-proof.'" ASERT-like retargets "are buildable and deployed in existing PoW systems" (ch02 §5.3); their operational existence does not prove MindChain's parameters stable under tensor-job bursts, proof latency, timestamp coalitions, or finality delays — that is E-FMM-01's job.

Program-internal. The competitive scorecard noosphere/10-competitive-audit.md places this mechanism among the consensus/useful-work verdicts; the ω-bounty proclamation and its mandatory boundary sentence are staged as launch moment W4 in noosphere/08-memetics-and-launch.md §3 ("retargeting fixes the block interval after a speedup; it does not by itself prove the new puzzle remains fair or useful").